Boundary element analysis of post-tensioned slabs

In this paper, the boundary element method is applied to carry out the structural analysis of post-tensioned flat slabs. The shear-deformable plate-bending model is employed. The effect of the pre-stressing cables is taken into account via the equivalent load method. The formulation is automated using a computer program, which uses quadratic boundary elements. Verification samples are presented, and finally a practical application is analyzed where results are compared against those obtained from the finite element method. The proposed method is efficient in terms of computer storage and processing time as well as the ease in data input and modifications.


Introduction
Flat slabs are desirable structural systems in engineering due to architectural needs and speed of construction. In order to construct flat slabs with large spans (about 12-15 m), pre-stressing cables are necessary (Post tensioning manual 2000). Several finite element method (FEM) (Zienkiewicz 1977) based packages are used to carry out the structural analysis of plate structures; among them are the SAP2000 (2006 and SAFE (2008), etc. However, special packages are designed to treat the presence of post-tensioned cables such as ADAPT (2007) and SAFE-PT (2008). All of these packages, no exception, are based on the finite element method.
The boundary element method (BEM) (Berrebia et al. 1984) has emerged as a powerful tool in engineering practice. The BEM for thin plates started with the work of Bézine (1978) and Stern (1979) for the direct formulation and by Tottenham (1979) for the indirect formulation. Van der Weeën (1982) was the first who applied the boundary element method to shear-deformable plate-bending problems according to Reissner (1947). Rashed (2005a, b) extended formulation of Van der Weeën (1982) to model flat plates over columns. It was presented in Ref. (Rashed 2005a, b;Nazief et al. 2010) that the Reissner platebending model is more refined and accurate in theory for thin slabs and thick foundation plates. To the author's best knowledge, none of these publications considered the presence of pre-stressing cables in flat slabs.
This paper aims to include the effect of post-tensioning cables within the boundary element formulation for flat slabs. It can be regarded as a new structural analysis tool for post-tensioned slabs. Unlike previous technology, which is mainly based on the FEM, the proposed method is based on the boundary element method. This will avoid any internal meshing problems and will guarantee high accuracy for both slabs and supporting elements. The proposed method is easy in data input and modifications. It is also fast in computer processing time and has less storage requirements. Therefore, practical projects results could be sent easily by e-mail. Cables are placed freely inside the slab with no meshing constraints, which provides robust tool for optimization and value engineering. Results of the proposed model are compared against those obtained from the FEM. This comparison proved the validity of the present formulation.

Boundary elements for flat slabs
Consider the slab shown in Fig. 1. According to Rashed (2005a, b) columns or internal walls are modeled using internal supporting cells with the real geometry of their cross sections. This ensures avoiding overshooting stress resultant values over supports. Three generalized forces are considered at each internal support: two bending moments in the two spatial directions (x 1 , x 2 ) as well as a shear force in the vertical direction (x 3 ). These generalized forces are considered to vary constantly over the column cross section (Rashed 2005a, b). A suitable boundary integral equation (in index form) can be written as follows (Rashed 2005a, b): where U ij (n,x) and T ij (n,x) are the fundamental solution kernels (Vander Weeën 1982), u j (x), t j (x) are the boundary displacements and tractions, C ij (n) is the jump term at the source point, n (= for n on smooth boundary), X C (y) denotes the column's y domain, y is a field point at the column center, S k (y) is the column's y stiffness, A(y) is the column's y area, q is the uniform domain loading and B(y) is a coefficient (equal to zero in case column stops below the considered floor and equal to one in case column continues above the considered floor). It has to be noted that long walls and cores can be treated as column segments. Beams can be also modeled in similar way but using special stiffness matrix. In order to solve the problem in Eq. (1), additional collocation scheme is carried out at each column center. In this case additional three integral equations similar to that of Eq. (1) could be written but with: n = y and C ij (y) = 1. These new equations together with Eq. (1) can solve the boundary value problem (Rashed 2005a, b).
After the boundary solution, values of the displacement and forces at column centers are also calculated. Generalized displacements at the internal point n can be obtained using Eq. (1) with C ij (y) = 1. The stress resultants at the internal point n are obtained as follows (Rashed 2005a, b): where the new kernels U ijk (n,x) and T ijk (n,x) are given in Vander Weeën (1982).

Proposed model for pre-stressing cables
In this section, the pre-stressing cables are introduced to the boundary element formulation presented in Sect. 3 using the equivalent load method (Lin 1963).

The equivalent load method
Consider a general cable profile shown in Fig. 2. This profile can be represented in terms of the following parabolic equation: where, a 1 , a 2 , and a 3 are constants to be determined from the cable boundary conditions. The coordinate L is measured from an arbitrary origin (see Fig. 1) along the cable and Z is the cable eccentricity in the x 3 direction measured from slab centerline (see Fig. 2). The derivatives of Eq. (4) w.r.t. the coordinate L are: Equivalent cable loads on each segment are represented (Lin 1963) by distributed load (W) along the cable length together with two concentrated forces (F 3 (L s ), F 3 (L e )) and two concentrated moments (F a (L s ), F a (L e )) at the segment start (L s ) and end (L e ) points (see Fig. 2). Equivalent distributed load is computed as follows (Lin 1963): Concentrated load values are computed as follows (Lin 1963): where, P is the pre-stressing force of the considered cable. Concentrated moment values due to the cable eccentricity are computed as follows (Lin 1963): where the a-direction is the perpendicular to the L-direction and the right hand vector notation is used to represent the bending moment. The F a ðL s Þ and F a ðL e Þ is going to be resolved into: F 1 (L s ), F 1 (L e ) and F 2 (L s ), F 2 (L e ) in the x 1 and x 2 directions. If cable profile has any local change in curvature, two segments are used to model this cable. Hence, additional concentrated load is added at the point of discontinuity, with a value equal to the summation of end concentrated loads for the two intersecting segments. Moreover, equivalent load for any continuous cable having variable segment profiles is calculated by dividing the cable into series of parabolic segments.

Boundary element implementations
In the boundary element model, each cable is divided into series of patch loading cells [n cells with centers located at (X 1 , X 2 … X n ) as shown in Fig. 3] to model the distributed load (W). The width of these cells is equal to the cable width (c). The locations of such points could be obtained easily using simple geometry relationships. Additional two square cells (of dimensions equal to c 9 c) are placed at the beginning (X s = X s (L s ) and at the end (X e = X e (L e )) of each cable to represent the concentrated loads (F 3 ) and moments (F a ). In order to account for such new cable loading additional terms have to be added to right hand side of the boundary integral Eq. (1) as follows: In addition, the following additional terms have to be added to the right hand side of Eq. (2) for the moment calculations: and the following terms have to be added to the right hand side of Eq. (3) for the shear calculations: where, X N is the center point of the subdivision number N (see Fig. 3), X S , X e are the start and the end points of the cable segment, respectively, F Ã 3 X N ð Þ is the vertical pressure for subdivision number N, and F Ã k X Q ð Þ is the moment per unit area when k = 1, 2 and it represents the vertical pressure F Ã 3 À Á when k = 3. Values of F Ã k can be obtained as follows: and in which ' cable is the horizontal distant from X s to X e . Similar to Eqs. 1, 2 and 3 in Sect. 3, it has to be notes that in Eq. (12) n has to be collocated at all boundary points plus at internal columns and walls centers as well as at beam cell centers. Whereas, in Eqs. (13) and (14), n has to be any internal point.

Numerical implementations
The previous boundary element formulation is implemented into computer code using quadratic boundary The considered cable segment The prestressing cable (Elevation view) Fig. 3 The proposed boundary element modeling of the cable segment equivalent loads elements. This code is called the PLPAK. Two main software tools are implemented to add the effect of the prestressing cables. The first tool is called the ''Cable Calculator'', which allows inputting the cable data using different formats and determine the constants a 1 , a 2 and a 3 from the cable geometry (recall Sect. 3.1). The second tool is the ''PTUpdater'' which changes the cable data into equivalent load as described in Sect. 3.1. It also updates the boundary element model with such equivalent loads as demonstrated in Sect. 3.2. Hence, the traditional steps of solution for boundary elements are carried out (SAP2000 2006).

Numerical verifications
The purpose of this section is to verify the proposed formulation presented in Sect. 3 for simple problems where analytical solutions are existed in the literature.

Load balancing of simply-supported slab own weight
In this example, the slab shown in Fig. 4 is considered. The slab has cross-section dimensions of 1.0 9 0.6 m. The material properties taken are E = 2.21 9 10 6 t/m 2 , t = 0 to allow comparison against results for the beam theory. The slab is pre-stressed with one cable of force  Fig. 4. The slab boundary is modeled (see Fig. 5) using 16 boundary elements. A simply supported boundary condition is employed. Such conditions are simulated using two column support of 1.0 9 0.1 m with zero rotational stiffnesses and high value of (10 10 ) for the axial stiffness. Eleven internal cells are used to represent the cable equivalent loading. The numbers of Gauss points used for integration purposes are ten. The total number of extreme points is 52. The results are calculated along a strip along the cable center line. Figure 5 demonstrates the deflection and bending moment distributions along the slab center line under its own weight only. Figure 6 demonstrates the same deflection and bending moment distributions under both own weight plus the balancing pre-stressing force. It can be seen that deflection approaches zero compared to the deflection distribution in Fig. 5. The bending moment in Fig. 6 approaches zero also; except near the end supports as such supports are not knife edge and has width of 0.1 m; therefore small negative moment is expected.

Comparison of central deflection against analytical values
The same example in Sect. 4.1 is reconsidered herein using different cable profile (see Table 1). The present boundary element results for the deflection at the mid span are shown in Table 1. It can be seen from Table 1 that the results for the central deflection are in excellent agreement with analytical values obtained from Ref. Michael and Denis (1997). The symbols used in Table 1 are: P is pre-stressing force, e, e c are the centerline eccentricity, e e is end eccentricity, E is modulus of elasticity, I is section moment of  inertia, b is the ratio of the distance from the harping point to the beam end, to the beam length. This ratio is equal to 1.4/4.9 in the considered case (Fig. 7).

Comparison of fixed end moments against analytical values
Using the same slab in Sect. 4.1, alternative cases are considered herein to verify values of the fixed end moments. The cable profiles shown in Table 2 are considered.
In this case the fixed-fixed boundary condition is employed. Such conditions are simulated within the boundary element model using the same previous columns but with very high value of (10 10 ) for the axial and the rotational stiffnesses in the two directions. The results of the proposed model fixed end moments together with the analytical values obtained from Michael and Denis (1997) are given in Table 2. It can be seen that the obtained results are in excellent agreement with analytical values. It is worth mentioning that in the last case, the end fixations are

Practical application
The previous simple cases (in Sect. 4) verified the present formulation. In this example, the slab shown in Fig. 8 is considered. The purpose of this example is to demonstrate that the present formulation can be used as an alternative to the existing finite element based software packages. The slab has maximum dimensions of 61 9 26 m with spans about 7 to 11 m and thickness of 0.24 m. The material properties taken are E = 2.1 9 10 6 t/m 2 , m = 0.16. The slab is pre-stressed with cables in X & Y directions as shown in Figs. 9 and 10, respectively. Cables spacing varies from 0.6 to 1.6 m and cable force are equal to 12 ton. Cable groups are used. Each group contains 2 to 5 cables. Cable layout and eccentricity are shown in Figs. 9 and 10, respectively. The slab is supported on group of irregular columns (cross section varies from 2 to 4 m 2 ) and central core as shown in Fig. 9. The floor height is 3 m. The slab boundary is modeled using the proposed boundary element models using 159 boundary elements and 4124 internal cells are used to represent the equivalent loading of cables as shown in Fig. 11. The number of  The same slab is considered using finite element analysis with 0.2 9 0.2 m mesh, columns are represented as 3D solids, shear walls and cores are represented using shell element. The used finite element model has 87,003 nodes and 22,098 four-node plate-bending elements as well as 48,990 solid elements as shown in Fig. 12. It has to be noted that results presented in this section will concentrate on slab results. Discussions on results for supporting elements are similar to those of slabs without pre-stressing cables which have been already considered by Rashed (2005a, b). Figures 13,14,15,16,17,18,19,20,21,22,23,24, 25 and 26 demonstrate the distribution of bending moment and deflection results along sections A, B, C, D, E, F and G in the considered slab (see Fig. 8). It can be seen that the  Table 3 demonstrates a comparison in terms of computer running time and computer storage requirements between the present (BEM) and (FEM). The superiority of the present formulation can be seen from this table.

Conclusions
The present paper developed a new boundary element formulation that account for the effect of pre-stressing cables in flat slabs. The equivalent load method is used to simulate the effect of pre-stressing cables. The formulation is automated and tested against simple cases and practical problems. The present formulation has many advantages over the existing finite element based codes in terms of data preparations, computer time and storage requirements. Analysis of pre-stressed foundation plates, punching calculations and pre-stressing losses could be easily considered using the proposed model. However, they will be considered as future research. The present method could also be regarded as a fast checking tool for results obtained from existing FEM-based software packages. It is also a promising tool for value engineering.