Strut-and-Tie models for ductility required R C members
S G Hong, Seoul National University, South Korea
S G Lee*, Seoul National University, South Korea
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Strut-and-Tie models for ductility required Remembers
S G Hong, Seoul National University, South Korea
S G Lee*, Seoul National University, South Korea
Abstract
This paper presents strut-and-tie models that can be used to determine strength
and deformation for ductile RC members. Traditional strut-and-tie models have been
applied to strength-based design while they cannot appropriately address the ductility
of members that is important to ensure safety under severe lateral loads. According
to the capacity design philosophy, a plastic hinge should occur at the bottom end of
columns at the ground story and at the ends of beams, where sufficient ductility
should be assured. The proposed strut-and-tie model is constituted to address this
condition and will be the basis for the calculation of deformation and correlated shear
strength as well. The behavior of each element of the strut-and-tie model is evaluated
based on the stress field. The deformation is obtained by combining the elongation of
tie element with the shortening of the strut element. The elongation of the tie element
depending on crack spacing and width is obtained from the bond-slip relationship.
The strut-and-tie model in this paper will provide useful tools in both the design and
evaluation of ductility-required RC members. Furthermore, it is expected to be
appropriate for newly developed seismic design methodology
Keywords: strut-and-tie model, deformation analysis, shear strength
1.
Introduction
Most current codes for seismic design have relied on strength-based design with selection of
appropriate system ductility. To meet the requirement for strength and ductility, most building codes,
such as ACI 318[1], provide the methodology for the dimensioning of members and detailing of
reinforcement. Recent researches on earthquake engineering, however, are tending toward
performance-based design so as to overcome the disadvantages of the current strength-based design
concepts, which have no direct relation between seismic load and performance of the structure such
as strength and ductility. Accordingly, appropriate member design methods, which can estimate the
behaviors of members, are required for the newly developed seismic design methodology such as
displacement-based deSign
design methods.
According to capacity design philosophy[2], plastic hinges are supposed to form at the bottom end
of columns at the ground story and at the ends of beams for desirable failure mechanisms. It is
necessary to ensure sufficient deformation capacity of such members. Brittle members such as beamcolumn joints should remain elastic at ultimate to assure seismic performance of the structure. For this
purpose, It is important to determine the deformation of members requiring ductility including plastic
hinge rotation. In particular, the rotation limit of the ends of the bottom columns should be determined
that may controls the lateral displacement capacity of the whole system.
Strut-and-tie models depict internal force flows represented by discrete compression and tension
elements joined together at nodes. In terms of the effective strength of each element, these models
317
have been successfully applied to the design of D- regions of RC members, where the strain and
stress are not linearly distributed. Even though it is recognized that strut-and-tie models have been
promising design tools in the D- regions of RC members, these strength-based models provide only
required strength of members for seismic design. To overcome this limitation, Marti [3] suggested a
tension chord model to calculate deformation of the tension tie element, which is applied to determine
the deformation limit of flexural members and extended to deal with shear problems. As another
application of strut-and-tie models to seismic loading, To et al. [4] suggested a strut-and-tie computer
model to analyze nonlinear behaviors of RC knee joints by truss elements within the Drain-2DX
program.
The objectives of this paper are to define the behaviors of elements the of strut-and-tie model and
to apply them to the deformability estimation of RC members. The strut-and-tie model to be presented
is to calculate the ultimate deformation and its corresponding shear strength. In this model, a simplified
bond-slip relationship is applied to the behavior of tie elements and concrete strut elements employ
the constitutive equations of cracked concrete in the published literature.
2.
Strut-and-Tie Modeling
Fig. 1 (a) shows RC flexural member subjected to shear and axial force. The model represents a
typical RC intermediate short column to provide an appropriate example for limited ductility.
The shear force V and the axial force P act on the inflection point of the member. The length
of the member between the inflection point and the end face is denoted by L. The deformation of the
member is mainly controlled by the deformation of the end region including plastic hinge. Thus, the
strut-and-tie model shown in Fig. 1 (b) is selected to focus on the behavior of the region, while the
other region above the end region is assumed as a rigid body.
Element forces T and C represent tie elements and strut elements, respectively. Subscript I
denotes association with the longitudinal element, and subscripts tr and d, transverse and diagonal
element, respectively. Fig. 1 (c) shows a statically admissible stress field that may determine the
geometric properties of truss elements.
Member deformation must include the effect of a jOint rotation determined by the slip of the
anchored steel bars and concrete block contraction. By assuming a joint region as an elastic state at
failure, the rotation is simply estimated.
Diagonal strut angle () is assumed as the angle of inclined crack at the yielding of main bars:
() = tan -I [
2L
AsJYl +2P
where Gcr
(AshEsGcr + ECGcrb)]
(1 )
s
=cracking stress of concrete; and s =spacing of transverse reinforcing bar.
T,3
t
TIT2
(),
T,2
T"
(a)
T
I
C d2,
T,rl
V
"v1\ /
. ( 12
'
I
~
T
C;, ... ~(II
(b)
(c)
Fig. 1 Modeling of columns (a) Force acting on the column (b) strut-and-tie model (c) stress field
3.
Element Force by Equilibrium
Truss models may be treated as determinate structures assuming yielding of components at
ultimate. Then, element forces are calculated by equilibrium equations. The force of each element is
expressed in terms of external shear force V and axial force P, as summarized in Table 1.
318
Table 1. Element Force by Equilibrium
4.
Tie Element
Element Force
Strut Element
Element Force
1;1
I
P
V--jd 2
CII
1
) +P
V (I
---cotB
jd 2
2
1;2
V(_I -.!.COtB)- P
jd 2
2
C'2
3
) +P
V (I
---cotB
jd 2
2
1;3
v(_1 -~COtB)- P
jd 2
2
Cdl
T,r
V
Cd2
V
~1+ CO~ B
V /sinB
Force Deformation Relationship of Each Element
4.1
Longitudinal Tie element
The longitudinal tie element is subjected to uniaxial tension with transverse cracks, used in the
flexural tension region. The area of the tie element is assumed as the longitudinal steel bar area As"
and the element length Ie is (jdcotB)/2 (in 1;1) and jdcotB (in 1;2,1;3)' Since the member
deformation is mainly dependent upon the extension of the tie element, their calculation models need
to be precise. The deformation of the tie element can be determined with crack spacing and crack
width, which are estimated by the bond stress - slip relationship between bar and cover concrete.
Minimum crack spacing is estimated as follows:
(2)
= minimum crack spacing; Ac,eff = effective
n = number of bars in tension chord; db = bar
J;
= tensile
where Smin
concrete area;
strength of
concrete;
diameter; and J;, = average bond stress.
Effective concrete area is selected as Ag /3, where Ag denotes gross sectional area. Suggesting to
assume that J;, / J; = 2 and S = I.5Smin gives a simple estimation of the crack spacing of longitudinal
tie element as follows:
Ag
S,=--
(3)
4mrdb
The equilibrium and compatibility conditions for a differential element as shown in Fig. 2 give a
differential equation for the bond - slip relationship [3],[5]:
ddx2~ =KsJ;"
Ks =4[1+ EsAs )
t AsU". +dfJ
t
(a)
AJc
~
(4)
tT
!.;~t 1~
+ 1+
~db,E.)
EcAeff / '
S,
Ash
j t
~ ~.t;,
~ ~
~ ~
xI
*T +!1T
Ash
(a)
(b)
(b)
(c)
Fig. 3 (a) Force acting on a crack spacing (b) bond
stress distribution (c) slip distribution
Fig. 2 (a) Force acting on a tension chord
length of dx (b) Equilibrium condition
319
Fig. 3 (a) shows the forces acting on a longitudinal tie element between two adjacent cracks.
Distribution of local bond stress J;, is assumed as a constant value that equals local bond strength,
as shown in Fig. 3 (b). Equating the sum of local bond stress to the vertical components of the diagonal
strut forces gives x o ' the length between a crack and a non-slip point:
Xo =
(~+lJ
(5)
SI
2 fb
where u=V/(mrdbljd) is a global bond stress developed by shear force; and
h
=1,(2- hl/i;,I) is a
local bond strength that is derived from the bond strength depending on steel strain [6].
Using the boundary condition, b
results in:
=0
at x = 0 and
Cs
= hi / Es
,
Cc
=0
at x
= xo '
the Eq.(4)
(6)
A slip in the opposite direction should be calculated in the same way so as to obtain a crack width.
Accordingly, the slip is distributed as shown in Fig. 3 (c), and the crack width is calculated by adding
two of the end slips:
W=hl
Es
SI-.!.Ksh(x~+(SI-x~)
2
(7)
Note that the second term of the right side of the above equation expresses the reduction of
deformation by a tension stiffening effect. Using the relations of c/ = W / S/ and AT,/ = lA, the forcedeformation relationship for a longitudinal tie element in elastic state is determined as follows:
AT,/
=
;J ~:
SI
-~Ksh (x; +(SI -X~)J
(8)
For yielding conditions, bond failure as well as bar yielding is considered:
At bar yielding condition: 7;1 ~ AsJYI
At bond failure:
u ~
J;, or
Xo ~
S/
The behavior after yielding is assumed to be perfectly plastic.
4.2
Longitudinal Strut Element
A longitudinal strut element is defined as a flexural compression element subjected to uniaxial
compression force. The area is assumed as 2/3 of be, an elastic triangular compression block area
at initial yielding of the tension steel bar:
2
Ac =-be
(9)
3
where
e=
-(Asli;,1 + P) + ~(AsJYI + p)2 + 2(AsJYI + P)dcybEc
cybEc
(10)
A stress-strain relationship of uniaxial concrete compression is selected as a parabolic shape:
"<=;;N~')-(::Jl
(11)
4.3
Transverse Tie Element
Transverse tie elements are supposed to carry the member shear force. According to the current
design equation for shear strength of shallow beams, the strength consist of the contribution of
transverse reinforcement, V.' and the concrete contribution, v" [1], [7]:
ASh fyh jd
+v"
scoW
where ASh =the cross sectional area of transverse reinforcing bars; and s =spacing of bars.
Vn =
320
(12)
(b)
(a)
(c)
(d)
Fig. 4 Force Deformation Relationship of Transverse Tie Element (a) Equilibrium and Deformed Shape
(b) Without Longitudinal element deformation (c) Crack Width change by Flexural Rotation (d) Crack Width
Change by Axial Deformation of Member
Fig. 4 (a) shows the shear mechanism in a diagonal crack at shear failure. It is assumed that all of
the transverse bars yield and the shear force carried by concrete is due to friction along the crack face.
The deformation of the transverse tie element is expressed in terms of diagonal crack spacing and
width using the relation Sir =Wd/Sd ,as follows:
~ =wdjd
S
T..
(13)
d
where Wd = diagonal crack width; and Sd = diagonal crack spacing approximated as follows:
Sd = Sx sinO+S1 cosO
SxS,
(14)
where Sx=bs/(4mrdbh ).
Neglecting the bond effect between the transverse reinforcing steel and concrete, the steel stress
is determined in terms of crack width:
(15)
The friction force along the crack is calculated using the equation proposed by Collins et al. [9],
which is derived, based on Walraven's work .
• =
c
where
w~,
0.18«
(MPa)
0.3+ 24wd/(a+16)
(16)
.c =the max. shear stress along the crack; and a =max. size of aggregate.
If there is not the effect of flexural deformation, the crack width at the maximum shear resistance,
can be determined from Eq. (13) by substituting yield stress iyh for steel stress Ish' Therefore,
the strength and corresponding deformation of the transverse tie element is calculated as follows:
T,r
= b jd
(iYhPh cot 0 + .c,
w~jd
~T.=--
..
Sd
atw", )
(17)
(18)
It is assumed that the relationship between force and deformation before the deformation in Eq
(18) is linear, neglecting large shear stiffness before the cracking. After the maximum, the element
force is deteriorated as crack width increases. Keeping the steel stress at a fixed yield stress !;,h'
concrete contribution decreases as described in Eq. (16).
The concrete contribution v" is considerably affected by member flexural behavior. Flexural
deformation components are depicted by tensile deformation at longitudinal tie ~st' and compressive
deformation longitudinal strut,
~sc:
~st = ST/Sd IcosO
(19)
~sc = &C,Sd IcosO
(20)
321
where
&r.
I
and
denote the longitudinal strain
&c
I
&r. '
&r.
12
13
and
&c
11
,&c12
in the cases of T'rl and
T,rz respectively. The longitudinal deformation components give the additional crack width so that all
of the transverse reinforcing bars at the crack face yield. Fig. 4 (b)-(d) show the change in yield crack
width.
~st-~'c' B
(21 )
SID
2cosB
2
Note that the deformation of the tie element is treated as a positive value in tension and that of the
strut element is positive in compression. Using this crack width at maximum shear resistance, Eq (17)
and (18) give transverse tie element behavior with flexural deformation.
f _
wdy
4.4
-
wdy
+ ~,,+~,c -
Diagonal Strut Element
A diagonal strut element is a discrete representation
of a diagonally cracked compression field subjected to
uniaxial compression. The strength of the compression
strut decreases as the transverse tensile strain increases.
The constitutive equation proposed in the Modified
Compression Field theory by Collins et al. [9] is used for
the stress strain relationship of the diagonal strut:
rr'~/'_H;J(;J]
O"c
fc '
f.._~_
(22)
Gc '
where
fz,max
= 0.8
~'70&1 ~ fc'
(MPa);
denote compressive stress and strain respectively;
&c'
= 0.002 ; and
&1
Fig. 5 Stress strain relationship of
cracked concrete in compression
and
&z
Gc
= ~T.. / jd .
The area of element C dZ is bjdjcosB. The area of CdI' representing the strut on a fan-shaped
region, however, cannot be obtained from the stress field. For convenience of a simple analysis, the
area of diagonal strut Cdl is assumed as bjdj(2cosB).
5.
Member Deformation
Member deformation is obtained by combining truss element deformation with jOint rotation. The
truss deformation is represented by the lateral displacement and the rotation at (3jdcotB)/2 from the
bottom end, as shown in Fig. 6 (a):
~truss =~r.
II
3cotB)
(2- +~r.12
cotB+~c cotB+~c
II
dl
WOeB
1
l+--+~c --+~T.
2
d2 sin B
..I
+~T.
..2
(23)
(24)
The jOint deformation is dominantly affected by the shear mechanism, as shown Fig. 6 (b). At the
member face of the joint, however, this deformation can be expressed by end rotation due to the
extrusion of anchored tension bar and the shortening of the compression concrete block, as shown in
---t
(a)
(b)
(c)
Fig. 6 Member Deformation (a) Truss Deformation (b) Joint Mechanism (c) Joint Rotation
322
Fig. 6 (c). In seismic design, a joint is desired to remain elastic at failure. By assuming the joint to be
elastic, the joint rotation is calculated as follows.
E>
_
(_1 _Ksd
joint -
)!!.EJ...(
bl
Es
8
4J"
7;1 )2_
ASl
(25)
4jd -3d
Therefore, the member deformation i.e. lateral displacement at inflection point is obtained as
below.
LlT = Lltruss + ( L -
3jdCOt(})
2
E>truss + L
(26)
E>joint
The plastic hinge rotation E>p can also be determined by dividing LlT by member length L.
6.
Analysis Procedure and Verification
A set of procedures for deformation analysis is proposed as follows:
Step 1: Assuming the yielding of longitudinal bars in tension, determine the geometric properties of
the strut-and-tie model, () and c from Eq. (1) and (10), respectively.
Step 2: Calculate the strength of member
which is the minimum value among the shear forces
V derived from Table 1 by substituting the element forces at yielding. Each element force at yielding is
described in chapter 4. Note that the longitudinal element deformation for transverse ties and diagonal
struts are assumed as 0 when the initial strength is calculated.
Step 3: Calculate all of the element forces at Vu from Table 1.
v.,
Step 4: Calculate all of the element deformations from chapter 4 with the element forces in Step 3.
Step 5: Calculate the member deformation LlT and E>p from Eq. (23), (24), (25), and (26).
Step 6: If the longitudinal tie element yields at Step 2, find the deformation of this element at
yielding of the transverse ties or the diagonal strut element. The longitudinal tie element extends with
maintaining the strength before any of the other elements yield. Then, calculate the member
deformation at that state by repeating Step 3 through Step 5. Note that this deformation is the limited
ductility of members. In the case where any other element does not yield, the member is sufficiently
ductile and analysis is preformed.
Step 7: If a strut element yields at Step 2 or Step 6, the member fails and the analysis is terminated.
Step 8: Repeat Step 3 through Step 5 with increasing the deformation of the transverse tie element.
This represents strength degradation after shear failure.
The results based on the proposed model are compared with those of experimental programs in
the literature. The data include two rectangular columns for this own study, and three circular and four
rectangular columns of Priestley et al. [10], [11]. All of the specimens are intermediately short columns
having the shear span ratio M IVh of 1.5 to 2. Most of the test results show the shear failure mode
after the yielding of the longitudinal main bars.
The comparison of strength shows a good agreement between the two results as shown in Fig. 7
(a). The results of limited ductility are represented in terms of end rotation. Some errors are found in
these results, as shown in Fig. 7 (b). In low ductility specimens, the strength is overestimated and the
2.0
2.0
1.8
1.8
Vthoory
1.6
V""P.
1.4
etlreo'l'
eexpo
..
1.2
1.0
1.4
1.2
.
0.8
1.6
1.0
0,8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.5
1.0
1.5
2.0
2.5
0.5
3.0
1.0
2.0
1.5
2.5
Ductility p( = ~u / ~y)
Ductility P (= ~u / ~y)
(b)
(a)
Fig. 7 Comparison with test results (a) Strength (b) Deformation at failure (Limited ductility)
323
3.0
deformation limit is underestimated. The case shows an opposite tendency in high ductility specimens.
The change of lever arm jd with ductility increase may reduce the error.
7.
Conclusions
This paper presents strut-and-tie models for ductility required members under shear force
combined with axial force and moment. Each element of a strut-and-tie model is defined in terms of
the force-deformation relationship and strength. The member strength is determined at the state of the
first failure of any element, while deformation is calculated by combining the deformation of all
elements. On the basis of the model, the following conclus.ions are drawn:
(1) The deformation of the longitudinal tie element obtained from the bond-slip relationship gives
an accurate estimation of elastic behavior. The assumption of a perfectly plastic behavior after yielding
for simplicity gives reasonable results of deformation estimation.
(2) Yielding of transverse tie elements or crushing of diagonal struts represents a decisive stage in
determining the shear strength. The force deformation relationship of a transverse tie element is
dependent on the diagonal crack spacing and crack width. Thus, the flexural deformation influencing
diagonal crack width should be considered to define behaviors, if transverse ties and diagonal struts
and thereby, the limited ductility of columns in shear is estimated. Based on these models, unfavorable
failure can be identified in the design of shear critical members.
(3) The proposed deformation model for applications to the other members needs consideration of
bond splitting of the longitudinal tie, and a refined model for the fan-shaped region. In addition, the
degradation of stiffness and strength under cyclic loading should be considered.
Acknowledgements
This work was partially supported by the Brain Korea 21 Project and funded by the Korea
Earthquake Engineering Research Center under project No. 2001-G0303, sponsored by KOSEF.
Opinions, findings, conclusions, and recommendations in this paper are those of the authors and do
not necessarily represent those of the sponsors.
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(ACI318-99/ ACI318R-99), American Concrete Institute, Farmington Hills, Michigan, 1999.
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Buildings, John Wiley & Sons, Incs., 1992, pp. 38-46
[3] Marti, P., Alvarez, M., Kaufmann, W. and Sigrist, V., Tension Chord Model for Structural
Concrete, ETH, ZOrich, Swiss
[4] To, N. H. T., Ingham, J. M. and Sritharan, S., Cyclic Strut & Tie Modeling of Simple Reinforced
Concrete Structures,
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