Simplified Non-Linear Time-History Analysis Based on the Theory of Plasticity

Earthquake Resistant Engineering Structures V 375

Simplified non-linear time-history analysis based on the Theory of Plasticity

J. L. Domingues Costa1, R. Bento2, V. Levtchitch3 & M. P. Nielsen1 1Department of Civil Engineering, The Technical University of Denmark 2ICIST – DECivil, IST, The Technical University of Lisbon, Portugal 3Frederick Institute of Technology, Nicosia, Cyprus

Abstract

This paper aims at giving a contribution to the problem of developing simplified non-linear time-history (NLTH) analysis of structures for which dynamical response is mainly governed by plastic deformations so as to be able to provide designers with sufficiently accurate results. The method to be presented is based on the Theory of Plasticity. Firstly, the formulation and the computational procedure to perform time-history analysis of a rigid-plastic single degree of freedom (SDOF) system are presented. The necessary conditions for the method to incorporate pinching as well as strength degradation are outlined. The procedure is applied to a typical SDOF system and results are compared with NLTH analysis commonly used for design purposes. Secondly, by means of the Virtual Work Principle, the definition of the equation of motion of a desired collapse mechanism of a multi degree of freedom (MDOF) system is presented. This equation is of the same type as in the SDOF case, and therefore the procedure presented in the first part of the paper may be used. The method is applied to a 4-story reinforced concrete frame structure. Results are compared to those derived by a conventional NLTH analysis and found to be encouraging. Keywords: non-linear time-history analysis, theory of plasticity, rigid-plastic material, collapse mechanism, virtual work principle.

1 Introduction

It is well known that the most powerful and rational technique to assess the dynamical response of a structure subjected to an earthquake is a NLTH analysis.

However, structural engineers are faced with difficulties when applying the different versions available nowadays of this tool. Even for simple structures designed to have localized inelastic behaviour, the analysis requires significant computational effort and there are difficulties in choosing the appropriate hysteretic behaviour at the plastic hinges regions. Therefore, simplification is needed. Plasticity Theory provides rational and efficient solutions concerning the analysis and design of structures that deform in the inelastic range, which is the case under severe earthquake motion. In this paper, Plasticity Theory is used to design and to assess the dynamical response of structures with the simplification of rigid-plastic behaviour for the plastic hinge regions and rigid behaviour for the remaining part of the structure. Special emphasis is put on reinforced concrete structures. Two main justifications for this may be exposed: The use of the extremum principles of the theory allows the choice of a suitable collapse mechanism where ductility demands may be realistically assigned to the plastic hinges regions. If properly designed, R/C structures may have significant levels of ductility, which means that it is reasonable to neglect the contribution of elastic deformations.

2 The rigid-plastic single-degree-of-freedom system

2.1 Formulation of the equation of motion

The equation of motion of a SDOF system solely subjected to base motion may be easily formulated expressing Newton’s second law in a coordinate system moving with the ground:

ma⋅−=−⋅rg() t Rt () ma () t (1) or, in a more convenient form, Rt() at()=− at () (2) rgm

Here t is the time, m is the mass of the SDOF system, ar(t) is the acceleration of the mass in the moving coordinate system, R(t) is the total force applied to the SDOF system and ag(t) is the ground acceleration. The values of ag(t) are derived from the accelerogram record. Here we shall assume that the only contribution to R(t) is the resultant force on the mass from internal forces. Any type of damping is disregarded as the only source of energy dissipation considered is due to plastic deformations. For illustrative purposes consider a typical SDOF system namely a fixed, vertical weightless column with length L, supporting a given mass, m, that can only move in the horizontal direction. The non-linear behaviour of the SDOF system is specified by the flexural behaviour of the plastic hinge in the cross- section located at the base, cross-section A, which is of the rigid-plastic type. For simplicity, the positive yield moment has the same absolute magnitude, MP, as the negative yield moment. The term θ is the relative rotation in the hinge.

Bending, shear and axial stiffnesses are infinite at any cross section of the column. M A m M P

L θ A

-M P A

Figure 1: Typical SDOF system with rigid-plastic behaviour.

Fig.2a) shows the sign convention adopted. The ground displacement and the displacement of the mass are positive to the right-hand side. The bending moment at the fixed end, MA(t), is positive when giving rise to tension in the right-hand side of the column. To express dynamical equilibrium, d’Alembert’s principle is used. In Fig. 2b) the free-body-diagram of the mass in the coordinate system moving with the ground is shown.

a) b) m

m ag(t) m m ar(t) M A (t) L L M A(t) +

Figure 2: a) SDOF system subjected to ground acceleration and b) corresponding free-body-diagram for the mass.

The dynamical equilibrium condition yields: Mt() A −⋅ma() t =⋅ ma () t (3) L gr M ()t Dividing eqn (3) by m, eqn 2 with Rt()= A is reached. L Finally, one has to associate the behaviour of cross-section A with the dynamical state of the system. To carry out this task, one has to identify the transition conditions between rigid and plastic behaviour. The system changes from rigid to plastic behaviour when the bending capacity is exhausted. From the sign convention expressed above and from the moment vs. rotation relationship shown in Fig. 1, one may easily conclude that exhaustion of the positive bending capacity at cross-section A implies positive rate of change of the rotations of the cross-section meaning that the mass is

WIT Transactions on The Built Environment, Vol 81, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) 378 Earthquake Resistant Engineering Structures V moving in the negative direction. Conversely, the exhaustion of the negative bending capacity of cross-section A, implies the mass to move in the positive direction. This is the same to say that the velocity of the mass in the moving coordinate system, vr(t), is positive when “negative plastic behaviour” at cross- section A takes place and is negative if “positive plastic behaviour” occurs. The system may change from plastic to rigid behaviour whenever the relative velocity of the mass is zero. The bending moment at the moment of transition corresponds to the bending capacity associated with the sign of the previous plastic deformations. When the system has rigid behaviour, MA(t) is within the extreme values MP or –MP, and therefore the rotation at cross-section A is constant and equal to the rotation at the transition moment. Obviously, the same applies to the relative displacement, dr(t). The complete formulation of the equation of motion for the SDOF system depicted in Fig. 1 may now be written.

vrr (t)== a (t) 0 If MAPP() t∈⇒ ][ -M ,M  d (t)= Constant  r (4) Mt() If M() t=≠ {} -M ,M and v (t) 0 ⇒ a (t)= A − a() t APPr rmL⋅ g It appears that the rigid-plastic approach to model the non-linear behaviour at the plastic hinge, yields an extremely simple formulation of the equation of motion as all the information necessary to compute the dynamical response is in the form of the set of expressions in (4). Additionally, it should be noted that when plastic behaviour takes place at the plastic hinge regions the effect of the base motion, given by ar(t), is determined by the absolute acceleration of the mass M given by the ratio ± P ,for the present case. This parameter may be of great mL⋅ usefulness to estimate the dynamical response of the system as the amount of plastic deformations has a direct correlation with the magnitude of this parameter. Moreover, as it will be shown in the next paragraph, it may also be associated with the transition conditions between different types of behaviour. Finally, it should be noted that the formulation for any other type of SDOF system with rigid-plastic behaviour, would yield similar expressions as (4) regardless the type of the internal force mechanism.

2.2 The computational procedure to perform a time-history analysis of a rigid plastic SDOF system

In the following, a step-by-step integration procedure for the computation of the dynamical response of a SDOF system to any type of base motion is presented. The response is evaluated at each successive time interval i with length ∆t sufficiently small to justify that ag(t) is assumed constant and equal to the average, ag,i, between the value at the beginning and at the end of that time interval, ag(ti) and ag(ti+∆t), respectively. Thus:

at+atgi() gi()+∆ t a() t== a for t∈ t; t +∆ t (5) ggi, 2 ii The dynamical response of the SDOF system considered in the previous section may then be evaluated at the time t:

MA (t)=M A,i +m⋅⋅ (a g,i -a g,i-1 ) L  i) If MA,i∈ ][ -M P ,M P⇒== v r (t) a r (t) 0 ⇒ d(t) d  rr,T=

⇒+∆∈⇒ if MAi() t t][ -MPP ,M time interval i+1 otherwise ⇒ condition ii)  MtA () a(t)rg=− a,i  mL⋅   MtA () ii)( If MA,i= {} -M P ,M P ⇒−⋅ vr (t)= ag,, i t− ti)+ v r i ⇒  mL⋅  Mt() ()tt− 2  A i d(t)=rg−⋅ a,,,ir +⋅−+vi() t ti dri  mL⋅ 2

⇒⇒⇒ if ττ :vrAA ( )=0 M ( τ )=M,i condition i): M A (ti +∆t)=MA,i otherwise ⇒ time interval i+1 (6) Here MA,i is the bending moment at time ti, dr,i is the relative displacement of the mass at time ti, vr,i is the relative velocity at time ti and dr,T is the relative displacement of the mass at the last time, T, at which the onset of rigid behaviour took place. M The usefulness of the parameter ± P referring to the absolute acceleration mL⋅ of the mass when the system has plastic behaviour, may be appreciated by the immediate conclusions one may take before proceeding to any type of calculations: If the system starts from rest, deformations occur first at the time at M which the ground acceleration exceeds the quantity P in one of the mL⋅ directions. A reverse of the motion of the mass takes place only if the ground acceleration changes direction and varies by an amount of at least M 2⋅ P . mL⋅

2.3 The consideration of pinching effects and strength degradation

Pinching affects the energy dissipation capacity of R/C members especially when these are subjected to loads, such as bending with axial load or high shear.

For the present idealization, pinching effects may be taken into account if one assumes that the plastic hinge is not able to resist deformations when the sign of the resisting forces changes. Therefore the residual deformation is “lost” and the path proceeds along the axis of the resisting forces until the load carrying capacity is exhausted once again, see Fig. 3. To include this effect in the computational procedure described in section 2.2 for the SDOF system of Fig. 1, the condition i) in (6) has to be changed according to (7). M A

M P

θ A

-M P

Figure 3: R/C column in bending with axial force, [4] and the subsequent rigid- plastic model taking into account the pinching effect.

MAA,ig,ig,i-1 (t)=M +m⋅⋅ (a -a ) L i) If MA,i∈⇒][ -M,M P P  ⇒ v(t)rr== a(t) 0 ⇒+∆∈⇒ if M() t t -M ,M Ai ][PP (7)

ìf MA,i⋅+∆>⇒= M A()0 t i t dr (t) d r,T ⇒⇒ time interval i+1 otherwise⇒= dr (t) 0 otherwise ⇒ condition ii)

Strength degradation in cyclic loading imposing significant inelastic deformations affects particularly R/C members subjected to biaxial flexure and/or to variable axial load such as corner columns in a frame exposed to high degrees of overturning moments. The quantification of this effect may be carried out if one keeps track of the number of cycles and choose an adequate function describing the loss of strength with the cycling. Details cannot be treated in this paper. However, once this function is chosen, strength degradation may be taken into account in the computational procedure in (6) by adding the condition:

M P,cc= fn() (8) where MP,c is the bending capacity after cycle c and nc is the number of cycles.

2.4 Example and comparison with elastoplastic and bilinear approaches

The procedure described above is applied to derive the dynamical response of a SDOF system of the same type as the one in Fig.1 subjected to the first 20s of the well-known N-S component of the El Centro Earthquake. The column

WIT Transactions on The Built Environment, Vol 81, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) Earthquake Resistant Engineering Structures V 381 supports a mass of 10 ton and the length of the column is 5m. The bending capacity is chosen to be such that the system’s plastic deformation ratio is 10% of the peak ground acceleration (PGA), which was 3.32 m/s2. M Therefore: PGA=⇒=⇒=3.32 m / s22P 0.332m / s M16.6 kNm . mL⋅ P In the following, the comparison with two types of SDOF systems with different bilinear hysteretic behaviour commonly used at the design level but with same bending capacity is presented: the first one with elastic-perfectly- plastic behaviour and the second one with elastic-plastic strain hardening behaviour. The latter has a strain-hardening branch which inclination is 5% of the inclination in the elastic branch. Two levels of stiffness were considered: one for which the natural period of vibration of the equivalent elastic SDOF system, Tel, equals 0.5s and another one with Tel =1.5s. Viscous damping is assumed to be 5%. The results in terms of displacement vs. time are shown in Fig.4. The effect of pinching and strength degradation has not been considered. The computational procedure to estimate the dynamical responses of the elastoplastic and bilinear SDOF system is of the format as proposed by Paz [1].

Figure 4: The displacement-history of the SDOF systems considered.

As may be seen, the displacement history of the bilinear systems deviates significantly at the time of peak response (t≈5.5s) according to the level of elastic stiffness considered. The use of the rigid-plastic model avoids complications regarding the choice of the level of the elastic stiffness throughout the dynamical response being the statical capacity the only parameter necessary to be defined. However, it should be noted that for this case, when large plastic deformations take place, the results corresponding to the rigid-plastic approach remarkably resemble the average of the other analyses in terms of displacement–history as well as peak displacement.

3 Application to the case of the MDOF system

The procedure described in the last section may also be applied to assess the dynamical response of a MDOF system. The first step is to design the structure according to the Theory of Plasticity, i.e. to choose a suitable collapse mechanism and the capacity at each plastic hinge region. (maybe better than: a set of plastic moments in the hinges). Then, the equation of motion is established by using the virtual work equation. Finally, the internal forces are determined for the external load and the inertia forces. These need in principal to be checked at any time in order to assure that plastic behaviour only takes place at the plastic hinge regions. Moreover, the overall analysis must include a set of accelerograms representative of the seismicity of the region. In the present approach, the collapse mechanism is treated as an assemblage of rigid bodies, which means that internal deformations may only take place at the plastic hinge regions. The constitutive relationships for the plastic hinges are of the rigid- plastic type. As for the SDOF system there are two types of behaviour: global rigid behaviour if the capacities of the plastic hinges are not reached, meaning no relative displacements in the system with respect to the ground and, conversely, global plastic behaviour, if the capacities of the plastic hinges are reached, implying relative displacements in the system with respect to the ground. Moreover, the displacement field has one degree of freedom as all deformations may be expressed in terms of a single displacement amplitude conveniently chosen. Therefore, the problem may be treated in the form of a generalized SDOF system.

∆ w4 w4

16 17 18 19 Floor wi (kN/m) φi 3m w3 w3 4 81,42 1 12 13 14 15 3 84,27 0,76 3m w2 w2 2 84,27 0,51 H=12,28m 8 9 10 11 1 85,25 0,27 3m w1 w1

4 5 6 7 hi 3,28m 1 2 3

4m 6m

Hinge # 1 2 3 4 5 6 7 8 9 10111213141516171819 Mp,- (kNm) -164 -246 -246 -112 -112 -251 -251 -110 -110 -249 -249 -110 -110 -249-249 -107 -107 -240 -240

Mp,+ (kNm) 164 246 246 86 86 193 193 85 85 191 191 8585 191 191 82 82185 185

Figure 5: Frame structure and the collapse mechanism.

Consider the case of a 4-story R/C frame structure shown in Fig. 5. This is a two-dimensional R/C frame based on the structure studied by Falcão [5]. It is evident, that the most suitable collapse mechanism is the side sway collapse mechanism at the right-hand side of in Fig. 5, i.e. the collapse mechanism

WIT Transactions on The Built Environment, Vol 81, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) Earthquake Resistant Engineering Structures V 383 involving the development of plastic hinges at both ends of the beams and at the base of the columns. Assuming that the horizontal displacements at the system are small, it is clear that the displacement at any point is proportional to a displacement amplitude, say the displacement at the top floor, ∆. The displacements at each floor are conveniently expressed by the quantity φi=hi/H, hi being the height of floor i and H the total height. The bending capacities at each plastic hinge are shown in the table at the bottom of Fig. 5. To derive the work equation of the generalized SDOF system in the moving coordinate system, one has to account for all the external and inertia forces producing work during a virtual displacement δ. These are shown in Fig. 6 and the sign convention used there is the same as in Fig. 2. In the virtual work equation the moments in the hinges are considered as external forces. When the virtual displacements are identical to the displacements in the collapse mechanism the work of the plastic moments is negative. The work equations for the system are shown in the system (9). 19 4 4 δ 2 M , jP ∑∑ rii )( ∑ gii tamtam 0)( if δδφδφ >=⋅⋅⋅−⋅⋅⋅−⋅− 0 j=1 H i=1 i=1 (9) 19 4 4 δ 2 M , jP ∑∑ rii )( ∑ gii tamtam 0)( if δδφδφ <=⋅⋅⋅+⋅⋅⋅+⋅− 0 j=1 H i=1 i=1 The work equation may be rearranged as in (11) introducing the terms: 4 m ⋅φ 41∑ ii 9 m∗ =⋅ mφ 2 , k=i=1 and M* = M (10) ∑∑ii 4 P Pj, ij==112 ∑mii⋅φ i=1 * M P r ta )( −= ⋅− g tak )( if δ > 0 * ⋅ Hm (11) * M P r ta )( += ⋅− g tak )( if δ < 0 m* ⋅ H

The system (11) is of the same form as eqn (3). This means that the dynamical response of the MDOF system corresponding to the chosen collapse mechanism may be treated using the procedure described in section 2. The procedure was applied to the frame in Fig. 5 using 5 accelerograms with PGA=0.5g generated artificially based on the 1976 Friuli earthquake. The results were compared with the ones derived from a standard NLTH analysis using the commercial programme SAP2000. In the latter analysis the flexural behaviour at the plastic hinges regions was modelled by bilinear constitutive relationships based on the bending capacities in Fig. 5 and a strain-hardening branch which inclination is 5% of the one in the elastic branch. The rest of the elements of the frame are assumed to remain in the elastic range.

The stiffness of the system was the same as the structure studied by Falcão and Bento [6]. Proportional damping to the mass and stiffness matrixes was assigned so that the damping of the first mode of vibration would be 5% of the critical value.

ar Virtual Displacement Inertial force External force at floor i at floor i at floor i δ 1617 18 19 φ4 x δ m4 x φ4 x ar m4 x ag

12 13 14 15 φ3 x δ m3 x φ3 x ar m3 x ag

8 9110 1φ2 x δ m2 x φ2 x ar m2 x ag

4 576 φ1 x δ m1 x φ1 x ar m1 x ag

12 3

Figure 6: Displacements and forces on the generalized SDOF system.

4th Floor

rd 3 Floor

nd 2 Floor

Standard NLTH 1st Floor analysis Rigid-plastic analysis

0 0,1 0,2 0,3 0,4 (m)

Figure 7: Displacement shape for the MDOF system.

Figure 7 shows the average story displacement curves normalized to the average maximum top displacement. There is no practical difference in terms of displacement shape and the rigid-plastic analysis yielded a maximum average displacement 18% less than the more refined NLTH analysis. Due to space limitation it is not possible to show the displacement history for the two types of analysis for each accelerogram. However, it may be mentioned that also in this respect the results showed good agreement for the two types of analysis.

4 Final remarks

The work presented in this paper demonstrated that the use of Plasticity Theory and the assumption of the rigid-plastic behaviour for the plastic hinges might be of great practical value to assess in a simplified manner the dynamical response of a structure subjected to strong earthquakes. The procedure explained yields results close to more sophisticated types of NLTH analysis when the elastic deformations are small and therefore, it should only be applied when the level of ductility of the structure is sufficiently high. The calculation procedure directly yields the relative rotations in the plastic hinges. These must be evaluated in relation to the rotation capacity available in order for the probability of final collapse to be estimated. Future developments of the method will be dealing with choice of collapse mechanism, P-∆ effects, pinching and the effects of strength degradation.

Acknowledgements

The authors would like to express their appreciation to M.Sc. Filipe Rodrigues in providing data for the study of the structure in section 3. Many thanks are also due to Fundação para a Ciência e Tecnologia – FCT, Portugal and the Otto Mønsteds Fond for their financial support.

References

[1] Paz, M., Structural Dynamics – Theory and Computation, 3rd Edition, Van Nostrand Reinhold, 1991 [2] Nielsen, M.P., Limit Analysis and Concrete Plasticity, 2nd Edition, CRC Press, 1998 [3] Neal, B.G., The Plastic Methods of Structural Analysis, 2nd Edition, Chapman and Hall, 1963 [4] Abrams, D., Influence of axial force variation on flexural behaviour of reinforced concrete columns, Structural Journal of the ACI, Vol. 84 (May- June), pp. 246-54, 1987 [5] Falcão, S., Performance based seismic design – Application to a reinforced concrete structure, M.Sc. Thesis presented to IST-UTL, Lisbon, 2002. [6] Falcão, S., Bento, R., Analysis Procedures for Performance-based Seismic Design, Proceedings of the 12th European Conference on Earthquake Engineering, EAEE, Elsevier, London, 2002.