2003:11
TECHNICAL REPORT
MARTIN NILSSON – Restraint Factors and Partial Coefficients – TECHNICAL REPORT
Restraint Factors and Partial
Coefficients for Crack Risk Analyses
of Early Age Concrete Structures
Preliminary Version
Diagrams and Tables
Martin Nilsson
Department of Civil and Mining Engineering
Division of Structural Engineering
2003:11 • ISSN: 1402 - 1536 • ISRN: LTU - TR -- 03/11 -- SE
TECHNICAL REPORT 2003:11
Restraint Factors and Partial
Coefficients for Crack Risk Analyses
of Early Age Concrete Structures
Diagrams and Tables
Martin Nilsson
Division of Structural Engineering
Department of Civil and Mining Engineering
Luleå University of Technology
SE - 971 87 Luleå
Sweden
Preface
This report contains all background data for the Doctoral thesis Restraint Factors and
Partial Coefficients for Crack Risk Analyses of Early Age Concrete Structure by Martin Nilsson. It contains the diagrams and tables with all data and calculation results necessary for
making the semi-analytical method presented in the thesis applicable.
Luleå in May 2003
Martin Nilsson
Abstract
Keywords:
V
VI
Table of Contents
Table of Contents
PART I
PREFACE ...............................................................................................................III
ABSTRACT .............................................................................................................V
TABLE OF CONTENTS...................................................................................... VII
1
INTRODUCTION .............................................................................. 1
1.1
1.2
2
THE SEMI-ANALYTICAL METHOD................................................ 5
2.1
2.2
2.3
2.4
2.5
2.6
2.7
3
Early age cracking .................................................................................. 1
Crack risk estimations............................................................................. 3
General formulation ............................................................................... 6
Effects of high walls................................................................................ 9
2.2.1 Basic resilience factors................................................................... 9
2.2.2 Resilience correction factors ........................................................11
Effects of slip failure in joints .................................................................13
Applicable expressions ...........................................................................15
Boundary restraint .................................................................................16
Effective width of slab ...........................................................................17
Decisive point .......................................................................................18
RESTRAINT VARIATION DETERMINED BY FINITE
ELEMENT METHOD CALCULATIONS ........................................ 19
VII
Restraint Factors and Partial Coefficients
4
HIGH WALL EFFECTS ..................................................................... 93
5
EFFECTIVE WIDTH OF SLAB ......................................................... 97
6
EFFECTS OF THE LOCATION OF WALLS ON SLABS ON
THE EFFECTIVE WIDTH OF SLAB ...............................................159
7
BASIC RESILIENCE CORRECTION FACTOR ...........................185
REFERENCES ...........................................................................................195
VIII
Introduction
1
Introduction
It is well known for contractors that due to volume changes in concrete structures large
stresses can arise if the movements of the structures are restrained. These restraint
stresses can be so large that they may cause extensive cracking, which give rise to
expensive repair and reduce both the durability and the function of the structures.
Therefore, today hardly any contractors overlook the effects of these early age stresses.
Still, and too often, newly cast concrete structures crack at many building sites.
Large resources are put into modern building to predict the restraint stresses and the
risk of cracking, and to design appropriate counter-measures to avoid possible cracking.
The measures, including safety limits, should be designed in the early productionplanning phase. However, if the conditions are changed for the actual casting, e.g. the
air and/or the concrete temperatures change, a new crack risk analysis is needed. Nevertheless, there is a need of fast and reliable tools to realize the analyses, especially if the
conditions are changed very late in the building process. Unfortunately, sometimes the
analysis tools and/or models are too simplified, which introduce uncertainties in the
results, an on the opposite, the tools can also be too complicated and time consuming.
As been indicated, one important part of every tool is to predict the degree of
restraint and, consequently, the restraint calculation should neither be too simplified
nor too complicated. In this paper an elastic and fairly simple model is presented for the
determination of the restraint in newly cast concrete structural elements on existing
ones and especially for the typical case wall on slab.
1.1
Early age cracking
During the hydration phase of a concrete structure, the chemical reaction between the
cement and the water in the concrete generates heat. Due to this heat, young structural
elements undergo temperature histories similar to what is shown in Figure 1.1a). During the heating phase, the structural elements expand and later when the chemical
reaction subsides, the concrete starts to contract when the heat development decreases
and the temperature starts to adjust to the surroundings. Meanwhile the concrete
1
Restraint Factors and Partial Coefficients
matures and its strength increases, see Figure 1.1b). In addition, for high performance
concretes (low water-to-cement ratios) significant autogenous shrinkage is present in
the same time as the thermal movements take place.
Temperature
The volume changes during the hydration phase can be hindered by adjoining
structures, foundations or by internal parts not undergoing the same volume changes.
This hindrance induces restraint stresses that may be so large that cracks possibly occur
if the stresses are larger than the tensile strength of the concrete, see Figure 1.1b).
Martin
Tmean
Time
Stress/Strength
a)
b)
Possible crack
Tensile strength
Time
Stress at partial restraint
Stress at 100% restraint
Figure 1.1 Example of the a) mean temperature and b) stress and strength development in a
hardening concrete element restrained at partial and totally (100 % degree).
The possible cracking of early age concrete structures can be divided into two
groups, namely cracking during the expansion and during the contraction phases,
respectively.
• Cracking during the expansion phase might happen both in the surface and/or
through the structural elements. Surface cracks in the expansion phase are induced in newly cast elements due to differences between the internal movements
within the elements, and through cracks expansion phase can be found in adjacent structural elements due to the difference in movements between the different elements. Expansion phase cracks arise shortly after casting, within a few
days, and tend to close by time. The influence by such cracks on the static capacity, the function and durability most be judged from case to case.
2
Introduction
• Cracking during the contraction phase are usually in the shape of through cracks
in newly cast elements. Depending on dimensions, environmental conditions
etc. they might arise weeks, months and in extreme cases even years after casting.
Cracks formed during the contraction phase are mostly through and lasting
cracks, which are induced in the most recently cast elements due to the hindrance of the thermal movements from adjacent structural elements.
1.2
Crack risk estimations
The estimation of the risk of cracking of early age concrete structures can be based on
five steps, see Figure 1.2. Firstly, the type of structure, the material proportions and
possible measures to avoid cracks have to be chosen. Secondly, the temperature development has to be determined, either by calculations, diagrams/databases or by measurements. Thirdly, the restraint situation has to be determined, that is both the boundary and the structural restraint. Fourthly, structural calculation of the stress or strain
ratios follows, i.e. the maximum tensile stress or strain is compared to the tensile
strength or the ultimate strain capacity, respectively. Fifthly the crack risk design is
based on so-called crack safety factors, the partial coefficients, which the stress or strain
ratios should not exceed. Recommendations and guidelines regarding this whole
process can be found in e.g. Emborg & Bernander (1994a) and Emborg et al. (2003).
1
Choice of structure, material
and possible measures
Temperature state by calculations,
2 diagrams/databases or measurements
Restraint conditions, both Martin
3 boundary and structural restraint
Loading
4 Structural calculation: ξ = Capacity
5 Crack risk design: ξ ≤
1
γr γ s
No
Yes
OK!
Figure 1.2 Description of the principal steps in estimations of the risk of cracking in early age
concrete structures. Nilsson (2000).
The first step in Figure 1.2, choice of structure, material and measures, is the primary base
of the estimation of risk of thermal cracking in early age concrete structures. Properly
chosen type of structure and dimensions along with suitable mix design of the concrete
3
Restraint Factors and Partial Coefficients
is the foundation of the avoidance of cracking. In addition, possible measures such as
cooling of hydrating parts, Bernander (1973 & 1998) and/or heating of adjacent older
parts, e.g. see Wallin et al. (1997), might be required. The second step in Figure 1.2
comprises the temperature development during the hydration phase. It has to be determined either by calculations, see e.g. Jonasson et al. (1994) and ConTeSt Pro (2003),
from diagrams/database, see Jonasson et al. (2001), or from measurements in real structures. From the temperature development, the stress and strength growth are obtained.
As the third step, the restraint condition has to be determined, including the restraint
within the studied structure, in the interface (joint) between parts of structures, from
adjacent structural members and ground/rock. In Emborg et al. (1997), several common cases of internal restraint situations are presented, and in e.g. Larson (1999 &
2000) the restraint from adjacent structures has been investigated. In Rostásy et al.
(2001) several engineering models are reported for the assessment of boundary restraint
in the phases of pre-design and design and execution. For the rotational bending restraint from adjacent ground materials, e.g. in Bernander (1993) and Nilsson (1998 &
2000), methods for the determination of the boundary restraint coefficient are given.
The determination of the stress/strength or strain/ultimate-strain situation, step four in
Figure 1.2, can be done by structural calculations, e.g. with ConTeSt Pro (2003) or JCI
(1992), by manual methods, see Löfquist (1946), Bernander (1982), Bernander &
Emborg (1994), Emborg & Bernander (1994b) and Larson (2000), or with help from
diagrams/databases, e.g. through Emborg et al. (1997). Comparing the stress/strength
or strain/ultimate-strain situations with stated partial coefficients for thermal cracking
problems forms the final step in the crack risk design. The partial coefficients - or crack
safety factors - are given in design codes, e.g. BRO 2002 (2002). The partial coefficients can be derived by probabilistic methods, see Nilsson (2000 & 2003).
Below, a semi-analytical elastic model will be presented for the determination of the
restraint in early age concrete walls cast on older slabs (step 3 in Figure 1.2). The model
is derived using the Compensation Plane theory, see JCI (1992) and Rostásy et al.
(1998), meaning plane sections remain plane after deformation in the structural analysis.
The model is supplemented with the effects of high walls and the effects of possible slip
failure in the joint between the young and the old concrete. In this report the model is
fitted to about 3000 three-dimensional elastic FEM calculations, deeper descriptions are
given, and models and engineering tools for each part in the model are introduced and
discussed. The approach presented in this paper is a modelling with respect to the
effects of high walls as only the deformations in the wall are taken into consideration as
in ACI (1973, 1990 & 1995), Stoffers (1978) and Emborg (1989).
4
The Semi-Analytical Method
2
The semi-analytical method
As been described and visualised in Figure 1.2, one of the crucial factors in the estimation of the risk of early age cracking is the determination of the degree of restraint
affecting a concrete structure. The restraint can be described as the hindrance of the
free movements of young concrete structural elements or parts of elements during the
hydration phase. If such elements or parts of elements are totally free to deform, when
they are exposed to thermal induced movements and/or shrinkage, no restraint stresses
will arise. On the opposite, if newly cast structural elements or parts of elements are
hindered to move by adjacent sections or stiff foundation materials, different degrees of
restraint stresses arise. E.g. in tunnel walls concreted on foundations and in walls on
foundation slabs restraint stresses arise. The degree of restraint can be determined in
different ways: e.g. by field measuring of actual strains in real structures and comparing
them to the strain at total fixation, see Heimdal et al. (2001a & b) and Larson (2003). It
can also be determined by finite element method calculations, see e.g. JCI (1992) and
Kanstad et al. (2001), or other more or less sophisticated theoretical models.
Below, a semi-analytical method will be presented that can be used as a fairly simple
engineering tool adaptable in many situations. The model is presented both as a general
formulation and as an applicable approach for the typical case wall on slab.
Generally, the restraint in a young concrete structural part, γR, can be defined as the
quota between the principal stress at the studied time, σ1, and the stress, σ0, at total
fixation (ε ≡ 0) by volume changes in the young concrete (shrinkage and thermal
strains):
γR =
σ1
σ0
(1)
Based on Eq. (1) with the use of the Compensation Plane theory, see Nilsson
(2003), the degree of restraint in the length direction of a structure can be expressed as
5
Restraint Factors and Partial Coefficients
γ R = δres δslip − γtR − γ ryR − γ Ryz
(2)
where
δres
δslip
γRt
γRry
γRrz
=
=
=
=
=
high wall effect, resilience, [-]
slip in joint effect, [-]
translational restraint part, [-]
rotational restraint part for rotation around the y-axis (the vertical axis), [-]
rotational restraint part for rotation around the z-axis (the horizontal, transverse axis), [-]
Restraint determined by the semi-analytical model in Eq. (2) depends partly on
effects of high walls in which the strains in the young parts do not vary linearly, partly
on possible slip failure in the joint between the young and the old concrete, partly on
three restraint parts, γRt, γRry and γRrz, determined by the geometric properties of the
structures as well as the boundary restraint situation from the adjacent foundation
material. The first part of Eq. (2), δresδslip, forms the semi part of the method due to
non-linear behaviour in high-wall structures. The second part, -γRt-γRry-γRrz, forms the
analytical part based on the Compensation Plane theory. The method includes adaptation to about 3000 3D elastic FEM calculations presented in chapter 3.
2.1
General formulation
A general expression is derived for the determination of the restraint variation in an
early age concrete part cast on an older one, see Nilsson (2003). The derivation is based
on the Compensation Plane theory (compare beam analysis) and uses an approach for
pre-stressed concrete beams, see Collins & Mitchell (1991). Under assumption of
uniform and elastic contraction in the whole young element and in the older element,
respectively, the restraint variation can be described by
γ R = δres δslip − (1 − γ RT )
−(1 − γ RR , y )
M RI , y (zcen − z )
∆ε0c ζE c 28I trans , y
N RI
0
∆εc ζE c 28 Atrans
− (1 − γ RR ,z )
M RI ,z ( ycen − y )
(3)
∆ε0c ζE c 28I trans ,z
where
NRI = compression force giving zero translational strain in the young concrete, [N]
∆ε0c = strain of applied volume changes in the young concrete (shrinkage and temperature induced strain), [-]
ζEc28 = modulus of elasticity of the young concrete at the studied time, [N/m2]
6
The Semi-Analytical Method
Atrans = transformed area of the cross section, [m2]
MRI,y = internal bending moment around the y-axis for obtaining zero curvature in
the xz-plane of the young concrete, [Nm]
MRI,z = internal bending moment around the z-axis for obtaining zero curvature in
the xy-plane of the young concrete, [Nm]
ycen = vertical location of the centroid of the transformed section relatively the joint,
[m]
y
= vertical co-ordinate from the joint and up-wards, [m]
zcen = horizontal location of the centroid of the transformed section relatively the
centre of the slab, [m]
z
= horizontal co-ordinate from the centre of the slab, [m]
Itrans,y = transformed second moment of inertia of the cross section for bending around
the y-axis, [m4]
Itrans,z = transformed second moment of inertia of the cross section for bending around
the z-axis, [m4]
γRT = translational boundary restraint, [-]
γRR,y = rotational boundary restraint for bending around the y-axis, [-]
γRR,z = rotational boundary restraint for bending around the z-axis, [-]
Note that this formulation is quite similar to the well-known Navier’s formula for
beam analyses.
The modulus of elasticity of the young concrete is expressed as a factor ζ times the
28-days value of the elasticity. This factor is introduced in order to take into account
that most often cracking of young concrete (through cracks in the cooling phase) takes
place fairly late in the hydration process but, for ordinary sized structures, not as late as
at 28 days of maturity. In Larson (2000) a small study of the factor ζ was presented
giving that ζ = 0.93 is a good estimation of the somewhat lower stiffness of the young
concrete during the contraction phase.
For simplicity at application, Eq. (3) is transformed to, see Nilsson (2003),
7
Restraint Factors and Partial Coefficients
γ R = δres δslip − (1 − γ RT )
δslip ∫ δresdAc +
Ac
E
Ea 28
λ
dA
ζE c 28 ∫Aa ,eff a ,eff
∫A dAc + ζEac2828 ∫A
c
−(1 − γ RR , y )(zcen
a , eff
dAa ,eff
Ea 28
λ
z ' dAa ,eff
Ac
ζE c 28 ∫Aa ,eff
− z)
E
2
2
∫Ac z ' dAc + ζEac2828 ∫Aa,eff z ' dAa,eff
−(1 − γ RR ,z )( ycen − y )
δslip ∫ z ' dAc +
δslip ∫ δres y′dAc +
Ac
E
Ea 28
λ
y′dAa ,eff
ζE c 28 ∫Aa ,eff
∫A y′ dAc + ζEac2828 ∫A
2
c
(4)
a , eff
y ′2dAa ,eff
where
= cross-section area of the young concrete, [m2]
Ac
Ea28 = 28 days modulus of elasticity of the adjacent older concrete, [N/m2]
λ
= factor describing the volume change in the old concrete relatively the volume
change in the young concrete, λ = ∆εa0/∆εc0, [-]
Aa,eff = effective cross-section area of the adjacent older concrete, [m2]
y’
= internal vertical lever arm to the total centroid for each part, [m]
As can be seen in Eqs. (3) and (4), the restraint variation depends on the crosssection areas of the structure, the geometric properties of the young part (high wall
effects meaning non-linearly varying strain), possible slip failure in the joint between
the young and the old parts, the modulus of elasticity of the young and the older
concrete, and the translational and rotational boundary restraint situation from the
foundation material, see Figure 2.1 for a brief description.
8
The Semi-Analytical Method
Slip reduction
δslip
Decisive restraint
γR
y
Young concrete
∆ε0<0
Older concrete
High wall effects
δres
y
Ac, ζEc28
z
x
Aa,eff, Ea28
x
z
Boundary restraint, γRR,y, γRR,z, γRT
Figure 2.1 Brief description of the including parts in the model for determination of the restraint
variation.
2.2
Effects of high walls
In not all early age concrete structures, the strains over the height of the young parts
vary linearly during the deformation. In structures with low length to height ratios the
strains vary non-linearly, which here is defined as effects of high walls. This means that
plane sections do not remain plane under deformation and that a simple application of
the Compensation Plane theory does not hold properly. In order to take into account
the non-linearity so-called resilience functions can be used for the determination of the
strain variations and thereby the restraint. The resilience is here based on a basic resilience factor that applies for fully base restrained walls, that is γRT = γRR,z = 1. The
rotational boundary restraint for rotation around the vertical y-axis is without any
further investigations in this model considered being zero, γRR,y = 0. For other base
restraint situations than γRT = γRR,z = 1, resilience correction factors are used, see subsection 2.2.2. Basic resilience factors for fully base restrained walls have previously been
presented in e.g. ACI (1973), Figure 2.2a), and Emborg (1989), Figure 2.2b).
The resilience factor is here determined as
0
δres = δres
δtransl δrot
(5)
where
δ0res = basic resilience factor, [-]
δtransl = resilience correction factor for translational boundary restraint, [-]
δrot = resilience correction factor for rotational boundary restraint, [-]
2.2.1 Basic resilience factors
High wall effects, basic resilience, referrer to structures with low length to height ratio
and that are totally restrained at the base. In accordance with the presentation in Figure
9
Restraint Factors and Partial Coefficients
2.2, the basic resilience is applicable for structures with length to height ratio smaller
than e.g. about ten, L/Hc < 10, in Figure 2.2a), or about seven, L/Hc < 7, in Figure
2.2b).
y/Hc [-]
1.0
8 9 10
5
0.8
L/Hc
4
3
0.6
0.4
6
7
2
1
0.2
0.0
-0.2 0.0 0.2
a) ACI (1973)
0.4
0.6
δ0res [-]
1.0
0.8
y
y/Hc [-]
1.0
0.8
0.6
0.4
Hc
L
Martin
5
4
L/Hc
6
7
3
2
1.6
1.4
1.2
1.0
0.2
0.0
-0.2 0.0 0.2
b) Emborg (1989)
0.4
0.6
0.8
δ0res [-]
1.0
Figure 2.2 Basic resilience factor δ0res at different distance from the base as function of the length
to height ratio. Modified from a) ACI (1973) and b) Emborg (1989).
The basic resilience curves in Figure 2.2a) originate from test data according to ACI
(1973) whereas the curves in Figure 2.2b) where determined by elastic twodimensional FEM calculations. The curves for L/Hc > 2 in Figure 2.2a) suggest “more”
resilience than the curves in Figure 2.2b), which means that by using the basic resil-
10
The Semi-Analytical Method
ience curves according to Figure 2.2b), higher restraint values will be obtained and
thereby higher crack risks than by the curves in Figure 2.2a). Probably this is an effect
of that the ACI-curves are partly based on measurements, while the curves according
to Emborg (1989) are calculated with a theoretically completely restrained wall base.
For the moment, the exact background for the ACI-curves has not been able to study
thoroughly.
Since an analytical way of describing the resilience functions is needed here together
with a more wide range of the length to height ratios L/Hc (<1 and >7, respectively),
the curves from Emborg (1989) are here used as a base and are completed with more
FEM calculations. Then all resulting curves are fitted to a polynomial function expressed as, see Nilsson (2003),
n
 y 
δ0res = ∑ ai 
H c 
i =0 
i
(6)
2.2.2 Resilience correction factors
The resilience correction factors are introduced, as been indicated above in Eq. (5), for
cases in which the boundary restraint is not total. They are determined as
0
δtransl = γ RT + (1 − γ RT ) δtransl
0
δrot = γ RR ,z + (1 − γ RR ,z ) δrot
(7)
where δ0transl and δ0rot are basic resilience correction factors for translation and rotation,
respectively. See Nilsson (2003) for more details on the determination of the basic
correction factors. Eq. (7) in Eq. (5) gives
(
0
0
δres = δres
γ RT + (1 − γ RT ) δtransl
) ( γ RR,z + (1 − γ RR ,z ) δrot0 )
(8)
For case of free translation and free rotation, that is no boundary restraint γRT =
γRR,z = 0, Eq. (8) becomes
0 0
0
δres = δres
δtransl δrot
(9)
For no boundary restraint the 3D effect is considered only by introducing an effective width of the slab, Ba,eff, which means that formally δ0translδ0rot ≡ 1 in this case. So,
the basic resilience correction factor for rotation is determined as the inverse of the
basic resilience correction factor for translation by
11
Restraint Factors and Partial Coefficients
δ0rot =
1
0
δtransl
(10)
This relation is used in the determination of the basic correction factor for bending,
see Nilsson (2003).
In Figure 2.3 the principles of the resilience correction factors are presented. The
figure is drawn for a point above the joint in a structure where δ0transl < 1 and consequently δ0rot > 1, see Eq. (10). In another point of the structure the opposite situation
might occur, i.e. δ0transl > 1 and δ0rot < 1. For γRT = 0 free translation prevails, and γRT
= 1 means no translation, and for γRR,z = 0 a structure is free to rotate and for γRR,z =
1 no rotation prevails. When γRT = 0 and γRR,z = 1 structures are subjected to pure
translation, and the opposite, for γRT = 1 and γRR,z = 0 only pure rotation is possible.
For other values of γRT and γRR,z the resilience correction factors are determined
according to the lines between free and no translation and free and no rotation, respectively.
FEM calculations are used, see chapter 3, for evaluation of the points at γRR,z = 0
and γRR,z = 1 at free translation (γRT = 0) yielding the resilience correction factor δ0transl
(left part of Figure 2.3). With the prerequisite that δ0translδ0rot ≡ 1 the corresponding
value of δ0rot is obtained (the point where γRT = γRR,z = 0 in the right part of Figure
2.3). Interpolation along the lines is then possible which saves enormous amounts of
calculations when establishing the model of resilience correction factors.
12
The Semi-Analytical Method
δtransl
δrot
δ0translδ0rot=1
Free translation
1
γRT
1
1
Pure rotation
Pure translation
No translation
1
1
1 γRR,z
1
1
1 γRR,z
Total base restraint,
δres=δ0res
1
γRT
Figure 2.3 Model description of resilience correction factors dependent on the boundary restraint
situation.
For symmetrical structures (no rotation around the y-axis) subjected to total base
restraint, γRT = γRR,z = 1, no correction is needed and the restraint is determined only
by the basic resilience factor according to, Eqs. (4) and (8) with γRT = γRR,z = 1,
0
γ R = δres
δslip
(11)
which also is shown in Figure 2.3 in the corners of the cubes where γRT = γRR,z =
δtransl = 1 and γRT = γRR,z = δrot = 1.
2.3
Effects of slip failure in joints
The restraint situation and thereby the cracking risk during the cooling phase of the
young concrete depends to a large extent on the connecting joints between the young
and the old parts. In Figure 2.4, the stress distribution before and after a possible slip
failure in the joint is briefly described in a wall on a slab during the cooling phase. If a
joint is very strong and contains a large amount of through reinforcement it is capable
to transfer large restraining forces from the older part, Figure 2.4a), implying high
horizontal stresses in the mid-section of the structure as well as high vertical and shear
stresses at the ends of the joint. On the opposite, see Figure 2.4b), if the joint is weaker
and does not contain much trough reinforcement, slip failure is possible and thereby
the stresses in the mid-section and in the ends of the joint are reduced and consequently the risk of cracking. In longer structures, effects of possible slip failure in joints
13
Restraint Factors and Partial Coefficients
are not as obvious as in shorter structures since the middle parts of such structures do
not respond to stress changes at the ends of the joints.
Due to the difficulty in knowing whether a joint will crack or not, decisions regarding possible measures as cooling of walls and heating of slabs will be quite uncertain.
However, if a slip failure in a joint is predictable, a considerable amount of money can
be saved. A predicted slip failure in a joint can be regarded as a controlled form of
cracking and thereby a kind of measure, which is preferable compared to un-controlled
cracking somewhere else in the young concrete. The usage of joint sealers and injection hoses in the joints are fairly simple measures to avoid leakage through any cracked
joint.
Today the knowledge about the behaviour of the joints between young and older
parts is limited. It is hard to determine how large forces that are transferred across a
joint and to determine if any slip failure will occur. However, in Nilsson et al. (1999)
and Nilsson (2000) three medium-scale tests of wall on slab cases are presented where
slip failures where detected. It was found that failures start at the ends of the joints and
progress in small steps towards the centre of the walls. The steps correspond to the
distance between the reinforcement going through the joint from the slab to the wall.
Further, in Bernander (2001) it is shown by basic classic theory of elasticity that for the
typical case wall on slab for reasonable structure lengths the occurrence of slip failure in
the joint is possible and probable.
y
C
L
∆ε<0
Wall
Slab
C
L
x
C
L
High restraint
effects
σxmax
a) Before slip failure in joint
High restraint
effects
τ
σxmax
σy
slip failure
τ
σy
b) After slip failure in joint
Figure 2.4 Principal description of the stresses in a structure a) before and b) after a possible slip
failure in a joint. Nilsson (2000). By possible slip failure in the joint the horizontal stresses in
the mid-section are reduced as well as the vertical and shear stresses in the end of the joint.
14
The Semi-Analytical Method
The effects of possible slip failure in joints are regarded by the slip failure factor δslip.
This factor are smaller than 1 and therefore reduces the restraint values, see Eq. (4). For
more details about values of the slip failure factor, see Nilsson (2003) and ConTeSt Pro
(2003).
2.4
Applicable expressions
The model presented above in Eq. (4) is general and applies to any structure with a
younger structural element cast on an older. For the case of a rectangular wall cast on a
rectangular slab, simple, straightforward and sufficiently accurate semi-analytical expression can be derived, see Nilsson (2003).
A simpler application of the semi-analytical model is derived if no volume change
takes place in the slab, if slip failure in the joint is negligible, if plane sections remain
plane (no effects of high walls), and if no translational nor rotational boundary restraint
is present, γRT = γRR,x = γRR,y = 0. Then the expression for the determination of the
restraint variation is changed from a semi-analytical expression to an analytical expression. That is, it is simplified to an expression exactly fulfilling the Compensation Plane
method according to the linear elastic theory along the centre of the wall. The expression reads
γR = 1 −
−
H c2
12
(
+ ycen
1
Ea 28 Aa ,eff
1+
ζE c 28 Ac
(
)
( ycen − y ) ycen − H c
2
2
2
A
E

a ,eff H a
H
a 28
− c + ζE
+ ycen + H a

A
2
12
c 28
c 
2
)
(
) 
2
(12)
2
−
Bc2 
+
12  zcen
Ba ,eff − Bc 

 zcen − ω

2


2
Ea 28 Aa ,eff  Ba2,eff

Ba ,eff − Bc 
2
−ω
 + ζE


+
z
cen

2
c 28 Ac  12

where
Hc
Ha
Bc
Ba,eff
=
=
=
=
height of wall, [m]
height of slab, [m]
width of wall, [m]
effective width of slab, [m]
ω is a coefficient describing the location of the wall on the slab in the z-direction. If
ω = 0, the wall is located in the middle of the slab, if ω = ±1, the wall is located at one
15
Restraint Factors and Partial Coefficients
of the sides of the slab, and for other values of ω the wall is located somewhere between the middle and the sides of the slab. ω is determined as
ω=
2u
Ba − B c
(13)
where u is the real horizontal distance in the z-direction between the centre of the slab
and the centre of the wall, [m].
Another situation, somewhat more complicated compared to above, is present when
slip failure in the joint is possible and if sections do not remain plane under deformation
(high wall effects), then the restraint in the wall is determined as, see Nilsson (2003),
n
i
n
 y 
γ R = δslip ∑ ai 
− δslip
H c 
i =0 
−δslip
a
∑ i +i 1
i =0
Ba ,eff H a
E
1 + a 28
ζE c 28 Bc H c
n
n

a
a 
( ycen − y )  ycen ∑ i − H c ∑ i 
i +1
i + 2
i =0
i =0

2
Ba ,eff H a  H 2
H c2
E
H
Ha
a
+ ycen − c + a 28
12
ζE c 28 Bc H c  12 + ycen + 2
2
(
)
(
) 
2
(14)
2
−δslip
Bc2 
+
12  zcen
Ba ,eff − Bc 

 zcen − ω

2


2
Ea 28 Ba ,eff H a  Ba2,eff

Ba ,eff − Bc 
2
−ω
 + ζE


+ zcen
B
H
2

c 28
c c  12

A more comprehensive application formulation of the semi-analytical model is
found if the boundary restraint may vary. In such cases, the resilience factor has to be
determined according to Eq. (8) with the resilience correction factors. These factors
can be determined from elastic FEM calculations with a method presented in chapter 3.
The determination of the resilience correction factors in its formulation is quite simple
but due to mathematical/numerical reasons, limitations in the modelling arise.
2.5
Boundary restraint
As been introduced earlier, the boundary restraint is divided into one translational part
and two rotational parts. The foundation material adjacent to deforming early age
concrete structures restrains the free movements of the structures. Depending on the
type of foundation material, the degree of compaction, different modulus of the subgrade and the geometry of the body resting on the ground etc., the boundary restraint
varies.
16
The Semi-Analytical Method
The free translation of a structure is counteracted by the friction and/or cohesion
properties of the foundation material. A low friction and/or low-cohesive foundation
material gives almost no resistance to the free translation, which means that the translational boundary restraint is almost zero, γRT ≈ 0.
The bending moment during the cooling phase tends to rotate the ends of structures
upward and the centre parts downward. The stiffness of the foundation material prescribes how much a structure may bend down into the ground. A soft material offers
almost no resistance to the free deformation of the structure, implying non or very little
rotational bending restraint, γRR,z ≈ 0. On the contrary, a structure that is founded on a
stiff ground, for example rock or dense gravel, can hardly bend at all, that is, the rotational boundary restraint is about 100 percent, γRR,z ≈ 1. If the stiffness of the foundation material is high and/or the structure is relatively long, the structure rotates anyhow
but lifts at its ends and rests on the ground only at intermediate parts of the structure.
This lifting of the ends is hindered by the dead weight of the structure, see Nilsson
(2000).
For structures founded on blasted rock, the restraint situation is much more complex. Interlocking between the concrete and the rock and influence from existing crack
zones in the rock, determination of the degree of restraint is complicated. However,
preliminary studies of the restraint from rock are given in Olofsson et al. (2001).
The axial force γRTNRI and the bending moments γRR,yMRI,y and γRR,zMRI,z in Eq.
(3) are caused by the restraint/counteraction from adjacent older concrete members
and/or more or less elastic foundations on the free movements of young concrete
structures. The amount of translation and/or rotation depends partly on the length of
the structures, partly on the friction and/or cohesion properties and stiffness of the
foundation material. Methods for determination of the amount of boundary restraint
can be found in Pettersson (1998), Rostásy et al. (2001) and Bernander (2001) and for
the rotational boundary restraint an elastic approach can be found in Bernander (1993)
and Nilsson (2000), see also Nilsson (2003).
2.6
Effective width of slab
For a relatively thin and low wall cast on a very wide slab, it is obvious that not the
whole width of the slab influences the movements of the wall. In such cases, only a
certain part of the slab – here defined as the effective width – is active and should be
used in the determination of the restraint in the assumed model. In Nilsson (2000), a
small study on the effective width was presented. It was found in some examples analysed by FEM that in order to obtain the same curvature at the bottom of the wall, the
width of the slab used in the semi-analytical model had to be different than the real
widths.
In the recently finished Brite-Euram Project IPACS (Improved Production of
Advanced Concrete Structures), about 3000 elastic FEM calculations of the restraint
17
Restraint Factors and Partial Coefficients
variation in rectangular walls on slabs were performed, see chapter 3. Within the present work, the results from these calculations have been used to determine the effective
width of the slabs that is needed in the semi-analytical model, Eq. (4), in order to
obtain the same restraint variation as in the FEM calculations. Furthermore, from these
results a method for the determination of the effective width is developed for geometric properties different from the ones used in the FEM calculations, see Nilsson (2003).
2.7
Decisive point
The cracking risk has of cause to be estimated at the point in a structure where the
origin of a crack is most probable. In elastic analyses of wall on slab cases, the highest
values of the restraint are located at the bottom of the wall. However, this is not the
location of the most critical point in real structures. In a wall on slab structure, the heat
condition is non-uniformly distributed over the height the wall, that is, the heat is
lower at the top and bottom of the wall than in the middle. Therefore, the decisive
point is not located at the bottom of the wall where some cooling from the slab is
present. Hence, the most critical point is located somewhere above the joint, but the
location varies from case to case. In many cases one wall thickness above the joint is a
good estimation of the location of the decisive point for a crack risk analysis, see
Olofsson et al. (2001) and Figure 2.5.
Temperature
Restraint
Loading
⇒
&
Bc
Bc
γR
T
Figure 2.5 Principle description of location of desicive point.
18
Decisive
point
σ
FEM Calculations
3
Restraint Variation Determined by Finite Element
Method Calculations
3.1
Introduction
For the semi-analytical method presented in Nilsson (2003), finite element method
(FEM) calculations have been chosen as a method of verifying and adjusting the
method. Exactly 2920 elastic three-dimensional FEM calculations of the restraint variation in walls on slabs have been performed. The majority of the calculations were produced within the Brite-Euram Project IPACS (Improved Production of Advanced
Concrete Structures), and the rest of the calculations were performed within the development of the semi-analytical method, see below for detailed description.
In all FEM calculations, the geometry of the structure has been varied as well as the
rotational boundary restraint. The calculations were performed with the Dutch software DIANA. In the calculations, the temperature within the walls was lowered uniformly and linear elastic calculations were performed.
3.2
Input
The geometry of the wall on slab cases has been varied according to the following.
Three different widths, Ba, and three different heights, Ha, of the slab have been used as
well as three different widths, Bc, and five different heights of the wall, Hc. Four different lengths of the structure, L, and three different locations of the wall on the slab, ω,
have been studied. Further, two different rotational boundary restraint situations have
been regarded, γRR,z. All the parameters and their values are listed in Table 3.1. The
coefficient ω describes the relative location of the wall on the slab. ω = 0 means that
the wall is centrically located on the slab and ω = ±1 means the wall is located at one of
the edges of the slab.
15
Restraint Factors and Partial Coefficients - Part II
Table 3.1 List of parameters and their values used in the Finite Element Method calculations
of the elastic restraint variations in the walls of wall-on-slab structures.
Ba [m]
Ha [m]
Bc [m]
Hc [m]
L [m]
γRR [-]
ω [-]
2, 4, 8
0.4, 1, 1.4
0.3, 1, 1.8
0.5, 1, 2, 4, 8
3, 5, 10, 18
0, 1
0, 0.5, 1
Num. of values
3
3
3
5
4
2
3
Hc
Values
Bc
Ha
Parameter
L
Ba
γRR
3.2.1 Material properties
In the calculations the modulus of elasticity of the slab was Ea28 = 30⋅109 N/m2 and
of the wall ζEc28 = 27.9⋅109 N/m2. The Poisson’s ratio ν = 0.2, the density ρ = 2350
kg/m3 and the thermal dilation coefficient of the wall α = 1⋅10-5°C-1.
3.2.2 Finite elements
The calculations have been performed using the eight-node isoparametric solid brick
element named HX24L, which is based on linear interpolation and Gauß integration.
3.2.3 Boundary conditions
Due to symmetry, only half of the length of the structures was studied. For the cases
with the walls located at the centre of the slabs, only one fourth of the structures was
studied due to the double symmetry. For all calculations two different rotational
boundary restraint cases have been studied, viz. free rotation γRR,z = 0 and totally hindered rotation γRR,z = 1, see Figure 3.1.
16
FEM Calculations
a)
b)
Figure 3.1 Description of the constraints in the FEM calculations for the boundary conditions a)
free rotation, γRR,z = 0, and b) totally hindered rotation, γRR,z = 1.
3.2.4 Loading
In all calculations the walls were subjected to a uniform temperature decrease of 10°C.
The imposed temperature load was applied in each element as a momentary change of
the temperature.
3.3
Number of calculations
The total number of possible calculations is
ncalc , possible = 3 ⋅ 3 ⋅ 3 ⋅ 5 ⋅ 4 ⋅ 2 ⋅ 3 = 3240
but for ω = 0.5 not all possible cases were calculated. Excluded were the cases with 1
and 1.8 m wide walls on 2 m wide slabs as well as 1 and 1.8 m wide walls on 8 m wide
and 0.4 m thick slabs. The number of excluded calculations is
nexcl = ( 3! ⋅ 2! + 1! ⋅ 2! ) ⋅ 5! ⋅ 4! ⋅ 2! = 320
Ha
Bc
Ha
Bc
Hc
L γ RR
which gives the total number of calculations
ncalc , performed = ncalc , possible − nexcl = 3240 − 320 = 2920
3.4
Post processing
3.4.1 Restraint variation
From the FEM calculations, the variations of the strain in the nodes along the line of
symmetry (middle of the walls of the structures), εx, εy and εz, were stored. These
17
Restraint Factors and Partial Coefficients - Part II
strains have been used to determine the restraint variation by calculating the ratio between the stresses in the longitudinal direction and the one-dimensional stress at total
fixation of the applied temperature decrease according to
ζE c 28 
ζE α∆T
ν
εx + ε y + εz )  − c 28
(
 εz +
1+ ν 
1 − 2ν
1 − 2ν

γR = 0 =
ζ
E
α∆
T
σ
c 28
σzz
(1)
This formula implies that if no external strains are present (εx = εy = εz = 0) the restraint is exactly 1/(1-2ν) = 1.67. Normally the maximum amount of restraint is defined being exactly 1, but with three-dimensional FEM calculations as reference to a
method applicable for one-dimensional crack risk analyses the restraint may be larger
than 1 depending on the Poisson’s ratio.
The values of the restraint determined by the FEM calculations have been plotted in
diagrams as function of the relative distance above the joint between the walls and
slabs, see Nilsson (2003). In each diagram, structures with the same cross-section but
different lengths and rotational boundary restraint situations are presented.
3.4.2 Diagrams of restraint variation
Below, the restraint distribution for each elastic FEM calculation is presented.
Figure 3.2 – Figure 3.10: Ba = 2m, Ha = 0.4, 1 and 1.4m, Bc = 0.3, 1 and 1.8m, ω = 0.
Figure 3.11 – Figure 3.19: Ba = 4m, Ha = 0.4, 1 and 1.4m, Bc = 0.3, 1 and 1.8m, ω =
0.
Figure 3.20 – Figure 3.28: Ba = 8m, Ha = 0.4, 1 and 1.4m, Bc = 0.3, 1 and 1.8m, ω =
0.
Figure 3.29 – Figure 3.31: Ba = 2m, Ha = 0.4, 1 and 1.4m, Bc = 0.3m, ω = 0.5.
Figure 3.32 – Figure 3.40: Ba = 4m, Ha = 0.4, 1 and 1.4m, Bc = 0.3, 1 and 1.8m, ω =
0.5.
Figure 3.41: Ba = 8m, Ha = 0.4m, Bc = 0.3m, ω = 0.5.
Figure 3.42 – Figure 3.47: Ba = 8m, Ha = 1 and 1.4m, Bc = 0.3, 1 and 1.8m, ω = 0.5.
Figure 3.48 – Figure 3.56: Ba = 2m, Ha = 0.4, 1 and 1.4m, Bc = 0.3, 1 and 1.8m, ω =
1.
Figure 3.57 – Figure 3.65: Ba = 4m, Ha = 0.4, 1 and 1.4m, Bc = 0.3, 1 and 1.8m, ω =
1.
Figure 3.66 – Figure 3.74: Ba = 4m, Ha = 0.4, 1 and 1.4m, Bc = 0.3, 1 and 1.8m, ω =
1.
18
FEM Calculations
B a = 2m, H a = 0.4m, B c = 0.3m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.2 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
19
Restraint Factors and Partial Coefficients
B a = 2m, H a = 0.4m, B c = 1m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.3 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
20
FEM Calculations
B a = 2m, H a = 0.4m, B c = 1.8m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.4 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
21
Restraint Factors and Partial Coefficients
B a = 2m, H a = 1m, B c = 0.3m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.5 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
22
FEM Calculations
B a = 2m, H a = 1m, B c = 1m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.6 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
23
Restraint Factors and Partial Coefficients
B a = 2m, H a = 1m, B c = 1.8m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.7 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
24
FEM Calculations
B a = 2m, H a = 1.4m, B c = 0.3m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.8 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
25
Restraint Factors and Partial Coefficients
B a = 2m, H a = 1.4m, B c = 1m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.9 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
26
FEM Calculations
B a = 2m, H a = 1.4m, B c = 1.8m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.10 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
27
Restraint Factors and Partial Coefficients
B a = 4m, H a = 0.4m, B c = 0.3m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.11 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
28
FEM Calculations
B a = 4m, H a = 0.4m, B c = 1m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.12 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
29
Restraint Factors and Partial Coefficients
B a = 4m, H a = 0.4m, B c = 1.8m and ω = 0.5
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.13 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
30
FEM Calculations
B a = 4m, H a = 1m, B c = 0.3m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.14 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
31
Restraint Factors and Partial Coefficients
B a = 4m, H a = 1m, B c = 1m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.15 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
32
FEM Calculations
B a = 4m, H a = 1m, B c = 1.8m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.16 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
33
Restraint Factors and Partial Coefficients
B a = 4m, H a = 1.4m, B c = 0.3m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.17 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
34
FEM Calculations
B a = 4m, H a = 1.4m, B c = 1m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.18 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
35
Restraint Factors and Partial Coefficients
B a = 4m, H a = 1.4m, B c = 1.8m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.19 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
36
FEM Calculations
B a = 8m, H a = 0.4m, B c = 0.3m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.20 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
37
Restraint Factors and Partial Coefficients
B a = 8m, H a = 0.4m, B c = 1m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.21 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
38
FEM Calculations
B a = 8m, H a = 0.4m, B c = 1.8m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.22 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
39
Restraint Factors and Partial Coefficients
B a = 8m, H a = 1m, B c = 0.3m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.23 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
40
FEM Calculations
B a = 8m, H a = 1m, B c = 1m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.24 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
41
Restraint Factors and Partial Coefficients
B a = 8m, H a = 1m, B c = 1.8m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.25 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
42
FEM Calculations
B a = 8m, H a = 1.4m, B c = 0.3m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.26 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
43
Restraint Factors and Partial Coefficients
B a = 8m, H a = 1.4m, B c = 1m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.27 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
44
FEM Calculations
B a = 8m, H a = 1.4m, B c = 1.8m and ω = 0
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.28 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
45
Restraint Factors and Partial Coefficients
B a = 2m, H a = 0.4m, B c = 0.3m and ω = 0.5
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.29 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
46
FEM Calculations
B a = 2m, H a = 1m, B c = 0.3m and ω = 0.5
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.30 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
47
Restraint Factors and Partial Coefficients
B a = 2m, H a = 1.4m, B c = 0.3m and ω = 0.5
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.31 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
48
FEM Calculations
B a = 4m, H a = 0.4m, B c = 0.3m and ω = 0.5
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.32 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
49
Restraint Factors and Partial Coefficients
B a = 4m, H a = 0.4m, B c = 1m and ω = 0.5
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.33 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
50
FEM Calculations
B a = 4m, H a = 0.4m, B c = 1.8m and ω = 0.5
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.34 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
51
Restraint Factors and Partial Coefficients
B a = 4m, H a = 1m, B c = 0.3m and ω = 0.5
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.35 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
52
FEM Calculations
B a = 4m, H a = 1m, B c = 1m and ω = 0.5
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.36 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
53
Restraint Factors and Partial Coefficients
B a = 4m, H a = 1m, B c = 1.8m and ω = 0.5
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.37 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
54
FEM Calculations
B a = 4m, H a = 1.4m, B c = 0.3m and ω = 0.5
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.38 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
55
Restraint Factors and Partial Coefficients
B a = 4m, H a = 1.4m, B c = 1m and ω = 0.5
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.39 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
56
FEM Calculations
B a = 4m, H a = 1.4m, B c = 1.8m and ω = 0.5
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.40 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
57
Restraint Factors and Partial Coefficients
B a = 8m, H a = 0.4m, B c = 0.3m and ω = 0.5
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.41 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
58
FEM Calculations
B a = 8m, H a = 1m, B c = 0.3m and ω = 0.5
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.42 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
59
Restraint Factors and Partial Coefficients
B a = 8m, H a = 1m, B c = 1m and ω = 0.5
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.43 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
60
FEM Calculations
B a = 8m, H a = 1m, B c = 1.8m and ω = 0.5
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.44 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
61
Restraint Factors and Partial Coefficients
B a = 8m, H a = 1.4m, B c = 0.3m and ω = 0.5
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.45 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
62
FEM Calculations
B a = 8m, H a = 1.4m, B c = 1m and ω = 0.5
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.46 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
63
Restraint Factors and Partial Coefficients
B a = 8m, H a = 1.4m, B c = 1.8m and ω = 0.5
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.47 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
64
FEM Calculations
B a = 2m, H a = 0.4m, B c = 0.3m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.48 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
65
Restraint Factors and Partial Coefficients
B a = 2m, H a = 0.4m, B c = 1m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.49 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
66
FEM Calculations
B a = 2m, H a = 0.4m, B c = 1.8m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.50 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
67
Restraint Factors and Partial Coefficients
B a = 2m, H a = 1m, B c = 0.3m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.51 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
68
FEM Calculations
B a = 2m, H a = 1m, B c = 1m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.52 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
69
Restraint Factors and Partial Coefficients
B a = 2m, H a = 1m, B c = 1.8m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.53 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
70
FEM Calculations
B a = 2m, H a = 1.4m, B c = 0.3m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.54 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
71
Restraint Factors and Partial Coefficients
B a = 2m, H a = 1.4m, B c = 1m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.55 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
72
FEM Calculations
B a = 2m, H a = 1.4m, B c = 1.8m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.56 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
73
Restraint Factors and Partial Coefficients
B a = 4m, H a = 0.4m, B c = 0.3m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.57 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
74
FEM Calculations
B a = 4m, H a = 0.4m, B c = 1m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.58 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
75
Restraint Factors and Partial Coefficients
B a = 4m, H a = 0.4m, B c = 1.8m and ω = 0.1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.59 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
76
FEM Calculations
B a = 4m, H a = 1m, B c = 0.3m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.60 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
77
Restraint Factors and Partial Coefficients
B a = 4m, H a = 1m, B c = 1m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.61 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
78
FEM Calculations
B a = 4m, H a = 1m, B c = 1.8m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.62 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
79
Restraint Factors and Partial Coefficients
B a = 4m, H a = 1.4m, B c = 0.3m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.63 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
80
FEM Calculations
B a = 4m, H a = 1.4m, B c = 1m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.64 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
81
Restraint Factors and Partial Coefficients
B a = 4m, H a = 1.4m, B c = 1.8m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.65 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
82
FEM Calculations
B a = 8m, H a = 0.4m, B c = 0.3m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.66 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
83
Restraint Factors and Partial Coefficients
B a = 8m, H a = 0.4m, B c = 1m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.67 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
84
FEM Calculations
B a = 8m, H a = 0.4m, B c = 1.8m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.68 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
85
Restraint Factors and Partial Coefficients
B a = 8m, H a = 1m, B c = 0.3m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.69 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
86
FEM Calculations
B a = 8m, H a = 1m, B c = 1m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.70 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
87
Restraint Factors and Partial Coefficients
B a = 8m, H a = 1m, B c = 1.8m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.71 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
88
FEM Calculations
B a = 8m, H a = 1.4m, B c = 0.3m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.72 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
89
Restraint Factors and Partial Coefficients
B a = 8m, H a = 1.4m, B c = 1m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.73 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
90
FEM Calculations
B a = 8m, H a = 1.4m, B c = 1.8m and ω = 1
0.8
0.8
y /Hc [-]
b) H c = 1m
1
y /Hc [-]
a) H c = 0.5m
1
0.6
0.4
0.6
0.4
0.2
0.2
0
-0.6 -0.2
0
-0.6 -0.2
1
1.4
1
d) H c = 4m
1
0.8
0.8
y /Hc [-]
y /Hc [-]
c) H c = 2m
0.2 0.6
γR [-]
0.6
0.4
0
-0.6 -0.2
0
-0.6 -0.2
e) H c = 8m
1.4
1
γRR
L
L
L
L
y /Hc [-]
0.8
0.6
0.4
1.4
0.2 0.6
γR [-]
1
1.4
0.4
0.2
1
1
0.6
0.2
0.2 0.6
γR [-]
0.2 0.6
γR [-]
=0
= 3m
= 5m
= 10m
= 18m
γRR
L
L
L
L
=1
= 3m
= 5m
= 10m
= 18m
0.2
0
-0.6 -0.2
0.2 0.6
γR [-]
1
1.4
Figure 3.74 Restraint variation for a) H c = 0.5, b) H c = 1, c) H c = 2, d) H c = 4 and e)
H c = 8m
91
High Wall Effects
4
High Wall Effects
4.1
Introduction
In Nilsson (2003) the concept of resilience were thoroughly described. In this paper
additional basic resilience curves will be presented that are needed for adjustment of the
semi-analytical method to the reference restraint variations determined by the FEM
calculations.
Below, the determination of four new basic resilience curves for base restrained
walls for L/Hc = 0.25, 0.5 and L/Hc = 10 and 40 are presented. The determination of
the new basic resilience curves has been performed by elastic 2D finite element method
calculations by use of the Dutch software DIANA.
4.2
Input
4.2.1 Parameters
In the calculations the length of the wall has been varied, L = 0.25, 0.5, 10 and 40m,
while the height was kept constant, Hc = 1m.
4.2.2 Material properties
The modulus of elasticity of the wall was ζEc28 = 22.5⋅109 Pa. The Poisson’s ratio was
ν = 0.2, the density ρ = 2350 kg/m3 and the thermal dilation coefficient α = 1⋅10-5°C1
.
4.2.3 Finite elements
The calculations were performed using eight-node quadrilateral isoparametric plane
stress elements named CQ16M, which is based on quadratic interpolation and Gauß
integration.
93
Restraint Factors and Partial Coefficients
4.2.4 Boundary conditions
Due to symmetry, only half of the lengths of the walls were studied. The base of the
wall was totally restrained both for bending and for translation, that is γRT = γRR,z = 1,
see Fel! Hittar inte referenskälla..
Hc=1m
CL
L=10m
Figure 4.1 Principle description of boundary conditions in the calculations, here γRR,z = 1 for
L/Hc = 10.
4.2.5 Loading
In all calculations the walls were subjected to a temperature decrease of magnitude
10°C. The temperature load was applied in each element as a momentary change.
4.3
Output
4.3.1 Strains
From the FEM calculations, the variations of the strain in the nodes along the line of
symmetry (middle of the walls of the structures), εx, εy and εz, were stored.
4.4
Post processing
4.4.1 Restraint variation
In the same way as for the FEM calculations of the restraint variation, see chapter 3, the
variations of the strain in the nodes along the line of symmetry (middle of the walls of
the structures), εx, εy and εz, were stored. The strains in each node in the mid-section
of the wall were then used to determine the restraint distribution. The determination is
done by calculating the ratio between the stresses in the longitudinal direction and the
free thermal stress of the applied temperature decrease according to
94
High Wall Effects
ζE c 28 
ζE α∆T
ν
( εx + εy + εz )  − 1c 28− 2ν
ε +
1 + ν  z 1 − 2ν
γR = 0 =
ζE c 28α∆T
σ
σzz
4.5
(3.1)
Result
The values of the restraint are then been plotted as function of the relative distances
above the bottom the walls, Figure 4.2, that is as new resilience curves.
y/Hc [-]
1.0
5
4
L/Hc
0.8
6
7
10
3
40
2
0.6
1.6
1.4
1.2
1.0
0.4
0.5
0.2
Martin 0.25
-0.2
0.0
0.2
0.4
0.6
0.8
δres [-]
1.0
Figure 4.2 Completed basic resilience factor δ0res, marked with bold lines, for L/Hc = 0.25,
0.5, 10 and 40 as function of the relative distance above the joint. Existing curves, marked with
thin lines, according to Emborg (1989).
For application purposes in the semi-analytical method the curves in Fel! Hittar
inte referenskälla. can be described by polynomials of seventh order according to
 y
δ0res 
 Hc

 y
 = a0 + a1  H

 c
2
3

 y 
 y 
 y 
 + a2  H  + a3  H  + a4  H 

 c
 c
 c
5
6
7
7
 y 
 y 
 y 
 y 
+ a5 
+ a6 
+ a7 
= ∑ ai 




 Hc 
 Hc 
 Hc 
i =0  H c 
with coefficients according to Table 4.1.
95
i
4
(2)
Restraint Factors and Partial Coefficients
Table 4.1 Coefficients for polynomials describing the resilience curves in Figure 4.2.
L/Hc
a0
a1
a2
0.25
0.5
1
1.2
1.4
1.6
2
3
4
5
6
7
10
40
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-14.43
-5.961
-3.112
-2.392
-1.907
-1.690
-1.238
-0.912
-0.641
-0.387
-0.206
-0.185
-0.019
0
73.76
7.786
1.913
0.227
-0.362
-0.304
-0.541
-0.041
0.131
0.036
-0.197
0.222
0.007
0
a3
Coefficients
a4
-167.6
152.5
12.14
-30.91
1.863
-1.746
3.072
-2.038
2.700
-1.588
1.907
-1.062
1.158
-0.441
0.189
0.054
0.026
0.063
0.132
-0.031
0.455
-0.202
-0.253
0.127
0
0
0
0
96
a5
a6
19.93
5.168
0
0
0
0
0
0
0
0
0
0
0
0
-117.5
24.24
0
0
0
0
0
0
0
0
0
0
0
0
a7
52.36
-13.46
0
0
0
0
0
0
0
0
0
0
0
0
Effective Width of Slab
5
Effective Width of Slab
5.1
Introduction
The effective width of adjacent slabs in early-age concrete wall-on-slab structures form
a basis in the semi-analytical method for the determination of the restraint variation in
walls in the wall-on-slab structures, see Nilsson (2003). The conception of effective
width of slab has been introduced in order to obtain the same restraint variation in
early-age concrete walls on slabs determined by the semi-analytical method as by 3D
elastic FEM calculations, see chapter 3. Below, values of the so-called effective width,
Ba,eff, corresponding to the values form the FEM-calculations are presented together
with coefficients for polynomials describing the variation of these values.
5.2
Determination of effective width of slab
The determination of the effective width of slab Ba,eff has been performed by varying
the width of the slab in the semi-analytical method and comparing the restraint distribution with the results from the FEM calculations. Using the least square method for
the difference in the restraint values over the height of the wall is expressed by
 n

 n
2
min  ∑ ∆i2  = min  ∑ ( γ R ,FEM ,i − γ R , semi,i ) 
 i =1 
 i =1

(4.1)
where i = 1, 2, … n = number of nodes in the FEM calculations. The values of the effective width of slab for all the different geometries in the FEM calculations, see above,
have been gathered in groups of nine structures with the same real width of the slab,
Ba, the same height of the wall, Hc, and the same length of the structure, L. The values
in each group were then sorted by a parameter Χ calculated as
97
Restraint Factors and Partial Coefficients
Χ=
ζE c 28 Ac L
0.93E c 28Bc H c L
BL
=
= 0.93 c
Ea 28 Aa H c
Ea 28Ba H a H c
Ba H a
(4.2)
where 0.93 is the value of the 28-days value of the modulus of elasticity of the concrete
in the wall. The 28-days values of the modulus of elasticity in the walls and the slabs
are set to be equal.
5.3
Modelling of effective width of slab
By plotting the obtained values of the effective width as a function of the parameter Χ,
fairly clear relations for each group of nine values of the effective width have been
found. In order to get applicable relations for the determination of the effective width
when using the semi-analytical method, functions based on polynomials of fourth order
according to
Ba ,eff = b0 + b1Χ + b2 Χ 2 + b3 Χ 3
(4.3)
have been established for the groups of nine values of the effective width. The establishment of the polynomials was done by use of regression analysis, where b0 – b3 are
the coefficients for the polynomials, see below.
5.4
Results
In the tables presented below, both the evaluated values of the effective width, and the
corresponding values obtained by the polynomials for the same values of Χ, are plotted.
In addition, the coefficients b0 – b3 of the polynomials for each set of data are given together with the numerical values of the effective width behind each graph.
The most values of the effective width have been possible to use in the determination of the polynomials, however, not all. Some of the values have been too far away
from the rest of the values in the groups of nine. Therefore, these values have been excluded in the establishing of the polynomials. The reason for this might originate from
the iteration procedure when determining the values of the effective width of slab. In
some cases several values of the effective width of slab gave almost the same restraint
variations and which of the values to choose have not been studied in detail within this
work.
Table 5.1 – Table 5.20: Ba = 2m, Hc = 0.5, 1, 2, 4 and 8m, L = 3, 5, 10 and 18m.
Table 5.21 – Table 5.40: Ba = 4m, Hc = 0.5, 1, 2, 4 and 8m, L = 3, 5, 10 and 18m.
Table 5.41 – Table 5.60: Ba = 8m, Hc = 0.5, 1, 2, 4 and 8m, L = 3, 5, 10 and 18m.
98
Effective Width of Slab
Table 5.1 Effective width for B a = 2m, H c = 0.5m and L = 3m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
Bc
[m]
1.8
1
1.8
1.8
1
0.3
1
0.3
0.3
H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [-]
[m]
[-]
[-]
[m]
[m]
0.5
3
0 3.24542 1.62271 6.2775 3.24846
0.003
0.5
3
0 2.89577 1.44788 3.4875 2.86252
-0.033
0.5
3
0 2.74028 1.37014
2.511 2.7513
0.011
0.5
3
0 2.44865 1.22433 1.79357 2.54042
0.092
0.5
3
0 2.29736 1.14868
1.395 2.35258
0.055
0.5
3
0 2.4176 1.2088 1.04625 2.1372
-0.280
0.5
3
0 1.97473 0.98736 0.99643 2.10215
0.127
0.5
3
0 1.72197 0.86099 0.4185 1.60833
-0.114
0.5
3
0 1.34576 0.67288 0.29893 1.48455
0.139
s ∆Ba,eff [m] =
0.13
b 0 = 1.139541 b 1 = 1.24226
-
b 2 = -0.30225 b 3 = 0.02515
-
3.5
3
2.5
Ba,eff [m]
Ba
[m]
2
2
2
2
2
2
2
2
2
2
1.5
1
0.5
0
0
1
2
3
4
X [-]
◆
B a,eff
■ B a,eff,calc
99
5
6
7
Restraint Factors and Partial Coefficients
Table 5.2 Effective width for B a = 2m, H c = 0.5m and L = 5m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
Bc
[m]
1.8
1
1.8
1.8
1
0.3
1
0.3
0.3
H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [-]
[m]
[-]
[-]
[m]
[m]
0.5
5
0 3.03101 1.5155 10.4625 3.03213
0.001
0.5
5
0 2.63727 1.31864 5.8125 2.61805
-0.019
0.5
5
0 2.46593 1.23296
4.185 2.50736
0.041
0.5
5
0 2.43446 1.21723 2.98929 2.38981
-0.045
0.5
5
0 2.22899 1.1145
2.325 2.30314
0.074
0.5
5
0 2.27501 1.13751 1.74375 2.21144
-0.064
0.5
5
0 2.19925 1.09963 1.66071
2.197
-0.002
0.5
5
0 2.01996 1.00998 0.6975 2.00175
-0.018
0.5
5
0 1.92378 0.96189 0.49821 1.95445
0.031
s ∆Ba,eff [m] =
0.04
b 0 = 1.824742 b 1 = 0.277406 b 2 = -0.0352
-
b 3 = 0.001884 -
3.5
3
2.5
Ba,eff [m]
Ba
[m]
2
2
2
2
2
2
2
2
2
2
1.5
1
0.5
0
0
◆
2
B a,eff
4
6
X [-]
■ B a,eff,calc
100
8
10
12
Effective Width of Slab
Table 5+B116.3 Effective width for B a = 2m, H c = 0.5m and L = 10m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
Bc
[m]
1.8
1
1.8
1.8
1
0.3
1
0.3
0.3
H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [-]
[m]
[-]
[-]
[m]
[m]
0.5 10
0 3.69881 1.84941 20.925 3.70441
0.006
0.5 10
0 3.09692 1.54846 11.625 3.00279
-0.094
0.5 10
0 2.52698 1.26349
8.37 2.68401
0.157
0.5 10
0 2.46147 1.23074 5.97857 2.48067
0.019
0.5 10
0 2.31342 1.15671
4.65 2.3884
0.075
0.5 10
0 2.61983 1.30991 3.4875 2.32364
-0.296
0.5 10
0 2.28047 1.14023 3.32143 2.31577
0.035
0.5 10
0 2.19827 1.09914
1.395 2.2532
0.055
0.5 10
0 2.20869 1.10434 0.99643 2.24749
0.039
s ∆Ba,eff [m] =
0.13
b 0 = 2.245345 b 1 = -0.00691 b 2 = 0.00937
-
b 3 = -0.00027 -
4
3.5
3
Ba,eff [m]
Ba
[m]
2
2
2
2
2
2
2
2
2
2.5
2
1.5
1
0.5
0
0
5
10
15
X [-]
◆
B a,eff
■ B a,eff,calc
101
20
25
Restraint Factors and Partial Coefficients
Table 5.4 Effective width for B a = 2m, H c = 0.5m and L = 18m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
Bc
[m]
1.8
1
1.8
1.8
1
0.3
1
0.3
0.3
H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [-]
[m]
[-]
[-]
[m]
[m]
0.5 18
0 3.01448 1.50724 37.665 3.02547
0.011
0.5 18
0 3.63595 1.81797 20.925 3.41779
-0.218
0.5 18
0 2.60019 1.3001 15.066 2.96013
0.360
0.5 18
0 2.48637 1.24318 10.7614 2.62007
0.134
0.5 18
0 2.37148 1.18574
8.37 2.46267
0.091
0.5 18
0 3.04091 1.52045 6.2775 2.35693
-0.684
0.5 18
0 2.29814 1.14907 5.97857 2.34483
0.047
0.5 18
0 2.1089 1.05445
2.511 2.27164
0.163
0.5 18
0 2.20298 1.10149 1.79357 2.27417
0.071
s ∆Ba,eff [m] =
0.30
b 0 = 2.311405 b 1 = -0.03381 b 2 = 0.00757
-
b 3 = -0.00016 -
4
3.5
3
Ba,eff [m]
Ba
[m]
2
2
2
2
2
2
2
2
2
2.5
2
1.5
1
0.5
0
0
◆
10
B a,eff
20
X [-]
■ B a,eff,calc
102
30
40
Effective Width of Slab
Table 5.5 Effective width for B a = 2m, H c = 1m and L = 3m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff
[m] [m] [m] [-]
[m]
1.8
1
3
0 3.90967
1
1
3
0 1.86808
1.8
1
3
0 5.64842
1.8
1
3
0 4.68473
1
1
3
0 4.71189
0.3
1
3
0 4.25136
1
1
3
0 3.75758
0.3
1
3
0 3.09827
0.3
1
3
0 2.24717
B a,eff /B a
[-]
1.95484
0.93404
2.82421
2.34236
2.35595
2.12568
1.87879
1.54913
1.12359
B a,eff,calc ∆B a,eff
X
[-]
[m]
[m]
6.2775 3.91165
0.002
3.4875 5.67273
2.511 5.52962
-0.119
1.79357 5.0148
0.330
1.395 4.55173
-0.160
1.04625 4.03167
-0.220
0.99643 3.94817
0.191
0.4185 2.80086
-0.297
0.29893 2.52061
0.273
s ∆Ba,eff [m] =
0.24
b 0 = 1.752022 b 1 = 2.737161 b 2 = -0.56408 b 3 = 0.029129 -
6
5
Ba,eff [m]
Ba
[m]
2
2
2
2
2
2
2
2
2
4
3
2
1
0
0
1
2
3
4
X [-]
◆
B a,eff
■ B a,eff,calc
103
5
6
7
Restraint Factors and Partial Coefficients
Table 5.6 Effective width for B a = 2m, H c = 1m and L = 5m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
1
5
0 7.03825 3.51913 10.4625 7.05253
0.014
1
1
5
0 4.73037 2.36519 5.8125 4.5099
-0.220
1.8
1
5
0 3.2207 1.61035
4.185 3.60093
0.380
1.8
1
5
0 2.98128 1.49064 2.98929 3.04483
0.064
1
1
5
0 2.90539 1.45269
2.325 2.79693
-0.108
0.3
1
5
0 2.75656 1.37828 1.74375 2.62405
-0.133
1
1
5
0 2.73991 1.36995 1.66071 2.60305
-0.137
0.3
1
5
0 2.47474 1.23737 0.6975 2.4347
-0.040
0.3
1
5
0 2.23879 1.1194 0.49821 2.41851
0.180
s ∆Ba,eff [m] =
0.19
b 0 = 2.408709 b 1 = -0.02592 b 2 = 0.093841 b 3 = -0.00468 -
8
7
6
Ba,eff [m]
Ba
[m]
2
2
2
2
2
2
2
2
2
5
4
3
2
1
0
0
2
◆
B a,eff
4
6
X [-]
■ B a,eff,calc
104
8
10
12
Effective Width of Slab
Table 5.7 Effective width for B a = 2m, H c = 1m and L = 10m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
1 10
0 6.43561 3.21781 20.925 6.45541
0.020
1
1 10
0 4.48492 2.24246 11.625 4.1769
-0.308
1.8
1 10
0 2.61215 1.30608
8.37 3.11613
0.504
1.8
1 10
0 2.35392 1.17696 5.97857 2.5111
0.157
1
1 10
0 2.36406 1.18203
4.65 2.2834
-0.081
0.3
1 10
0 2.63462 1.31731 3.4875
2.166
-0.469
1
1 10
0 2.23571 1.11785 3.32143 2.15623
-0.079
0.3
1 10
0 2.1083 1.05415
1.395 2.18759
0.079
0.3
1 10
0 2.05838 1.02919 0.99643 2.23035
0.172
s ∆Ba,eff [m] =
0.28
b 0 = 2.397641 b 1 = -0.21297 b 2 = 0.046537 b 3 = -0.00129 -
7
6
5
Ba,eff [m]
Ba
[m]
2
2
2
2
2
2
2
2
2
4
3
2
1
0
0
5
10
15
X [-]
◆
B a,eff
■ B a,eff,calc
105
20
25
Restraint Factors and Partial Coefficients
Table 5.8 Effective width for B a = 2m, H c = 1m and L = 18m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
1 18
0 6.68589 3.34295 37.665 6.71047
0.025
1
1 18
0 5.02599
2.513 20.925 4.60431
-0.422
1.8
1 18
0 2.7087 1.35435 15.066 3.3924
0.684
1.8
1 18
0 2.39378 1.19689 10.7614 2.67461
0.281
1
1 18
0 2.44489 1.22245
8.37 2.39527
-0.050
0.3
1 18
0 3.14314 1.57157 6.2775 2.24392
-0.899
1
1 18
0 2.26748 1.13374 5.97857 2.23034
-0.037
0.3
1 18
0 2.03464 1.01732
2.511 2.24133
0.207
0.3
1 18
0 2.09999 1.04999 1.79357 2.2861
0.186
s ∆Ba,eff [m] =
0.45
b 0 = 2.46929
-
b 1 = -0.13178 b 2 = 0.017031 b 3 = -0.00028 -
8
7
6
Ba,eff [m]
Ba
[m]
2
2
2
2
2
2
2
2
2
5
4
3
2
1
0
0
10
◆
B a,eff
20
X [-]
■ B a,eff,calc
106
30
40
Effective Width of Slab
Table 5.9 Effective width for B a = 2m, H c = 2m and L = 3m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
2
3
0 2.09094 1.04547 6.2775 2.09682
0.006
1
2
3
0 1.21522 0.60761 3.4875 1.13198
-0.083
1.8
2
3
0 0.60195 0.30097
2.511 0.71576
0.114
1.8
2
3
0 0.3603 0.18015 1.79357 0.45392
0.094
1
2
3
0 0.35543 0.17771
1.395 0.33649
-0.019
0.3
2
3
0 0.46709 0.23355 1.04625 0.25505
-0.212
1
2
3
0 0.22257 0.11128 0.99643 0.24525
0.023
0.3
2
3
0 0.15217 0.07608 0.4185 0.16943
0.017
0.3
2
3
0 0.10231 0.05116 0.29893 0.16326
0.061
s ∆Ba,eff [m] =
0.10
b 0 = 0.163725 b 1 = -0.04109 b 2 = 0.136131 b 3 = -0.01283 -
2.5
2
Ba,eff [m]
Ba
[m]
2
2
2
2
2
2
2
2
2
1.5
1
0.5
0
0
1
2
3
4
X [-]
◆
B a,eff
■ B a,eff,calc
107
5
6
7
Restraint Factors and Partial Coefficients
Table 5.10 Effective width for B a = 2m, H c = 2m and L = 5m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
2
5
0 2.66981 1.33491 10.4625 2.67628
0.006
1
2
5
0 1.95815 0.97907 5.8125 1.87526
-0.083
1.8
2
5
0 1.25009 0.62505
4.185 1.31969
0.070
1.8
2
5
0 0.72711 0.36356 2.98929 0.93689
0.210
1
2
5
0 0.75888 0.37944
2.325 0.75338
-0.006
0.3
2
5
0 1.09754 0.54877 1.74375 0.61804
-0.479
1
2
5
0 0.44718 0.22359 1.66071 0.60097
0.154
0.3
2
5
0 0.32347 0.16174 0.6975 0.45177
0.128
0.3
2
5
0 3.76753 1.88377 0.49821 0.43347
s ∆Ba,eff [m] =
0.22
b 0 = 0.409204 b 1 = 0.016266 b 2 = 0.067394 b 3 = -0.00461 -
4
3.5
3
Ba,eff [m]
Ba
[m]
2
2
2
2
2
2
2
2
2
2.5
2
1.5
1
0.5
0
0
◆
2
B a,eff
4
6
X [-]
■ B a,eff,calc
108
8
10
12
Effective Width of Slab
Table 5.11 Effective width for B a = 2m, H c = 2m and L = 10m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff
[m] [m] [m] [-]
[m]
1.8
2 10
0 3.07809
1
2 10
0 2.58295
1.8
2 10
0 3.97886
1.8
2 10
0 3.64951
1
2 10
0 2.89014
0.3
2 10
0 2.14035
1
2 10
0 2.72705
0.3
2 10
0 2.46441
0.3
2 10
0 2.48734
B a,eff /B a
B a,eff,calc ∆B a,eff
X
[-]
[-]
[m]
[m]
1.53905 20.925 3.07543
-0.003
1.29147 11.625 5.22688
1.98943
8.37 4.13196
0.153
1.82476 5.97857 3.27637
-0.373
1.44507
4.65 2.87786
-0.012
1.07018 3.4875 2.61338
0.473
1.36353 3.32143 2.58364
-0.143
1.2322
1.395 2.42191
-0.042
1.24367 0.99643 2.43698
-0.050
s ∆Ba,eff [m] =
0.24
b 0 = 2.560249 b 1 = -0.18892 b 2 = 0.068207 b 3 = -0.00277 -
6
5
Ba,eff [m]
Ba
[m]
2
2
2
2
2
2
2
2
2
4
3
2
1
0
0
5
10
15
X [-]
◆
B a,eff
■ B a,eff,calc
109
20
25
Restraint Factors and Partial Coefficients
Table 5.12 Effective width for B a = 2m, H c = 2m and L = 18m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff
[m] [m] [m] [-]
[m]
1.8
2 18
0 3.02406
1
2 18
0 2.55465
1.8
2 18
0 4.5096
1.8
2 18
0 3.65949
1
2 18
0 3.09027
0.3
2 18
0 2.12413
1
2 18
0 2.59748
0.3
2 18
0 2.07528
0.3
2 18
0 2.14559
B a,eff /B a
B a,eff,calc ∆B a,eff
X
[-]
[-]
[m]
[m]
1.51203 37.665 3.01765
-0.006
1.27733 20.925 6.01703
2.2548 15.066 4.63932
0.130
1.82975 10.7614
3.495
-0.164
1.54514
8.37 2.92535
-0.165
1.06206 6.2775 2.51255
0.388
1.29874 5.97857 2.46205
-0.135
1.03764
2.511 2.07461
-0.001
1.07279 1.79357 2.04771
-0.098
s ∆Ba,eff [m] =
0.19
b 0 = 2.075673 b 1 = -0.05607 b 2 = 0.023592 b 3 = -0.00057 -
7
6
5
Ba,eff [m]
Ba
[m]
2
2
2
2
2
2
2
2
2
4
3
2
1
0
0
10
◆
B a,eff
20
X [-]
■ B a,eff,calc
110
30
40
Effective Width of Slab
Table 5.13 Effective width for B a = 2m, H c = 4m and L = 3m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
4
3
0 3.34296 1.67148 6.2775 3.34656
0.004
1
4
3
0 1.80678 0.90339 3.4875 1.75649
-0.050
1.8
4
3
0 1.1318 0.5659
2.511 1.19769
0.066
1.8
4
3
0 0.76999
0.385 1.79357 0.83041
0.060
1
4
3
0 0.65029 0.32515
1.395 0.64972
-0.001
0.3
4
3
0 0.66642 0.33321 1.04625 0.50838
-0.158
1
4
3
0 0.46031 0.23016 0.99643 0.48959
0.029
0.3
4
3
0 0.28979 0.14489 0.4185 0.30021
0.010
0.3
4
3
0 0.22882 0.11441 0.29893 0.26809
0.039
s ∆Ba,eff [m] =
0.07
b 0 = 0.199395 b 1 = 0.201075 b 2 = 0.098535 b 3 = -0.00808 -
4
3.5
3
Ba,eff [m]
Ba
[m]
2
2
2
2
2
2
2
2
2
2.5
2
1.5
1
0.5
0
0
1
2
3
4
X [-]
◆
B a,eff
■ B a,eff,calc
111
5
6
7
Restraint Factors and Partial Coefficients
Table 5.14 Effective width for B a = 2m, H c = 4m and L = 5m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
4
5
0 2.6865 1.34325 10.4625 2.6932
0.007
1
4
5
0 1.87443 0.93721 5.8125 1.78669
-0.088
1.8
4
5
0 1.20939 0.6047
4.185 1.29456
0.085
1.8
4
5
0 0.75611 0.37806 2.98929 0.93616
0.180
1
4
5
0 0.74496 0.37248
2.325 0.7486
0.004
0.3
4
5
0 0.99274 0.49637 1.74375 0.59553
-0.397
1
4
5
0 0.46995 0.23497 1.66071 0.57468
0.105
0.3
4
5
0 0.33401 0.16701 0.6975 0.3554
0.021
0.3
4
5
0 0.23306 0.11653 0.49821 0.31592
0.083
s ∆Ba,eff [m] =
0.17
b 0 = 0.227428 b 1 = 0.162126 b 2 = 0.032297 b 3 = -0.00241 -
3
2.5
Ba,eff [m]
Ba
[m]
2
2
2
2
2
2
2
2
2
2
1.5
1
0.5
0
0
◆
2
B a,eff
4
6
X [-]
■ B a,eff,calc
112
8
10
12
Effective Width of Slab
Table 5.15 Effective width for B a = 2m, H c = 4m and L = 10m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
4 10
0 2.42792 1.21396 20.925 2.43195
0.004
1
4 10
0 2.09615 1.04807 11.625 2.04467
-0.051
1.8
4 10
0 1.92608 0.96304
8.37 1.93481
0.009
1.8
4 10
0 1.51505 0.75752 5.97857 1.73012
0.215
1
4 10
0 1.54108 0.77054
4.65 1.5484
0.007
0.3
4 10
0 1.79326 0.89663 3.4875 1.34027
-0.453
1
4 10
0 1.10185 0.55093 3.32143 1.30641
0.205
0.3
4 10
0 0.92155 0.46077
1.395 0.82961
-0.092
0.3
4 10
0 0.55788 0.27894 0.99643 0.71013
0.152
s ∆Ba,eff [m] =
0.20
b 0 = 0.377146 b 1 = 0.359657 b 2 = -0.02623 b 3 = 0.000656 -
3
2.5
Ba,eff [m]
Ba
[m]
2
2
2
2
2
2
2
2
2
2
1.5
1
0.5
0
0
5
10
15
X [-]
◆
B a,eff
■ B a,eff,calc
113
20
25
Restraint Factors and Partial Coefficients
Table 5.16 Effective width for B a = 2m, H c = 4m and L = 18m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
4 18
0 2.48602 1.24301 37.665 2.47522
-0.01
1
4 18
0 2.20126 1.10063 20.925 2.29728
0.10
1.8
4 18
0 2.38417 1.19209 15.066 2.2831
-0.10
1.8
4 18
0 2.33759 1.16879 10.7614 2.23494
-0.10
1
4 18
0 2.13082 1.06541
8.37 2.18521
0.05
0.3
4 18
0
2.129 1.0645 6.2775 2.12447
0.00
1
4 18
0 1.9946 0.9973 5.97857 2.11433
0.12
0.3
4 18
0 1.9439 0.97195
2.511 1.96636
0.02
0.3
4 18
0 2.02714 1.01357 1.79357 1.92817
-0.10
s ∆Ba,eff [m] =
0.08
b 0 = 1.820088 b 1 = 0.065481 b 2 = -0.00299 b 3 = 4.56E-05 -
3
2.5
Ba,eff [m]
Ba
[m]
2
2
2
2
2
2
2
2
2
2
1.5
1
0.5
0
0
◆
10
B a,eff
20
X [-]
■ B a,eff,calc
114
30
40
Effective Width of Slab
Table 5.17 Effective width for B a = 2m, H c = 8m and L = 3m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
8
3
0 5.86543 2.93272 6.2775 5.86667
0.001
1
8
3
0 3.88075 1.94038 3.4875 3.87855
-0.002
1.8
8
3
0 2.77233 1.38616
2.511 2.69771
-0.075
1.8
8
3
0 1.61042 0.80521 1.79357 1.85815
0.248
1
8
3
0 1.58116 0.79058
1.395 1.43414
-0.147
0.3
8
3
0 1.03822 0.51911 1.04625 1.10116
0.063
1
8
3
0 1.20098 0.60049 0.99643 1.05703
-0.144
0.3
8
3
0 0.54572 0.27286 0.4185 0.62038
0.075
0.3
8
3
0 0.56824 0.28412 0.29893 0.54947
-0.019
s ∆Ba,eff [m] =
0.12
b 0 = 0.405659 b 1 = 0.396815 -
b 2 = 0.29224
b 3 = -0.03455 -
7
6
5
Ba,eff [m]
Ba
[m]
2
2
2
2
2
2
2
2
2
4
3
2
1
0
0
1
2
3
4
X [-]
◆
B a,eff
■ B a,eff,calc
115
5
6
7
Restraint Factors and Partial Coefficients
Table 5.18 Effective width for B a = 2m, H c = 8m and L = 5m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
8
5
0 4.19798 2.09899 10.4625 4.20694
0.009
1
8
5
0 3.21577 1.60788 5.8125 3.09842
-0.117
1.8
8
5
0 2.06626 1.03313
4.185 2.17715
0.111
1.8
8
5
0 1.24162 0.62081 2.98929 1.51257
0.271
1
8
5
0 1.24054 0.62027
2.325 1.18004
-0.060
0.3
8
5
0 1.31362 0.65681 1.74375 0.92378
-0.390
1
8
5
0 0.8614 0.4307 1.66071 0.89036
0.029
0.3
8
5
0 0.50434 0.25217 0.6975 0.57327
0.069
0.3
8
5
0 0.44747 0.22373 0.49821 0.52604
0.079
s ∆Ba,eff [m] =
0.18
b 0 = 0.439842 b 1 = 0.124716 b 2 = 0.10066
-
b 3 = -0.00747 -
Ba,eff [m]
Ba
[m]
2
2
2
2
2
2
2
2
2
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
◆
2
B a,eff
4
6
X [-]
■ B a,eff,calc
116
8
10
12
Effective Width of Slab
Table 5.19 Effective width for B a = 2m, H c = 8m and L = 10m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
8 10
0 2.61144 1.30572 20.925 2.61703
0.006
1
8 10
0 2.40606 1.20303 11.625 2.32835
-0.078
1.8
8 10
0 2.02944 1.01472
8.37 2.08845
0.059
1.8
8 10
0 1.57226 0.78613 5.97857 1.78908
0.217
1
8 10
0 1.57896 0.78948
4.65 1.56451
-0.014
0.3
8 10
0 1.76347 0.88174 3.4875 1.32846
-0.435
1
8 10
0 1.13721 0.5686 3.32143 1.29151
0.154
0.3
8 10
0 0.84196 0.42098
1.395 0.79862
-0.043
0.3
8 10
0 0.5507 0.27535 0.99643 0.68101
0.130
s ∆Ba,eff [m] =
0.19
b 0 = 0.361771 b 1 = 0.338997 b 2 = -0.01906 b 3 = 0.000383 -
3
2.5
Ba,eff [m]
Ba
[m]
2
2
2
2
2
2
2
2
2
2
1.5
1
0.5
0
0
5
10
15
X [-]
◆
B a,eff
■ B a,eff,calc
117
20
25
Restraint Factors and Partial Coefficients
Table 5.20 Effective width for B a = 2m, H c = 8m and L = 18m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
8 18
0 2.22789 1.11394 37.665
2.227
-0.001
1
8 18
0 2.0884 1.0442 20.925 2.05867
-0.030
1.8
8 18
0 2.02514 1.01257 15.066 2.06355
0.038
1.8
8 18
0 1.88082 0.94041 10.7614 1.97709
0.096
1
8 18
0 1.87538 0.93769
8.37 1.87712
0.002
0.3
8 18
0 1.95502 0.97751 6.2775 1.75144
-0.204
1
8 18
0 1.66636 0.83318 5.97857 1.73025
0.064
0.3
8 18
0 1.55892 0.77946
2.511 1.41814
-0.141
0.3
8 18
0 1.18613 0.59306 1.79357 1.33705
0.151
s ∆Ba,eff [m] =
0.11
b 0 = 1.107003 b 1 = 0.139564 b 2 = -0.00647 b 3 = 9.44E-05 -
2.5
2
Ba,eff [m]
Ba
[m]
2
2
2
2
2
2
2
2
2
1.5
1
0.5
0
0
◆
10
B a,eff
20
X [-]
■ B a,eff,calc
118
30
40
Effective Width of Slab
Table 5.21 Effective width for B a = 4m, H c = 0.5m and L = 3m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
Bc
[m]
1.8
1
1.8
1.8
1
0.3
1
0.3
0.3
H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [-]
[m]
[-]
[-]
[m]
[m]
0.5
3
0 4.53471 1.13368 3.13875 4.53675
0.002
0.5
3
0 4.09464 1.02366 1.74375 4.08499
-0.010
0.5
3
0 3.91785 0.97946 1.2555 3.86076
-0.057
0.5
3
0 3.35497 0.83874 0.89679 3.48338
0.128
0.5
3
0 3.02915 0.75729 0.6975 3.1633
0.134
0.5
3
0 3.35219 0.83805 0.52313 2.80517
-0.547
0.5
3
0 2.41047 0.60262 0.49821 2.7475
0.337
0.5
3
0 2.11805 0.52951 0.20925 1.9474
-0.171
0.5
3
0 1.56678 0.3917 0.14946 1.74952
0.183
s ∆Ba,eff [m] =
0.25
b 0 = 1.201945 b 1 = 3.925077 b 2 = -1.79122 b 3 = 0.28011
Ba,eff [m]
Ba
[m]
4
4
4
4
4
4
4
4
4
-
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
X [-]
◆
B a,eff
■ B a,eff,calc
119
2.5
3
3.5
Restraint Factors and Partial Coefficients
Table 5.22 Effective width for B a = 4m, H c = 0.5m and L = 5m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
Bc
[m]
1.8
1
1.8
1.8
1
0.3
1
0.3
0.3
H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [-]
[m]
[-]
[-]
[m]
[m]
0.5
5
0 4.71408 1.17852 5.23125 4.71296
-0.001
0.5
5
0 4.35959 1.0899 2.90625 4.37573
0.016
0.5
5
0 4.37306 1.09326 2.0925 4.36406
-0.009
0.5
5
0 4.31824 1.07956 1.49464 4.19798
-0.120
0.5
5
0 3.83457 0.95864 1.1625 4.01516
0.181
0.5
5
0 3.9736 0.9934 0.87188 3.78864
-0.185
0.5
5
0 3.59322 0.89831 0.83036 3.75065
0.157
0.5
5
0 3.31686 0.82921 0.34875 3.19457
-0.122
0.5
5
0 2.96735 0.74184 0.24911 3.05079
0.083
s ∆Ba,eff [m] =
0.13
b 0 = 2.643835 b 1 = 1.775173 b 2 = -0.58331 b 3 = 0.06109
Ba,eff [m]
Ba
[m]
4
4
4
4
4
4
4
4
4
-
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
◆
1
B a,eff
2
3
X [-]
■ B a,eff,calc
120
4
5
6
Effective Width of Slab
Table 5.23 Effective width for B a = 4m, H c = 0.5m and L = 10m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
Bc
[m]
1.8
1
1.8
1.8
1
0.3
1
0.3
0.3
H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [-]
[m]
[-]
[-]
[m]
[m]
0.5 10
0
6.145 1.53625 10.4625 6.15213
0.007
0.5 10
0 5.55543 1.38886 5.8125 5.44363
-0.112
0.5 10
0 4.90126 1.22532
4.185 5.1087
0.207
0.5 10
0 4.97099 1.24275 2.98929 4.89103
-0.080
0.5 10
0 4.59198 1.14799
2.325 4.79026
0.198
0.5 10
0 5.15678 1.2892 1.74375 4.71789
-0.439
0.5 10
0 4.6071 1.15177 1.66071 4.70892
0.102
0.5 10
0 4.51872 1.12968 0.6975 4.63364
0.115
0.5 10
0 4.62515 1.15629 0.49821 4.62533
0.000
s ∆Ba,eff [m] =
0.20
b 0 = 4.616753 b 1 = -0.00106 b 2 = 0.037819 b 3 = -0.00226 -
7
6
5
Ba,eff [m]
Ba
[m]
4
4
4
4
4
4
4
4
4
4
3
2
1
0
0
2
◆
B a,eff
4
6
X [-]
■ B a,eff,calc
121
8
10
12
Restraint Factors and Partial Coefficients
Table 5.24 Effective width for B a = 4m, H c = 0.5m and L = 18m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
Bc
[m]
1.8
1
1.8
1.8
1
0.3
1
0.3
0.3
H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [-]
[m]
[-]
[-]
[m]
[m]
0.5 18
0 7.1923 1.79808 18.8325 7.2014
0.009
0.5 18
0 6.45107 1.61277 10.4625 6.2828
-0.168
0.5 18
0 5.01652 1.25413
7.533 5.39045
0.374
0.5 18
0 5.0106 1.25265 5.38071 4.85234
-0.158
0.5 18
0 4.68322 1.17081
4.185 4.64868
-0.035
0.5 18
0 5.90112 1.47528 3.13875 4.54776
0.5 18
0 4.64477 1.16119 2.98929 4.54011
-0.105
0.5 18
0 4.5626 1.14065 1.2555 4.5952
0.033
0.5 18
0 4.59628 1.14907 0.89679 4.64314
0.047
s ∆Ba,eff [m] =
0.17
b 0 = 4.82479
-
b 1 = -0.25414 b 2 = 0.059377 b 3 = -0.00208 -
8
7
6
Ba,eff [m]
Ba
[m]
4
4
4
4
4
4
4
4
4
5
4
3
2
1
0
0
5
◆
B a,eff
10
X [-]
■ B a,eff,calc
122
15
20
Effective Width of Slab
Table 5.25 Effective width for B a = 4m, H c = 1m and L = 3m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
1
3
0 4.45482 1.11371 3.13875 4.45641
0.002
1
1
3
0
1.814
0.4535 1.74375 8.53396
1.8
1
3
0 7.77525 1.94381 1.2555 7.67025
-0.105
1.8
1
3
0
6.109 1.52725 0.89679 6.48938
0.380
1
1
3
0 6.09581 1.52395 0.6975 5.66936
-0.426
0.3
1
3
0 6.2889 1.57223 0.52313 4.87052
1
1
3
0 4.49044 1.12261 0.49821 4.75081
0.260
0.3
1
3
0 3.77597 0.94399 0.20925 3.2739
-0.502
0.3
1
3
0 2.55926 0.63982 0.14946 2.95043
0.391
s ∆Ba,eff [m] =
0.37
b 0 = 2.119736 -
b 1 = 5.65342
b 2 = -0.59342 b 3 = -0.30922 -
Ba,eff [m]
Ba
[m]
4
4
4
4
4
4
4
4
4
9
8
7
6
5
4
3
2
1
0
0
0.5
1
1.5
2
X [-]
◆
B a,eff
■ B a,eff,calc
123
2.5
3
3.5
Restraint Factors and Partial Coefficients
Table 5.26 Effective width for B a = 4m, H c = 1m and L = 5m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
1
5
0 9.67028 2.41757 5.23125 9.68434
0.014
1
1
5
0 6.73081 1.6827 2.90625 6.52727
-0.204
1.8
1
5
0 5.47839 1.3696 2.0925 5.78893
0.311
1.8
1
5
0 5.08371 1.27093 1.49464 5.21518
0.131
1
1
5
0 4.9149 1.22873 1.1625 4.85895
-0.056
0.3
1
5
0 4.90633 1.22658 0.87188 4.5146
-0.392
1
1
5
0 4.37511 1.09378 0.83036 4.46248
0.087
0.3
1
5
0 4.00661 1.00165 0.34875 3.79441
-0.212
0.3
1
5
0 3.31989 0.82997 0.24911 3.63987
0.320
s ∆Ba,eff [m] =
0.24
b 0 = 3.22552
-
b 1 = 1.747749 b 2 = -0.35094 b 3 = 0.048337 -
12
10
Ba,eff [m]
Ba
[m]
4
4
4
4
4
4
4
4
4
8
6
4
2
0
0
◆
1
B a,eff
2
3
X [-]
■ B a,eff,calc
124
4
5
6
Effective Width of Slab
Table 5.27 Effective width for B a = 4m, H c = 1m and L = 10m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
1 10
0 9.66861 2.41715 10.4625 9.6984
0.030
1
1 10
0 7.04648 1.76162 5.8125 6.60479
-0.442
1.8
1 10
0 4.72968 1.18242
4.185 5.4278
0.698
1.8
1 10
0 4.51548 1.12887 2.98929 4.7518
0.236
1
1 10
0 4.52391 1.13098
2.325 4.48286
-0.041
0.3
1 10
0 5.13169 1.28292 1.74375 4.32543
-0.806
1
1 10
0 4.35234 1.08808 1.66071 4.30951
-0.043
0.3
1 10
0 4.1766 1.04415 0.6975 4.25921
0.083
0.3
1 10
0 3.99805 0.99951 0.49821 4.28221
0.284
s ∆Ba,eff [m] =
0.43
b 0 = 4.394842 b 1 = -0.30814 b 2 = 0.169079 b 3 = -0.00871 -
12
10
Ba,eff [m]
Ba
[m]
4
4
4
4
4
4
4
4
4
8
6
4
2
0
0
◆
2
B a,eff
4
6
X [-]
■ B a,eff,calc
125
8
10
12
Restraint Factors and Partial Coefficients
Table 5.28 Effective width for B a = 4m, H c = 1m and L = 18m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
1 18
0 10.511 2.62775 18.8325 10.5385
0.027
1
1 18
0 8.06308 2.01577 10.4625 7.61462
-0.448
1.8
1 18
0 4.86888 1.21722
7.533 5.70364
0.835
1.8
1 18
0 4.57195 1.14299 5.38071 4.62626
0.054
1
1 18
0 4.64018 1.16004
4.185 4.25314
-0.387
0.3
1 18
0 5.98164 1.49541 3.13875 4.10216
1
1 18
0 4.43577 1.10894 2.98929 4.09577
-0.340
0.3
1 18
0 4.33253 1.08313 1.2555 4.33909
0.007
0.3
1 18
0 4.22036 1.05509 0.89679 4.46952
0.249
s ∆Ba,eff [m] =
0.42
b 0 = 4.92988
-
b 1 = -0.62506 b 2 = 0.12835
-
b 3 = -0.00421 -
12
10
Ba,eff [m]
Ba
[m]
4
4
4
4
4
4
4
4
4
8
6
4
2
0
0
◆
5
B a,eff
10
X [-]
■ B a,eff,calc
126
15
20
Effective Width of Slab
Table 5.29 Effective width for B a = 4m, H c = 2m and L = 3m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
2
3
0 2.22145 0.55536 3.13875 2.22649
0.005
1
2
3
0 1.24111 0.31028 1.74375 1.1707
-0.070
1.8
2
3
0 0.66574 0.16644 1.2555 0.75922
0.093
1.8
2
3
0 0.41383 0.10346 0.89679 0.49779
0.084
1
2
3
0 0.38827 0.09707 0.6975 0.37687
-0.011
0.3
2
3
0 0.48798
0.122 0.52313 0.28915
-0.199
1
2
3
0 0.24671 0.06168 0.49821 0.27815
0.031
0.3
2
3
0 0.17146 0.04287 0.20925 0.18224
0.011
0.3
2
3
0 0.11436 0.02859 0.14946 0.1703
0.056
s ∆Ba,eff [m] =
0.09
b 0 = 0.153538 b 1 = 0.047025 b 2 = 0.447699 b 3 = -0.08037 -
2.5
2
Ba,eff [m]
Ba
[m]
4
4
4
4
4
4
4
4
4
1.5
1
0.5
0
0
0.5
1
1.5
2
X [-]
◆
B a,eff
■ B a,eff,calc
127
2.5
3
3.5
Restraint Factors and Partial Coefficients
Table 5.30 Effective width for B a = 4m, H c = 2m and L = 5m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
2
5
0 3.82117 0.95529 5.23125 3.82548
0.004
1
2
5
0 2.37753 0.59438 2.90625 2.32691
-0.051
1.8
2
5
0 1.57224 0.39306 2.0925 1.59589
0.024
1.8
2
5
0 0.96457 0.24114 1.49464 1.12671
0.162
1
2
5
0 0.87638 0.21909 1.1625 0.91415
0.038
0.3
2
5
0 1.20191 0.30048 0.87188 0.76589
-0.436
1
2
5
0 0.5844 0.1461 0.83036 0.74798
0.164
0.3
2
5
0 0.5136 0.1284 0.34875 0.6088
0.095
0.3
2
5
0 5.98551 1.49638 0.24911 0.59733
s ∆Ba,eff [m] =
0.19
b 0 = 0.597754 b 1 = -0.08889 b 2 = 0.360772 b 3 = -0.04317 -
7
6
5
Ba,eff [m]
Ba
[m]
4
4
4
4
4
4
4
4
4
4
3
2
1
0
0
1
◆
B a,eff
2
3
X [-]
■ B a,eff,calc
128
4
5
6
Effective Width of Slab
Table 5.31 Effective width for B a = 4m, H c = 2m and L = 10m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff
[m] [m] [m] [-]
[m]
1.8
2 10
0 6.92473
1
2 10
0 5.49334
1.8
2 10
0 7.85947
1.8
2 10
0 6.44297
1
2 10
0 5.6502
0.3
2 10
0 4.61085
1
2 10
0 5.34196
0.3
2 10
0 5.10854
0.3
2 10
0 4.92789
b 0 = 5.38135
B a,eff /B a
[-]
1.73118
1.37333
1.96487
1.61074
1.41255
1.15271
1.33549
1.27713
1.23197
B a,eff,calc ∆B a,eff
X
[-]
[m]
[m]
10.4625 6.92367
-0.001
5.8125 10.1517
4.185 7.92567
0.066
2.98929 6.28862
-0.154
2.325 5.56768
-0.083
1.74375 5.12488
0.514
1.66071 5.07916
-0.263
0.6975 4.94136
-0.167
0.49821 5.01574
0.088
s ∆Ba,eff [m] =
0.24
-
b 1 = -1.00668 b 2 = 0.569488 b 3 = -0.04389 -
12
10
Ba,eff [m]
Ba
[m]
4
4
4
4
4
4
4
4
4
8
6
4
2
0
0
◆
2
B a,eff
4
6
X [-]
■ B a,eff,calc
129
8
10
12
Restraint Factors and Partial Coefficients
Table 5.32 Effective width for B a = 4m, H c = 2m and L = 18m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff
[m] [m] [m] [-]
[m]
1.8
2 18
0 7.02136
1
2 18
0 5.89008
1.8
2 18
0 7.64881
1.8
2 18
0 5.88874
1
2 18
0 5.35803
0.3
2 18
0 5.14097
1
2 18
0 4.7429
0.3
2 18
0 4.43267
0.3
2 18
0 4.30676
B a,eff /B a
[-]
1.75534
1.47252
1.9122
1.47219
1.33951
1.28524
1.18572
1.10817
1.07669
B a,eff,calc ∆B a,eff
X
[-]
[m]
[m]
18.8325 7.02266
0.001
10.4625 9.46179
7.533 7.56495
-0.084
5.38071 6.08842
0.200
4.185 5.37884
0.021
3.13875 4.88068
-0.260
2.98929 4.82126
0.078
1.2555 4.4004
-0.032
0.89679 4.38446
0.078
s ∆Ba,eff [m] =
0.14
b 0 = 4.470224 b 1 = -0.20199 b 2 = 0.123636 b 3 = -0.00561 -
Ba,eff [m]
Ba
[m]
4
4
4
4
4
4
4
4
4
10
9
8
7
6
5
4
3
2
1
0
0
◆
5
B a,eff
10
X [-]
■ B a,eff,calc
130
15
20
Effective Width of Slab
Table 5.33 Effective width for B a = 4m, H c = 4m and L = 3m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
4
3
0 3.4903 0.87258 3.13875 3.4924
0.002
1
4
3
0 1.8167 0.45417 1.74375 1.78967
-0.027
1.8
4
3
0 1.23368 0.30842 1.2555 1.26094
0.027
1.8
4
3
0 0.86169 0.21542 0.89679 0.90675
0.045
1
4
3
0 0.70676 0.17669 0.6975 0.72459
0.018
0.3
4
3
0 0.71856 0.17964 0.52313 0.57459
-0.144
1
4
3
0 0.50349 0.12587 0.49821 0.55391
0.050
0.3
4
3
0 0.33868 0.08467 0.20925 0.32857
-0.010
0.3
4
3
0
0.247 0.06175 0.14946 0.28542
0.038
s ∆Ba,eff [m] =
0.06
b 0 = 0.183034 b 1 = 0.658262 b 2 = 0.181748 b 3 = -0.0177
-
4
3.5
3
Ba,eff [m]
Ba
[m]
4
4
4
4
4
4
4
4
4
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
X [-]
◆
B a,eff
■ B a,eff,calc
131
2.5
3
3.5
Restraint Factors and Partial Coefficients
Table 5.34 Effective width for B a = 4m, H c = 4m and L = 5m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
4
5
0 3.49139 0.87285 5.23125 3.49696
0.006
1
4
5
0 2.17738 0.54435 2.90625 2.10795
-0.069
1.8
4
5
0 1.45631 0.36408 2.0925 1.51241
0.056
1.8
4
5
0 0.92944 0.23236 1.49464 1.09218
0.163
1
4
5
0 0.85463 0.21366 1.1625 0.87418
0.020
0.3
4
5
0 1.06683 0.26671 0.87188 0.69625
-0.371
1
4
5
0 0.56521 0.1413 0.83036 0.67196
0.107
0.3
4
5
0 0.39874 0.09969 0.34875 0.41435
0.016
0.3
4
5
0 0.29351 0.07338 0.24911 0.36721
0.074
s ∆Ba,eff [m] =
0.15
b 0 = 0.259834 b 1 = 0.399578 b 2 = 0.130584 b 3 = -0.01695 -
4
3.5
3
Ba,eff [m]
Ba
[m]
4
4
4
4
4
4
4
4
4
2.5
2
1.5
1
0.5
0
0
◆
1
B a,eff
2
3
X [-]
■ B a,eff,calc
132
4
5
6
Effective Width of Slab
Table 5.35 Effective width for B a = 4m, H c = 4m and L = 10m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
4 10
0 4.45949 1.11487 10.4625 4.45589
-0.004
1
4 10
0 3.6572 0.9143 5.8125 3.72237
0.065
1.8
4 10
0 3.26844 0.81711
4.185 3.10847
-0.160
1.8
4 10
0 2.35019 0.58755 2.98929 2.5151
0.165
1
4 10
0 2.30389 0.57597
2.325 2.1292
-0.175
0.3
4 10
0 2.84108 0.71027 1.74375 1.75686
1
4 10
0 1.51014 0.37753 1.66071 1.70096
0.191
0.3
4 10
0 1.17321 0.2933 0.6975 1.00135
-0.172
0.3
4 10
0 0.75527 0.18882 0.49821 0.84457
0.089
s ∆Ba,eff [m] =
0.15
b 0 = 0.43402
-
b 1 = 0.851059 b 2 = -0.05469 b 3 = 0.000965 -
Ba,eff [m]
Ba
[m]
4
4
4
4
4
4
4
4
4
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
◆
2
B a,eff
4
6
X [-]
■ B a,eff,calc
133
8
10
12
Restraint Factors and Partial Coefficients
Table 5.36 Effective width for B a = 4m, H c = 4m and L = 18m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
4 18
0 4.97389 1.24347 18.8325 4.95035
-0.024
1
4 18
0 4.53438 1.1336 10.4625 4.84669
0.312
1.8
4 18
0 5.15347 1.28837
7.533 4.77396
-0.380
1.8
4 18
0 5.18848 1.29712 5.38071 4.71783
-0.471
1
4 18
0 4.45559 1.1139
4.185 4.68727
0.232
0.3
4 18
0 4.50787 1.12697 3.13875 4.66154
0.154
1
4 18
0 4.23703 1.05926 2.98929 4.65797
0.421
0.3
4 18
0 4.39963 1.09991 1.2555 4.61907
0.219
0.3
4 18
0 5.07964 1.26991 0.89679 4.61171
-0.468
s ∆Ba,eff [m] =
0.35
b 0 = 4.594543 b 1 = 0.018077 b 2 = 0.001241 b 3 = -6.4E-05 -
6
5
Ba,eff [m]
Ba
[m]
4
4
4
4
4
4
4
4
4
4
3
2
1
0
0
5
◆
B a,eff
10
X [-]
■ B a,eff,calc
134
15
20
Effective Width of Slab
Table 5.37 Effective width for B a = 4m, H c = 8m and L = 3m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
8
3
0 7.52916 1.88229 3.13875 7.52478
-0.004
1
8
3
0 3.16405 0.79101 1.74375 3.23068
0.067
1.8
8
3
0 2.48075 0.62019 1.2555 2.37389
-0.107
1.8
8
3
0 1.87162 0.4679 0.89679 1.80936
-0.062
1
8
3
0 1.40664 0.35166 0.6975 1.49414
0.087
0.3
8
3
0 1.21909 0.30477 0.52313 1.20697
-0.012
1
8
3
0 1.07558 0.26889 0.49821 1.16465
0.089
0.3
8
3
0 0.68952 0.17238 0.20925 0.64042
-0.049
0.3
8
3
0 0.53108 0.13277 0.14946 0.52259
-0.008
s ∆Ba,eff [m] =
0.07
b 0 = 0.210592 b 1 = 2.177615 b 2 = -0.63585 b 3 = 0.218078 -
8
7
6
Ba,eff [m]
Ba
[m]
4
4
4
4
4
4
4
4
4
5
4
3
2
1
0
0
0.5
1
1.5
2
X [-]
◆
B a,eff
■ B a,eff,calc
135
2.5
3
3.5
Restraint Factors and Partial Coefficients
Table 5.38 Effective width for B a = 4m, H c = 8m and L = 5m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
8
5
0 5.91779 1.47945 5.23125 5.92223
0.004
1
8
5
0 3.16034 0.79009 2.90625 3.10603
-0.054
1.8
8
5
0 2.19951 0.54988 2.0925 2.24003
0.041
1.8
8
5
0 1.53113 0.38278 1.49464 1.65393
0.123
1
8
5
0 1.29441 0.3236 1.1625 1.34849
0.054
0.3
8
5
0 1.44961 0.3624 0.87188 1.09378
-0.356
1
8
5
0 0.94658 0.23665 0.83036 1.05838
0.112
0.3
8
5
0 0.64787 0.16197 0.34875 0.66647
0.019
0.3
8
5
0 0.53192 0.13298 0.24911 0.58982
0.058
s ∆Ba,eff [m] =
0.14
b 0 = 0.405061 b 1 = 0.721647 b 2 = 0.081242 b 3 = -0.00336 -
7
6
5
Ba,eff [m]
Ba
[m]
4
4
4
4
4
4
4
4
4
4
3
2
1
0
0
1
◆
B a,eff
2
3
X [-]
■ B a,eff,calc
136
4
5
6
Effective Width of Slab
Table 5.39 Effective width for B a = 4m, H c = 8m and L = 10m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
8 10
0 4.69334 1.17334 10.4625 4.7012
0.008
1
8 10
0 3.67452 0.91863 5.8125 3.58183
-0.093
1.8
8 10
0 3.00272 0.75068
4.185 3.04676
0.044
1.8
8 10
0 2.21193 0.55298 2.98929 2.51188
0.300
1
8 10
0 2.13357 0.53339
2.325 2.1467
0.013
0.3
8 10
0 2.44224 0.61056 1.74375 1.7807
-0.662
1
8 10
0 1.45725 0.36431 1.66071 1.7246
0.267
0.3
8 10
0 1.02364 0.25591 0.6975 0.99799
-0.026
0.3
8 10
0 0.68196 0.17049 0.49821 0.82915
0.147
s ∆Ba,eff [m] =
0.28
b 0 = 0.377098 b 1 = 0.951548 b 2 = -0.09061 b 3 = 0.003743 -
Ba,eff [m]
Ba
[m]
4
4
4
4
4
4
4
4
4
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
◆
2
B a,eff
4
6
X [-]
■ B a,eff,calc
137
8
10
12
Restraint Factors and Partial Coefficients
Table 5.40 Effective width for B a = 4m, H c = 8m and L = 18m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
8 18
0 4.30416 1.07604 18.8325 4.31244
0.008
1
8 18
0 3.98057 0.99514 10.4625 3.87449
-0.106
1.8
8 18
0 3.78944 0.94736
7.533 3.87111
0.082
1.8
8 18
0 3.39028 0.84757 5.38071 3.63455
0.244
1
8 18
0 3.3236 0.8309
4.185 3.36901
0.045
0.3
8 18
0 3.61892 0.90473 3.13875 3.03822
-0.581
1
8 18
0 2.76061 0.69015 2.98929 2.98264
0.222
0.3
8 18
0 2.38666 0.59666 1.2555 2.16736
-0.219
0.3
8 18
0 1.6548 0.4137 0.89679 1.95623
0.301
s ∆Ba,eff [m] =
0.28
b 0 = 1.358258 b 1 = 0.724847 b 2 = -0.06646 b 3 = 0.001928 -
Ba,eff [m]
Ba
[m]
4
4
4
4
4
4
4
4
4
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
◆
5
B a,eff
10
X [-]
■ B a,eff,calc
138
15
20
Effective Width of Slab
Table 5.41 Effective width for B a = 8m, H c = 0.5m and L = 3m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
Bc
[m]
1.8
1
1.8
1.8
1
0.3
1
0.3
0.3
H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [-]
[m]
[-]
[-]
[m]
[m]
0.5
3
0 5.57236 0.69654 1.56938 5.5751
0.003
0.5
3
0 4.50793 0.56349 0.87188 4.49212
-0.016
0.5
3
0 4.21287 0.52661 0.62775 4.14832
-0.065
0.5
3
0 3.51618 0.43952 0.44839 3.69147
0.175
0.5
3
0 3.19406 0.39926 0.34875 3.32602
0.132
0.5
3
0 3.55334 0.44417 0.26156 2.92571
-0.628
0.5
3
0 2.49016 0.31127 0.24911 2.86176
0.372
0.5
3
0 2.15021 0.26878 0.10463 1.98232
-0.168
0.5
3
0 1.57197 0.1965 0.07473 1.76626
0.194
s ∆Ba,eff [m] =
0.29
b 0 = 1.170022 b 1 = 8.534326 b 2 = -7.62918 b 3 = 2.535842 -
6
5
Ba,eff [m]
Ba
[m]
8
8
8
8
8
8
8
8
8
4
3
2
1
0
0
0.5
◆
B a,eff
1
X [-]
■ B a,eff,calc
139
1.5
2
Restraint Factors and Partial Coefficients
Table 5.42 Effective width for B a = 8m, H c = 0.5m and L = 5m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
Bc
[m]
1.8
1
1.8
1.8
1
0.3
1
0.3
0.3
H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [-]
[m]
[-]
[-]
[m]
[m]
0.5
5
0 7.12232 0.89029 2.61563 7.12376
0.001
0.5
5
0 6.00085 0.75011 1.45313 5.99053
-0.010
0.5
5
0 5.81135 0.72642 1.04625 5.80272
-0.009
0.5
5
0 5.48561 0.6857 0.74732 5.4591
-0.027
0.5
5
0 4.91022 0.61378 0.58125 5.14602
0.236
0.5
5
0 5.30321 0.6629 0.43594 4.78126
-0.522
0.5
5
0 4.38077 0.5476 0.41518 4.72143
0.341
0.5
5
0 4.06076 0.50759 0.17438 3.86858
-0.192
0.5
5
0 3.47086 0.43386 0.12455 3.65246
0.182
s ∆Ba,eff [m] =
0.25
b 0 = 3.04641
-
b 1 = 5.258137 b 2 = -3.23662 b 3 = 0.696702 -
8
7
6
Ba,eff [m]
Ba
[m]
8
8
8
8
8
8
8
8
8
5
4
3
2
1
0
0
0.5
◆
B a,eff
1
1.5
X [-]
■ B a,eff,calc
140
2
2.5
3
Effective Width of Slab
Table 5.43 Effective width for B a = 8m, H c = 0.5m and L = 10m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
Bc
[m]
1.8
1
1.8
1.8
1
0.3
1
0.3
0.3
H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [-]
[m]
[-]
[-]
[m]
[m]
0.5 10
0 10.638 1.32975 5.23125 10.6374
-0.001
0.5 10
0 9.78965 1.22371 2.90625 9.76191
-0.028
0.5 10
0 8.85391 1.10674 2.0925 9.05199
0.198
0.5 10
0 9.06459 1.13307 1.49464 8.64875
-0.416
0.5 10
0 8.32997 1.04125 1.1625 8.51103
0.181
0.5 10
0 9.41666 1.17708 0.87188 8.45871
0.5 10
0 8.36797
1.046 0.83036 8.45716
0.089
0.5 10
0 8.44133 1.05517 0.34875 8.56387
0.123
0.5 10
0 8.76418 1.09552 0.24911 8.61748
-0.147
s ∆Ba,eff [m] =
0.20
b 0 = 8.804552 b 1 = -0.91005 b 2 = 0.65841
-
b 3 = -0.0798
-
12
10
Ba,eff [m]
Ba
[m]
8
8
8
8
8
8
8
8
8
8
6
4
2
0
0
◆
1
B a,eff
2
3
X [-]
■ B a,eff,calc
141
4
5
6
Restraint Factors and Partial Coefficients
Table 5.44 Effective width for B a = 8m, H c = 0.5m and L = 18m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
Bc
[m]
1.8
1
1.8
1.8
1
0.3
1
0.3
0.3
H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [-]
[m]
[-]
[-]
[m]
[m]
0.5 18
0 13.0236 1.62795 9.41625 13.0365
0.013
0.5 18
0 12.0976 1.51221 5.23125 11.8513
-0.246
0.5 18
0 9.97478 1.24685 3.7665 10.594
0.619
0.5 18
0 10.3446 1.29308 2.69036 9.87709
-0.468
0.5 18
0 9.52187 1.19023 2.0925 9.6374
0.116
0.5 18
0 11.6332 1.45415 1.56938 9.55458
0.5 18
0 9.72368 1.21546 1.49464 9.55383
-0.170
0.5 18
0 9.53143 1.19143 0.62775 9.77905
0.248
0.5 18
0 9.99636 1.24955 0.44839 9.88498
-0.111
s ∆Ba,eff [m] =
0.33
b 0 = 10.2499
-
b 1 = -0.98072 b 2 = 0.384002 b 3 = -0.02638 -
14
12
10
Ba,eff [m]
Ba
[m]
8
8
8
8
8
8
8
8
8
8
6
4
2
0
0
2
4
6
X [-]
◆
B a,eff
■ B a,eff,calc
142
8
10
Effective Width of Slab
Table 5.45 Effective width for B a = 8m, H c = 1m and L = 3m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff
[m] [m] [m] [-]
[m]
1.8
1
3
0 4.61855
1
1
3
0 1.83966
1.8
1
3
0 8.34861
1.8
1
3
0 6.37423
1
1
3
0 6.40255
0.3
1
3
0 6.70942
1
1
3
0 4.61879
0.3
1
3
0 3.83067
0.3
1
3
0 2.56513
B a,eff /B a
[-]
0.57732
0.22996
1.04358
0.79678
0.80032
0.83868
0.57735
0.47883
0.32064
B a,eff,calc ∆B a,eff
X
[-]
[m]
[m]
1.56938 4.62044
0.002
0.87188 9.31336
0.62775 8.22125
-0.127
0.44839 6.83473
0.461
0.34875 5.9052
-0.497
0.26156 5.02094
0.24911 4.89014
0.271
0.10463 3.31184
-0.519
0.07473 2.97492
0.410
s ∆Ba,eff [m] =
0.41
b 0 = 2.124383 b 1 = 11.43074 b 2 = -0.38246 b 3 = -3.75164 -
Ba,eff [m]
Ba
[m]
8
8
8
8
8
8
8
8
8
10
9
8
7
6
5
4
3
2
1
0
0
◆
0.5
B a,eff
1
X [-]
■ B a,eff,calc
143
1.5
2
Restraint Factors and Partial Coefficients
Table 5.46 Effective width for B a = 8m, H c = 1m and L = 5m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
1
5
0 11.4137 1.42671 2.61563 11.4318
0.018
1
1
5
0 8.69367 1.08671 1.45313 8.45324
-0.240
1.8
1
5
0 7.30532 0.91317 1.04625 7.58297
0.278
1.8
1
5
0 6.42589 0.80324 0.74732 6.78706
0.361
1
1
5
0 6.27798 0.78475 0.58125 6.25114
-0.027
0.3
1
5
0 6.61076 0.82634 0.43594 5.71231
-0.898
1
1
5
0
5.282 0.66025 0.41518 5.6294
0.347
0.3
1
5
0 4.85737 0.60717 0.17438 4.54505
-0.312
0.3
1
5
0 3.81649 0.47706 0.12455 4.2901
0.474
s ∆Ba,eff [m] =
0.44
b 0 = 3.601955 b 1 = 5.827895 b 2 = -2.49997 b 3 = 0.541488 -
14
12
10
Ba,eff [m]
Ba
[m]
8
8
8
8
8
8
8
8
8
8
6
4
2
0
0
◆
0.5
B a,eff
1
1.5
X [-]
■ B a,eff,calc
144
2
2.5
3
Effective Width of Slab
Table 5.47 Effective width for B a = 8m, H c = 1m and L = 10m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
1 10
0 13.1275 1.64093 5.23125 13.1523
0.025
1
1 10
0 11.1001 1.38751 2.90625 10.7117
-0.388
1.8
1 10
0 8.41888 1.05236 2.0925 9.11103
0.692
1.8
1 10
0 7.96576 0.99572 1.49464 8.08208
0.116
1
1 10
0 8.02368 1.00296 1.1625 7.6315
-0.392
0.3
1 10
0 9.26141 1.15768 0.87188 7.33577
1
1 10
0 7.53077 0.94135 0.83036 7.30217
-0.229
0.3
1 10
0 7.31092 0.91386 0.34875 7.09606
-0.215
0.3
1 10
0 6.7094 0.83868 0.24911 7.10013
0.391
s ∆Ba,eff [m] =
0.39
b 0 = 7.189392 b 1 = -0.59592 b 2 = 0.984902 b 3 = -0.12484 -
14
12
10
Ba,eff [m]
Ba
[m]
8
8
8
8
8
8
8
8
8
8
6
4
2
0
0
◆
1
B a,eff
2
3
X [-]
■ B a,eff,calc
145
4
5
6
Restraint Factors and Partial Coefficients
Table 5.48 Effective width for B a = 8m, H c = 1m and L = 18m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
1 18
0 15.5901 1.94876 9.41625 15.5994
0.009
1
1 18
0 13.5862 1.69827 5.23125 13.485
-0.101
1.8
1 18
0 9.31703 1.16463 3.7665 11.0104
1.8
1 18
0 8.97046 1.12131 2.69036 9.48256
0.512
1
1 18
0 9.13624 1.14203 2.0925 8.88753
-0.249
0.3
1 18
0 11.5736
1.4467 1.56938 8.57607
1
1 18
0 8.85904 1.10738 1.49464 8.55002
-0.309
0.3
1 18
0 8.73075 1.09134 0.62775 8.64099
-0.090
0.3
1 18
0 8.53296 1.06662 0.44839 8.76007
0.227
s ∆Ba,eff [m] =
0.29
b 0 = 9.227897 b 1 = -1.32758 b 2 = 0.65496
-
b 3 = -0.04695 -
Ba,eff [m]
Ba
[m]
8
8
8
8
8
8
8
8
8
18
16
14
12
10
8
6
4
2
0
0
2
4
6
X [-]
◆
B a,eff
■ B a,eff,calc
146
8
10
Effective Width of Slab
Table 5.49 Effective width for B a = 8m, H c = 2m and L = 3m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
2
3
0 2.26428 0.28303 1.56938 2.26925
0.005
1
2
3
0 1.26279 0.15785 0.87188 1.19343
-0.069
1.8
2
3
0 0.68382 0.08548 0.62775 0.7755
0.092
1.8
2
3
0 0.4253 0.05316 0.44839 0.50916
0.084
1
2
3
0 0.39777 0.04972 0.34875 0.38539
-0.012
0.3
2
3
0 0.49073 0.06134 0.26156 0.29508
-0.196
1
2
3
0 0.2524 0.03155 0.24911 0.2837
0.031
0.3
2
3
0 0.17329 0.02166 0.10463 0.18324
0.010
0.3
2
3
0 0.11468 0.01433 0.07473 0.1703
0.056
s ∆Ba,eff [m] =
0.09
b 0 = 0.151001 b 1 = 0.128782 b 2 = 1.780646 b 3 = -0.63889 -
2.5
2
Ba,eff [m]
Ba
[m]
8
8
8
8
8
8
8
8
8
1.5
1
0.5
0
0
◆
0.5
B a,eff
1
X [-]
■ B a,eff,calc
147
1.5
2
Restraint Factors and Partial Coefficients
Table 5.50 Effective width for B a = 8m, H c = 2m and L = 5m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
2
5
0 3.88258 0.48532 2.61563 3.88207
-0.001
1
2
5
0 2.41019 0.30127 1.45313 2.42342
0.013
1.8
2
5
0 1.60788 0.20098 1.04625 1.56304
-0.045
1.8
2
5
0 1.05684 0.13211 0.74732 1.10204
0.045
1
2
5
0 0.91698 0.11462 0.58125 0.95754
0.041
0.3
2
5
0 1.20967 0.15121 0.43594 0.91658
-0.293
1
2
5
0 0.67865 0.08483 0.41518 0.91807
0.239
0.3
2
5
0 8.80415 1.10052 0.17438 1.08825
0.3
2
5
0 6.95745 0.86968 0.12455 1.16188
s ∆Ba,eff [m] =
0.16
b 0 = 1.410127 b 1 = -2.37698 b 2 = 3.172617 b 3 = -0.72737 -
Ba,eff [m]
Ba
[m]
8
8
8
8
8
8
8
8
8
10
9
8
7
6
5
4
3
2
1
0
0
◆
0.5
B a,eff
1
1.5
X [-]
■ B a,eff,calc
148
2
2.5
3
Effective Width of Slab
Table 5.51 Effective width for B a = 8m, H c = 2m and L = 10m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff
[m] [m] [m] [-]
[m]
1.8
2 10
0 10.027
1
2 10
0 7.08013
1.8
2 10
0 11.8767
1.8
2 10
0 10.5586
1
2 10
0 9.70552
0.3
2 10
0 8.95734
1
2 10
0 9.17628
0.3
2 10
0 9.0395
0.3
2 10
0 8.25542
B a,eff /B a
[-]
1.25337
0.88502
1.48459
1.31983
1.21319
1.11967
1.14703
1.12994
1.03193
B a,eff,calc ∆B a,eff
X
[-]
[m]
[m]
5.23125 10.0269
0.000
2.90625 13.6512
2.0925 11.9025
0.026
1.49464 10.4496
-0.109
1.1625 9.72244
0.017
0.87188 9.1911
0.234
0.83036 9.1256
-0.051
0.34875 8.6097
-0.430
0.24911 8.56851
0.313
s ∆Ba,eff [m] =
0.22
3
X [-]
4
b 0 = 8.582846 b 1 = -0.41669 b 2 = 1.507215 b 3 = -0.2628
-
16
14
12
Ba,eff [m]
Ba
[m]
8
8
8
8
8
8
8
8
8
10
8
6
4
2
0
0
◆
1
B a,eff
2
■ B a,eff,calc
149
5
6
Restraint Factors and Partial Coefficients
Table 5.52 Effective width for B a = 8m, H c = 2m and L = 18m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
2 18
0 19.7785 2.47232 9.41625 19.7758
-0.003
1
2 18
0 14.3336 1.7917 5.23125 14.3909
0.057
1.8
2 18
0 12.1194 1.51492 3.7665 11.9635
-0.156
1.8
2 18
0 10.2877 1.28596 2.69036 10.4236
0.136
1
2 18
0 9.77344 1.22168 2.0925 9.72941
-0.044
0.3
2 18
0 11.5878 1.44848 1.56938 9.24611
1
2 18
0 9.12508 1.14064 1.49464 9.18775
0.063
0.3
2 18
0 8.90048 1.11256 0.62775 8.7335
-0.167
0.3
2 18
0 8.58168 1.07271 0.44839 8.69554
0.114
s ∆Ba,eff [m] =
0.12
b 0 = 8.694375 b 1 = -0.15315 b 2 = 0.357675 b 3 = -0.02298 -
25
20
Ba,eff [m]
Ba
[m]
8
8
8
8
8
8
8
8
8
15
10
5
0
0
2
4
6
X [-]
◆
B a,eff
■ B a,eff,calc
150
8
10
Effective Width of Slab
Table 5.53 Effective width for B a = 8m, H c = 4m and L = 3m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
4
3
0 3.4554 0.43192 1.56938 3.45731
0.002
1
4
3
0 1.85242 0.23155 0.87188 1.82832
-0.024
1.8
4
3
0 1.2726 0.15907 0.62775 1.29497
0.022
1.8
4
3
0 0.88688 0.11086 0.44839 0.93182
0.045
1
4
3
0 0.72777 0.09097 0.34875 0.74341
0.016
0.3
4
3
0 0.72574 0.09072 0.26156 0.58753
-0.138
1
4
3
0 0.51381 0.06423 0.24911 0.56599
0.052
0.3
4
3
0 0.34319 0.0429 0.10463 0.3306
-0.013
0.3
4
3
0 0.24755 0.03094 0.07473 0.28541
0.038
s ∆Ba,eff [m] =
0.06
b 0 = 0.178074 b 1 = 1.380781 b 2 = 0.756729 b 3 = -0.19443 -
4
3.5
3
Ba,eff [m]
Ba
[m]
8
8
8
8
8
8
8
8
8
2.5
2
1.5
1
0.5
0
0
◆
0.5
B a,eff
1
X [-]
■ B a,eff,calc
151
1.5
2
Restraint Factors and Partial Coefficients
Table 5.54 Effective width for B a = 8m, H c = 4m and L = 5m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
4
5
0 3.56314 0.44539 2.61563 3.56834
0.005
1
4
5
0 2.21811 0.27726 1.45313 2.15342
-0.065
1.8
4
5
0 1.50654 0.18832 1.04625 1.55911
0.053
1.8
4
5
0 0.98787 0.12348 0.74732 1.13727
0.149
1
4
5
0 0.89594 0.11199 0.58125 0.91646
0.021
0.3
4
5
0 1.08368 0.13546 0.43594 0.73452
-0.349
1
4
5
0 0.60506 0.07563 0.41518 0.70952
0.104
0.3
4
5
0 0.43135 0.05392 0.17438 0.44087
0.010
0.3
4
5
0 0.31854 0.03982 0.12455 0.39072
0.072
s ∆Ba,eff [m] =
0.14
b 0 = 0.274611 b 1 = 0.876585 b 2 = 0.46181
-
b 3 = -0.12063 -
4
3.5
3
Ba,eff [m]
Ba
[m]
8
8
8
8
8
8
8
8
8
2.5
2
1.5
1
0.5
0
0
◆
0.5
B a,eff
1
1.5
X [-]
■ B a,eff,calc
152
2
2.5
3
Effective Width of Slab
Table 5.55 Effective width for B a = 8m, H c = 4m and L = 10m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
4 10
0 5.63097 0.70387 5.23125 5.63317
0.002
1
4 10
0 4.29626 0.53703 2.90625 4.30118
0.005
1.8
4 10
0 3.80837 0.47605 2.0925 3.64278
-0.166
1.8
4 10
0 2.65204 0.33151 1.49464 3.02829
0.376
1
4 10
0 2.53552 0.31694 1.1625 2.62755
0.092
0.3
4 10
0 3.16526 0.39566 0.87188 2.2374
-0.928
1
4 10
0 1.67024 0.20878 0.83036 2.17846
0.508
0.3
4 10
0 1.32199 0.16525 0.34875 1.43184
0.110
0.3
4 10
0 15.1963 1.89954 0.24911 1.26214
s ∆Ba,eff [m] =
0.432
b 0 = 0.813484 b 1 = 1.872837 b 2 = -0.29349 b 3 = 0.021334 -
16
14
12
Ba,eff [m]
Ba
[m]
8
8
8
8
8
8
8
8
8
10
8
6
4
2
0
0
◆
1
B a,eff
2
3
X [-]
■ B a,eff,calc
153
4
5
6
Restraint Factors and Partial Coefficients
Table 5.56 Effective width for B a = 8m, H c = 4m and L = 18m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
4 18
0 9.49183 1.18648 9.41625 9.49218
0.000
1
4 18
0 8.35005 1.04376 5.23125 8.34738
-0.003
1.8
4 18
0 10.5735 1.32169 3.7665 8.01893
1.8
4 18
0 10.9298 1.36623 2.69036 8.25349
1
4 18
0 8.66153 1.08269 2.0925 8.63519
-0.026
0.3
4 18
0 8.64606 1.08076 1.56938 9.14814
0.502
1
4 18
0 9.71595 1.21449 1.49464 9.23637
-0.480
0.3
4 18
0 10.6081 1.32601 0.62775 10.5623
-0.046
0.3
4 18
0 10.8595 1.35743 0.44839 10.9113
0.052
s ∆Ba,eff [m] =
0.29
b 0 = 11.90633 b 1 = -2.42093 b 2 = 0.461307 b 3 = -0.02458 -
12
10
Ba,eff [m]
Ba
[m]
8
8
8
8
8
8
8
8
8
8
6
4
2
0
0
2
4
6
X [-]
◆
B a,eff
■ B a,eff,calc
154
8
10
Effective Width of Slab
Table 5.57 Effective width for B a = 8m, H c = 8m and L = 3m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
8
3
0 6.1521 0.76901 1.56938 6.1479
-0.004
1
8
3
0 3.21732 0.40217 0.87188 3.28117
0.064
1.8
8
3
0 2.56676 0.32085 0.62775 2.46441
-0.102
1.8
8
3
0 1.92845 0.24106 0.44839 1.86859
-0.060
1
8
3
0 1.44636 0.18079 0.34875 1.52932
0.083
0.3
8
3
0 1.23315 0.15414 0.26156 1.22362
-0.010
1
8
3
0 1.09319 0.13665 0.24911 1.1791
0.086
0.3
8
3
0 0.69686 0.08711 0.10463 0.64363
-0.053
0.3
8
3
0 0.53135 0.06642 0.07473 0.5278
-0.004
s ∆Ba,eff [m] =
0.07
b 0 = 0.229326 b 1 = 4.084103 b 2 = -1.25737 b 3 = 0.674185 -
7
6
5
Ba,eff [m]
Ba
[m]
8
8
8
8
8
8
8
8
8
4
3
2
1
0
0
0.5
◆
B a,eff
1
X [-]
■ B a,eff,calc
155
1.5
2
Restraint Factors and Partial Coefficients
Table 5.58 Effective width for B a = 8m, H c = 8m and L = 5m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
8
5
0 5.33953 0.66744 2.61563 5.34327
0.004
1
8
5
0 3.22008 0.40251 1.45313 3.17443
-0.046
1.8
8
5
0 2.31151 0.28894 1.04625 2.34653
0.035
1.8
8
5
0 1.65773 0.20722 0.74732 1.75427
0.097
1
8
5
0 1.38321 0.1729 0.58125 1.4377
0.054
0.3
8
5
0 1.47958 0.18495 0.43594 1.17071
-0.309
1
8
5
0 1.03045 0.12881 0.41518 1.13345
0.103
0.3
8
5
0 0.71438 0.0893 0.17438 0.71968
0.005
0.3
8
5
0 0.58236 0.0728 0.12455 0.63877
0.056
s ∆Ba,eff [m] =
0.12
b 0 = 0.444435 b 1 = 1.512633 b 2 = 0.394643 b 3 = -0.09822 -
6
5
Ba,eff [m]
Ba
[m]
8
8
8
8
8
8
8
8
8
4
3
2
1
0
0
0.5
◆
B a,eff
1
1.5
X [-]
■ B a,eff,calc
156
2
2.5
3
Effective Width of Slab
Table 5.59 Effective width for B a = 8m, H c = 8m and L = 10m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
8 10
0 5.52784 0.69098 5.23125 5.53407
0.006
1
8 10
0 4.18161 0.5227 2.90625 4.11923
-0.062
1.8
8 10
0 3.44833 0.43104 2.0925 3.42508
-0.023
1.8
8 10
0 2.45595 0.30699 1.49464 2.77851
0.323
1
8 10
0 2.31966 0.28996 1.1625 2.3572
0.038
0.3
8 10
0 2.62878 0.3286 0.87188 1.94722
-0.682
1
8 10
0 1.59695 0.19962 0.83036 1.88529
0.288
0.3
8 10
0 1.11749 0.13969 0.34875 1.10103
-0.016
0.3
8 10
0 0.79384 0.09923 0.24911 0.9228
0.129
s ∆Ba,eff [m] =
0.29
b 0 = 0.451665 b 1 = 1.966562 b 2 = -0.30774 b 3 = 0.022468 -
6
5
Ba,eff [m]
Ba
[m]
8
8
8
8
8
8
8
8
8
4
3
2
1
0
0
1
◆
B a,eff
2
3
X [-]
■ B a,eff,calc
157
4
5
6
Restraint Factors and Partial Coefficients
Table 5.60 Effective width for B a = 8m, H c = 8m and L = 18m.
Ha
[m]
0.4
0.4
1
1.4
1
0.4
1.4
1
1.4
B c H c L γRR B a,eff B a,eff /B a
B a,eff,calc ∆B a,eff
X
[m] [m] [m] [-]
[m]
[-]
[-]
[m]
[m]
1.8
8 18
0 7.32203 0.91525 9.41625 7.33423
0.012
1
8 18
0 6.43761 0.8047 5.23125 6.3045
-0.133
1.8
8 18
0 6.05075 0.75634 3.7665 6.07836
0.028
1.8
8 18
0 5.04811 0.63101 2.69036 5.51739
0.469
1
8 18
0 4.88962 0.6112 2.0925 4.9862
0.097
0.3
8 18
0 5.46908 0.68363 1.56938 4.36202
-1.107
1
8 18
0 3.76737 0.47092 1.49464 4.25943
0.492
0.3
8 18
0 3.06908 0.38363 0.62775 2.79563
-0.273
0.3
8 18
0 2.00912 0.25114 0.44839 2.42482
0.416
s ∆Ba,eff [m] =
0.50
b 0 = 1.385826 b 1 = 2.502261 b 2 = -0.42352 b 3 = 0.023881 -
8
7
6
Ba,eff [m]
Ba
[m]
8
8
8
8
8
8
8
8
8
5
4
3
2
1
0
0
2
◆
B a,eff
4
X [-]
■ B a,eff,calc
158
6
8
10
Location of Walls on Slabs
6
Effects of the Location of Walls on Slabs on the Effective Width of Slab
6.1
Introduction
The location of walls on slabs influences the degree of restraint. A wall located at the
centre of a slab is naturally subjected to a different degree of restraint than a wall located at one of the edges of the slab, see 2hapter 3. For structures with walls not located at the centre of the slab, the same type of calculations as above have been carried
trough with one additional adjustment of the effective width of the slab. In the determination of the effective width of slab, it was found that the values vary depending on
the location of the walls on the slabs
6.2
Effects of location of wall on slab on the effective width
In the semi-analytical method presented in this thesis, the only model parameter to adjust the geometry in the analytical part is the effective width of the slab. In the FEM
calculations of the restraint variations in the wall on slab structures the location of the
walls on the slabs was altered, see Chapter 2. First centrically located walls on slabs
were calculated, then walls located at the edges of the slabs and finally walls located in
the middle between the centre and the edges of the slabs, see Figure 6.1.
159
Restraint Factors and Partial Coefficients – Part II
a) ω=0
b) ω=0.5
c) ω=1
Figure 6.1 Description of relative location of walls on slabs a) ω = 0, b) ω = 0.5 and c) ω =
1.
For the same cross section but different locations of the wall on the slab the effective
widths of the slabs have been determined, see chapter 5, and then compared. The relation between the effective widths of the slabs of the un-symmetrically located walls and
the symmetrically located walls has been determined from the results in Chapter 5, that
is
Ba ,eff (ω ≠ 0)
Ba ,eff ( ω = 0)
(1)
These values are presented below and are used in the semi-analytical method for the
determination of the effective width of the slab that should be used to determine the
restraint variation in un-symmetrically located walls on slabs.
6.3
Results
Not all possible cross-sections with the walls in the middle between the centres and the
edges of the slabs have been calculated by FEM for the restraint variation, see chapter
3. Some cases are excluded, which means that the effective width has not been determined for these cases. In these cases, the ratio of effective width, Eq. (1), has been determined as the mean value of the ratios with the wall at the centre of the slab and with
the wall at one of the edges. These cases are marked in the tables below by use of italic
letters on the values of the ratio of effective width, see e.g. Table 6.2.
Table 6.1 – Table 6.15: Ba = 2m, Bc = 0.3, 1 and 1.8m, Hc = 0.5, 1, 2, 4 and 8m.
Table 6.16 – Table 6.30: Ba = 4m, Bc = 0.3, 1 and 1.8m, Hc = 0.5, 1, 2, 4 and 8m.
Table 6.31 – Table 6.45: Ba = 8m, Bc = 0.3, 1 and 1.8m, Hc = 0.5, 1, 2, 4 and 8m.
160
Effects of Location of Walls on Slabs
Table 6.1 B a,eff /B a for B a = 2m, H c = 0.5m and B c = 0.3m
Ba
[m]
2
2
2
2
2
2
2
2
2
2
2
2
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
Bc
[m]
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
H c L γRR
B a,eff [m]
[m] [m] [-] ω=0 ω=0.5 ω=1
0.5
3
0 2.4176 2.8739 3.8889
0.5
5
0 2.275 2.7728 3.6347
0.5 10
0 2.6198 3.145 3.9194
0.5 18
0 3.0409 3.5321 4.2396
0.5
3
0 1.722 1.804 1.8342
0.5
5
0
2.02 2.081 2.0789
0.5 10
0 2.1983 2.2844 2.2784
0.5 18
0 2.1089 2.3213 2.3083
0.5
3
0 1.3458 1.4106 1.327
0.5
5
0 1.9238 1.9378 1.7944
0.5 10
0 2.2087 2.2327 2.0772
0.5 18
0 2.203 2.2319 2.0801
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 1.1888 1.6086
1 1.2188 1.5976
1 1.2005 1.4961
1 1.1615 1.3942
1 1.0477 1.0652
1 1.0302 1.0292
1 1.0392 1.0365
1 1.1007 1.0945
1 1.0482 0.9861
1 1.0073 0.9327
1 1.0109 0.9405
1 1.0131 0.9442
Table 6.2 B a,eff /B a for B a = 2m, H c = 0.5m and B c = 1m
Ba
[m]
2
2
2
2
2
2
2
2
2
2
2
2
H a B c H c L γRR
B a,eff [m]
[m] [m] [m] [m] [-] ω=0 ω=0.5
0.4
1 0.5
3
0 2.8958
0.4
1 0.5
5
0 2.6373
0.4
1 0.5 10
0 3.0969
0.4
1 0.5 18
0 3.6359
1
1 0.5
3
0 2.2974
1
1 0.5
5
0 2.229
1
1 0.5 10
0 2.3134
1
1 0.5 18
0 2.3715
1.4
1 0.5
3
0 1.9747
1.4
1 0.5
5
0 2.1993
1.4
1 0.5 10
0 2.2805
1.4
1 0.5 18
0 2.2981
-
161
ω=1
6.7736
5.7264
6.0116
6.4027
2.4364
2.2061
2.1972
2.2585
1.3783
1.7654
1.8181
1.8334
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 1.67 2.3392
1 1.586 2.1713
1 1.471 1.9412
1 1.38 1.7609
1 1.03 1.0605
1 0.995 0.9897
1 0.975 0.9498
1 0.976 0.9524
1 0.849 0.698
1 0.901 0.8027
1 0.899 0.7973
1 0.899 0.7978
Restraint Factors and Partial Coefficients
Table 6.3 B a,eff /B a for B a = 2m, H c = 0.5m and B c = 1.8m
Ba
[m]
2
2
2
2
2
2
2
2
2
2
2
2
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
Bc
[m]
1.8
1.8
1.8
1.8
1.8
1.8
1.8
1.8
1.8
1.8
1.8
1.8
H c L γRR
B a,eff [m]
[m] [m] [-] ω=0 ω=0.5
0.5
3
0 3.2454
0.5
5
0 3.031
0.5 10
0 3.6988
0.5 18
0 3.0145
0.5
3
0 2.7403
0.5
5
0 2.4659
0.5 10
0 2.527
0.5 18
0 2.6002
0.5
3
0 2.4487
0.5
5
0 2.4345
0.5 10
0 2.4615
0.5 18
0 2.4864
-
ω=1
1.4372
1.4033
1.4812
1.6075
2.4473
1.7984
1.817
1.8556
1.778
1.7896
1.8002
1.8085
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.721 0.4428
1 0.731 0.463
1
0.7 0.4005
1 0.767 0.5333
1 0.947 0.8931
1 0.865 0.7293
1 0.86 0.719
1 0.857 0.7137
1 0.863 0.7261
1 0.868 0.7351
1 0.866 0.7314
1 0.864 0.7274
Table 6.4 B a,eff /B a for B a = 2m, H c = 1m and B c = 0.3m
Ba
[m]
2
2
2
2
2
2
2
2
2
2
2
2
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5 ω=1
0.3
1
3
0 4.2514 8.0047 13.174
0.3
1
5
0 2.7566 5.5474 9.5164
0.3
1 10
0 2.6346 5.0371 8.2126
0.3
1 18
0 3.1431 5.4436 8.4933
0.3
1
3
0 3.0983 3.7298 5.1714
0.3
1
5
0 2.4747 2.848 3.7417
0.3
1 10
0 2.1083 2.4554 3.1263
0.3
1 18
0 2.0346 2.5212 3.1876
0.3
1
3
0 2.2472 2.6726 3.6229
0.3
1
5
0 2.2388
2.44 2.9719
0.3
1 10
0 2.0584 2.2394 2.6167
0.3
1 18
0
2.1 2.2871 2.6685
162
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 1.8829 3.0988
1 2.0124 3.4523
1 1.9119 3.1172
1 1.7319 2.7022
1 1.2038 1.6691
1 1.1508 1.512
1 1.1646 1.4829
1 1.2391 1.5667
1 1.1893 1.6122
1 1.0899 1.3274
1 1.088 1.2713
1 1.0891 1.2707
Effects of Location of Walls on Slabs
Table 6.5 B a,eff /B a for B a = 2m, H c = 1m and B c = 1m
Ba
[m]
2
2
2
2
2
2
2
2
2
2
2
2
H a B c H c L γRR
B a,eff [m]
[m] [m] [m] [m] [-] ω=0 ω=0.5
0.4
1
1
3
0 1.8681
0.4
1
1
5
0 4.7304
0.4
1
1 10
0 4.4849
0.4
1
1 18
0 5.026
1
1
1
3
0 4.7119
1
1
1
5
0 2.9054
1
1
1 10
0 2.3641
1
1
1 18
0 2.4449
1.4
1
1
3
0 3.7576
1.4
1
1
5
0 2.7399
1.4
1
1 10
0 2.2357
1.4
1
1 18
0 2.2675
-
ω=1
1.6594
1.9224
1.9841
1.949
9.3125
5.6955
4.2767
4.3299
6.2868
4.0504
2.8685
2.8825
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.944 0.8883
1 0.703 0.4064
1 0.721 0.4424
1 0.694 0.3878
1 1.488 1.9764
1 1.48 1.9603
1 1.405 1.8091
1 1.385 1.771
1 1.337 1.6731
1 1.239 1.4783
1 1.142 1.2831
1 1.136 1.2712
Table 6.6 B a,eff /B a for B a = 2m, H c = 1m and B c = 1.8m
Ba
[m]
2
2
2
2
2
2
2
2
2
2
2
2
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5
1.8
1
3
0 3.9097
1.8
1
5
0 7.0383
1.8
1 10
0 6.4356
1.8
1 18
0 6.6859
1.8
1
3
0 5.6484
1.8
1
5
0 3.2207
1.8
1 10
0 2.6122
1.8
1 18
0 2.7087
1.8
1
3
0 4.6847
1.8
1
5
0 2.9813
1.8
1 10
0 2.3539
1.8
1 18
0 2.3938
-
163
ω=1
0.7886
0.9125
0.8699
0.8461
15.32
8.9081
6.778
6.8952
10.088
5.428
3.7164
3.7379
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.601 0.2017
1 0.565 0.1297
1 0.568 0.1352
1 0.563 0.1266
1 1.856 2.7122
1 1.883 2.7659
1 1.797 2.5948
1 1.773 2.5456
1 1.577 2.1534
1 1.41 1.8207
1 1.289 1.5788
1 1.281 1.5615
Restraint Factors and Partial Coefficients
Table 6.7 B a,eff /B a for B a = 2m, H c = 2m and B c = 0.3m
Ba
[m]
2
2
2
2
2
2
2
2
2
2
2
2
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5 ω=1
0.3
2
3
0 0.4671 0.4307 0.2559
0.3
2
5
0 1.0975 0.6494 0.4577
0.3
2 10
0 2.1404 0.7004 0.4698
0.3
2 18
0 2.1241 0.6897 0.4646
0.3
2
3
0 0.1522 0.1507 0.1034
0.3
2
5
0 0.3235 0.2858 0.2471
0.3
2 10
0 2.4644 4.2483 6.8341
0.3
2 18
0 2.0753 3.7844 6.0465
0.3
2
3
0 0.1023 0.0926 0.0573
0.3
2
5
0 3.7675 5.1805 7.9696
0.3
2 10
0 2.4873 3.3521 4.8617
0.3
2 18
0 2.1456 2.9092 4.2038
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.922 0.5479
1 0.5917 0.4171
1 0.3272 0.2195
1 0.3247 0.2187
1 0.9901 0.6798
1 0.8836 0.7638
1 1.7239 2.7731
1 1.8236 2.9136
1 0.9055 0.5596
1 1.375 2.1153
1 1.3477 1.9546
1 1.3559 1.9593
Table 6.8 B a,eff /B a for B a = 2m, H c = 2m and B c = 1m
Ba
[m]
2
2
2
2
2
2
2
2
2
2
2
2
H a B c H c L γRR
B a,eff [m]
[m] [m] [m] [m] [-] ω=0 ω=0.5
0.4
1
2
3
0 1.2152
0.4
1
2
5
0 1.9581
0.4
1
2 10
0 2.5829
0.4
1
2 18
0 2.5547
1
1
2
3
0 0.3554
1
1
2
5
0 0.7589
1
1
2 10
0 2.8901
1
1
2 18
0 3.0903
1.4
1
2
3
0 0.2226
1.4
1
2
5
0 0.4472
1.4
1
2 10
0 2.7271
1.4
1
2 18
0 2.5975
-
164
ω=1
0.393
0.4809
0.5194
0.5237
0.1528
0.279
0.4577
0.4885
0.0934
0.1567
8.3689
7.4028
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.662 0.3234
1 0.623 0.2456
1 0.601 0.2011
1 0.602 0.205
1 0.715
0.43
1 0.684 0.3677
1 0.579 0.1584
1 0.579 0.1581
1 0.71 0.4198
1 0.675 0.3503
1 2.034 3.0689
1 1.925
2.85
Effects of Location of Walls on Slabs
Table 6.9 B a,eff /B a for B a = 2m, H c = 2m and B c = 1.8m
Ba
[m]
2
2
2
2
2
2
2
2
2
2
2
2
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5
1.8
2
3
0 2.0909
1.8
2
5
0 2.6698
1.8
2 10
0 3.0781
1.8
2 18
0 3.0241
1.8
2
3
0 0.6019
1.8
2
5
0 1.2501
1.8
2 10
0 3.9789
1.8
2 18
0 4.5096
1.8
2
3
0 0.3603
1.8
2
5
0 0.7271
1.8
2 10
0 3.6495
1.8
2 18
0 3.6595
-
ω=1
0.6528
0.9258
1.0114
0.9992
0.2279
0.4872
0.8675
0.9295
0.1369
0.2638
0.6416
1.004
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.656 0.3122
1 0.673 0.3468
1 0.664 0.3286
1 0.665 0.3304
1 0.689 0.3786
1 0.695 0.3897
1 0.609 0.218
1 0.603 0.2061
1 0.69 0.3799
1 0.681 0.3628
1 0.588 0.1758
1 0.637 0.2744
Table 6.10 B a,eff /B a for B a = 2m, H c = 4m and B c = 0.3m
Ba
[m]
2
2
2
2
2
2
2
2
2
2
2
2
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5 ω=1
0.3
4
3
0 0.6664 0.6039 0.3954
0.3
4
5
0 0.9927 0.7638 0.4784
0.3
4 10
0 1.7933 1.0401 0.6466
0.3
4 18
0 2.129 1.0939 0.6795
0.3
4
3
0 0.2898 0.2876 0.2347
0.3
4
5
0 0.334 0.3007 0.2244
0.3
4 10
0 0.9215 0.5407 0.4065
0.3
4 18
0 1.9439 0.5983 0.4348
0.3
4
3
0 0.2288 0.228 0.2035
0.3
4
5
0 0.2331 0.2216 0.1839
0.3
4 10
0 0.5579 0.3824 0.2697
0.3
4 18
0 2.0271 0.4603 0.3554
165
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.9062 0.5933
1 0.7694 0.4819
1
0.58 0.3606
1 0.5138 0.3192
1 0.9924 0.8099
1 0.9003 0.6718
1 0.5867 0.4411
1 0.3078 0.2237
1 0.9966 0.8895
1 0.9507 0.789
1 0.6855 0.4834
1 0.2271 0.1753
Restraint Factors and Partial Coefficients
Table 6.11 B a,eff /B a for B a = 2m, H c = 4m and B c = 1m
Ba
[m]
2
2
2
2
2
2
2
2
2
2
2
2
H a B c H c L γRR
B a,eff [m]
[m] [m] [m] [m] [-] ω=0 ω=0.5
0.4
1
4
3
0 1.8068
0.4
1
4
5
0 1.8744
0.4
1
4 10
0 2.0961
0.4
1
4 18
0 2.2013
1
1
4
3
0 0.6503
1
1
4
5
0 0.745
1
1
4 10
0 1.5411
1
1
4 18
0 2.1308
1.4
1
4
3
0 0.4603
1.4
1
4
5
0 0.4699
1.4
1
4 10
0 1.1019
1.4
1
4 18
0 1.9946
-
ω=1
0.615
0.5149
0.5395
0.5765
0.3177
0.3458
0.5368
0.6137
0.23
0.2457
0.4532
0.5771
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.67 0.3404
1 0.637 0.2747
1 0.629 0.2574
1 0.631 0.2619
1 0.744 0.4886
1 0.732 0.4642
1 0.674 0.3483
1 0.644 0.288
1 0.75 0.4996
1 0.761 0.5228
1 0.706 0.4113
1 0.645 0.2894
Table 6.12 B a,eff /B a for B a = 2m, H c = 4m and B c = 1.8m
Ba
[m]
2
2
2
2
2
2
2
2
2
2
2
2
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5
1.8
4
3
0 3.343
1.8
4
5
0 2.6865
1.8
4 10
0 2.4279
1.8
4 18
0 2.486
1.8
4
3
0 1.1318
1.8
4
5
0 1.2094
1.8
4 10
0 1.9261
1.8
4 18
0 2.3842
1.8
4
3
0
0.77
1.8
4
5
0 0.7561
1.8
4 10
0 1.515
1.8
4 18
0 2.3376
-
166
ω=1
0.9754
1.1533
1.2487
1.2861
0.4069
0.6244
1.1145
1.2911
0.2896
0.3977
0.9034
1.1899
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.646 0.2918
1 0.715 0.4293
1 0.757 0.5143
1 0.759 0.5173
1 0.68 0.3595
1 0.758 0.5163
1 0.789 0.5786
1 0.771 0.5415
1 0.688 0.3761
1 0.763 0.526
1 0.798 0.5963
1 0.755 0.509
Effects of Location of Walls on Slabs
Table 6.13 B a,eff /B a for B a = 2m, H c = 8m and B c = 0.3m
Ba
[m]
2
2
2
2
2
2
2
2
2
2
2
2
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5 ω=1
0.3
8
3
0 1.0382 0.7443 0.4799
0.3
8
5
0 1.3136 0.9198 0.5295
0.3
8 10
0 1.7635 1.2103 0.5746
0.3
8 18
0 1.955 1.2661 0.6751
0.3
8
3
0 0.5457 0.4944 0.3842
0.3
8
5
0 0.5043 0.4893 0.3739
0.3
8 10
0 0.842 0.6545 0.4339
0.3
8 18
0 1.5589 0.892 0.5903
0.3
8
3
0 0.5682 0.425 0.3477
0.3
8
5
0 0.4475 0.3948 0.3181
0.3
8 10
0 0.5507 0.4657 0.2938
0.3
8 18
0 1.1861 0.7021 0.5017
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.7169 0.4623
1 0.7002 0.4031
1 0.6863 0.3258
1 0.6476 0.3453
1 0.9059 0.704
1 0.9703 0.7413
1 0.7773 0.5153
1 0.5722 0.3787
1 0.7479 0.6119
1 0.8823 0.7109
1 0.8457 0.5334
1 0.592 0.423
Table 6.14 B a,eff /B a for B a = 2m, H c = 8m and B c = 1m
Ba
[m]
2
2
2
2
2
2
2
2
2
2
2
2
H a B c H c L γRR
B a,eff [m]
[m] [m] [m] [m] [-] ω=0 ω=0.5
0.4
1
8
3
0 3.8808
0.4
1
8
5
0 3.2158
0.4
1
8 10
0 2.4061
0.4
1
8 18
0 2.0884
1
1
8
3
0 1.5812
1
1
8
5
0 1.2405
1
1
8 10
0 1.579
1
1
8 18
0 1.8754
1.4
1
8
3
0 1.201
1.4
1
8
5
0 0.8614
1.4
1
8 10
0 1.1372
1.4
1
8 18
0 1.6664
-
167
ω=1
1.5938
0.8748
0.4582
0.5022
0.9587
0.5869
0.5329
0.611
0.7941
0.4854
0.4639
0.5986
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.705 0.4107
1 0.636 0.272
1 0.595 0.1905
1 0.62 0.2405
1 0.803 0.6063
1 0.737 0.4731
1 0.669 0.3375
1 0.663 0.3258
1 0.831 0.6612
1 0.782 0.5635
1 0.704 0.4079
1 0.68 0.3592
Restraint Factors and Partial Coefficients
Table 6.15 B a,eff /B a for B a = 2m, H c = 8m and B c = 1.8m
Ba
[m]
2
2
2
2
2
2
2
2
2
2
2
2
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5
1.8
8
3
0 5.8654
1.8
8
5
0 4.198
1.8
8 10
0 2.6114
1.8
8 18
0 2.2279
1.8
8
3
0 2.7723
1.8
8
5
0 2.0663
1.8
8 10
0 2.0294
1.8
8 18
0 2.0251
1.8
8
3
0 1.6104
1.8
8
5
0 1.2416
1.8
8 10
0 1.5723
1.8
8 18
0 1.8808
-
ω=1
2.8049
2.7051
1.8155
1.5486
1.3654
1.1881
1.4221
1.5247
1.1083
0.8431
1.1294
1.4109
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.739 0.4782
1 0.822 0.6444
1 0.848 0.6952
1 0.848 0.6951
1 0.746 0.4925
1 0.787 0.575
1 0.85 0.7007
1 0.876 0.7529
1 0.844 0.6882
1 0.84 0.6791
1 0.859 0.7183
1 0.875 0.7502
Table 6.16 B a,eff /B a for B a = 4m, H c = 0.5m and B c = 0.3m
Ba
[m]
4
4
4
4
4
4
4
4
4
4
4
4
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
Bc
[m]
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
H c L γRR
B a,eff [m]
[m] [m] [-] ω=0 ω=0.5 ω=1
0.5
3
0 3.3522 3.7523 4.1529
0.5
5
0 3.9736 4.286 4.6485
0.5 10
0 5.1568 5.5036 5.9573
0.5 18
0 5.9011 6.2885 6.6948
0.5
3
0 2.118 2.2526 1.976
0.5
5
0 3.3169 3.2979 2.7561
0.5 10
0 4.5187 4.4957 4.0492
0.5 18
0 4.5626 4.6262 4.2557
0.5
3
0 1.5668 1.6823 1.4193
0.5
5
0 2.9673 2.9583 2.3305
0.5 10
0 4.6252 4.5264 3.7523
0.5 18
0 4.5963 4.5882 3.8959
168
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 1.1193 1.2389
1 1.0786 1.1698
1 1.0672 1.1552
1 1.0656 1.1345
1 1.0635 0.9329
1 0.9943 0.8309
1 0.9949 0.8961
1 1.0139 0.9327
1 1.0737 0.9059
1 0.997 0.7854
1 0.9786 0.8113
1 0.9982 0.8476
Effects of Location of Walls on Slabs
Table 6.17 B a,eff /B a for B a = 4m, H c = 0.5m and B c = 1m
Ba
[m]
4
4
4
4
4
4
4
4
4
4
4
4
H a B c H c L γRR
B a,eff [m]
[m] [m] [m] [m] [-] ω=0 ω=0.5 ω=1
0.4
1 0.5
3
0 4.0946 5.9053 7.5213
0.4
1 0.5
5
0 4.3596 6.1023 7.8762
0.4
1 0.5 10
0 5.5554 7.3096 9.0178
0.4
1 0.5 18
0 6.4511 8.1679 9.5218
1
1 0.5
3
0 3.0291 3.2321 2.8406
1
1 0.5
5
0 3.8346 3.9149 3.749
1
1 0.5 10
0 4.592 4.7742 4.6656
1
1 0.5 18
0 4.6832 4.9202 4.745
1.4
1 0.5
3
0 2.4105 2.5017 1.689
1.4
1 0.5
5
0 3.5932 3.5472 3.0116
1.4
1 0.5 10
0 4.6071 4.5969 4.0817
1.4
1 0.5 18
0 4.6448 4.6995 4.1366
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 1.4422 1.8369
1 1.3997 1.8066
1 1.3157 1.6232
1 1.2661 1.476
1 1.067 0.9378
1 1.021 0.9777
1 1.0397 1.016
1 1.0506 1.0132
1 1.0378 0.7007
1 0.9872 0.8381
1 0.9978 0.886
1 1.0118 0.8906
Table 6.18 B a,eff /B a for B a = 4m, H c = 0.5m and B c = 1.8m
Ba
[m]
4
4
4
4
4
4
4
4
4
4
4
4
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
Bc
[m]
1.8
1.8
1.8
1.8
1.8
1.8
1.8
1.8
1.8
1.8
1.8
1.8
H c L γRR
B a,eff [m]
[m] [m] [-] ω=0 ω=0.5 ω=1
0.5
3
0 4.5347 2.4607 1.4819
0.5
5
0 4.7141 2.5406 1.4572
0.5 10
0 6.145 2.5342 2.3946
0.5 18
0 7.1923 2.5353 2.1454
0.5
3
0 3.9179 3.9501 3.8974
0.5
5
0 4.3731 4.3988 4.3358
0.5 10
0 4.9013 5.1039 4.9966
0.5 18
0 5.0165 5.2474 5.0264
0.5
3
0 3.355 3.1996 2.0015
0.5
5
0 4.3182 4.0727 3.3029
0.5 10
0 4.971 4.8797 4.2132
0.5 18
0 5.0106 4.9656 4.1993
169
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.5426 0.3268
1 0.5389 0.3091
1 0.4124 0.3897
1 0.3525 0.2983
1 1.0082 0.9948
1 1.0059 0.9915
1 1.0413 1.0195
1 1.046 1.002
1 0.9537 0.5966
1 0.9431 0.7649
1 0.9816 0.8476
1 0.991 0.8381
Restraint Factors and Partial Coefficients
Table 6.19 B a,eff /B a for B a = 4m, H c = 1m and B c = 0.3m
Ba
[m]
4
4
4
4
4
4
4
4
4
4
4
4
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5 ω=1
0.3
1
3
0 6.2889 9.4695 14.107
0.3
1
5
0 4.9063 7.1646 11.33
0.3
1 10
0 5.1317 7.0849 10.559
0.3
1 18
0 5.9816 7.9108 11.215
0.3
1
3
0 3.776 4.4555 5.4901
0.3
1
5
0 4.0066 4.2041 4.6484
0.3
1 10
0 4.1766 4.3881 4.9022
0.3
1 18
0 4.3325 4.6635 5.3388
0.3
1
3
0 2.5593 3.0655 3.8262
0.3
1
5
0 3.3199 3.4655 3.6541
0.3
1 10
0 3.9981 4.0763 4.2823
0.3
1 18
0 4.2204 4.3999 4.7672
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 1.5057 2.2431
1 1.4603 2.3093
1 1.3806 2.0576
1 1.3225 1.8749
1
1.18 1.454
1 1.0493 1.1602
1 1.0506 1.1737
1 1.0764 1.2323
1 1.1978 1.495
1 1.0439 1.1007
1 1.0196 1.0711
1 1.0425 1.1296
Table 6.20 B a,eff /B a for B a = 4m, H c = 1m and B c = 1m
Ba
[m]
4
4
4
4
4
4
4
4
4
4
4
4
H a B c H c L γRR
B a,eff [m]
[m] [m] [m] [m] [-] ω=0 ω=0.5 ω=1
0.4
1
1
3
0 1.814 1.2365 0.473
0.4
1
1
5
0 6.7308 1.3858 0.5698
0.4
1
1 10
0 7.0465 1.3536 0.5732
0.4
1
1 18
0 8.0631 1.3145 0.5486
1
1
1
3
0 6.0958 8.2755 9.9683
1
1
1
5
0 4.9149 6.0494 7.7018
1
1
1 10
0 4.5239 5.5874 6.9924
1
1
1 18
0 4.6402 5.7882 6.9457
1.4
1
1
3
0 4.4904 5.8068 6.6482
1.4
1
1
5
0 4.3751 4.8827 5.6088
1.4
1
1 10
0 4.3523 4.8235 5.4181
1.4
1
1 18
0 4.4358 5.0063 5.4303
170
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.6817 0.2608
1 0.2059 0.0847
1 0.1921 0.0813
1 0.163 0.068
1 1.3576 1.6353
1 1.2308 1.567
1 1.2351 1.5457
1 1.2474 1.4969
1 1.2932 1.4805
1 1.116 1.282
1 1.1083 1.2449
1 1.1286 1.2242
Effects of Location of Walls on Slabs
Table 6.21 B a,eff /B a for B a = 4m, H c = 1m and B c = 1.8m
Ba
[m]
4
4
4
4
4
4
4
4
4
4
4
4
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5 ω=1
1.8
1
3
0 4.4548 2.456 0.7725
1.8
1
5
0 9.6703 2.7072 0.8618
1.8
1 10
0 9.6686 2.6568 0.8082
1.8
1 18
0 10.511 2.5698 0.7849
1.8
1
3
0 7.7752 11.556 15.802
1.8
1
5
0 5.4784 7.5162 10.212
1.8
1 10
0 4.7297 6.4167 8.1516
1.8
1 18
0 4.8689 6.6025 8.0535
1.8
1
3
0 6.109 8.023 10.447
1.8
1
5
0 5.0837 5.7961 7.0306
1.8
1 10
0 4.5155 5.167 5.7459
1.8
1 18
0 4.5719 5.2952 5.6144
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.5513 0.1734
1 0.2799 0.0891
1 0.2748 0.0836
1 0.2445 0.0747
1 1.4862 2.0323
1 1.372 1.864
1 1.3567 1.7235
1 1.3561 1.6541
1 1.3133 1.7102
1 1.1401 1.383
1 1.1443 1.2725
1 1.1582 1.228
Table 6.22 B a,eff /B a for B a = 4m, H c = 2m and B c = 0.3m
Ba
[m]
4
4
4
4
4
4
4
4
4
4
4
4
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5 ω=1
0.3
2
3
0 0.488 0.4636 0.282
0.3
2
5
0 1.2019 0.6954 0.4414
0.3
2 10
0 4.6109 0.7501 0.4416
0.3
2 18
0 5.141 0.7329 0.4397
0.3
2
3
0 0.1715 0.184 0.1148
0.3
2
5
0 0.5136 0.3195 0.2899
0.3
2 10
0 5.1085 6.5116 9.2614
0.3
2 18
0 4.4327 5.8353 8.2499
0.3
2
3
0 0.1144 0.1121
0.06
0.3
2
5
0 5.9855 7.1896
9.73
0.3
2 10
0 4.9279 5.5558 6.9832
0.3
2 18
0 4.3068 4.9911 6.3143
171
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1
0.95 0.5779
1 0.5786 0.3672
1 0.1627 0.0958
1 0.1426 0.0855
1 1.0729 0.6698
1 0.6221 0.5644
1 1.2746 1.8129
1 1.3164 1.8612
1 0.9805 0.5247
1 1.2012 1.6256
1 1.1274 1.4171
1 1.1589 1.4661
Restraint Factors and Partial Coefficients
Table 6.23 B a,eff /B a for B a = 4m, H c = 2m and B c = 1m
Ba
[m]
4
4
4
4
4
4
4
4
4
4
4
4
H a B c H c L γRR
B a,eff [m]
[m] [m] [m] [m] [-] ω=0 ω=0.5 ω=1
0.4
1
2
3
0 1.2411 1.2352 0.4369
0.4
1
2
5
0 2.3775 1.9343 0.6276
0.4
1
2 10
0 5.4933 2.1935 0.7421
0.4
1
2 18
0 5.8901 2.1713 0.7259
1
1
2
3
0 0.3883 0.3873 0.1631
1
1
2
5
0 0.8764 0.8738 0.3356
1
1
2 10
0 5.6502 1.3055 0.7035
1
1
2 18
0 5.358 1.3456 0.7123
1.4
1
2
3
0 0.2467 0.2331 0.0986
1.4
1
2
5
0 0.5844 0.8387 0.1793
1.4
1
2 10
0 5.342 8.2467 12.025
1.4
1
2 18
0 4.7429 7.2986 10.184
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.9952 0.3521
1 0.8136 0.264
1 0.3993 0.1351
1 0.3686 0.1232
1 0.9975 0.4201
1 0.9971 0.383
1 0.2311 0.1245
1 0.2511 0.1329
1 0.945 0.3997
1 1.4352 0.3068
1 1.5438 2.251
1 1.5388 2.1472
Table 6.24 B a,eff /B a for B a = 4m, H c = 2m and B c = 1.8m
Ba
[m]
4
4
4
4
4
4
4
4
4
4
4
4
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5 ω=1
1.8
2
3
0 2.2215 2.1424 0.6151
1.8
2
5
0 3.8212 2.9718 0.7988
1.8
2 10
0 6.9247 3.4615 0.7741
1.8
2 18
0 7.0214 3.4599 0.7513
1.8
2
3
0 0.6657 0.6553 0.1338
1.8
2
5
0 1.5722 1.4628 0.1534
1.8
2 10
0 7.8595 2.4058 2.5839
1.8
2 18
0 7.6488 2.494 2.642
1.8
2
3
0 0.4138 0.3848 0.1392
1.8
2
5
0 0.9646 1.1924 0.255
1.8
2 10
0 6.443
11.7 0.4692
1.8
2 18
0 5.8887 10.256 0.5803
172
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.9644 0.2769
1 0.7777 0.2091
1 0.4999 0.1118
1 0.4928 0.107
1 0.9843 0.201
1 0.9304 0.0976
1 0.3061 0.3288
1 0.3261 0.3454
1
0.93 0.3363
1 1.2362 0.2644
1 1.8159 0.0728
1 1.7417 0.0985
Effects of Location of Walls on Slabs
Table 6.25 B a,eff /B a for B a = 4m, H c = 4m and B c = 0.3m
Ba
[m]
4
4
4
4
4
4
4
4
4
4
4
4
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5 ω=1
0.3
4
3
0 0.7186 0.6282 0.4391
0.3
4
5
0 1.0668 0.8699 0.5329
0.3
4 10
0 2.8411 1.6591 0.637
0.3
4 18
0 4.5079
2.4 0.6439
0.3
4
3
0 0.3387 0.3248 0.2511
0.3
4
5
0 0.3987 0.3606 0.2824
0.3
4 10
0 1.1732 14.631 0.3952
0.3
4 18
0 4.3996 12.485 0.4052
0.3
4
3
0 0.247 0.2574 0.2184
0.3
4
5
0 0.2935 0.2792 0.2385
0.3
4 10
0 0.7553 0.4023 0.3283
0.3
4 18
0 5.0796 0.4449 0.3453
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.8742 0.611
1 0.8154 0.4995
1 0.584 0.2242
1 0.5324 0.1428
1 0.9589 0.7413
1 0.9043 0.7083
1 12.47 0.3368
1 2.8378 0.0921
1 1.0422 0.8843
1 0.9514 0.8125
1 0.5327 0.4347
1 0.0876 0.068
Table 6.26 B a,eff /B a for B a = 4m, H c = 4m and B c = 1m
Ba
[m]
4
4
4
4
4
4
4
4
4
4
4
4
H a B c H c L γRR
B a,eff [m]
[m] [m] [m] [m] [-] ω=0 ω=0.5 ω=1
0.4
1
4
3
0 1.8167 1.8096 0.6906
0.4
1
4
5
0 2.1774 2.0135 0.721
0.4
1
4 10
0 3.6572 2.4873 0.8011
0.4
1
4 18
0 4.5344 2.6487 0.8036
1
1
4
3
0 0.7068 0.7156 0.3504
1
1
4
5
0 0.8546 0.8042 0.4494
1
1
4 10
0 2.3039 1.6101 0.7319
1
1
4 18
0 4.4556 1.8837 1.8281
1.4
1
4
3
0 0.5035 0.518 0.247
1.4
1
4
5
0 0.5652 0.5378 0.3187
1.4
1
4 10
0 1.5101 1.1434 0.6097
1.4
1
4 18
0 4.237 1.5364 0.7831
173
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.9961 0.3801
1 0.9247 0.3311
1 0.6801 0.2191
1 0.5841 0.1772
1 1.0125 0.4959
1 0.941 0.5259
1 0.6989 0.3177
1 0.4228 0.4103
1 1.0289 0.4905
1 0.9515 0.5639
1 0.7572 0.4038
1 0.3626 0.1848
Restraint Factors and Partial Coefficients
Table 6.27 B a,eff /B a for B a = 4m, H c = 4m and B c = 1.8m
Ba
[m]
4
4
4
4
4
4
4
4
4
4
4
4
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5 ω=1
1.8
4
3
0 3.4903 5.6198 0.8739
1.8
4
5
0 3.4914 7.6789 0.8647
1.8
4 10
0 4.4595 6.561 0.7765
1.8
4 18
0 4.9739 6.7053 0.7804
1.8
4
3
0 1.2337 1.249 0.4057
1.8
4
5
0 1.4563 1.3237 0.568
1.8
4 10
0 3.2684 2.2912 0.8975
1.8
4 18
0 5.1535 2.9276 0.9983
1.8
4
3
0 0.8617 0.8749 0.2937
1.8
4
5
0 0.9294 0.8618 0.3907
1.8
4 10
0 2.3502 1.7214 0.7607
1.8
4 18
0 5.1885 2.595 0.9498
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 1.6101 0.2504
1 2.1994 0.2477
1 1.4713 0.1741
1 1.3481 0.1569
1 1.0124 0.3288
1 0.909
0.39
1 0.701 0.2746
1 0.5681 0.1937
1 1.0153 0.3408
1 0.9272 0.4204
1 0.7324 0.3237
1 0.5001 0.1831
Table 6.28 B a,eff /B a for B a = 4m, H c = 8m and B c = 0.3m
Ba
[m]
4
4
4
4
4
4
4
4
4
4
4
4
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5 ω=1
0.3
8
3
0 1.2191 0.7506 0.5042
0.3
8
5
0 1.4496 0.9703 0.5767
0.3
8 10
0 2.4422 1.7689 0.7649
0.3
8 18
0 3.6189 2.4082 0.9214
0.3
8
3
0 0.6895 0.5042 0.3864
0.3
8
5
0 0.6479 0.5363 0.4043
0.3
8 10
0 1.0236 0.7432 0.4857
0.3
8 18
0 2.3867 1.2133 0.5664
0.3
8
3
0 0.5311 0.4342 0.3509
0.3
8
5
0 0.5319 0.447 0.3555
0.3
8 10
0 0.682 0.5396 0.4002
0.3
8 18
0 1.6548 0.7992 0.476
174
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.6157 0.4136
1 0.6693 0.3978
1 0.7243 0.3132
1 0.6655 0.2546
1 0.7312 0.5604
1 0.8277 0.624
1 0.7261 0.4745
1 0.5083 0.2373
1 0.8176 0.6607
1 0.8404 0.6682
1 0.7912 0.5868
1 0.4829 0.2877
Effects of Location of Walls on Slabs
Table 6.29 B a,eff /B a for B a = 4m, H c = 8m and B c = 1m
Ba
[m]
4
4
4
4
4
4
4
4
4
4
4
4
H a B c H c L γRR
B a,eff [m]
[m] [m] [m] [m] [-] ω=0 ω=0.5 ω=1
0.4
1
8
3
0 3.164 2.4281 1.613
0.4
1
8
5
0 3.1603 2.769 1.4519
0.4
1
8 10
0 3.6745 2.8155 0.8096
0.4
1
8 18
0 3.9806 2.6835 0.6336
1
1
8
3
0 1.4066 1.5129 1.034
1
1
8
5
0 1.2944 1.3359 0.8128
1
1
8 10
0 2.1336 1.7341 0.809
1
1
8 18
0 3.3236 2.2072 0.8805
1.4
1
8
3
0 1.0756 1.2234 0.8374
1.4
1
8
5
0 0.9466 0.9765 0.6695
1.4
1
8 10
0 1.4573 1.2203 0.6799
1.4
1
8 18
0 2.7606 1.8804 0.837
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.7674 0.5098
1 0.8762 0.4594
1 0.7662 0.2203
1 0.6742 0.1592
1 1.0755 0.7351
1 1.0321 0.6279
1 0.8128 0.3792
1 0.6641 0.2649
1 1.1375 0.7785
1 1.0316 0.7073
1 0.8374 0.4666
1 0.6811 0.3032
Table 6.30 B a,eff /B a for B a = 4m, H c = 8m and B c = 1.8m
Ba
[m]
4
4
4
4
4
4
4
4
4
4
4
4
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5 ω=1
1.8
8
3
0 7.5292 4.3861 2.7432
1.8
8
5
0 5.9178 4.684 1.7749
1.8
8 10
0 4.6933 3.6438
0.68
1.8
8 18
0 4.3042 3.3155 0.603
1.8
8
3
0 2.4807 2.6955 1.3567
1.8
8
5
0 2.1995 2.2196 1.0427
1.8
8 10
0 3.0027 2.3683 0.9604
1.8
8 18
0 3.7894 2.6877 1.0156
1.8
8
3
0 1.8716 2.136 1.1071
1.8
8
5
0 1.5311 1.5625 0.8175
1.8
8 10
0 2.2119 1.7596 0.848
1.8
8 18
0 3.3903 2.3582 1.0253
175
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.5825 0.3643
1 0.7915 0.2999
1 0.7764 0.1449
1 0.7703 0.1401
1 1.0866 0.5469
1 1.0092 0.4741
1 0.7887 0.3198
1 0.7093 0.268
1 1.1412 0.5915
1 1.0205 0.5339
1 0.7955 0.3834
1 0.6956 0.3024
Restraint Factors and Partial Coefficients
Table 6.31 B a,eff /B a for B a = 8m, H c = 0.5m and B c = 0.3m
Ba
[m]
8
8
8
8
8
8
8
8
8
8
8
8
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
Bc
[m]
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
H c L γRR
B a,eff [m]
[m] [m] [-] ω=0 ω=0.5 ω=1
0.5
3
0 3.5533 4.2856 4.0294
0.5
5
0 5.3032 5.5788 4.6649
0.5 10
0 9.4167 9.041 7.4675
0.5 18
0 11.633 11.579 10.525
0.5
3
0 2.1502 2.4595 1.9344
0.5
5
0 4.0608 4.2026 2.804
0.5 10
0 8.4413 8.053 5.5377
0.5 18
0 9.5314 9.3727 7.4993
0.5
3
0 1.572 1.7902 1.3848
0.5
5
0 3.4709 3.652 2.3603
0.5 10
0 8.7642 8.3265 5.1598
0.5 18
0 9.9964 9.686 6.8925
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 1.2061 1.134
1 1.052 0.8796
1 0.9601 0.793
1 0.9953 0.9047
1 1.1439 0.8996
1 1.0349 0.6905
1 0.954 0.656
1 0.9834 0.7868
1 1.1389 0.881
1 1.0522
0.68
1 0.9501 0.5887
1 0.969 0.6895
Table 6.32 B a,eff /B a for B a = 8m, H c = 0.5m and B c = 1m
Ba
[m]
8
8
8
8
8
8
8
8
8
8
8
8
H a B c H c L γRR
B a,eff [m]
[m] [m] [m] [m] [-] ω=0 ω=0.5
0.4
1 0.5
3
0 4.5079
0.4
1 0.5
5
0 6.0008
0.4
1 0.5 10
0 9.7897
0.4
1 0.5 18
0 12.098
1
1 0.5
3
0 3.1941
1
1 0.5
5
0 4.9102
1
1 0.5 10
0
8.33
1
1 0.5 18
0 9.5219
1.4
1 0.5
3
0 2.4902
1.4
1 0.5
5
0 4.3808
1.4
1 0.5 10
0 8.368
1.4
1 0.5 18
0 9.7237
-
176
ω=1
7.5406
8.2136
11.78
14.413
2.8486
3.9541
6.793
8.5188
1.6956
3.1573
6.0027
7.643
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 1.336 1.6727
1 1.184 1.3687
1 1.102 1.2033
1 1.096 1.1914
1 0.946 0.8918
1 0.903 0.8053
1 0.908 0.8155
1 0.947 0.8947
1 0.84 0.6809
1 0.86 0.7207
1 0.859 0.7173
1 0.893 0.786
Effects of Location of Walls on Slabs
Table 6.33 B a,eff /B a for B a = 8m, H c = 0.5m and B c = 1.8m
Ba
[m]
8
8
8
8
8
8
8
8
8
8
8
8
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
Bc
[m]
1.8
1.8
1.8
1.8
1.8
1.8
1.8
1.8
1.8
1.8
1.8
1.8
H c L γRR
B a,eff [m]
[m] [m] [-] ω=0 ω=0.5
0.5
3
0 5.5724
0.5
5
0 7.1223
0.5 10
0 10.638
0.5 18
0 13.024
0.5
3
0 4.2129
0.5
5
0 5.8113
0.5 10
0 8.8539
0.5 18
0 9.9748
0.5
3
0 3.5162
0.5
5
0 5.4856
0.5 10
0 9.0646
0.5 18
0 10.345
-
ω=1
1.4852
1.6094
1.9039
1.9084
3.9237
4.8836
8.1801
9.8203
2.0075
3.7182
7.0779
8.69
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.633 0.2665
1 0.613 0.226
1 0.589 0.179
1 0.573 0.1465
1 0.966 0.9314
1 0.92 0.8404
1 0.962 0.9239
1 0.992 0.9845
1 0.785 0.5709
1 0.839 0.6778
1 0.89 0.7808
1 0.92
0.84
Table 6.34 B a,eff /B a for B a = 8m, H c = 1m and B c = 0.3m
Ba
[m]
8
8
8
8
8
8
8
8
8
8
8
8
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5 ω=1
0.3
1
3
0 6.7094 10.398 13.851
0.3
1
5
0 6.6108 8.6043 11.444
0.3
1 10
0 9.2614 10.14 12.198
0.3
1 18
0 11.574 12.679 14.884
0.3
1
3
0 3.8307 4.7921 5.3836
0.3
1
5
0 4.8574 5.2031 4.7025
0.3
1 10
0 7.3109 7.102 6.1988
0.3
1 18
0 8.7307
8.77 8.7494
0.3
1
3
0 2.5651 3.2219
3.74
0.3
1
5
0 3.8165 4.1344 3.6785
0.3
1 10
0 6.7094 6.5336 5.4867
0.3
1 18
0 8.533 8.4566 8.1943
177
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 1.5498 2.0644
1 1.3016 1.7311
1 1.0948 1.3171
1 1.0955 1.2861
1 1.251 1.4054
1 1.0712 0.9681
1 0.9714 0.8479
1 1.0045 1.0021
1 1.2561 1.458
1 1.0833 0.9639
1 0.9738 0.8178
1 0.991 0.9603
Restraint Factors and Partial Coefficients
Table 6.35 B a,eff /B a for B a = 8m, H c = 1m and B c = 1m
Ba
[m]
8
8
8
8
8
8
8
8
8
8
8
8
H a B c H c L γRR
B a,eff [m]
[m] [m] [m] [m] [-] ω=0 ω=0.5
0.4
1
1
3
0 1.8397
0.4
1
1
5
0 8.6937
0.4
1
1 10
0
11.1
0.4
1
1 18
0 13.586
1
1
1
3
0 6.4026
1
1
1
5
0 6.278
1
1
1 10
0 8.0237
1
1
1 18
0 9.1362
1.4
1
1
3
0 4.6188
1.4
1
1
5
0 5.282
1.4
1
1 10
0 7.5308
1.4
1
1 18
0 8.859
-
ω=1
0.4733
0.5839
0.8406
0.9225
9.9743
7.9859
9.3413
10.992
6.6527
5.7963
7.4311
9.1909
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.629 0.2573
1 0.534 0.0672
1 0.538 0.0757
1 0.534 0.0679
1 1.279 1.5579
1 1.136 1.2721
1 1.082 1.1642
1 1.102 1.2031
1 1.22 1.4404
1 1.049 1.0974
1 0.993 0.9868
1 1.019 1.0375
Table 6.36 B a,eff /B a for B a = 8m, H c = 1m and B c = 1.8m
Ba
[m]
8
8
8
8
8
8
8
8
8
8
8
8
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5
1.8
1
3
0 4.6186
1.8
1
5
0 11.414
1.8
1 10
0 13.127
1.8
1 18
0 15.59
1.8
1
3
0 8.3486
1.8
1
5
0 7.3053
1.8
1 10
0 8.4189
1.8
1 18
0 9.317
1.8
1
3
0 6.3742
1.8
1
5
0 6.4259
1.8
1 10
0 7.9658
1.8
1 18
0 8.9705
-
178
ω=1
0.7726
0.8721
0.8858
0.8992
15.823
10.792
11.662
13.162
10.453
7.457
8.8846
10.411
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.584 0.1673
1 0.538 0.0764
1 0.534 0.0675
1 0.529 0.0577
1 1.448 1.8952
1 1.239 1.4773
1 1.193 1.3853
1 1.206 1.4127
1 1.32 1.6398
1 1.08 1.1605
1 1.058 1.1154
1 1.08 1.1605
Effects of Location of Walls on Slabs
Table 6.37 B a,eff /B a for B a = 8m, H c = 2m and B c = 0.3m
Ba
[m]
8
8
8
8
8
8
8
8
8
8
8
8
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5 ω=1
0.3
2
3
0 0.4907 0.466 0.2748
0.3
2
5
0 1.2097 0.6875 0.4406
0.3
2 10
0 8.9573
1.9 0.4372
0.3
2 18
0 11.588 0.7095 0.4322
0.3
2
3
0 0.1733 0.198 0.1113
0.3
2
5
0 8.8041 0.3267 0.2913
0.3
2 10
0 9.0395 9.6897 10.951
0.3
2 18
0 8.9005 9.7367 11.428
0.3
2
3
0 0.1147 0.1241 0.0591
0.3
2
5
0 6.9574 8.4756 9.7899
0.3
2 10
0 8.2554 8.4381 8.4158
0.3
2 18
0 8.5817 8.8583 9.4059
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.9497 0.5599
1 0.5683 0.3642
1 0.2121 0.0488
1 0.0612 0.0373
1 1.1427 0.6422
1 0.0371 0.0331
1 1.0719 1.2114
1 1.0939 1.284
1 1.0825 0.5151
1 1.2182 1.4071
1 1.0221 1.0194
1 1.0322 1.096
Table 6.38 B a,eff /B a for B a = 8m, H c = 2m and B c = 1m
Ba
[m]
8
8
8
8
8
8
8
8
8
8
8
8
H a B c H c L γRR
B a,eff [m]
[m] [m] [m] [m] [-] ω=0 ω=0.5
0.4
1
2
3
0 1.2628
0.4
1
2
5
0 2.4102
0.4
1
2 10
0 7.0801
0.4
1
2 18
0 14.334
1
1
2
3
0 0.3978
1
1
2
5
0 0.917
1
1
2 10
0 9.7055
1
1
2 18
0 9.7734
1.4
1
2
3
0 0.2524
1.4
1
2
5
0 0.6787
1.4
1
2 10
0 9.1763
1.4
1
2 18
0 9.1251
-
179
ω=1
0.437
0.6506
1.6038
1.4676
0.1633
0.346
0.9786
1.0233
0.0987
0.1826
15.255
14.66
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.673 0.3461
1 0.635 0.2699
1 0.613 0.2265
1 0.551 0.1024
1 0.705 0.4105
1 0.689 0.3774
1 0.55 0.1008
1 0.552 0.1047
1 0.696 0.3911
1 0.635 0.2691
1 1.331 1.6624
1 1.303 1.6066
Restraint Factors and Partial Coefficients
Table 6.39 B a,eff /B a for B a = 8m, H c = 2m and B c = 1.8m
Ba
[m]
8
8
8
8
8
8
8
8
8
8
8
8
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5
1.8
2
3
0 2.2643
1.8
2
5
0 3.8826
1.8
2 10
0 10.027
1.8
2 18
0 19.779
1.8
2
3
0 0.6838
1.8
2
5
0 1.6079
1.8
2 10
0 11.877
1.8
2 18
0 12.119
1.8
2
3
0 0.4253
1.8
2
5
0 1.0568
1.8
2 10
0 10.559
1.8
2 18
0 10.288
-
ω=1
0.6155
0.8347
1.0414
1.1541
0.2277
0.4644
0.9539
1.1691
0.1394
0.2608
1.4195
1.6997
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.636 0.2718
1 0.607 0.215
1 0.552 0.1039
1 0.529 0.0584
1 0.667 0.333
1 0.644 0.2888
1 0.54 0.0803
1 0.548 0.0965
1 0.664 0.3277
1 0.623 0.2468
1 0.567 0.1344
1 0.583 0.1652
Table 6.40 B a,eff /B a for B a = 8m, H c = 4m and B c = 0.3m
Ba
[m]
8
8
8
8
8
8
8
8
8
8
8
8
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5 ω=1
0.3
4
3
0 0.7257 0.6339 0.4354
0.3
4
5
0 1.0837 0.8878 0.5317
0.3
4 10
0 3.1653 31.565 0.6425
0.3
4 18
0 8.6461 46.763 0.6494
0.3
4
3
0 0.3432 0.3381 0.2463
0.3
4
5
0 0.4314 0.3843 0.2853
0.3
4 10
0 1.322 0.5475 34.369
0.3
4 18
0 10.608 0.5721 27.405
0.3
4
3
0 0.2476 0.2672 0.2128
0.3
4
5
0 0.3185 0.3039
0.24
0.3
4 10
0 15.196 18.578 24.676
0.3
4 18
0 10.859 13.425 18.579
180
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.8734 0.5999
1 0.8193 0.4906
1 9.9723 0.203
1 5.4086 0.0751
1 0.9852 0.7178
1 0.891 0.6614
1 0.4141 25.998
1 0.0539 2.5834
1 1.0792 0.8595
1 0.9541 0.7535
1 1.2225 1.6238
1 1.2362 1.7108
Effects of Location of Walls on Slabs
Table 6.41 B a,eff /B a for B a = 8m, H c = 4m and B c = 1m
Ba
[m]
8
8
8
8
8
8
8
8
8
8
8
8
H a B c H c L γRR
B a,eff [m]
[m] [m] [m] [m] [-] ω=0 ω=0.5
0.4
1
4
3
0 1.8524
0.4
1
4
5
0 2.2181
0.4
1
4 10
0 4.2963
0.4
1
4 18
0 8.3501
1
1
4
3
0 0.7278
1
1
4
5
0 0.8959
1
1
4 10
0 2.5355
1
1
4 18
0 8.6615
1.4
1
4
3
0 0.5138
1.4
1
4
5
0 0.6051
1.4
1
4 10
0 1.6702
1.4
1
4 18
0 9.716
-
ω=1
0.6905
0.754
2.0953
2.1272
0.3506
0.4647
1.0717
1.3562
0.2474
0.3288
0.8328
1.1335
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.686 0.3728
1 0.67 0.3399
1 0.744 0.4877
1 0.627 0.2548
1 0.741 0.4817
1 0.759 0.5187
1 0.711 0.4227
1 0.578 0.1566
1 0.741 0.4814
1 0.772 0.5435
1 0.749 0.4986
1 0.558 0.1167
Table 6.42 B a,eff /B a for B a = 8m, H c = 4m and B c = 1.8m
Ba
[m]
8
8
8
8
8
8
8
8
8
8
8
8
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5
1.8
4
3
0 3.4554
1.8
4
5
0 3.5631
1.8
4 10
0 5.631
1.8
4 18
0 9.4918
1.8
4
3
0 1.2726
1.8
4
5
0 1.5065
1.8
4 10
0 3.8084
1.8
4 18
0 10.574
1.8
4
3
0 0.8869
1.8
4
5
0 0.9879
1.8
4 10
0 2.652
1.8
4 18
0 10.93
-
181
ω=1
0.8739
0.9234
1.1654
1.3469
0.4062
0.5923
1.1559
3.1646
0.2938
0.4066
0.9642
1.3794
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.626 0.2529
1 0.63 0.2591
1 0.603 0.207
1 0.571 0.1419
1 0.66 0.3192
1 0.697 0.3932
1 0.652 0.3035
1 0.65 0.2993
1 0.666 0.3313
1 0.706 0.4116
1 0.682 0.3636
1 0.563 0.1262
Restraint Factors and Partial Coefficients
Table 6.43 B a,eff /B a for B a = 8m, H c = 8m and B c = 0.3m
Ba
[m]
8
8
8
8
8
8
8
8
8
8
8
8
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5 ω=1
0.3
8
3
0 1.2332 0.753 0.5038
0.3
8
5
0 1.4796 0.9818 0.5766
0.3
8 10
0 2.6288 2.2097 0.8019
0.3
8 18
0 5.4691 33.529 1.1133
0.3
8
3
0 0.6969 0.5075 0.3858
0.3
8
5
0 0.7144 0.5528 0.4048
0.3
8 10
0 1.1175 0.7822 0.4919
0.3
8 18
0 3.0691 1.6088 0.5677
0.3
8
3
0 0.5314 0.4372 0.3496
0.3
8
5
0 0.5824 0.4639 0.3559
0.3
8 10
0 0.7938 0.5694 0.413
0.3
8 18
0 2.0091 0.8166 0.4714
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.6107 0.4085
1 0.6636 0.3897
1 0.8406 0.305
1 6.1306 0.2036
1 0.7283 0.5536
1 0.7739 0.5667
1 0.6999 0.4402
1 0.5242 0.185
1 0.8228 0.658
1 0.7966 0.6112
1 0.7173 0.5202
1 0.4064 0.2346
Table 6.44 B a,eff /B a for B a = 8m, H c = 8m and B c = 1m
Ba
[m]
8
8
8
8
8
8
8
8
8
8
8
8
H a B c H c L γRR
B a,eff [m]
[m] [m] [m] [m] [-] ω=0 ω=0.5
0.4
1
8
3
0 3.2173
0.4
1
8
5
0 3.2201
0.4
1
8 10
0 4.1816
0.4
1
8 18
0 6.4376
1
1
8
3
0 1.4464
1
1
8
5
0 1.3832
1
1
8 10
0 2.3197
1
1
8 18
0 4.8896
1.4
1
8
3
0 1.0932
1.4
1
8
5
0 1.0304
1.4
1
8 10
0 1.597
1.4
1
8 18
0 3.7674
-
182
ω=1
1.6129
1.5422
2.0578
2.5754
1.0342
0.842
1.2624
1.8718
0.8376
0.6916
0.9283
1.5847
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.751 0.5013
1 0.739 0.4789
1 0.746 0.4921
1
0.7 0.4001
1 0.858 0.715
1 0.804 0.6088
1 0.772 0.5442
1 0.691 0.3828
1 0.883 0.7662
1 0.836 0.6711
1 0.791 0.5813
1 0.71 0.4206
Effects of Location of Walls on Slabs
Table 6.45 B a,eff /B a for B a = 8m, H c = 8m and B c = 1.8m
Ba
[m]
8
8
8
8
8
8
8
8
8
8
8
8
Ha
[m]
0.4
0.4
0.4
0.4
1
1
1
1
1.4
1.4
1.4
1.4
B c H c L γRR
B a,eff [m]
[m] [m] [m] [-] ω=0 ω=0.5
1.8
8
3
0 6.1521
1.8
8
5
0 5.3395
1.8
8 10
0 5.5278
1.8
8 18
0 7.322
1.8
8
3
0 2.5668
1.8
8
5
0 2.3115
1.8
8 10
0 3.4483
1.8
8 18
0 6.0508
1.8
8
3
0 1.9285
1.8
8
5
0 1.6577
1.8
8 10
0 2.456
1.8
8 18
0 5.0481
-
183
ω=1
1.4066
0.5325
0
0
1.3566
1.097
1.3471
1.6703
1.1074
0.8571
1.1374
1.5744
B a,eff /B a [-]
ω=0 ω=0.5 ω=1
1 0.614 0.2286
1 0.55 0.0997
1
0.5
0
1
0.5
0
1 0.764 0.5285
1 0.737 0.4746
1 0.695 0.3907
1 0.638 0.276
1 0.787 0.5743
1 0.759 0.517
1 0.732 0.4631
1 0.656 0.3119
Basic Resilience Correction Factors
7
Basic Resilience Correction Factors
7.1
Introduction
In cases where the boundary restraint is not negligible, according to Nilsson et al.
(2003) the high wall effects, resilience, depend on the degree of boundary restraint and
is calculated as
0
δres = δres
δtransl δrot
(1)
0
δtransl = γ RT + (1 − γ RT ) δtransl
(2)
0
δrot = γ RR ,z + (1 − γ RR ,z ) δrot
(3)
with
where
δ0res = basic resilience factor, see Figure 3.3, [-]
δtransl = resilience correction factor for translational boundary restraint, [-]
δrot = resilience correction factor for rotational boundary restraint, [-]
In cases with free translation and free rotation, that is no degree of boundary restraint γRT = γRR,z = 0, Eq. (1) becomes
0 0
0
δres = δres
δtransl δrot
(4)
For no boundary restraint 3D effects are considered only by introducing an effective
width of the slab Ba,eff, see chapter 4, which means that formally δ0translδ0rot ≡ 1 in this
185
Restraint Factors and Partial Coefficients
case. So, the basic resilience correction factor for rotation is determined as the inverse
of the basic resilience correction factor for translation by
δ0rot =
1
(5)
0
δtransl
This relation is used in the determination of the basic correction factor for bending
for structures subjected to rotational boundary restraint, see below.
The last and worst case rises at total boundary restraint, γRT = γRR,y = γRR,z = 1 that
with Eq. (1) in the general formulation of the semi-analytical expression, see Nilsson et
al. (2003) gives
0
γ R = δres
(6)
which exactly corresponds to the methodology in ACI (1995) for base restraint walls.
7.1.1 Determination of basic resilience correction for translational boundary restraint
The determination of the basic resilience correction factors for translational boundary
restraint has been done by evaluating the restraint distributions from the results of the
FEM calculations with γRT = 0, γRR,z = 1 and δslip = 1, see Nilsson (2003). The high
wall effects are calculated from Eq. (1) giving
 y 
0  y  0
δres = δres
 H  δtransl  H 
 c
 c
(7)
 y 
 y 
0
is calculated according to Chapter 3. δtransl
where δ0res 

 H  is then determined
 Hc 
 c
by using the formula of the semi-analytical method with γRT = 0, γRR,z = 1, δslip = 1
and Eq. (7)
0
 y 
0  y  0
γ R = δres
 H  δtransl  H  −
 c
 c
 y 
 y 
∫A δres  H c  δtransl  H c  dAc
186
0
c
Atrans
(8)
Basic Resilience Correction Factors
As the distribution of the restraint factor, γR,j, is known from the FEM calculations,
 yj 
0
the values of δtransl

 can be established point-by-point (j = 1, …, n) using Eq. (8)
 Hc 
expressed as
 yj 
0
δtransl

=
 H c  δ0  y j
res 
 Hc
γR, j

1
−
 Atrans
∫
δ0
Ac res
 y
H
 c

 dAc

(9)
which by introducing the polynomials for describing the resilience gives
 yj
0
δtransl

 Hc

=

γR, j
i
y 
A
∑ ai  Hj  − A c
trans
c 
i =0 
7
7
(10)
a
∑ i +i 1
i =0
For each set of restraint variations obtained in the FEM calculations with the rotational boundary restraint γRR,z = 1, see chapter 3, the variations of δ0transl are determined by use of Eq. (10). However, there is one limitation due to arisen numerical
problems when the denominator in Eq. (10) reaches zero and then change sign, which
is the case for length to height ratios smaller than about 2.2. This problem is also
vaguely indicated in JCI (1992). A second way of avoiding these numerical problems is
by skipping the interpolation and always choosing data on the safe..
7.2
Results
Below diagrams of the variations of the denominator in Eq. (10) are shown as function
of the area ratio Ac/Atrans for structures with length to height ratios corresponding to
these for the basic resilience curves, see chapter 5. It can be seen that with larger area
ratios the denominator changes sign and consequently δ0transl changes sign, which makes
Eq. (10) impossible to use, as δ0transl reaches infinity.
Figure 7.1 – Figure 7.14: L/Hc = 0.25, 0.5, 1, 1.2, 1.4, 1.6, 2, 3, 4, 5, 6, 7, 19 and 40.
187
Restraint Factors and Partial Coefficients
y/Hc [-]
L/Hc = 0.25
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.6 -0.4 -0.2
n
Ac
Atrans
0
0.2
0.4
0.6
0.8
1
0
 yj 
Ac
 −
 c  Atrans
∑ ai  H
i =0
0.2
i
0.4
n
0.6
a
∑ i +i 1
0.8
1
[ −]
i =0
Figure 7.1 Denominator of Eq. (10) as function of the ratio Ac/Atrans for L/Hc = 0.25.
y/Hc [-]
L/Hc = 0.5
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.6 -0.4 -0.2
n
Ac
Atrans
0
0.2
0.4
0.6
0.8
1
0
 yj 
Ac
 −
 c  Atrans
∑ ai  H
i =0
0.2
i
0.4
n
0.6
a
∑ i +i 1
0.8
1
[-]
i =0
Figure 7.2 Denominator of Eq. (10) as function of the ratio Ac/Atrans for L/Hc = 0.5.
188
Basic Resilience Correction Factors
y/Hc [-]
L/Hc = 1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.6 -0.4 -0.2
n
Ac
Atrans
0
0.2
0.4
0.6
0.8
1
0
 yj 
Ac
 −
 c  Atrans
∑ ai  H
i =0
0.2
i
0.4
n
0.6
a
∑ i +i 1
0.8
1
[-]
i =0
Figure 7.3 Denominator of Eq. (10) as function of the ratio Ac/Atrans for L/Hc = 1.
y/Hc [-]
L/Hc = 1.2
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.6 -0.4 -0.2
n
Ac
Atrans
0
0.2
0.4
0.6
0.8
1
0
 yj 
Ac
 −
 c  Atrans
∑ ai  H
i =0
0.2
i
0.4
n
0.6
a
∑ i +i 1
0.8
1
[-]
i =0
Figure 7.4 Denominator of Eq. (10) as function of the ratio Ac/Atrans for L/Hc = 1.2.
189
Restraint Factors and Partial Coefficients
y/Hc [-]
L/Hc = 1.4
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.6 -0.4 -0.2
n
Ac
Atrans
0
0.2
0.4
0.6
0.8
1
0
 yj 
Ac
 −
 c  Atrans
∑ ai  H
i =0
0.2
i
0.4
n
0.6
a
∑ i +i 1
0.8
1
[-]
i =0
Figure 7.5 Denominator of Eq. (10) as s function of the ratio Ac/Atrans for L/Hc = 1.4.
y/Hc [-]
L/Hc = 1.6
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.6 -0.4 -0.2
n
Ac
Atrans
0
0.2
0.4
0.6
0.8
1
0
 yj 
Ac
 −
 c  Atrans
∑ ai  H
i =0
0.2
i
0.4
n
0.6
a
∑ i +i 1
0.8
1
[-]
i =0
Figure 7.6 Denominator of Eq. (10) as function of the ratio Ac/Atrans for L/Hc = 1.6.
190
Basic Resilience Correction Factors
y/Hc [-]
L/Hc = 2
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.6 -0.4 -0.2
n
Ac
Atrans
0
0.2
0.4
0.6
0.8
1
0
 yj 
Ac
 −
 c  Atrans
∑ ai  H
i =0
0.2
i
0.4
n
0.6
a
∑ i +i 1
0.8
1
[-]
i =0
Figure 7.7 Denominator of Eq. (10) as function of the ratio Ac/Atrans for L/Hc = 2.
y/Hc [-]
L/Hc = 3
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.6 -0.4 -0.2
n
Ac
Atrans
0
0.2
0.4
0.6
0.8
1
0
 yj 
Ac
 −
 c  Atrans
∑ ai  H
i =0
0.2
i
0.4
n
0.6
a
∑ i +i 1
0.8
1
[-]
i =0
Figure 7.8 Denominator of Eq. (10) as function of the ratio Ac/Atrans for L/Hc = 3.
191
Restraint Factors and Partial Coefficients
y/Hc [-]
L/Hc = 4
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.6 -0.4 -0.2
n
Ac
Atrans
0
0.2
0.4
0.6
0.8
1
0
 yj 
Ac
 −
 c  Atrans
∑ ai  H
i =0
0.2
i
0.4
n
0.6
a
∑ i +i 1
0.8
1
[-]
i =0
Figure 7.9 Denominator of Eq. (10) as function of the ratio Ac/Atrans for L/Hc = 4.
y/Hc [-]
L/Hc = 5
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.6 -0.4 -0.2
n
Ac
Atrans
0
0.2
0.4
0.6
0.8
1
0
 yj 
Ac
 −
 c  Atrans
∑ ai  H
i =0
0.2
i
0.4
n
0.6
a
∑ i +i 1
0.8
1
[-]
i =0
Figure 7.10 Denominator of Eq. (10) as function of the ratio Ac/Atrans for L/Hc = 5.
192
Basic Resilience Correction Factors
y/Hc [-]
L/Hc = 6
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.6 -0.4 -0.2
n
Ac
Atrans
0
0.2
0.4
0.6
0.8
1
0
 yj 
Ac
 −
 c  Atrans
∑ ai  H
i =0
0.2
i
0.4
n
0.6
a
∑ i +i 1
0.8
1
[-]
i =0
Figure 7.11 Denominator of Eq. (10) as function of the ratio Ac/Atrans for L/Hc = 6.
y/Hc [-]
L/Hc = 7
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.6 -0.4 -0.2
n
Ac
Atrans
0
0.2
0.4
0.6
0.8
1
0
 yj 
Ac
 −
 c  Atrans
∑ ai  H
i =0
0.2
i
0.4
n
0.6
a
∑ i +i 1
0.8
1
[-]
i =0
Figure 7.12 Denominator of Eq. (10) as function of the ratio Ac/Atrans for L/Hc = 7.
193
Restraint Factors and Partial Coefficients
y/Hc [-]
L/Hc = 10
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.6 -0.4 -0.2
n
Ac
Atrans
0
0.2
0.4
0.6
0.8
1
0
 yj 
Ac
 −
 c  Atrans
∑ ai  H
i =0
0.2
i
0.4
n
0.6
a
∑ i +i 1
0.8
1
[-]
i =0
Figure 7.13 Denominator of Eq. (10) as function of the ratio Ac/Atrans for L/Hc = 10.
y/Hc [-]
L/Hc = 40
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.6 -0.4 -0.2
n
Ac
Atrans
0
0.2
0.4
0.6
0.8
1
0
 yj 
Ac
 −
 c  Atrans
∑ ai  H
i =0
0.2
i
0.4
n
0.6
a
∑ i +i 1
0.8
1
[-]
i =0
Figure 7.14 Denominator of Eq. (10) as function of the ratio Ac/Atrans for L/Hc = 40.
194
References
References
ACI (1973). Effect of Restraint, Volume Change, and Reinforcement on
Cracking of Massive Concrete. ACI Committee 207. ACI Journal / July 1973.
Title No. 70-45. pp. 445-470.
ACI (1990). Effect of Restraint, Volume Change, and Reinforcement on
Cracking of Massive Concrete. ACI Committee 207. ACI Materials Journal /
May-June 1990. Title No. 87-M31. pp. 271-295.
ACI Committee 207 (1995). Effect of Restraint, Volume Change, and
Reinforcement on Cracking of Massive Concrete. ACI Committee 207. ACI
207.2R-95.
Bernander, S (1973). Cooling by Means of Embedded Cooling Pipes.
Applications in Connection with the Construction of the Tingstad Tunnel,
Göteborg. Nordisk Betong, n:o 2-73. pp. 21-31. (In Swedish).
Bernander, S (1982). Temperature Stresses in Early Age Concrete Due to
Hydration. In: Proceedings from International Conference on Concrete at Early Ages,
RILEM, held in Paris, France on 6-8 April 1982. Vol II. pp. 218-221.
Bernander, S (1993). Balk på elastiskt underlag åverkad av ändmoment M1 (Beam on
Resilient Ground Loaded by Bending End Moments M1). Göteborg, Sweden:
ConGeo AB. Notes and calculations with diagrams. (In Swedish).
Bernander, S (1998). Practical Measures to Avoiding Early Age Thermal
Cracking in Concrete Structures. In: Prevention of Thermal Cracking in Concrete at
Early Ages. Ed. by Springenschmidt, R. London, UK: E & FN Spon. RILEM
Report 15. pp. 255-314. ISBN 0-419-22310-X.
195
Restraint Factors and Partial Coefficients
Bernander, S (2001). Analysis of Deformations in an Elastic Halfspace due to
Horisontal Loading. Göteborg, Sweden: Con-Geo AB. Notes and calculations
with diagrams. pp. 47.
Bernander, S & Emborg, M (1994). Temperaturförhållanden och
sprickbegränsning i grova betongkonstruktioner (Temperature conditions and
crack limitation in mass concrete structures). In: Betonghandbok – Arbetsutförande,
projektering och byggande (Concrete Handbook – Performance, projecting and
building). Solna, Sweden: AB Svensk Byggtjänst. pp. 639-666. ISBN: 91-7332586-4.
BRO 2002 (2002). Bro 2002. Allmän teknisk beskrivning för broar, 4
Betongkonstruktioner. (General Technical Description for Bridges, 4. Concrete
structures). Borlänge, Sweden: Swedish National Road Administration. (In
Swedish).
Collins, M, P & Mitchell, D (1991). Prestressed Concrete Structures. Englewood
Cliffs, New Jersey, U.S.A.: Prentice-Hall Inc. pp. 766. ISBN 0-13-691635-X.
ConTeSt Pro (2003). Users manual - Program for Temperature and Stress
Calculations in Concrete. Developed by JEJMS Concrete AB in co-operation with
Luleå University of Technology, Cementa AB and Peab AB. Luleå, Sweden:
Luleå University of Technology. (In progress May 2003).
Emborg, M & Bernander, S (1994a). Assessment of the Risk of Thermal
Cracking in Hardening Concrete. In: Journal of Structural Engineering (ASCE),
Vol 120, No 10, October 1994. New York, U.S.A. pp. 2893-2912.
Emborg, M & Bernander, S (1994b). Thermal stresses computed by a method
for manual calculation. In: Proceedings of the International RILEM Symposium on
Thermal Cracking in Concrete at Early Ages. Munich, Germany, Oct. 10-12 1994.
London, England: E & FN Spon. RILEM Proceedings 25. pp. 321-328. ISBN:
0 419 18710 3.
Emborg, M (1989). Thermal Stresses in Concrete Structures at Early Ages.
Luleå, Sweden: Division of Structural Engineering, Luleå University of
Technology. Doctoral Thesis 1989:73D. pp. 285.
Emborg, M, Bernander, S, Ekefors, K, Groth, P & Hedlund, H (1997).
Temperatursprickor
i
betongkonstruktioner
Beräkningsmetoder
för
hydratationsspänningar och diagram för några vanliga typfall (Temperature Cracks in
Concrete Structures - Calculation Methods for Hydration Stresses and Diagrams
for some Common Typical Examples). Luleå, Sweden: Luleå University of
196
References
Technology. Technical Report 1997:02. 100 pp. ISSN 1402-1536. (In
Swedish).
Emborg, M, Bernander, S, Jonasson, J-E & Nilsson M (2003). Avoidance of early
age cracking - principles and recommendations. Luleå, Sweden: Luleå University of
Technology, Department of Civil and Mining Engineering. IPACS Report. pp.
ISBN-91-89580-80-X. (In progress).
Heimdal, E, Kanstad, T & Kompen, R (2001a). Maridal Culvert, Norway - Field
test I. Luleå, Sweden: Luleå University of Technology, Department of Civil and
Mining Engineering. IPACS-report BE96-3843/2001:73-7. pp. 69. ISBN 9189580–73–7.
Heimdal, E, Kanstad, T & Kompen, R (2001b). Maridal Culvert, Norway - Field
test II. Luleå, Sweden: Luleå University of Technology, Department of Civil
and Mining Engineering. IPACS-report BE96-3843/2001:74-5. pp. 20. ISBN
91-89580–74–5.
JCI (1992). A Proposal of a Method of Calculating Crack Width due to Thermal Stress
(1992). Tokyo, Japan: Japan Concrete Institute, Committee on Thermal Stress
of Massive Concrete Structures. JCI Committee Report. pp. 106.
Jonasson, J-E, Emborg, M & Bernander, S (1994). Temperatur,
mognadsutveckling och egenspänningar i ung betong (Temperature, Maturity
Growth and Eigenstresses in Young Concrete). In: Betonghandbok - Material.
(Concrete Handbook - Material). Edition 2. Stockholm, Sweden: AB Svensk
Byggtjänst and Cementa AB. pp. 547-607. ISBN 91-7332-709-3. (In Swedish).
Jonasson, J-E, Wallin, K, Emborg, M, Gram, A, Saleh, I, Nilsson, M, Larson,
M & Hedlund, H (2001). Temperatursprickor i betongkonstruktioner – Handbok med
diagram för sprickriskbedömning inclusive åtgärder för några vanliga typfall. Del D och E.
(Temperature cracks in concrete structures – Handbook with diagrams for crack
risk judgement including measures for som typical cases. Part D and E). Luleå,
Sweden: Division of Structural Engineering, Luleå University of Technology,
Technical Report 2001:14. ISSN: 1402-1536, ISRN: LTU – TR – 01/14 –
SE.
Kanstad, T, Bosnjak, D & Øverli, J A (2001). 3D Restraint Analyses of Typical
Structures with Early Age Cracking Problems. Luleå, Sweden: Luleå University of
Technology, Department of Civil and Mining Engineering. IPACS-report
BE96-3843/2001:32-X. ISBN 91-89580-32-X. pp. 27.
197
Restraint Factors and Partial Coefficients
Larson, M (1999). Evaluation of Restraint from Adjoining Structures. Luleå, Sweden:
Luleå University of Technology, Department of Civil and Mining Engineering.
IPACS-report BE96-3843/2001:57-5. ISBN 91-89580–57–5. pp. 23.
Larson, M (2000). Estimation of Crack Risk in Early Age Concrete. Luleå, Sweden:
Division of Structural Engineering, Luleå University of Technology. Licentiate
Thesis 2000:10. pp. 170.
Larson, M (2003). Thermal Crack Estimation in Early Age Concrete – Models and Methods
for Practical Application. Doctoral Thesis 2003:20. pp. 190. ISBN: 91-86580-06-0.
Löfquist, B (1946). Temperatureffekter i hårdnande betong (Temperature Effects in
Hardening Concrete). Stockholm, Sweden: Royal Hydro Power
Administration. Technical Bulletins, Serie B, No 22. pp. 195. (In Swedish).
Nilsson, M (1998). Inverkan av tvång i gjutfogar och i betongkonstruktioner på elastiskt
underlag (Influence of Restraint in Casting Joints and in Concrete Structures on
Elastic Foundation). Luleå, Sweden: Luleå University of Technology, Division
of Structural Engineering. Master Thesis 1998:090 CIV. pp. 61. (In Swedish).
Nilsson, M (2000). Thermal Cracking of Young Concrete – Partial
Coefficients, Restraint Effects and Influence of Casting Joints. Luleå, Sweden:
Division of Structural Engineering, Luleå University of Technology. Licentiate
Thesis 2000:27. pp. 267. http://epubl.luth.se/1402-1757/2000/27/LTU-LIC0027-SE.pdf.
Nilsson, M (2003). Restraint Factors and Partial Coefficients for Crack Risk Analyses
of Early Age Concrete Structures. Luleå, Sweden: Division of Structural
Engineering, Luleå University of Technology. Doctoral Thesis 2003:19. pp.
170. ISBN: 91-86580-05-2.
Nilsson, M, Jonasson, J-E, Wallin, K, Emborg, M, Bernander, S & Elfgren, L
(1999). Crack Prevention in Walls and Slabs - The Influence of Restraint. In:
Innovation in Concrete Structures, Design and Construction. Proceedings of the
International Conference held at the University of Dundee, Scotland, UK on 8-10
September 1999. Ed. by R. K. Dhir & M. R. Jones. London, UK: Thomas
Telford Publishing. pp. 461-471. ISBN: 0 7277 2824 5.
Olofsson, J, Uhlán, M & Hedlund, H (2001). Slab cast on rock ground - Model for
restraint estimation. Luleå, Sweden: Luleå University of Technology, Department
of Civil and Mining Engineering. IPACS-report BE96-3843/2001:66-4. ISBN
91-89580-66-4. pp. 34.
198
References
Olofsson, J, Uhlán, M & Hedlund, H (2001). 2D and 3D Restraint Analyses,
Typical Structure – Wall on Slab. Luleå, Sweden: Luleå University of Technology,
Department of Civil and Mining Engineering. IPACS-report BE963843/2001:64-8. ISBN 91-89580-64-8. pp. 81.
Pettersson, D (1998). Stresses in Concrete Structures from Ground Restraint. Lund,
Sweden: Department of Structural Engineering, Lund Institute of Technology.
Report TVBK-1014. pp. 112.
Rostásy, F, S, Gutsch, A-W & Krauß, M (2001). Engineering models for the
assessment of restraint of slabs by soil and in piles in the early age of concrete. Luleå,
Sweden: Luleå University of Technology, Department of Civil and Mining
Engineering. IPACS-report BE96-3843/2001:59-1. ISBN 91-89580-59-1. pp.
135.
Rostásy, F S, Tanabe, T & Laube, M (1998). Assessment of External Restraint.
In: Prevention of Thermal Cracking in Concrete at Early Ages. Ed. by R.
Springenschmid. London, England: E & FN Spon. RILEM Report 15. Stateof-the Art Report by RILEM Technical Committee 119, Avoidance of
Thermal Cracking in Concrete at Early Ages. pp. 149-177. ISBN: 0 419 22310
X.
Stoffers, H (1978). Cracking due to Shrinkage and Temperature Variations in
Walls. Heron, vol. 23, no. 23. pp. 5-68.
Wallin, K, Emborg, M & Jonasson, J-E (1997). Värme ett alternativ till kyla (Heat
an alternative to cold). Luleå, Sweden: Division of Structural Engineering, Luleå
University of Technology. Technical Report 1997:15. pp. 168. (In Swedish).
199