6 Seismic Response and Nonlinear Analysis of

Concrete Structures

179

TRANSIENT RESPONSE ANALYSIS OF STRUCTURAL SYSTEMS WITH NONLINEAR BEHAVIOR?

M. A. BHAT-D and K. S. PDTER University of California, Berkeley, CA 94720, U.S.A.

(Received 11 May 1980)

Abstract-This paper presents a mixed algorithm for integration of equations of motion for structural systems having evolutionary type models for cyclic behavior of their constituent elements. The global equa- tions of motion are integrated using Newmark’s method while the internal resisting forces are calculated using an explicit, fourth order Runge-Kutta scheme with the option of usmg B time-step smaller than that used for Newmark’s method. The algorithm also takes advantage of the spatially localized nonlinear nature of the problem. in the case where nonlinearity is concentrated in discrete parts of the structure. As a numerical example. ~~hq~k~indu~d response of a three-story steel frame tested on an Earthquake Simulator, is presented.

1. lNTRODWTION

A number of models has been employed to specify the for~~efo~~on re~tions~p for inelastic structural elements under cyclic loading. Two of the most common are the bilinear and the Ramberg-Osgood models. The bilinear model exhibits sharp transition from elastic to inelastic states. Kinematic or isotropic hardening rules are used for unloading and rei~din~ The model fails to represent actual material behavior and is computation- ally quite ine&cient because it requires one to keep track of all stiffness transition points.

The Ramberg-Osgood model coupled with Masing’s rule for unloading and reloading gives a continuous t~n~tion from elastic to inelastic states. This model is quite adequate for steel members but suffers from other limitations. For example, it is not possible to include isotropic hardening, stiffness degradation, etc. From a computational viewpoint, it is a very d&cult model to use, because it specifies deformation as a function of force and therefore dete~ina~on of forces given deformations requires iterative techniques. Moreover, Matzen and McNiven [I] have pointed out that the model as presented originally is not suitable for random earthquake-type excitations. At least thirteen new rules have been added to make it applicable to this case, making the model even harder to use. Figure 1 shows sample hysteresis loops generated by a Ramberg- Osgood model in which different parts requiring special attention have been identified.

Recently a series of models utilizing internal vari- ables has been proposed for cyclic behavior of struc- tural elements 123. These models take the form of rate- type evolutionary equations and are sufficiently general to include strain-hardening stiffness degradation. etc. Computations are facilitated since the same set of equations govern initial loading unloading and re- loading and the models behave well in case of arbitrary excitations. Since these models are in the form of differential equations. conventional methods for inte- gration of global structural dynamics equations cannot be used directly. This paper presents a mixed algorithm

t’This research was supported by the National Science Foundation Grant ENV76-04264.

18

for integration of equations of motion for the structural systems having evolutionary type models for cyclic behavior of their ~nstituent elements. The gfobal equations of motion are integrated using Newmarks method while the intemal resisting forces are calculated using an explicit, fourth order Runge-Kutta scheme with the option of using a time-step smaller than that used for Newmark’s method. The algorithm also takes advantage of the spatially localized nonlinear nature of the problem, in the case where non-linearity is con- centrated in discrete parts of the structure, such as in rigid frames.

As a numerical example, earthquake-induced re- sponse of a three-story steel frame inning nonlinear energy-absorbing devices is presented. The frame chosen was tested on the Earthquake Simulator at the University of California, Berkeley. In addition, the sig- nificance of controlled, localized energy dissipation in ea~hquake-re~s~nt design of structures is illustrated.

2. A MODEL FOR HYSmRETIC BEHAVlOR

OF NONLINEAR ELEMENTS

In Ref. [Z-J. a number of models for describing dyster- etic behavior of nonhnear elements are presented. The particular rate-independent model to be used for nonlinear elements in this study is given by the follow- ing equations :

where, F(r) is the generalized force. S(r) is generalized defo~ation and &tlis the defo~ation rate of the non- linear element. F,, 6,. e and n are material parameters. Physically, Fe is generalized yield force. 6, is a general- ized yield displacement. K, = F,&, the initial stiffness, x is a constant which controls slope after yielding (I(, =K,,[cr/l +a]f, and n is taken as an odd integer which controls the sharpness of transition from the elastic to the inelastic region. As n+cc the mode1 approaches a bilinear model. Typical loops generated

182 M. A. BHATTI and K. S. PETER

DISPLKEMENT

Fig. 1. Sample hysteresis loops from Ramberg-Osgood model

by this model under deformation varying smusoidally freedom structural system subjected to earthquake with time are shown in Fig. 2. ground motion can be written as follows [3] :

It should be pointed out that the above model is just one of a class of models for inelastic behavior. This particular choice was made for the immediate applica- tion of the present work to optimal design of frames with energy-absorbing devices. More complicated models, such as models exhibiting stiffness degrada- tion, etc., can be obtained by introducing more param- eters into the basic model, as explained in Ref. [2]. These models can be introduced into the present formulation without any di!Ikulty.

Mii(t)+Crift)+ F(t)= -Mti&) (3)

where u(t)= [u,(t), u,(t), . . , u&)lT is the nodal point displacement vector ; i(t)= nodal point velocity vector ; ii(t)=nodal point acceleration vector; M-mass matrix of the system, ME W” x RN: C =structural damping matrix, CeRNxRN; F=nodal force vector, FeRN; r =earthquake influence coetficient vector, c e WN. This vector represents displacements at nodal degrees of freedom resulting from a unit support displacement. For example, r 41. 1, , lJT for an N story shear

frame (with one degree of freedom at each story) sub- jected to horizontal ground motion and ii&}= ground acceleration time history.

3. EQUATIONS OF MOTION FOR THE SYSTEM

Equations of motion for a discrete, N degree-of-

t - 0.60

I 060

DISPLACEMENT

Fig. 2. Hysteresis loops generated by rate-independent model under sinusoidal excitation.

Transient response anslysis of structural systems with non&ear behavior- 183

1. NUMERiCAL SOLUTiON OF THE ~~~~L CRAYONS OF MOTlON

The equations of motion (3) are sofved u~~~~~~ with the exact sotution uftf, *r) and @t) a~~~~a~ by u, ti, snd ii, respectively, at discrete time intervais, The step-by-step integration procedures start with the known initial conditions and march forward in time giving the saIution at discrete points in time. The process for a nonlinear system has two distinct phases. The first phase is the lktitition phase, in which the equations are iinearized about the cutest state by retaining only first-order terms of a Taylor series expansion, Estimates of the solution at the next step are then obtained by using these linearized equations. The second phase is the state determinatiun phase, in which the internal forces in equilibrium with the new state of motion are calculated. If the discrepancy between these internal forces and the external applied loads is within some tdemnce fevei, the scdution is asxepted and the process repeated for the next step. Otherwise, a N~o~~aphson sype iteration is used until the unbalanced forces are within acceptable &nits,

In this study, estimates of the solution are obtained using Newmark’s method and internal forces in the nonlinear elements are computed using a four-ord~ ~~n~~~ut~ scheme. Details of the process are given below.

The equations of motion (3) at time z = t-t At can be written as

Mi&-&&+F,=f, (44)

where P, +tl -M&(t).

Define the increments in acceleration, velocity, disw placement and force occurring in the time increment ASt by

A+&-$

A,ti,=&--rl,

Au,=n,--ug

AF,=F,-P$ 61

Substituting these expressions in eqn t4), the incre- mental form of the equations of motion is obtained as foliows:

w

An implicit, singie-step, two parameter family cf integration operators described by Newmark 141 is used for the numerical integration of the equations of motion. The method assumes that the increments in velocity and ac&eration are related to the increment in displacement an& the state of motion at time I, as foliows:

where At is time step of ~ta~ation and yr fi are integraw tion ~ramet~s. A “constant average acceleration”

operator, which is unconditionally stable for linear problems, is obtained with j--l/4 and y-I/h A %ear accekation” operatur is obtained with /I- l/6 and p= i/Z.

Substituting (7) and (8) into the incremental equa- tions of motion (6) and simplifying gives

eAn,=R: (9

where

The most expensive part of the ~n~~~~ process is the solution of the above set of linear equations. Fortunately, for spatially localized nonfin= problems, it is not neccs$ary to form and decompose the whole matrix e at each step The su~~~~ technique is used to separate effectiv6fy the not&near part from the linear part of the probkm as lohows;

Partition the displacement vector such that dis- placements ~~~~n~ng to the iosilinear degrees of freedom are separated from the remaining dispke- merits:

rAnE-i Aq=LAuNj

where AuN IC: incremental displacements corresponding to the nonliucar degrees of freedom and Aua=incre- mental displacements ~~~~n~ng to the rest of the system.

Partition @ and R: allay, as foM~ws:

The first submatrix equation gives:

KLEA~E+ KENAuN=RE

OF

A~~=K~~~~-K~~A~~~ fll)

The second submatrix equation in eqn (10) gives:

K,,AuE + KNNAn’ = RN, (121

Substitute eqn (IIt into eqn (121:

KNEK;d [R”: -KX,AnN-j + K,,Att” = RN,

Define

184 M. A. BtiATrI and K S. RSTER

Once the Au’ are known, AuL are calculated from eqn (11).

The computational steps can be summarized in the following aIgorithm.

In the beginning of the integration loop (i) form I&,, K,, K,, = K:,.,.

(ii) trianguianze K,, {iii} obtain Q by forward reduction and back sub-

stitution from K,,Q== -K,, (iv) form QT and the product K,,Q.

At each time step of integration. (9 form K,, at the current step,

(ii) form load vectors RE and RN, (iii) solve [E,,Q + KNN]A@‘= RN + QTRE for Ann”, liv) obtain Au” by forward reduction and back sub-

stitution from

Kr6AuUL = RE - K,,A&“.

4.3 Computation of internal resisting forces After the increments in the displacements and veloc-

ities are obtained, the next step is to compute the internal resisting forces in equilibrium with this new state of motion. The internal forces in the linear elements are obtained simply by multiplying the current displacement by the appropriate stiffnesses of these elements. Computation of forces in the nonlinear elements, however. is not so simple. because of lack of an algebraic expression for their force-deformation be- havior, which is described by a set of first-order differential equations. These differential equations must be integrated numerically to obtain the internal forces in the nonlinear elements. An explicit fourth- order Runge-Kutta scheme, with the option of using a smaller time step than the one used in Newmark’s method, is used in this study. An explicit scheme is favored over an implicit scheme because of the added complexity of an implicit scheme, which would involve an additional iteration cycle. The details of the process are given below.

To integrate fork-deformation equations of non- linear elements from time t to time ~=t + At, some assumptions regarding the variation of acceleration, velocity and displacement during the time interval (t,r) are needed. Since the Newmark’s linear acceleration method has been demonstrated to be quite effective for solving nonlinear structural dynamic problems [S], it seems reasonable to assume linear variation in the acceleration during the time interval. This implies quatratic variation of velocity and cubic variation of displacement. These vanations are shown in Fig. 3.

The force corresponding to the ith nonlinear degree of freedom is given by eqns (I ) and (2).

where 6,(x) is the deformatton corresponding to the ith degree of freedom. The deformations ark related to nodal displacements by a transformation matrix which depends upon the type of structural system. For example. for a shear frame S,=u,--u,_ 1, while more

Fig. 3. Vanatlons in acceleration. velocity and dtsplacement during time interval [t. r= r + bt].

complicated expressions are required for other types of frames. Combining eqns (14) and ( 15) then

x E [O, At]. 116)

Equation (161 is integrated by employing a fourth-order Runge-Kutta method with time step Ax, where Ax SAt. and initial condition Fi(O)= Fit).

The following caicufaiions advance the solution from ~~-+x~+~=x~+Ax.

K,=Ax K, &x,)--~~~(.Y~)I F,W

(a+ 11~ 0

K,=Ax K, [

ci~~,+:Axt-/61x,+:Ax~l

(cy ~ 1 ,F,W + $1 _g8,~xK+~Ax) ”

K,=Ax K, [

i F,, a0 II 6,(.u,+~A~)-id~.~,+~Ax)Jx

1

fa+ lf,W+jG 6,Jx,+$Ax) ’ F, -% so II

K&=Ax K, S,(X,+AX)-_I~,(X~+AX)I x

F&x,)+ K, (r-+11 F 0

where

Transient response analysis of structural systems with nonlinear behavior 185

then F~w,+,)=Plxd+~(K,+2K2+2K,+K,).

(17)

4.4 Algorit~for integration of the equations of motion The process of numerical integration of the equations

of motion (3) can now be summarized in the following algorithm.

A. Initial calculations Data: Integration parameters, j?, y; Tie steps, At

and Ax; Convergence tolerance parameter, TOL; Structural property matrices, Ks, M and C; Parameters of hysteretic model for nonlinear elements, F,, 6, a and n. Compute the constants Step 1:

step 2 :

step 3:

as=; a,=At G-1 1 J

Initialize the state of motion, i.e. specify II@ i, and ii,. Form structural property matrices, M and C. Partition the stiffness matrix as explained in eqn (lo), triangularize I(,, and form Q.

B. For each time step Step 4: Form K: and RF

K:=a,M+a,C+K’ R~=P~+M[a,u,+a,ii,]+C[a,i,+a&]

where P,*=P,-[Mii,+C&+F,] and P,= -Mriids).

Step 5 : Solve ” K:Au, = R:

Step 6 : for Ap, using the algorithm given previously. Update the state of motion at 7 = t + At

ij=ii,+a,Au,--a&--a@, Ij=i,+a,Au, -a&, -a&i, u, = u, + Au,.

Step 7 : Compute the internal resisting forces, F, in equilibrium with the current state, as explained previously.

Step 8 : Compute the unbalanced force at. time 7

Step 9 :

f=P,-[M&+Cii,+F,].

Compute Ilq12. the Euclidean norm of f. If llfllt <TOL, no iteration is needed in this step. Go to Step 4 for the next step calculations, else proceed to Step 10.

C. Iterurion within a time step Step 10: Compute K:=a,M+a,C+K,. Step 11: Solve K:Gu,=f for 6u, Step 12: Update the state of motion

new &=&+a,&

new u, = i, + a,&

new u, = u, + 6u,.

Step 13: Compute the unbalance as in Step 8. See if convergence criterion of Step 9 is satisfied. If yes, go to Step 4 for next time step. Else go to Step 10.

5. NUMERICAL W

As an example, the technique is applied to compute earthquake-induced response of a steel frame with an earthquake isolation system, tested on the Earthquake simulator at the Earthquake Engineering Research Center, University of California, Berkeley. The teat structure is a three-story steel frame with added masSeS at each floor as shown in Fig. 4. The structure is supported vertically by specially designed rubber bearings whose properties are spcci&d. The bearings also provide nominal shear resistance. An energy. absorbing device is linked to the base of the structure. as shown. This device acts as a hysteretic passive con- troller, supplying a time-dependent horizontal force to the base. Details of the test configuration and the results can be found in reference [a]. The tests show that for strong earthquakes the energy&sorbing device yields and absorbs amounts of energy equivalent to as much as 350/, of critical viscous damping. Thus the frame itself is left without damage.

In the following sections, equations of motion for the test frame are given and the response is computed using the algorithm given in section 4. It is assumed that the frame itself remains elastic, so that the only non- linearity is in the energy-absorbing device.

5.1 Equarions of motion for the testfrmne As shown in Fig. 4, the structural system consists of

an assemblage of beam and column sections. Masses are assumed to be lumped at the floor levels and rotary inertia is neglected. Axial deformations in both beams and columns are neglected. Thus, the frame has 12 degrees of freedom; one lateral and two rotational degrees of freedom per floor. Rotational degrees of freedom may be eliminated by appropriate partitioning of the mass, damping and stiffness matrices associated with the discretized structural model of the frame. The resulting equations of motion can be written [3]:

Mii(t)+Ci(t)+K”u(t)+F(t)= -Mrii,(t) (18)

moo LB

w Fig. 4. Steel test frame.

186 M A. Bwrn and K. S. PWER

where KE = stiffness matrix of the frame including the rubber bearings but not the energy-absorbing device; F=force in the energy-absorbing device. The only nonzero entry in this vector corresponds to the degree of freedom at which the energy-absorbing device is connected. In this case, only F, will be nonzero and rr=[llll].

The structural property matrices, K”, M and C can be calculated from the material and section properties shown in Fig. 4. Thus, the lateral, elastic stiffness matrix of the complete structure, including the stiffness of the rubber bearings is (units are kip-inches):

-66.64 23.04 -2.18

KE= 144.40 -96.50 18.74 122.43 -48.97

34.21 I

The mass matrix of the structure corresponding to the lateral degrees of freedom is (in kip-inch units):

r 0.02438 1 M=

0.02438 0.625 14

Rayleigh damping is assumed in constructing the damping matrix :

C=aM+/?KE. (19)

The coefficients a and /I are computed from:

l-l 1

where w, and o2 are first and second mode frequencies, and 5, and cz are the respective critical damping ratios

in these modes. The damping matrix for the present structures, assuming {I =3?;, and Sz= lo,, is given below.

L

0.0279 -0.0332 0.0115 0.0014

c= 0.0768 -0.048 1 0.0093

0.0660 - 0.0244 SYMMETRIC I 0.0226

(14)

The force in the device is computed from eqns I 1) and (2), which for this application take the form:

S(t)=a c u,(t) _ F,(t) 1 . L”0 PO_1

The tangent stiffness matrix K, at any time IS obtained by adding SF,(t)/&, to Kz, as follows: Equation (21) gives

P,(r)= Koti,(tl [ 1 -sign @ -S(l)>‘]

where

Thus.

sign = 1 if ti&) > 0 = - 1 if ri,(t)<O.

y=K,[I-sigtf(y -s(t)]],

5.2 Numerical results The hysteresis model presented in eqns (1) and (2)

contains four parameters, namely, F,, a,, z and n. The values for these parameters must be chosen so that the experimental response closely matches with the pre- dicted response. From the test results on the mild steel energy-absorbing devices [7], the following set of

m < 3.0

: VI 0

>

z-3.0 I- I

67 -6.0

0 3 6 9 12 15 I8 I- m

VI 0

z.C - 5.0

TIME ~SEC0H0S1

Fig 5. First story shear time history.

: ii.5 w I

m 0

F-

g-z.5 c In

-5.0 0 3 6 9 12 15

P

z N

TIME ~SECONOSl

Fig. 6. Second story shear time history

Transient response analysis of structural systems with nonlinear behavior

Time, set Figure 7 Third story shear tic History

TlfWt, set ---1nMt design --Optimal design

Fig. 8. Base displa~ent time history.

-4 -2 0 2 4

DISPLACEMENT (INCHES1

Fig. 9. Energy-absorber hysteresis loops (initial design).

parameters is obtained :

F,=S.O (i,=O.ll 1=0.064 n=l (22)

Since the purpose of an earthquake isolation system is to minimize forces in the structure, other sets of param- eters could be obtained which minimize some function of these forces. In Ref. [Sl an optimal design problem was formulated in which the device parameters were adjusted so that the sum of story shears in the frame were minimized. The optimal values of the parameters

were found to be:

F, = 4.337 &, = 0.2503 r = 0.05831 II = 1. (23)

The response of the structure, subjected to modified El Centro 194ONS (see Ref. [S]). is computed by using the above two sets of parameters. The parameters given in eqn (22) are labeled “initial design” and those in eqn (23) are labeled as ‘optimal design’. The story shears and base displacement time histories are shown in Figs. 5-8. The energy absorber hysteresis loops are

188 M A. BriArrr and K. S. PISTER

0 4.0 cn a

Y U 0 W u

lx 0

L-4.0

-3.0 -1.5 0 1.5 3.0

DISPLACEMENT (TNCIIESI

Fig. 10. Energy-absorber hysteresis loops (optimal desrgn).

shown in Figs. 9 and 10. The response was computed with a time-step of 0.01 for the Newmark’s method and 0.005 far the Rungs-Kutta method. Only one iteration within each time-step was required in the large motion area and none in the rest. The algorithm was very stable and was not very sensitive to the global time-step as long as internal resisting forces were computed with reasonable accuracy.

It is interesting to note that the story shears obtained by using parameters of eqn (23) are much lower than those obtained by using vahzes of eqn (22). This shows that by proper design of the isolation system, conaider- able reductions in the forces could be achieved in the structdre.

6. CONCLUSIONS

An algorithm for integration of the equations of motion for structural systems having evolutionary type models for cyclic behavior of their constituent elements is presented. One of the principal advantages of the rate type of the models, obtained from using internal vari- ables, is the generality they offer. For example, while it is not possible to account for stiffness or yield force degradation in a Ramberg+Osgood model; it is very easy to incorporate a new internal variable, described by a suitable equation, to account for these effects. Work is underway to include stiffness degrading models for analyzing concrete structures.

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REFERENCES

V. C. Matzen and H. D. McNiven, Investtgauon of the inelastic characteristics of a single-story steel structure using system identification and shaking table experi- ments. Rep. No. EERC 76-20, Earthquake Engineering Research Center, Umversity of California, Berkeley, Aug. 1976. H. Ozdemir. Nonlinear transient dynamic anaiysrs of yieiding structures. Ph.D. Dissertation, Division of Structural Engineering and Structural Mechanics, De- partment of Civil Engineering, Univ~~ty of Califorma, Eerkeiey, 1976. R. W. Clough and J. Penden, Dynamzcs ojStructures. McGraw-Hill, New York (1975). N. M. Newmark, A method of computanon for struc- tural dynamics. .I. Engng Mech. Div.. AXE 85, (EM3) 67-94 (1959). D. P. Mondkar and G. H. Powell, Static and dynamic analysis of nonlinear structures. Rep. No. EERC 75-10, Earthquake En~n~~ng Research Center, Universrty of Caliiornia, Berkeley, March 1975. J. M. Kelly, J. M. Eklinger and C. J. Derham A practical soft-story earthquake isolation system. Rep. No. UCi3/ EERC-77127, Earthquake Engineering Research Center, University of California, Berkeley, Nov. 1977. J. M. Kelly and D. F. Tsrtoo, The development of energy- absorbing devices for aseismic base isolation system. Rep. No. UCB/EERC-78/01, Earthquake Engineering Research Center, University of Califoinia, Berkeley, Jan. 1978. M. A. Bhatti, Optimal destgn of hxahzed nor&near systems with dual performance criteria under earthquake excitations. Rep. No. UCB/EERC-B/IS, Earthquake Engineering Research Center, University of Caliiornia. Berkeley, July 1979.