Ham C. Chen” , Nasir I. Munir” and Jayanth N. Kudvaf Aircraft Division, Northrop Corporation

One Northrop Avenue, 3852/MF, Hawthorne, CA 91250-3277

Abstract

Special finite elements are developed using the hybrid displacement finite element method for efficient evaluation of stress concentration around a hole in complex structures. The special finite elements are designed to embody the stress concentration effects of a hole in the element interior and to simultaneously maintain the displacement compatibility with conventional finite elements along the element boundary. The complex variable formulation is used to derive a special set of trial functions which satisfy all the governing equations and the traction-free boundary conditions at the hole for the construction of the special finite elements. Several numerical examples are presented to show that the use of special finite elements to model critical regions around a hole, together with conventional finite elements to model other regions away from the hole, is not only very convenient but also highly accurate.

“X’ “y’ oxy QB QF

displacement matrix displacement vector displacement components rectangular coordinates complex variable analytic trial functions special finite element boundary conventional finite element boundary hole boundary displacement boundary traction boundary element boundary material constant shear modulus Poisson’s ratio hybrid variational functional stress vector stresses special finite element region conventional finite element region

omenclature Introductioo

a j , bj b E

i

n n

r S T t

X’ Y

tx, ‘y

complex coefficients unlaown parameters vector Young’s modulus finite element matrices imaginary number element stiffness matrix shape functions matrix unit normals matrix components of an unit normal vector nodal displacements radius of hole stress matrix traction matrix traction vector components of traction vector

* Engineering Specialist, Member j. Manager

Copyright 0 1993 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved

Holes or cutouts are often required in many practical structures for access, services, and other reasons. Although the finite element method is predominantly used to analyze most real-world structures, the use of conventional finite elements to analyze structures with holes is still not too practical. This is because, to account for the stress concentration and redistribution caused by the presence of holes, refined meshes which include several transitions of elements and exceptionally small elements around the holes are usually required to obtain decent solutions. Therefore, a more convenient and efficient approach is desired especially in a production environment, where design modifications and changes are frequently made.

To reduce the time-consuming and tedious efforts involved in constructing refined meshes, one can resort to the local-global analysis approach, as described in reference 1. In this approach, a global analysis is first performed where a relatively coarse mesh is used throughout the domain. After that, more detailed stress

7 86

distributions around a hole are obtained by a separate local analysis which involves only a small region around the hole modeled by a much refined mesh. The load distribution for the local region is obtained from interpolation of the global solution. Although an improved solution can often be achieved with the local analysis, this approach has some limitations in the solution accuracy. The reason is that the interaction between the local region and other regions is only crudely accounted for during the global analysis. Besides, the load distribution for the local region is obtained from the already approximate global solution by interpolation, thereby introducing further approximation in the final solution.

In this paper, a more accurate and convenient approach is described for analyzing structures with holes. In this approach, specially formulated finite elements are used to model the region around a hole while conventional finite elements are still used for other regions away from the hole. Here, the special finite elements are developed by taking into account not only the stress concentration characteristics of the hole but also the displacement compatibility with adjacent conventional finite elements. Therefore, a special element with only a few finite element nodes can be used to accurately represent a local region with a hole, and, more importantly, this special element can be directly tied to adjoining conventional finite elements representing other regions without introducing any discontinuity or incompatibility in the model. In contrast to the local- global analysis, the present approach includes the local stress concentration effects directly in the global model, and no additional refined local analysis is needed. Thus, the present approach is more accurate and efficient than the local-global analysis approach.

The use of specially formulated finite elements to model critical regions where stress concentration or stress singularity is of a concern has been reported by several researchers [2-51. However, this approach has not been widely used possibly due to the complexity involved in the formulation. In the following, we first summarize the basic formulations of the hybrid displacement finite element method [6], which is to be used to construct the special finite elements. Later we describe how to use the complex variable formulation developed by Muskhelishvili [7] to find a special set of trial functions which reflect the local stress concentration characteristics for the construction of the special finite elements. Finally, example problems are analyzed and the numerical results are presented to demonstrate the effectiveness and practical usefulness of the special finite elements developed herein.

A special finite element which embodies local stress concentration effects of a hole and maintains the displacement compatibility with adjacent conventional finite elements can be constructed with the hybrid displacement finite element method [6]. There are two major reasons for choosing the hybrid displacement finite element method. First, the prescribed boundary conditions on the region under consideration are satisfied overall in a variational (energy minimization) sense in the hybrid displacement finite element method while they are only satisfied at selective points in other methods, such as boundary collocation or least square. As a result, the hybrid method generally tends to yield more accurate results than the other methods. Second and more importantly, unlike the boundary element methods, the hybrid displacement finite element method results in a symmetric stiffness matrix. Therefore, the resulting special elements can be used as a superelement and naturally connected to conventional finite elements for analyzing complex structures with material and/or geometric inhomogeneities. In the following, the essential formulations of the hybrid displacement finite element method applicable to the present problem are briefly summarized.

Consider Figure 1 in which a field of conventional finite elements surrounds a region R, , in which a special finite element with a hole is to be prescribed. In the region RB the stress field s and displacement field u are defined in terms of a finite set of parameters b. That is,

787

(1)

(2)

where U and S contain specially derived functions which satisfy all the governing equations of equilibrium and compatibility. The traction field t can be expressed in terms of the stresses field O by

is a matrix given by

b = T b (3)

in which (n to the element boundary.

n ] are the components of the unit normal X' Y

Along the external boundary TB, where the special element connects to the conventional finite elements, the displacement field fi is defined in terms of the nodal displacements given by the parameters q. That is,

ii = L q ( 5 )

where L is an interpolation function matrix defined only along the external boundary, and should be chosen so that

is the same for two elements over their common boundaries, In the present case, the matrix L is made up of conventional displacement shape functions so that compatibility is maintained between the special finite element and its adjacent conventional finite elements.

With all the components defined as above, the hybrid functional for the region R, can be denoted by

dT - 1/2 &dSZ (6) QB

where (3, E represents the stress, strain in the region RB and t is the traction on the boundary TB . Note the first term involves the coupling of the special finite element with its adjacent conventional finite elements.

As the assumed fields in the region a, satisfy all the governing equations the volume integral in equation (6) can be replaced by a boundary integral as

(7)

On substitution of discretized expressions (l), (3) and ( 5 ) into equation (6), one obtains the following equation:

Since the b can be independently assumed from the nodal displacements vector q, one can set

to obtain

or

in which

dT G = I,,

Substituting equation (11) into equation (8) to eliminate the b and letting

one obtains the symmetric stiffness matrix

for the region R, . The special finite element constructed this way can be conveniently used to join any other variationally compatible elements, such as conventional finite elements or special elements themselves.

Note that the integration in the formulation of the element stiffness matrix are performed along the element boundary rB instead of on the element domain a,, as in the boundary element method. For this reason, the hybrid displacement method can be categorized as a boundary solution method, and the special element thus

78 8

constructed may be referred to as boundary solution elements 181.

erivation of Trial

As mentioned earlier, an essential step in the construction of a special finite element for a hole region is to find a special set of trial functions which satisfy all the governing equations and reflect the local stress concentration characteristics. To achieve this, the complex variable formulation of Muskhelishvili [7] is used herein. In this formulation, the solution to a plane stress problem amounts to determining two complex analytic functions $ (z) and y (z) such that the prescribed boundary conditions are satisfied. The variable z is given by

z = x + i y (15)

where i is the imaginary number. In terms of these two analytic functions @(z) and y(z), one can express the stresses and displacements as follows

cry+ 8, = 2 [ @'(z) + 7' (z) ] (16)

by - 8, + i 2 oxy = 2 [z$"(z) + y'(z)] (17)

in which p = EY2(1+U); K = (3- U)/(l+ U); E and U are, respectively, Young's modulus and Poisson's ratio; ()' denotes differentiation; and (> is the complex conjugate. The boundary conditions are given by

-

with Tu representing the boundary where displacements are prescribed, and Tt representing the boundary where tractions are prescribed.

To facilitate the treatment of the boundary conditions on the hole, the following conformal mapping is performed so that the hole boundary is mapped to an unit circle,

where r is the radius of the circular hole. In terms of this new variable 5, the expressions for the stresses and displacements are as follows

The boundary conditions are

In general, it is impossible to find a closed form formulation for @( 5 ) and y( 5 ) for arbitrary geometry and boundary conditions. A commonly used method for obtaining approximate solutions is to consider the finite series expansion of the functions. That is, it is usually assumed that

and the unknown complex coefficients a j and bj are determined by imposing the boundary conditions which are to be satisfied only approximately in general.

However, in order to include the stress concentration effects of a hole in the solution a priori, we impose the condition that the selected functions, $( 5 ) and y( 5 ), exactly satisfy the traction-free condition on the hole. This can be achieved in the following way. Using the expression given by (26) we can obtain

From this and the fact that 5 = e-' on the hole boundary, we can have

7 89

Expressions in equation (29) are used to derive the displacement and stress fields from equations (22) through (24). Since the boundary conditions on the hole are already satisfied by the two functions, discretization of the hole boundary is not needed. In addition, fewer terms in the series expansion will be required to achieve solution convergence than using the two independent stress functions as given in equation (27), and the resulting stress concentration factors will also be much more accurate.

To evaluate the performance of the special elements, three circular hole elements CHOLE4, CHOLESL, and CHOLESQ are constructed, as shown in Figure 2. The four-noded element CHOLE4 and the eight-noded element CHOLESL use a linear displacement field along the element sides, so they are to be used with linear finite elements, such as the standard three-noded triangular and four-noded quadrilateral elements, The eight-noded element CHOLE8Q uses a quadratic displacement field along the element sides, so it is to be used with quadratic finite elements, such as the standard six-noded triangular and eight-noded quadrilateral elements.

In the present development, N = M = 2 is used for CHOLE4 and N = M = 4 is used for CHOLESL and CHOLESQ, where M and N are the number of positive and negative subscript terms in equation (29). Note that the a0 and Im[ a l l terms correspond to rigid body modes and contribute no stress; therefore, they are discarded in the hybrid displacement finite element formulation. That is, there are 7 terms in the unknown coefficients vector b for CHOLE4, and 15 terms for CHOLE8L and CHOLE8Q.

The special finite elements are used to solve several numerical examples in which stress concentration is present. Besides the special finite elements with a hole as described above, conventional displacement finite elements are also used to model the structure. In the following, unless stated otherwise, the values E = 1, 2) = 0.3, and thickness = 1 are used.

A plate with a small circular hole under normal tension is an ideal example to verify the accuracy of the special elements since the exact elasticity solution is known for the classical problem of an infinite plate with a circular hole [9, 101. The plate considered has a dimension of 15 by 15. The hole is at the center of the plate and has a radius of 0.25. Uniform tension of unit value is applied on two opposite sides of the plate. The plate is modeled with a single special finite element for the hole region and a number of conventional finite elements for the remaining region, as shown in Figure 3.

el 1

The computed stresses at location A and B are presented in Table 1 . It is seen that the use of the special elements and a small number of conventional finite elements yields solutions in excellent agreement with the analytical solution from the theory of elasticity.

790

Plate with Two Circular Holes A plate with two circular holes under biaxial

tension is considered to examine the coupling between a special finite element with itself besides other conventional finite elements. The plate has a dimension of 100 by 100. The distance between the two holes is 20 and the radius of the holes is 5. Uniform tension of unit value is applied on all sides of the plate. Two special finite elements, together with some conventional finite elements, are used to model the plate; the models are shown in Figure 4.

with el 1

ode1 3

ig. Plate with Two Holes and

The computed stresses at locations A, B and C are presented in Table 2. It is seen from Table 2 that the solutions given by the three different finite element models are all very close to the reference solution [3]. If only conventional finite elements are used to analyze this problem, then exceptionally fine mesh would be required to accurately model the interaction between the holes and the solution obtained would also be much more sensitive to mesh changes.

Table 2 Stresses in Plate with Two Holes

Cantilever Beam with a Circular Hole The third example used to examine the

performance of the special elements is a cantilever beam with a circular hole under three loading conditions: extension (P), shear (Q), and moment (M), all of unit value and applied at the free end. The cantilever beam has a length of 3.6 and a width of 1. The hole is at the center of the beam and has a diameter of 0.1. Three models of the plate using a single special finite element and some conventional finite elements are shown in Figure 5.

The stress at the designated location A is reported in Table 3. It is seen that the finite element solutions obtained are in good agreement with the reference solutions [lo] for the extension and moment cases. There is no reference solution for the shear case.

Table 3 Stress at Location A of the Cantilever Beam with a Hole under Different Loading Conditions

791

ntil

A thin plate with a central circular hole subjected to uniaxial uniform shortening is used to compare the effectiveness and efficiency of the special finite elements and conventional finite elements. The plate is 20 in. long, 10 in. wide, and 0.1 in. thick. The diameter of the hole is 2 in. and material properties are: E = 10.0 x lo6 psi and

2) = 0.3. The CSM Testbed finite element software system [ll] is used to calculate the stress concentration factor (SCF), and a typical finite element mesh is shown in Figure 6. The results obtained with the ES4EX43 assumed-stress hybrid element of the CSM Testbed is listed in Table 4.

Table 4 Stress Concentration Factor

/

with Conventional

It is seen that a highly refined mesh with more than five hundred degrees of freedom is needed to achieve an accurate solution. In contrast, the use of a special finite element together with a small number of conventional finite elements is sufficient to give a satisfactory solution, as shown in Figure 7 and Table 5.

792

el I

el II

el 111

el IV

pecial Finite Element and Conventional Finite Elements

Concluding Remarks

problems with complex geometry and material inhomogeneities, where the analytical solutions are very difficult or impossible.

eferences

1. 3. B. Ransom and N. F. Knight, Jr., 'Global/local Stress Analysis of Composite Panels', NASA Technical Memorandum 101622,1989.

2. P. Tong, T. H. H. Pian, and S. Larsy, 'A hybrid element approach to crack problems in plane elasticity', Int. J. Num. Methods Eng., 7, pp. 297-308, 1973.

3. R. Piltner, 'Special finite elements with holes and internal cracks', Int. J. Num. Methods Eng., 21, pp.

4. J. Jirousek, 'Implementation of local effects into conventional and non-conventional finite element formulations', in P. Ladeveze (ed.), Local effects in the Analysis of Structures, Elsevier, New York, pp.

5. J. Zhao and H. Shan, 'Stress analysis around holes in orthotropic plates by the subregion mixed finite element method', Computers and Structures, 41, pp.

6. P. Tong, 'New displacement hybrid finite element models for solid continua', Int. J. Nunz. Methods Eng.,

7. N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Nordhoff, Groningen Holland, 1953.

8. 0. C. Zienkiewicz, The Finite Element Method, 3rd Edition, McGraw-Hill, London, 1977.

9. S. Timoshenko and J. N. Goodier, Theory of Elasticity, 2nd Edition, McGraw-Hill, New York, 1951.

10. R. E. Peterson, Stress Concentration Factors, John Wiley & Sons, New York, 1974.

11. C . B. Stewart, Compiler: The Computational Structural Mechanics Testbed User's Manual, NASA

1471-1485, 1985.

279-298, 1985.

105-108 1991.

, pp. 73-83, 1970.

Th4-100644,1989.

It is shown by numerical examples that the use of special finite elements to analyze structures with holes is both convenient and effective. Therefore, the developed special finite elements can be routinely used to solve

79 3