Transactions on Modelling and Simulation vol 18, 1997 WIT Press, ISSN X

Transcription

1 Boundary element analysis of masonry structures Y. F. Rashed*, M. H. Abdalla and M. A. R. Youssef Department of Structural Engineering, Cairo University, Giza, *Present address: Wessex nstitute of Technology, Southampton Abstract n this paper, the boundary element method is used to model the non-linear behaviour of masonry. Cracking, debonding and crushing failure modes are considered. However, the material non-linearity is ignored. nitial stresses, based on a developed algorithm, are used to represent the failure modes. The present model uses an incremental iterative solution procedure to track the failure at each loading stage. A masonry wall under vertical loading is analyzed using the present model. The results are compared to the existing experimental and finite element results to show the accuracy and the validity of the present model.. ntroduction Masonry in structural engineering is used in load bearing elements such as bearing walls, or as infill as in building partitions. Field observations and research indicated that masonry members contribute significantly to the overall resisting capacity of the building system. Therefore, researchers have investigated different aspects of masonry behaviour employing experimental techniques (see for example [2]) and numerical techniques such as the finite element method (see for example [3]). Alessandri and Brebbia [4] used the no-tension solution algorithm developed by Vinturini and Brebbia [5] to analyze masonry walls via the boundary element technique. They considered soil-like cracking assuming that masonry has no resistance in tension. However, this technique is only suitable for infinite domains and it shows a lot of difficulty in modeling problems that have many failure zones in different places as shown by Rashed [6]. n this work, the boundary element method is used to model the non-linear behaviour of the masonry. Cracking, debonding and crushing failure modes are considered. nitial stresses, based on a developed algorithm, are used to represent the failure modes. The proposed procedures are implemented in an incrementaliterative solution technique to track the failure at each loading stage. 2. Boundary element formulation For an elastic body of domain $ with boundary T, the integral representation of the governing Navier differential equations between the source point x' G located on a smooth boundary and thefieldpoint x G F is:

2 24 Boundary Elements where: up,pp = boundary values of the displacements and tractions U*p,P*p, 0p^ = the two-point Kelvin-fundamental solution kernels [] (Tp^ = initial stresses. in which, a,/3,7 =,2, X G ft is internal field point and the integration sign f- denotes Cauchy principal value integral. The corresponding internal stress tensor cr*/3 at internal point X' G ft can be computed by differentiating equation () with respect to the coordinate of the source point X' and substituting in Hooke's law to give: () -f vn (2) where the new ( )* are given in ref. []. After the discretisation the corresponding algebraic equations to equations () and (2) can be written as follows, ] (3) ] (4) where the symbol {[ ]} denotes vector of sub matrices, the symbol [{ }] denotes matrix of vectors and the symbol [[ ]] denotes matrix of sub matrices. 3. The proposed model The masonry wall is composed of bricks bonded together by layers of mortar. At first, the wall is treated as a two-dimensional plane stress problem with homogenized properties. Then, three failure modes are checked according to the place of the check point; weather it is located in the bricks or in the mortar or at the interface between the bricks and mortar. These three failure modes are, the crushing of bricks or mortar, the cracking of either bricks or mortar and the debonding along the brick-mortar interface. Figure shows the state of the stresses before and after the each failure mode. The used failure criteria in the present work are also give in Figure. The analysis of the masonry is carried out in the following steps:

3 Boundary Elements 25. The problem is discretised. Herein, constant boundary elements are used to discretise the boundary and constant internal cells are used to discretise the predicted failure zone. The problem is also covered by internal points to check the validity of the problem discretisation scheme (resume the analysis in case of any failure was happened outside the discretised domain). 2. The load is applied in increments. For each increment, the failure points inside the domain are detected and classified according to the corresponding failure criteria. 3. nitial stresses, which are predicted using the developed algorithm (see the coming section), are applied at each failed point to redistribute the stress field according to the occurred failure mode. 4. The problem is analyzed for the next load increment. The redistribution of the stress field is done first for the failed points corresponding to the previous load increment. Hence, the newly failed points for the current stage are detected and dealt with. 5. Steps 3 and 4 are repeated until the full load is reached or a case of failure instability is detected, where unlimited number of the redistribution procedures are required. 4. Determination of the nitial stresses Consider an internal cell with the domain part denoted by HS subjected to unknown initial stresses cr, where the effect of these initial stresses are known in the final solution (such as zero tensile stresses in the direction perpendicular to a crack). Equation (4) can be rewritten in the following form: where W = X} + (Y] + [Z}}(* } (5) f^,4,^ (7), ^4,^4-^ (8) in which [cr] is prescribed and [<r ] is unknown. n order to solve the inverse problem to find [cr ], the following iterative procedures are proposed:. Compute [Y] and [[Z]] noting that they are independent of (a \. 2. Assume initial values for [v ] (e.g. [<r ] = [0]). 3. Carry out the standard BEM analysis using equation (3).

4 26 Boundary Elements 4. Compute the approximate value of [X] using equation (6). 5. Compute the value of the matrix [C] where: [C] = [<r]-[x]-(y] (9) 6. Substitute in equation (8), to get the new value of [a ] as: 7. Repeat the former steps starting from step 3 to get the new values of {u}, {p} using the computed approximate values of [cr ], until [<r ] converge to a constant values. (0) 5. Numerical example A masonry wall of 7SCm x 83Cm (see Figure 2) is analyzed using the present model. The wall has the following properties of materials: Brick compressive strength = 30 Brick tensile strength = 0.5 Mortar compressive strength = 20 Mortar tensile strength = 9.5 Brick modulus of elasticity = 90 Mortar modulus of elasticity = 75 Wall modulus of elasticity = 88 Figure 3 shows the boundary element discretisation. Figure 4 shows the internal cells discretisation. Figure 5 shows the additional internal points in which the internal functions are evaluated and also it checks for the validity of the internal cells mesh by ensuring that there is no failure point outside the cell discretisation zone. Table : Denoting numbers for the loading stages Denoting number Percentage of failure load Failure Figure 6 shows the crack pattern at different loading stages (give in Table ), when the loaded area is 0. of the wall width. The wall failure mode was predicted using the proposed model as successive crushing under the applied load which matches the experimental results in [3]. However, the used integration scheme (4 Gauss points per element and 4 x 4 for internal cells)results in the appearance of the near-top surface cracks due to the near-singular problems which do not appear in the experimental tests [3]. These cracks releases the stresses in the

5 Boundary Elements 27 wall and prevent the central crack from propagating to the bottom part of the analyzed wall. Figures 7 and 8 show a comparison between the model solution and the elastic solution for the normal displacement along the wall vertical sides and top edge respectively. t can be seen that central crack releases the displacement on the top zone of the wall. Figure 9 shows a comparison between the elastic solution and the proposed model solution for the horizontal stresses along the central vertical line. t can be seen that the tension zones in the elastic solution change to zero-stress or compression zones in the model solution. Figure 0 shows a comparison among the experimental data [3], Finite element analysis [3] and the proposed model analysis for the failure loads for different loaded area ratio. The finite element model () uses smeared crack modeling. Model (2) uses discretised crack modeling. As can be seen that the proposed BEM model shows a good agreement with the experimental data. 6. Conclusions A two-dimensional plane stress boundary element model is developed for analysis of masonry structures. The proposed model accounts for the different failure modes of masonry. A developed algorithm is presented to solve the inverse problem of finding the values of unknown initial stresses when their effects are known. This algorithm is used to predict the required values of the initial stresses that needed to redistribute the stress field according the mode of failure which occurred. The model results are presented via analyzing masonry wall under patch loading. The results for the failure loads are compared to the published experimental and finite element results, and show good agreement. References [] Brebbia, C.A., Telles, J.C.F. and Wrobel, L.C., Boundary element techniques: Theory and Applications in Engineering, Springer- Verlag, Berlin- Heidelberg, (984). [2] Hamid, A. A. and Chukwanenye, A.O., Compression behaviour of concrete masonry prisms, J. o/sfrwc. Er#., A^CE, 2(3), , (986). [3] All, S. and Page, A.W., Finite element model for masonry subjected to concentrated loads, J. o/afrwc. En#., A^CE, 4(8), , (989). [4] Strength of masonry walls under static horizontal loads: boundary element analysis and experimental tests, C. Alessandri and Brebbia C.A., Em?. Analt/sz'g, 4(3), 8-34, (987). [5] Venturing W.S. and Brebbia, C.A., The boundary element method for the solution of no-tension materials, n Boundary element methods, Brebbia, C.A. (Ed.), Springer- Verlag, (98). [6] Hashed, Y.F., A new non-linear boundary element model for brick masonry walls, M.Sc. Thesis. Cairo university, (993).

6 28 Boundary Elements b u Cl g.2 j 3 3 W) u m

7 Boundary Elements 29 ^ o w # H «"O c3o CQ 4 :4 O a

8 30 Boundary Elements c o o vi 8 S3UJ [BUOZUOH i i \ - L_ "! *- i $ i J_. r - i i i i j! i M 4 j \ J ' \ \ - t f // \ \ / / / c 4 8

9 Crashing Boundary Elements 3 4 X JUF F Figure 6-Crack pattern. Horizontal displacement (/Omm) mill Tf Jt.,tT Horizontal displacement (/0mm) ) Elastic Solution Model Solution Figure 7-Displacements along the wall vertical sides.

10 Horizontal stresses (Kg/cnT) Compression f Tension CO to W -T a(/) ' a "3 o Elastic Solution Model Solution Elastic Solution Model Solution Figure 8- Displacements along the wall top boundary. Figure 9-Splitting horizontal stress.

11 Boundary Elements 33 ^ 40 J2 # Experimental Finite element -model Finite element -model 2 BEM present model O Loaded area ratio Figure 0-Failure loads.