Kaliszky, S.

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Periodica Polytechnica (Budapest University of Technology and Economics)

LIMIT ANALYSIS OF LARGE PANEL BUILDINGS AND THE SUBGRADE By S. KALISZKY Department of Civil Engineering Mechanics, Technical University, Budapest Received: :\iarch 26, 1981 1. The model of the structure Since the joints made in situ are generally the weakest parts of a prefabricated structme, in the present analysis of large panel (LP) buildings it is assumed that in plastic limit state merely the joints of the panels undergo plastic deformation. Thus, in the model of the LP structme the panels can be considered as rigid elements, while the joints are replaced by plastic springs acting in tension, compression and shear (Fig. 1) [1]. Considering a wall of a LP building, the equilibrium of the ith rigid element (Fig. 2a) subject to ex-ternal loads and spring forces "' [N" N'" T 1 Sik == 1 tic lllk ik_ resp., can be expressed as (1) Here k 1, 2, 3, 4 refers to the edges of the element, and o -1 1 1 0 0 0 1 -1-1 0 r~rl 1 0 0 0 1 -1 -1 0 0 o -1 Si2 bi ai al -ai b l b i -bi ai ,-ai ai b l Si3 814 Fig. 1. Model of the structure and the subgrade 166 KALISZKY ~ _____ 2_b~i ______ ~ Fig. 2. Discrete elements a) structure, b) subgrade Denoting the yield forces of the springs in tension, compression and shear by N"/k, NU; and Tu, resp., and neglecting the interaction between the springs, the yield conditions of the edges have the form Here o 0 1 --1 o 0 ~ ], 1 --1 o o k = 1,2,3,4. * -- [7\7+ 7\;- 7\7+ 7\T- TT] rih - l"ik J.'ili l'ik l'ik ill. if{' 2. The model of the suhgl'ade (2) The ideal plastic subgrade (soil) is subdivided into discrete prismatical elements which transmit normal, shear and couple forces to each other along their contact surfaces (Fig. 1) [2]. Considering the jth element (Fig. 2b) sub-ject to its own weight and to contact forces resp., the equilibrium equations can be \VTitten as (3) Here k = 1, 2, 3, 4 refers to the boundary surfaces, c and Yj denote the thickness and the specific weight, resp., of the element, and 1 0 o 1 --aj 1 1 o o o 0 1 0 b j --1 o --1 0-1 0 o 0 0-1 o -aj -1 0 b 0], Pj = [Pj1l· ° pj2 1 pj3 pj4 LARGE PA);"EL BUILDINGS 167 Applying the linearized Tresca yield condition to the stress state at the centre o of the element we obtain the inequality Cj pj- Cij < O. (4) Here Cj=1 -1 3 9 b j 3bj 9 3 9 b j 3bj 9 -1 - -1 8 8aj aj 8aj 8aj 8 8aj aj 8aj 8aj 1 3 9 b j 3b· 9 3 9 b j 3bJ 9 __J 1 .. 8 8aj aj 8aj 8aj 8 8aj aj 8aj 8aj 1 3 9 3bJ 9 1 3 9 b j 3bj 9 8 8aj aj 8aj 8aj 8 8aj a· J 8aj 8aj 1 3 9 -b j 3bj 9 1 3 9 ht 3bj 9 L 8 8aj aj 8a, J 8aj 8 8aj aj 8aj 8aj -1 aj = 4·b j caj [1 1 1 1] and aj denotes the yield stress of the jth element of the subgrade. 3. Limit analysis Using the models described aboye, statically admissible stress fields can he constructed for the LP structure and the subgrade which makes possihle the unified limit analysis of the whole system. Considering a wall subject to a one-parameter load q = mqo, the equilihrium and the yield conditions have the form G*s + mqo = 0, Hs-l':::::O, (5) resp., while the same conditions for the sub grade are B*p + q = 0, CP-Ci:::::O. (6) Since sand p are related by the statical boundary conditions of the contact line between the structure and the subgrade, (5) and (6) make up a system of linear equations and inequalities the solution of which, together "\.,,-ith the extremal condition m = maxI (7) leads to linear programming. 168 KALISZKY 4. Applications The method described suits the analysis of the structural response of LP buildings under abnormal loading conditions. Assuming ineffective (missing) panels or sub grade elements the effect of gas explosion or of cavitation of the soil can be investigated. Besides, the discrete models can be applied to the approximate limit analysis of LP structures subject to dynamic pressure. Summary Simple discrete models are pre,ented for the limit analysis of prefabricated large panel buildings resting on 3. plastic subgrade. The model of the structure consists of rigid panels inter-connected along their edges by pla~tic springs, while the mbgrade is subdivided into prism at-ical elements which transmit normal, shear and couple forces to each other. Using these discrete models and linear(ized) yield conditions, statically admissible stress fields can be con-structed for the structure and the subgrade. This makes possible the unified limit analysis of the whole system by the use of linear programming. References 1. KALISZKY. S .. WOLF. K.: Analysis of Panel Buildings by the Use of Rigid Panel 1!odcl. PeriodicaPolyte~hnica Civii Eng. Vo!. 23. (1979) No. 3. ~ 2. K.HISZKY, S.: Statically Admissible Stress Fields in Plane Plastic Problems. Bu!. Acad. Po!. Sci. Vo!. XXVII. No. 5/6. (1979). 3. KALISZKY, S.; Analy.sis of Large Panel Structures by the Use of Discrete Models. (In Hun-garian.) Epites-Epiteszettudomany Vo!. XI. (1979) No. 3-4. Professor Sandor KALISZKY, D. Techn. Sci., H-1521, Budapest. 

LIMIT ANALYSIS OF LARGE PANEL BUILDINGS AND THE SUBGRADE

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LIMIT ANALYSIS OF LARGE PANEL BUILDINGS AND THE SUBGRADE

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Periodica Polytechnica (Budapest University of Technology and Economics)