Kaliszky, S.

Nédli, P.

Tornyos, Á.

English

Periodica Polytechnica (Budapest University of Technology and Economics)

PROGRESSIVE COLLAPSE ANALYSIS OF LARGE PANEL BUILDINGS S. KALISZKY, P. NEDLI and A. TORNYOS Department of Civil Engineering Mechanics, Technical University, H-1521 Budapest Received May 8, 1984 Summary Paper deals with the dynamic elasto-plastic response of large panel structures when they are locally damaged by a high intensity pressure or impact loading such as gas explosion or vehicle collision. The analysis is based on a model, which consists of an assembly of rectan-gular rigid plates interconnected along their edges by elasto-plastic springs acting in tension, compression and shear. A detailed parametric study based on a cantilever mechanism is presented. The results show the influence of the different parameters on the response of large panel structures under abnormal loading conditions. Basic concepts When a large panel (LP) building is locally damaged by a high intensity pressure or impact load one or more of the panels can be perfectly destroyed and this accident might cause the progressive collapse of the entire structure or a large part of it. The problem is to determine the static and dynamic response of the damaged building. Since the in situ joints are generally the weakest parts of a prefabricated structure in the present analysis it is assumed, that merely the joints undergo deformations. Thus, in the proposed model of the LP structure the panels are considered as perfectly rigid elements, while the joints are replaced by springs acting in tension, compression and shear (Fig. 1) [1]. The springs are assumed linear elastic-perfectly plastic with a limited deformation capacity (Fig. 2). For the strength and deformation char-acteristics in tension and in compression different values can be taken into account. Quasi~static analysis The goal of the quasi-static analysis is to determine the load carrying capacity of the damaged structure under quasi-static conditions. This is a simple problem of limit analysis which can be solved by mathematical pro-gramming [2]. When, however, the joints have limited deformation capacity a step-by-step load history analysis has to be conducted and after each step the deformations of the joints have to be compared with their capacities. 3 34 S. KALISZKY et al. ~'II..,jIm,..,=...;:;:::;:.,..~~.cI~ r:.....L~":':':'L-'.L~~N.L Fig. 1. :Hodel of the structure Fig. 2. Model of the joints A parametric study has shown that the tensional joints along the 'vertical edges of the panels have the most significant influence on the load carrying capacity of the damaged structure [3]. Dynamic analysis When in a wall of a LP structure one of the panels looses its load carry-ing capacity the vertical reaction force of this panel suddenly diminishes to zero (Figs 3 and 4). For the analysis of this dynamic effect the kinematical approximation can be used [4]. The idea of this approach is to impose a sta-tionary, kinematically admissible, displacement field (e.g. the yield mechanism of the quasi-static solution) on the structure. In this manner a one-degree-of-freedom elasto-plastic system is obtained, the dynamic response of which can be followed by a step-by-step procedure. Next, this will be illustrated by a simple example. Solntion based on a cantilever mechanism Figure 5 sho"ws the cantilever mechanism of a LP wall where n panels above the damaged panel rotate as a rigid body by angle tP about point O. (The elastic, quasi-static analysis of this mechanism has bcen published else-COLLAPSE ANALYSIS OF LARGE PANEL BUILDINGS Fig. 3. Model of the dynamic analysis FA Fa F=FOC1-t1to)e;c:t; if O~ t~ to F:;:;: J J if t >to Fig. 4. The dynamic reaction force 35 where [5].) Considering Fig. 2. the forces of the hOl'izontal springs (reinforce-ments) can be expressed as follows. a) In case of loading (.Jid.J i > 0): I o} = {I, ~f Ll i < .Jei 0, If .Ji > .Jei Ni = 07 ki + of N oi, where . oP - {I, If Llei < Ll i s;;: LlOi i - 0, if Ll i > LlOi b) In case of unloading (LlidLl i < 0): dNi = ki d.J i . Here Ll i = y/l> and dNi and dLl i are the force and strain increments of the springs. The differential equation of motion has the form [2 o}kiy?J<f> = 2 of NoiYi -~ [G -F(t)]. i i 2 This equation can be solved by the Wilson - e numerical integration method. In our parametric study the fixed values a = 5.4 m, h = 2.8 m, H = n X 3* 36 S. KALISZKY et al. Fig. 5. The cantilever mechanism ,NO=1i.O : N:J=lt..O UO=142 , : r---- ... --, . , I , ' t : " t' , . " 0.2 0.3 04 0.5 0.6 1 ... 1.0 t , s Fig. 6. Quasi-static and dynamic response of the structure x 2.8 m, Ll Oi = 0.02 m, Fo = G = n X 800 kN, m = n X 80 000 kg have been assumed, while the other data have been considered variable parameters. Case 1.: NOi = No is a variable parameter, while the other data are: n = 10, ki = 5 . 104 kNm- 1, Fo = G = nx800 kN and F(t) = 800 n (1 - t/to)e-71 , if 0 ::;:: t::;:: 0.35 sec F(t) = 0, if t > 0.35 sec Figure 6 shows the quasi-static and dynamic response of the structure at differ-ent values of No. The minimum value of No at which the load carrying capacity of the damaged structure is large enough to support its weight under quasi-static conditions is (Ng)min = 140.26 kN. At lower values of No!J> increases v.-ithout limits, while at higher values after reaching a certain rotation the cantilever ceases to move. Under d'ynamic conditions the minimum value of No, at which!J> does not increase indefinitely, is (N~)min = 149 kN. When No> (Ng)min the cantilever undertakes elastoplastic or/and elastic vibrations. When No is larger than 200.4 kN and 271.3 kN, the response of the structure COLLAPSE AiYAL YSIS OF LARGE PASEL BUILDD\'GS 37 I No=lS0kN I -- E;cstic deforma.tions __ __ : Etcs~o.- p:csti~ defo;r:'''Lc:ion~ I ... 02 0.3 S4 0.5 0.5 0.7 0.8 0.9 1.0 t ,s Fig. 7. Plastification of the joints Fig. 8. (No)min in terms of the level of the damaged panel is purely elastic under quasi-static and dynamic loading, respectively. Finally, Fig. 7 shows the plastification of the joints when No = 150 kN. ,1i nowhcre exceeded the strain capacity of the joints 'which was ,10 = 0.02 m. Case 2.: Now to is the variable parameter and has the values 0.1; 0.2; 0.3; 0.4 and 0.5 sec. All the other data are identical 'with those of case 1. Evidently the quasi-static results are the same as in case 1. According to the dynamic analysis, the smaner is to the stronger joints are needed to support the loads. When e.g. to is 0.3 sec and 0.1 sec, then (Ng)min is 160 kN and 180 kN, respectively. Case 3.: When k i = 2.5 . 104 kNm -1 and an the other parameters have the values of case 1, then (Ng)min remains unchanged (140.26 kN), the dynamic response of the structure is, however, different and (Ng)min has a higher value (159.5 kN). Case 4.: Let us assume, that the level of the damaged panel is variable i.e. the number of the levels above the damaged panel varies from n = 1 to n = 10. Then, using the same data as in case 1, the minimum values of No which ·are needed to support the structure under quasi-static and dynamic conditions are plotted in Fig. 8. It is to be seen that the higher is the level 38 S. KALISZKY et al. of the damaged panel the higher is the necessary minimum value of No' Thus, on higher levels stronger horizontal joints (reinforcements) are needed and they are more efficient. Remarks The investigation of other one- and multi-degree-of-freedom yield mecha-nisms and the elasto-plastic static and dynamic analysis of space problems are in progress. References 1. KALISZKY. S.-WOLF, K.: Analysis of panel buildings by the use of rigid panel model. Periodica Polytechnic a Civ. Eng. 23 (1979) 2. KALISZKY, S.: Limit analysis of large panel buildings by the use of discretc models. Proc. HJTAll Symposium. Physical non-linearities in structural analysis. France (1980) 3. KALISZKY, S. et al.: Analysis of large panel structures under abnormal loading. (In Hungar-ian) Report. Technical University, Budapest (1982) 4. KALISZKY, S.: Kinematical method for the analysis of inelastic structures under dynamic loading. Advances in :Mechanics 4. No. 1. (1981) 5. Scm.'LTz, D. M.-BuRNETT, E. F. P.-FINETL, M.: Report 4: A design approach to general structural integrity. Design and construction oflarge panel concrete structures. Depart-ment of HUD. Washington D. C. (1977). (PCA publication EB 94.) ~:~fp~:~ ;;::~. KALISZKY 1J Arpad TORNYOS H-1521 Budapest 

PROGRESSIVE COLLAPSE ANALYSIS OF LARGE PANEL BUILDINGS

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PROGRESSIVE COLLAPSE ANALYSIS OF LARGE PANEL BUILDINGS

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