Application of the matrix force method to the BEM … · BEM analysis of plates with internal...

Application of the matrix force method to the BEM analysis of plates with internal constraints

Y . F. Rashed Wessex Institute of Technology, UK (On leave from the Faculty of Engineering, Cairo University, Giza, Egypt).

Abstract

This paper presents a new method for the analysis of plates in bending with internal constraints. The proposed method can be regarded as an extension of the well-known force method in matrix analysis of structures. The proposed solution is performed through two phases: the released plate phase, in which the plate is released from all internal constraints and solved using the Boundary Element Method (BEM). The effect of internal constraints is considered in the second phase, where a series of the unit virtual loads is placed instead of the unknown redundant reactions at internal constraints. The flexibility matrix is formed by satisfying the compatibility of deformations at the internal locations of supports. Hence the corresponding system of equations is solved for the unknown redundant forces at intemal supports. The final solution of the problem is consisted of the summation of two phases: the released plate phase and the cases of virtual unit loads phase. Two numerical examples are analysed. The results are compared to those of analytical models to demonstrate the accuracy and the validity of the present formulation.

1 Introduction

Plates with internal constraints (supports) are commonly used in building structures, such as flat plate floors. The analysis of such structure started using simplified empirical approaches by dividing the continuum into field and column strips to allow the analysis using series of discrete skeletal frames. The frame model is highly uneconomical, as it is highly conservative. The appearance of the new computer systems simplified the use of more sophisticated numerical techniques in structural analysis. The Finite Difference Method (FDM), was the

Transactions on Modelling and Simulation vol 27, © 2001 WIT Press, www.witpress.com, ISSN 1743-355X

first technique used. The main idea of the FDM is to convert the governing differential equations to a system of algebraic equations. Despite the simplicity of the FDM, it suffers from serious disadvantages, such as: low accuracy, difficulty in modelling irregular boundaries and the high computational requirements due to the need of the discretisation of the over all problem domain. These disadvantages made the FDM becoming not practical. Nowadays, the Finite Element Method (FEM) [ l ] completely replaced the FDM and widely spread in most of design offices and companies. There are many commercial codes based on the FEM such as, SAP90, SAP2000, COSMOS, etc. The main idea of the FEM is to locally solve the governing differential equations in its integral form. The accuracy of the FEM is adequate for most structural applications. However, the FEM still requires the discretisation of the overall problem domain. Moreover internal supporting batches, such as, columns or walls need special approximation in the modelling.

In the last twenty years, the Boundary Element Method (BEM) [2] has emerged as a superior numerical tool to solve engineering applications. The main advantage of the BEM is the boundary only discretisation of the considered problem. Unlike the FEM, the BEM gives continuous field solution for unknown identities, such as, displacements and stresses. In other words, the BEM ensures the compatibility and equilibrium everywhere inside the continuum. Modelling plates in bending using the BEM is introduced by Bezine [3] and Stem [4] for the classical thin plate theory. Vander Weeen [5] applied the BEM for thick plate theory. Since that time, many researchers have studied the extension of the above formulations to many different applications. Modelling internal supports in the BEM requires special modification to the standard BEM formulation. This is mainly because all the considered unknowns in the BEM are located on the boundary. In the literature, there are two techniques to deal with internal supports. The first is to use the technique of multi regions, i.e. to split the original problem into series of regions and hence the internal support can be placed on the interface boundary between the divided regions. This technique is used by Kamiya and Saito [6] for elasticity problem. However this technique forces the unknowns to vary according to the variation of the chosen shape function along the interfaces. Moreover it requires high computational effort as it introduces new unknowns along the interface boundaries. Another approach was developed also in Ref.[6]. This technique is based on the treatment of the internal support as unknown force, then introduce additional equation by satisfying the displacement compatibility at the location of the internal supports. This technique is efficient but requires substantial modification to the standard BEM codes. This technique was applied to solve two-dimensional elastostatics with internal supports Kamiya and Saito [6]. They recommended using the later technique better than the multi-region technique from the view of accuracy. Providakis and Toungelidis [7] and Katsikadelis et al. [g] used the later technique to solve dynamic problems of thin plates with internal supports. It has to be noted that in both of the above techniques, it is difficult to model the actual shape of the internal supports.


In this paper, a new technique for the analysis of plates with internal supports is presented. The present formulation is based on two phases. In the first phase, the plate is released f?om all internal supports and solved using the BEM. The second phase consists of as many cases of virtual loads as the number of internal supports. Hence the compatibility of deformations are enforced at the internal supporting locations. The flexibility matrix is formed and the system of equations can be solved for the unknown redundant forces at internal supports. The final solution of the problem is the summation of the formerly mentioned two phases. Two numerical examples are analysed. The results are compared to those of analytical and finite element models to demonstrate the accuracy and the validity of the present formulation.

2 The proposed approach

Consider an arbitrary plate shown in Fig. 1 of thickness h in the X, space. The X,-

x2 plane is assumed to be located at the middle surface where x3=0. The generalised displacements are denoted by U,, where, U , denotes rotations and u3 is the transverse deflection in the x 3 direction. Throughout this paper, the tensor notation is used. Greek indices vary from 1 to 2 whereas, Roman indices from 1 to 3. The plate is internally supported by n supports of different types, including area supports, line supports and point supports. Each support has reaction force Rk where k=l+n. The solution of this problem is done through two phases as follows:

Fig. 1 : The actual plate structure.


2.1 Phase 1: the released plate phase (k=O)

The plate is released fiom all internal supports as shown in Fig. 2. Then the problem can be easily solved using the standard BEM. Without losing generality, herein, the plate bending theory according to Reissner[9] is considered. The direct boundary integral equation can be written as follows:

where U,, and c, are the two-point fundamental solution kernels (see Ref.[5]), x_' and g denote source and field points respectively. The integral sign with the dash denotes Cauchy principal value integral. C is a jump term equal 0 if x_' is outside the boundary, 'h if& is on a smooth boundary and 1 if g' is internal point. F,@)@') is the body force term due to the external applied load which is given by:

r 1

. . . . . . . . . ( 2 ) Where 9 is the shear factor, ; is Poisson's ratio, is the applied external domain loading, including body forces and X is an internal field point. It has to be noted that for k=O: d0)=f2 is the plate domain, r(OJ=ris the plate boundary and t,@)('x) denotes the external loading on the boundary.

For the sake of simplicity, it is assumed that the interaction between the internal support and plate is in the x3 direction only (i.e. the internal supports behave as column link members or springs). The deflections at the internal support centres can be computed using Eq. 1 with C=l to give:


6, = u : ~ ' ( A , ) k = O and r n = l + n

where A, denotes the centre point of the support m and dmk denotes the deflection at the support centre A, due to the loading scheme of the phase k.

2.2 Phase 2: the virtual work cases (&=Gm)

In this phase, all of the applied loads (whether on the boundary or inside the domain) is removed £tom the released plate. Hence a virtual unit load is placed instead of the unknown redundant reaction at the first support centre ( A l ) as shown in Fig. 3 (this virtual load case is represented by k=l). The plate can be solved using the standard BEM in Eq. l with k=l and the displacements at the internal support centres can be computed using Eq. 3 with k= l .

The former steps can be repeated when k2, by placing the virtual unit load at the second support centre (A2). Similar to the case when R - l , the plate can be solved using Eq. 1 and the relevant internal displacements at the internal support centres can be computed using Eq.3 for k2.

Again the former steps can be repeated for each unknown redundant reaction until the case for the support number n (see Fig. 4) is processed.

Fig. 3: The first virtual load case (R=]).


Fig. 4: The n' virtual load case (k=n).

2.3 Compatibility of displacements

Applying the compatibility of displacements at each location of the internal support centres (Ak), the following equation can be obtained:

Where [&,l is the flexibility matrix, {Rk) is the vector of the unknown redundant forces and {Sd) is the vector of the displacements at the internal supporting locations in the released plate due to the original external loading (phase 1).

2.4 Forced deformations

One of the advantages of the present formulation is it can account for forced deformations such as settlement at the location of internal supports also it allows taking into account the flexibility of internal supports. In this case, Eq. 5 can be modified as follows:


Boundary Element Technology XIV 245

Where, dm) denotes the settlement at the location of the support m, C& denotes the Kronecker delta symbol and dm) (K@""=~m)afm)/lfm), E denotes Young's modulus, a is the area and I is the length of the internal support) is the axial stifhess of the internal support m.

2.5 Final solution

After adding the appropriate forced deformation of internal supports, the former system of equations in Eq. 6 can be solved to get the redundant {Rk}. Hence, the final solution of the problem can be written as:

Where ( 0 ) can be the boundary or internal generalised displacement or traction. The superscript V) denotes quantities of the final solution.

2.6 Failure tracing

Another advantage of the present formulation is it allows tracing the plate structural behaviour clue to failure of one or more internal supports. This can be done easily by changing the value of K('") in Eq. 6 until completely removing the failed internal suppcn-t (by removing one row and one column of the flexibility matrix). It has to be noted that, the failure can be traced without re-meshing or re- analysis of the overall problem.

3 Numerical modelling

As it can be seen that the proposed formulation needs the solution of many cases for a certain problem. This makes it inefficient. However, by another inspection of the methodology of the proposed formulation, it can be seen easily that all phases and cases can be solved simultaneously according to the following strategy:

The corresponding matrix form for Eq. 1 can be written as follows:

Where [W and [q are the well-known BEM influence matrices [2], N and NE denote the number of boundary nodes and elements respectively. It is assumed that quadratic isoparametric boundary elements are used in the above equation. {U) and {t) are the vectors of boundary displacernents and tractions respectively.


Equation 7 represents the solution for phase 1. A similar set of equations (n equations) can be written for the solution of the virtual load cases in phase 2, for example, when the virtual unit load applied at the first location of internal support, the corresponding matrix form of the integral equation can be written as follows:

Similarly, if the virtual unit load applied at the nm location of internal support, The following equation can be written:

It can be seen that, both phase 1 and all of the virtual load cases in phase 2 can be solved simultaneously. This can be done easily by combining Eqs. 8, 9 and 10 together to give:

+ IF(') p . . . F ( n ) $ N x ( n + l )

It has to be noted that in Eq. 11 the original boundary and domain loading will be considered in the same analysis with the applied virtual unit loads. However, in computing the {p)} vector, each phase and state will be placed as separate column, i.e. the (p) vector now is a matrix of dimension 3Nx(n+l). It has to be noted that the first column in [p)] is the corespondent column for phase 1.

As it can be seen that, the solution can be camed out only one time. Then the required internal displacernents to set up the flexibility matrix can be computed in a similar way as in Eq. 3.

4 Stress resultants at internal points

After computing the final values of the boundary displacements and tractions, bending moment and shear force stress resultants can be evaluated using the standard BEM as follows:


in which the kernels U,@ and Tfi and its relevant derivatives are given in by Rashed [10]. It has to be noted that the last integral in Eq. 12 is interpreted in Cauchy principal value sense and special treatment is required to compute such integrals (see Ref.[lO] for details).

It is also worth noting that the rule in Eq. 7 is also valid and can be used to compute the internal stress resultants.

Fig. 5: The plate model.

Fig. 6: The corresponding beam model.


248 Bounda~:v Elernetlt Techtlology XIV

5 Applications

In this section the formulation presented in this paper is tested and the results are compared to those of analytical and finite element solutions. Curved quadratic isoparametric elements are used. The position of the nodes are left free. Constant internal cells are used to simulate the virtual unit load cases. The virtual pressure over cells can be computed as follows:

(Virtual Pressure) P,,, = 1 (Virtual unit load)

Area of the Cell

Regular integrals are evaluated using Gauss-Legendre scheme with 10 points for boundary element and 10x10 points for internal cells. Weakly singular integrals are evaluated using the non-linear cuordinate transformation proposed by Telles [ l I]. Cauchy principal value integrals in Eq. I me indirectly evaluated using the generalised rigid body displacements (see Ref.[2]).

5.1 Comparison to analytical solution of the beam theory

Consider the plate shown in Fig. 5. The plate is a square of side length 4m and fixed fiom one side and left fiee from the other three sides. The plate is internally supported by a line support parallel to the fixed side at an offset of 3m. A total load of 5ton is applied along its fiee edge. The plate has thickness of 0.2m. Poisson's ratio is taken to be zero to allow the comparison of the results to those obtained from the beam theory for the corresponding beam shown in Fig. 6. In order to analyse the present plate with the proposed BEM technique, the plate boundary is discretised using 16 quadratic boundary elements and the internal support is modelled using one internal cell of width O.lm (see Fig. 5) to avoid stress singularity arise from knife line loads.

Table 1 : Value of the reaction at the internal support for the beam model.

Table 1 shows a comparison between the analytical value and the present BEM solution value for the reaction at the internal support. The results for the deflection, bending moment and shearing forces are plotted together with the analytical results obtained from the beam theory in Figs. 7, 8 and 9 respectively. It can be seen that the results are in excellent agreement. It has to be noted that at the centre line of the internal support the BEM solution for the shear (see Fig. 9) gives average value (1.2%) of the shear before and after the support, as the support is modelled as small area of distributed pressure.

Analytical Present BEM

Reaction in the internal support (ton) 7.5000000 7.4941476


Bo~rtldary Elernerrr Technology XI\' 249

Fig. 7: Deflection (m) X E1 along the plate centre line.

1 -

X 0.0 1

Present BEM

Distance along the plate centre line

g -1.00'0 .;: -2.0 0 o -3.0 % -4.0

-5.0

Fig. 8: Bending moment (m.t.) along the plate centre line.

0.5 1.0 1.5 2. - Distance along the - - - - Analytical

4 0 - Present BEM

-4.0 L

Fig. 9: Shear force (t.) along the plate centre line.

k -t-----i

3 0 3 5 4 0

-Analytical W 3 0 0 4 0 1 . Present BEM & 2 0

1 0 8 0 0 C m -'O0.'0

-2 0

Distance alongt he plate center l$ - - +-- - + ---t- i - ---+-----

0 5 1 0 1 5 2 0 2 5

-3 0 'F - - - - - -


5.2 Comparison to analytical solutions of the classical plate theory

In this example, a simply supported circular plate with radius a is analysed. The analytical solution for such a problem based on the thin plate theory can be obtained using direct integration in the polar co-ordinate then applying suitable boundary conditions, to give:

M, = ~ k 3 + v)c , r2 + {C, + ( 1 + v)C4} + (1 + v)C3 h r ]

M, = D[(I + 3 v ) ~ , r + {C, (I + v ) + 6, } + ( 1 + v ) ~ , ln r ]

where:

in which R is the reactive force in the internal support. In order to analyse such a problem with the proposed boundary element technique, the full plate is discretised using 16 boundary element and 32 nodes. Poisson's ratio ; is taken to be 0.3 and E=1x106 t/m2. The plate radius a=4m and thickness is taken to be 0 . 0 5 ~ 0 . 2 m . The internal supported is modelled using unit load applied at the plate centre. Another BEM model is set up for the purpose of comparison. In this model only quarter of the plate is solved using the standard BEM (12 boundary elements and 26 boundary nodes). The symmetry boundary conditions along the symmetry lines (using the so-called guided boundary condition) are enforced. The corner point, which represents the plate centre, is modelled using continuous boundary


elements to ensure the continuity of the displacements at the plate centre. A fully restraint boundary conditions is used to represent the internal support at that corner node.

Table 2: Value of the reaction at the internal support for the circular plate example.

Table 2 shows the values of the internal support reactions for both BEM models. It can be seen that the flexibility model is more accurate even than the standard BEM quarter model.

Quarter* 19.8293 1.7495

R (ton) % Error

L X Present -2.5E-03

*~nalytical Value (Reference value) of R=20.1824 tons (computed using Eq.20).

Present* 19.9624 1 .090 1

Fig. 10: Deflection (m) for the circular plate.


252 Buurldar:v Elenlet~t Technology XIV

1 SE-03 - Analytical

a^ 1.0FA3 2

5.0Er04 X Present 0

.H z 0.0E+00 c.r

,.m, X 3 -1.OEr03 d

-1.5503

-2.0503

Fig. 1 1 : Radial rotation (rad.) for the circular plate.

Fig. 12: Radial bending moment (m.t) for the circular plate.

-2.5

-2 C:

E V -1.5

2 -1 cd 3 -0.5 .H 3 0 .

0.5

1

-

- - Analytical

- o Quarter

- X Present

-


- Analytical

o Quarter

x Present

Fig. 13: Radial bending moment (m.t.) for the circular plate.

- Analjtical

0 Quarter

X Present

Fig. 14: Shear force (t.) for the circular plate.

Figures 10 to 14 show comparisons between the results of the two BEM models and the analytical results. It can be seen that excellent agreements are demonstrated between the present BEM formulation and the results of the analytical solution.


6 Conclusions

In this paper a new method for analysing plates with intemal constraints has been presented. The new technique can be regarded as the extension of the well-known force method for analysing statically indeterminate structures. First the plate was released from intemal supports and solved using the BEM. The effect of the internal supports were taken into account using separate cases of virtual work. Hence the flexibility matrix is formed and the compatibility of displacement equations was solved for the unknown redundant reactions at internal supports. The following conclusions may be drown from the implementation and the application of the present technique:

l . The new technique can be used to analyse internally supported plates with few modifications introduced on the standard BEM codes.

2. The present technique is efficient, as it only require the discretisation of the plate boundary.

3. The accuracy of the proposed technique is high. 4. The proposed technique has a limitation, as it cannot model plate only

supported internally.

The present formulation can be extended to solve two- and three-dimensional elasticity problems with internal supports. Also, it could be used as an easy tool to find the optimum location of internal supports.

References

Zienkiewicz, O.C. (1977), "The Finite Element Method, 3rd ed"., McGraw-Hill, UK. Brebbia, C.A., and Dominguez, J. (1992), "Boundary Elements: An Introductory Course," CMP, McGraw-Hill, UK. Bezine, G. (1978), "Boundary integral formulation for plate flexure with arbitrary boundary conditions," Mech. Res. Comm., Vo1.5 No.4,pp. 197- 206. Stern, M. (1979), "A general boundary integral formulation for the n~unerical solution of plate bending problems," Int. J. Solids Structures, Vol. 15, pp. 769-782. Vander Weeen, F. (1982), "Application of the boundary integral equation method to Reissner's plate model," Int. J. Nurn. Methods Engineering, Vol. 18, pp. 1-10. Karniya, N. and Saito, K. (1986),'The boundary element method for elastostatics with internal constraints," Microsoftware for Engineers, Vo1.2 No.4, pp. 233-237. Providakis, C.P. and Toungelidis, G. (1998), "A D/BEM approach to the transient response analysis of elastoplastic plates with internal supports," Engineering computations, Vol. 5 No.4, pp. 50 1-5 1 1.


[8] Katsikadelis, J. T., Sapountzakis, E. J., Zorba, E. G. (1990), "A REM approach to static and dynamic analysis of plates with internal supports," Computational mechanics, Vo1.7 No. l , pp. 3 1-42.

[9] Reissner, E. (1947), "On bending of elastic plates, Quart. Applied Mathematics," Vol. 5, pp.55-68.

[l01 Rashed, Y.F. (2001), "Application of the Green first identity to transform domain integrals to the boundary for plate bending problems", to appear in the international conference for BEM 23, Greece.

[l l ] Telles, J.C.F. (1987), "A self-adaptive cu-ordinate transformation for efficient numerical evaluation of general boundary element integrals," Jnt. J. Num. Methods Engineering, Vo1.24, pp. 959-973.


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