COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Limit State Analysis of Seismically Excited 3D R/C Beam Bearing
Structures
N. Kaufmann *
Bauhausstraße 4, 99423 Weimar, Germany
Email: [email protected]
Abstract In the paper a calculation model is presented which enables the analysis of seismically excited 3d r/c frame
works using non-linear optimization algorithms. In this connection as criterion for evaluation the ultimate limit state
(shakedown state) the adaptive load factor pA is defined. The plasticity conditions of a cross section as function of
the elastic internal stress-state can be described linear as well as non-linear. With an example of a 3d structure will
be quantified the influence both of the formulation of the plasticity conditions and of the kind of elastic dynamic
determination of internal stress state on the adaptive ultimate load.
Key words: Adaptive analysis, non-linear optimization, non-linear plasticity conditions, 3d r/c structures
INTRODUCTION
The capacity design method is an generally accepted model for the analysis of seismically excited structures, which
is also considered in international standards. Based on this method in [1] a calculation model is presented which
enables the analysis of plane r/c frameworks using linear and quadratic optimization algorithms. In this connection
as criterion for evaluation the ultimate limit state the adaptive load factor pA is defined. This factor is characterising
the shakedown state of structures. Hence, on the one hand, energy dissipation through inelastic deformations is
possible and, on the other hand, because of a limitation of the plastic work an accumulation of plastic strains is
excluded.
The determining of permissible actions results independent of the number of load cycles and the load history,
because after the formation of a stable distribution of residual forces every load action with an intensity p ≤ pA causes
purely elastic deformations. Also in the shakedown state a maximum value of elastic or plastic deformations and the
inelastic strain energy is calculable directly, i.e. without the estimation of all possible time histories of excitation.
Based on this conception an extended non-linear optimization model for the limit state analysis of 3d r/c structures
will be presented. Furthermore the consideration of non-linear and consequently nearly realistic plasticity conditions
is possible.
This kind of analysis is important provided that an reduction of the structural model in plane frameworks is impossible or not useful. Particularly in case of non-symmetrical structures and irregular distributions of stiffness or masses
3d calculations are necessary.
FOUNDATIONS OF THE THEORY OF ADAPTIVE STRUCTURES
The presented calculation model for the adaptive state analysis is based on bilinear stress-, strain-, or internal forcedeformation relationships for the description of the linear-elastic-ideal-plastic material behaviour. The second basis
of the calculation model is an extreme-value principle that uses the possibility of dividing the entire stress or the
entire deformation state into a linear elastic part and into an irreversible residual-stress part, respectively [2]:
σ = σ E +σ R
(1)
u = uE + uR
(2)
⎯ 1127 ⎯
By that, the elastic part of the solution fulfils the equilibrium conditions
AT σ E − m u&&E − c u& E + f = 0
∈V
(3)
as well as the static boundary conditions
AsT σ E − g s = 0
∈Sf
(4)
and are calculable with known methods of the linear-elastic dynamic analysis separated from the non-linear analysis.
An extreme-value problem is given for the calculation of the desired residual stresses, representing an inherent stress
state, if the effective intensity p does not exceed the adaptive ultimate load pA:
1
σ R D −1σ R dV → Minimum
∫
2
(5)
AsT σ R = 0
∈Sf
(6a)
AT σ R = 0
∈V
(6b)
F σ R + p F σ E − S0 ≤ 0
∈V
(6c)
Considering the restrictions (6) the adaptive ultimate load pA can be determined from the following extreme-value
problem:
p → Maximum
(7)
Eqs. (6a) and (6b), respectively, represent equilibrium conditions in space and on the surface and (6c) describes the
plasticity conditions.
Analogously, an extreme-value problem for the calculation of the adaptive parameter of resistance rA can be
formulated from a given intensity p.
The extreme-value problems (5) and (7) are transformed, by a suitable discretisation, into optimization problems for
the calculation of the residual stress state and the ultimate adaptive load pA, or the adaptive resistance parameter rA.
If a constant distribution of longitudinal forces and torsion moment as well as a linear distribution of bending moments is assumed for each element of the structure, the discretisation can be carried out corresponding to Figure 1
and the residual internal forces are summarized in the vector sR:
[
s R = N R1 M RT 1 M Ry 11 M Ry 12 M Rz 11 M Rz 12 K N Rn M RTn M Ryn1 M Ryn 2 M Rzn1 M Rzn 2
Mzi1 Myi1 Qzi1
Ni1
]
T
(8)
Qyi1
MTi1
MTi2
Ni2
Qyi2 2
Qzi2 Myi2 Mzi2
1
x
li
y
z
Figure 1: Definition of the residual internal forces on beam element i
The optimization problems for the calculation of the adaptive bearing behaviour of 3d framed beam structures are
schematically shown in Table 1.
⎯ 1128 ⎯
Table 1 Summary of the problem matrices
Objective function
Determination of residual
internal forces
Determination of the
adaptive load factor
Determination of the
parameter of resistance
p ≤ pA
p = pA
r = rA
1
p → MAXIMUM
r → MINIMUM
A G sR = 0
A G sR = 0
A P s R + p b Ev ≤ b P − b Eg
A P s R − r b P ≤ − b Eg + b Ev
f s Eg + p s Ev + s R ≤ b P
f s Eg + s Ev + s R − r b P ≤ 0
2
Equilibrium condition
sTR Q s R → MINIMUM
A G sR = 0
Plasticity conditions
(
- linear
A P s R ≤ b P − b Eg + p b Ev
- non-linear
f s Eg + p s Ev + s R ≤ b P
(
)
)
(
)
(
)
(
)
The matrix AG contains the coefficients of the equilibrium conditions for all the nodes of the structure. The
linearized plasticity conditions are contained in matrix AP. From this follows, that the plasticity conditions are only
controlled at the ends of the beam elements. Vector bE is the vector of the internal forces derived from linear-elastic
computation of live load v and dead load g. Analogously, vector s contains the internal forces in case of non-linear
plasticity conditions.
FORMULATION OF THE PLASTICITY CONDITIONS OF A R/C CROSS SECTION
In [3] the formulation of the plasticity conditions considering the interaction between the longitudinal force N and
bending moments My and Mz of an 3d r/c cross section is presented.
Corresponding to Fig. 2, the plasticity conditions are formulated as function of decisive forces s0, characterizing the
one-axial bearing capacity of the cross section. So, using a minimal amount of conditions, a realistical description of
the cross section capacity is possible.
a)
b)
-N
-N
ND
1
1
(ND, MDy, MDz)
fD (s)
6
(Nbpz, 0, Mbpz)
2
2
Nk
(Nbpy, Mbpy, 0)
4
(Nbny, Mbny, 0)
My
5
(Nbnz, 0, Mbnz)
MD0pz
Mk1z
3
Mz
MZ0py Mk1y
5
(NZ, MZy, MZz)
MD0py
My
MZ0pz
fZ (s)
NZ
3
Mz
(a) Determining of the reference points; (b) Maximal moments M in dependence on the function fD (s)
Figure 2: Extreme combinations s0
The variable exponents αN and αM enable a non-linear or simplified linear description of the two- or threedimensional formulated plasticity conditions. The determining of the exponents is dependent on the shape of a r/c
cross section and the technical standard. Additional a convex enclosure of the space of permitted internal forces s is
important, which is to guarantee for all adaptive analysis problems.
In general, the plasticity condition fD (s) describes the pressure failure of a cross section and is defined as function of
the forces s representing reference point 1 and the balance points.
⎯ 1129 ⎯
⎛ N − N b ⎞ ⎛ M y − M Dy ⎞
⎟
⎟⎟ + ⎜
f D = ⎜⎜
⎜
⎟
⎝ N D − N b ⎠ ⎝ M by − M Dy ⎠
α My
⎛ M − M Dz
+ ⎜⎜ z
⎝ M bz − M Dz
⎞
⎟⎟
⎠
α Mz
≤ 1.0
(9)
Because in case of rational exponents equation (9) is defined only for positive bending moments My and Mz,
respectively, a formulation of fD (s) for all possible combinations of My and Mz is necessary. Hence, a complete
description of the failure surface consist of four variations of equation (9). Exemplary for positive bending moments
(quadrant 1) the following equation is valid:
⎛ N ⎞ ⎛ M y − M Dy ⎞
⎟
⎟⎟ + ⎜
f D1 = ⎜⎜
⎜M
⎟
−
N
M
Dy ⎠
⎝ D ⎠ ⎝ D0 py
α My
⎛ M z − M Dz
+⎜
⎜M
⎝ D0 pz − M Dz
⎞
⎟
⎟
⎠
α Mz
≤ 1.0
(10)
The plasticity condition fZ (s), describing the tension failure of a cross section, is defined as a cone function of
reference point 3 and the balance points. Analogously to condition fD (s) the formulation of fZ (s) requires a
differentiation according to the bending moment state. Equation (11) represents the plasticity condition in tension
space for quadrant 1:
⎛ N − NZ
f Z 1 = − ⎜⎜
⎝ − NZ
⎞
⎟⎟
⎠
αN
⎛ M y − M Zy
+⎜
⎜M
⎝ Z 0 py − M Zy
⎞
⎟
⎟
⎠
α My
⎛ M z − M Zz
+⎜
⎜M
⎝ Z 0 pz − M Zz
⎞
⎟
⎟
⎠
α Mz
+ 1.0 ≤ 1.0
(11)
Neglecting one of the terms of general functions fD (s) and fZ (s), respectively, the two-dimensional interaction conditions N / My, N / Mz, My / Mz result.
The variable exponents in equations (9-11) are influencing the curvature of the plasticity conditions. The
determination of αN and αM, respectively, is influenced of the shape of the r/c cross section, the geometrical
reinforcement ratio, the quality of the reinforcement bars, the reinforcement arrangement across the r/c section and
the used standard.
ADAPTIVE ANALYSIS OF A 3D BEAM BEARING STRUCTURE
The 3d frame shown in Figure 3 will be taken as an illustration of the analysis of seismically excited reinforced concrete structures. With this example will be quantified both of the influence of the linear-elastic dynamic calculation
method and of the formulation of the plasticity restrictions on the results of the adaptive analysis.
b =1/27cm)
Querschnittsausbildung
Cross
sections (d1/2 = b1/2 =(d71/2=cm)
7
6
Columns
Stütze
2
3
4
ASE1 ASy1 ASE2
ASz1
ASz2
50
y
8
5.0
ASE4 Az ASE3
Sy2
[cm]
50
4.0
1
4.0
5
X
4.0
Y
4.0
Beams
Riegel
[m]
70
y
Z
z
50
Material
[cm]
B 25
BSt 500
Reinforcement
ASy1 ASy2 ASz1 ASz2 ASE1 ASE2 ASE3 ASE4
[cm2]
Columns
10.0 10.0 10.0 10.0 5.0 5.0 5.0 5.0
Beams
10.0 15.0 5.0 5.0 5.0 5.0 5.0 5.0
Figure 3: Geometry and cross sections of the seismically excited r/c frame
⎯ 1130 ⎯
In accordance with Figure 2 the formulation of the plasticity conditions is based on the German standard DIN 1045.
Following variants will be analysed:
Variant 1a:
Non-linear interaction condition N / My / Mz
Variant 1b:
Linear interaction condition N / My / Mz
Variant 2:
Plane linear interaction conditions N / My and N / Mz
In case the analysis of a rectangular r/c cross section is based on German standard DIN 1045 in [4] the values αN =
1.25 and αMy = αMz = 1.40 are recommended for a reinforcement quality BSt 500. In contrast to this in case of the
European standard EC 2 and identically reinforcement quality the exponents αN = 1.15 and αMy = αMz = 1.25 are
valid.
The linear-elastic state of internal forces sE results of the program system ETABS considering a time-history method
and also method of response spectrum.
2
Spektralbeschleunigung
in m/s
Bodenbeschleunigung in m/s?
The r/c structure is excited by the south-east component of the TAFT-earthquake 1952, an earthquake with a period
of 54.18 s (time interval of the recording 0.2 s). The ground accelerations, given with a maximal value of 1.76 m/s2,
are initiated as horizontal excitations. Fig. 4 shows the accellogram and the response spectrum of the TAFT-earthquake, respectively.
1,5
1
0,5
0
-0,5
-1
-1,5
7
6
5
4
3
2
1
0
0
-2
0,5
1
1,5
2
Periode in s
0
5
10
15
20
25 30
Zeit in s
35
40
45
50
55
Figure 4: TAFT-earthquake 1952
Using the time-history method the elastic state sE is calculated for each time step. Because of the formulation of the
plasticity conditions of the adaptive analysis as function of the hole elastic state a reduction of the elastic solution sE
to extreme combinations se is recommendable ensuring a restricted number of plasticity conditions.
In case of linear plasticity conditions the extreme values are to be determined relative to the individual plasticity
restrictions. Including non-linear plasticity conditions one extreme value is to assign to the pressure restriction
fD (s) and the tension restriction fZ (s), respectively. The non-linear plasticity conditions for all possible bending moment combinations are to be formulated as function of the both extreme values se. The determination of the decisive
combinations se is based on the criterion of minimal load intensity p shown in Fig. 5.
f(min p ⋅ sel ) = bp
Figure 5: Determination of the decisive combinations of internal forces (interaction conditions N / My)
⎯ 1131 ⎯
In contrast to time-history method the response spectrum method gives a priori the (amount wise) extreme values of
the bending moments and the longitudinal force, by which the time case of the solution is lost. The stressed areas in
the interaction diagram of the elastic internal forces result from reflection about the axes, and are doubly symmetric.
In case of the three-dimensional stress state follows a cuboid shaped limitation of the space of elastic internal forces,
including the extreme solution based on the time-history method. This causes a reduction of the space of statically
permissible residual stress part sR, and hence, in contrast to the time-history analysis, a reduction of the adaptive
ultimate load pA.
This is to confirm by the results of the adaptive analysis, summarized in Table 2. The difference in the adaptive load
factor pA calculated on the basis of the time-history method and the response spectrum method lies between 5 % and
18 %.
Table 2 Adaptive ultimate limit load intensity pA (load direction Θ = 0°)
Variant Time-history analysis
Response spectrum
1a
5.089
4.438
1b
4.476
3.639
2
5.128
4.889
For both cases of calculation and variant 1a the internal bending moment states MRy caused by the redistribution
(residual internal forces) are shown in Fig. 6.
-13.352
-1.960
(2)
4.096
2.227
(1)
-0.966
-178.719
-76.266
(3)
26.542
0.028
6.595
(5)
26.542
0.028
-8.825
-5.437
(6)
17.448
0.419
161.069
65.391
(4)
(1)
-62.148
-1.559
0.429
(7)
-22.472
0.219
0.0
X
Y
Z
Z
Zeitintegration (p = pA = 5.089)
Antwortspektrum
4.438)
Time-history (p(p==ppA == 5.089)
Response spectrum (p = pA = 4.438)
A
Figure 6: Residual moment MRy [kNm] for variant 1a (Θ = 0°)
In addition to the method of the calculation of the elastic internal forces the formulation of the plasticity conditions
(variant 1 and 2) is influencing the adaptive results. Assuming the 3d interaction N / My / Mz, in case of the linearized
plasticity conditions (variant 1b) the load intensity pA has lower values (12 % and 18 %, respectively) in comparison
with the non-linear restrictions (variant 1a).
The results in Table 2 lead one to assume that using a simplified description of the load bearing capacity of a r/c
cross section based on separated interaction conditions N / My and N / Mz (variant 2) also leads to satisfying results.
The calculated load intensities pA for a excitation direction θ = 45°, summarized in Table 3, make it clear that this
variant is unsuitable. The adaptive ultimate load pA lies 47 % and 87 %, respectively, over the values of variants 1a
and 1b, i.e. the calculated load bearing capacity of the cross sections of the structure is unrealistically high.
Table 3 Adaptive load intensity pA (Θ = 45°)
Variant
Response spectrum method
1a
4.810
1b
3.772
2
7.061
⎯ 1132 ⎯
SUMMARY
The represented calculation model enables the adaptive analysis of seismically excited 3d r/c structures using optimization algorithms. The model is based on statically formulated extreme principles including an integral formulation of plasticity conditions in internal force space.
It could be shown that the use of response spectrum method enables an adequate exactly and certainly determination
of the adaptive bearing behaviour of a r/c structure. Because of the separation of the non-linear analysis from the
elastic dynamic calculation the lost of the time case of the solution is not a disadvantage.
The adaptive calculation based on linearized 3d interaction conditions is exactly enough. Non-linear restrictions
require a higher numerical determination. In contrast to this, the consideration of a simplified separated description
of the plasticity conditions N / My and N / Mz is unsuitable.
The following aspects are recommendable to the adaptive analysis:
(1) 3d modelling of a structure
(2) determination of the elastic solution based on the response spectrum method
(3) including linearized plasticity conditions N / My / Mz
REFERENCES
1.
Schüler H. Zur Analyse und zur Bemessung adaptiver Tragwerke aus Stahlbeton unter dynamischen Einwirkungen. Dissertation, Bauhaus-Universität Weimar, Germany, 1997.
2.
Hampe E, Raue E, Timmler HG, Saad M. Nonlinear bearing behaviour of adaptive reinforced concrete
structures. in Dynamics of civil engineering structures, A.A. Balkema, Rotterdam, Brookfield, 1996.
3.
Kaufmann N. Nichtlineare Analyse räumlicher Rahmentragwerke aus Stahlbeton. DAfStb-Forschungskolloquium: Beiträge zum 37. Forschungskolloquium des Deutschen Ausschusses für Stahlbeton an
der Bauhaus-Universität Weimar, 1999 (in German).
4.
Kaufmann N. Physikalisch nichtlineare Analyse dreidimensionaler Stabtragwerke aus Stahlbeton mit der
Methode der mathematischen Optimierung. Dissertation, Bauhaus-Universität Weimar, Shaker Verlag,
Aachen, Germany, 2003 (in German).
⎯ 1133 ⎯