INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 1, No 3, 2010

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1 Prbabilistic Analysis f Lateral Displacements f Shear in a 20 strey building Samir Benaissa 1, Belaid Mechab 2 1 Labratire de Mathematiques, Université de Sidi Bel Abbes BP 89, Sidi Bel Abbes 22000, Algerie. 2 LMPM, Department f Mechanical Engineering, Université de Sidi Bel Abbes BP 89, Sidi Bel Abbes 22000, Algerie. benaissamir@yah.fr ABSTRACT Lateral displacements f shear wall (SWII) analyzed by finite element methd. Keywrds: Prbabilistic, vibratin strengthened shear wall, Finite element methd, Cmpsite plates 1. Intrductin In many areas arund the wrld, reinfrced cncrete (RC) buildings designed using cdes that are nw knwn t prvide inadequate safety under seismic frces are ptential hazards. In such areas, the number f RC buildings built prir t 1980 utnumbers thse that are built accrding t the newer cdes. Therefre, these structurally deficient buildings shuld be retrfitted t withstand earthquake frces in cmpliance with mdern design cdes. The lateral lads that arise frm the effects f wind and earthquakes, are resisted by shear walls. These latter, are usually built ver the whle height f the buildings, as series f walls cupled by beams r slabs r as a central cre structure. Pagnini and Slari used the first rder and the secnd rder techniques in cmputing the acceleratin respnse f a square plan multistry steel building excited by the wind lading. Obviusly, the main purpse f these methds is t determine the secnd rder statistical quantities f dynamic structural respnse, such as the mean, the standard deviatin, etc. As far as the prbability density functin (PDF) f the respnse is cncerned, mst f these methds will be met with difficulty. Extensive testing has shwn that externally bnded carbn fiber reinfrced plymer (CFRP) laminates are particularly suited fr imprving the shrt term behavir f deficient reinfrced cncrete beams and slabs. Als nte that this technique f reparatin has been als used in aircraft structures. 373

2 Lateral displacements f shear wall (SWII) analyzed by finite element methd. 2. Gemetrical and mechanical prperties The numerical values f the gemetrical and material parameters were presented in Table 1 and Figure 1 fr shear walls structures: SWII.. Table 1: Material prperties f the shear walls in the cnvergence studies SWII SWII Mass density q (kg/m 2 ) 2400 Yung s mdulus E 11 = E 22 (GN m 2 ) 15 Shear mdulus G 12 (GN m 2 ) 6.25 Figure1: Shear wall structure SWII un strengthened 3. Numerical results and discussin Numerical results are presented in this study analyzed by finite element methd fr shear walls strengthened by a bnded cmpsite thin plate. We cnsider in the first example the free vibratin fr typical unstrengthened shear wall structure. Results f methd are listed in Table 2 and cmpared with thse f Syngellakis and Papulia. It can be seen frm the cmparisn that there is very little difference in the three first natural frequency results. 374

3 Table 2: Cnvergence and cmparisn studies f first five natural frequencies (Hz) fr shear wall type SWII N. mde Present study (FEM) Syngellakis and Papulia The dimensins f shear wall structure SWII Strengthened and unstrengthened are shws in Figure 2 the results presented in Figure3 Figure2: Shear wall structure SWII Strengthened (a), unstrengthened (b) 375

4 15 12 Height (in) unstrengthened strengthened 0 0,00 0,25 0,50 0,75 1,00 1,25 1,50 1,75 2,00 2,25 2,50 lateral displacement (10 3 x in) Figure3: Lateral displacements f shear wall (SWII). Figure 3 Shws the Effect f the bending and vibratin analysis f shear wall structures strengthened by bnded cmpsite plates and unstrengthened can be bserved. The Lateral displacement f shear wall (SWII) is prprtinally with height. 4. Prbabilistic analysis 4.1. Prbabilistic results and discussins Lateral displacements f shear wall (SWII) analyzed by finite element methd In this paper a methd is presented t calculate the prbability f a prblem, f which ne r mre f the mdel parameters are randm variables, i.e. statistically determined. Mathematically this can be stated as: def def F ( j ) = Pr U ( X ) < j = ƒ ( x ) dx (14) U U ( X ) < j x 376

5 f(x) the Jint Prbability Density Functin (JPDF) f the vectr f randm variables X, such as material parameters, dimensins and lads. The displacement U shuld be evaluated by the prbability r the prbability density. ƒ ( j ) = df U U ( j ) / dj Where F u (j 0 ) is the cumulative prbability distributin functin f U and ƒ x (x) is the knwn jint prbability density functin f X. Figure4 presents similar results f prbability Gaussian distributins f randm variables. U: Lateral displacements f shear wall structure (SWII). Prbability Density f U 1,0 0,8 0,6 0,4 0,2 Mean Gauss distributins 0,0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 U (10 3 x in) 4. Cnclusins Figure 4: Presents similar results f prbability density f U Lateral displacements f shear wall (SWII) analyzed by finite element methd. The Effect f the bending and vibratin analysis f shear wall structures strengthened by bnded cmpsite plates and un strengthened can be bserved. The Lateral displacement f shear wall (SWII) is prprtinally with height. Prbability density functins f U see Figure 4 presents similar results f prbability Gaussian distributins f randm variables. 377

6 References 1. Pagnini, L.C., Slari, G., Service ability criteria fr wind induced acceleratin and damping uncertainties. J.Wind Eng.Ind. Aerdyn.74,pp Shahaway MA, Beitelman T, Arckiasamy M, Swrirajan R. Experimental investigatin n structural repair and strengthening. f damaged prestressed cncrete slabes utilizing external bnded carbn laminates. Cmps B 1996;27:pp Shen H S, Chen Y, Yang J. Bending and vibratin characteristics f a strengthened plate under varius bundary cnditins. Eng Struct 2003; 25:pp Megueni A, Tunsi A, Bachir Buidjra B, Serier B. The effect f a bnded hygrthermal aged cmpsite patch n the stress intensity factr fr cracked metallic structures. Cmps Struct 2003:pp Baker AA, Callinan MJ, Jnes R, Williams JG. Repair f Mirage aircraft using BFRP crack patching technlgy. Ther Appl Fract Mech 1984;2:pp Syngellakis S, Papulia KD. A transfer matrix apprach t free vibratins f cupled shear walls. Eng Struct 1987;9:pp