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.iPfi}

It will be found that where the loads are known, {Pa}, [i.e. prescribed nodal forces (and moments, in beam applications)], the corresponding displacements will be unknown, {/?„}, and where the displacements are known, {pp}, (i.e. prescribed nodal displacements), the forces, {Pp}, (and moments, in beam applications), usually the reactions, will be unknown.

9.12. Solution for displacements and reactions

A solution for the unknown nodal displacements, {pa}, is obtained from the upper partition of eqn. (9.96)

Rearranging

To obtain a solution for the unknown nodal displacements, {pa}, it is only necessary to invert the submatrix Pre-multiplying the above equation by [A-»«]-' (and using the matrix relation, [^„„¡-'[ioa] = [/], the unit matrix), will yield the values of the unknown nodal displacements as

If all the prescribed displacements are zero, i.e. {pß} — {0}, the above reduces to

The unknown reactions, {Pß}, can be found from the lower partition of eqn. (9.96)

Again, if all the prescribed displacements are zero, the above reduces to

Bibliography

1. R.T. Fenner, Finite Element Methods for Engineers, Macmillan Press, London, 1996.

2. E. Hinton and D.R.J. Owen, An Introduction to Finite Element Computations, Pineridge Press, Swansea, 1979.

3. R.K. Liversley, Finite Elements; An Introduction for Engineers, Cambridge University Press, 1983.

4. NAFEMS, Guidelines to Finite Element Practice, DTI, NEL, Glasgow, 1992.

5. NAFEMS, A Finite Element Primer, DTI, NEL, Glasgow, 1992.

6. S.S. Rao, The Finite Element Method in Engineering, Pergamon Press, Oxford, 1989.

7. J.N. Reddy, An Introduction to the Finite Element Method, McGraw-Hill, 1993.

8. K.C. Rockey, et al., The Finite Element Method; A Basic Introduction, Crosby Lockwood Staples, London, 1975.

9. R.L. Sack, Matrix Structural Analysis, PWS-Kent, 1989.

10. O.C. Zienkiewicz, The Finite Element Method, McGraw-Hill, London, 1988.

11. C.A. Brebbia and A J. Ferrante, Computational Methods for the Solution of Engineering Problems, Pentech Press, London, 1986.

12. HKS ABAQUS, User's Manual, 1995.

13. SDRC I-DEAS Finite Element Modelling, User's Guide, 1991.

14. K. Thomas, Effects of geometric distortion on the accuracy of plane quadratic isoparametric finite elements, Guidelines for finite element idealisation, Meeting Preprint 2504, ASCE, pp. 161-204, 1975.

Examples

Example 9.1

Figure 9.31 shows a planar steel support structure, all three members of which have the same axial stiffness, such that AE/L = 20 MN/m throughout. Using the displacement based finite element method and treating each member as a rod:

(a) assemble the necessary terms in the structural stiffness matrix;

(b) hence, determine, with respect to the global coordinates (i) the nodal displacements, and (ii) the reactions, showing the latter on a sketch of the structure and demonstrating that equilibrium is satisfied.

90 kN

90 kN

72 kN

72 kN

Solution

(a) Figure 9.32 shows suitable node, dof. and element labelling. Lack of symmetry prevents any advantage being taken to reduce the calculations. None of the members are redundant and hence the stiffness contributions of all three members need to be included. All three elements will have the same stiffness matrix scalar, i.e. (AE/L)(a) = (AE/Lfb) = (AE/L)(C) = AE/L

90 kN

90 kN

72 kN

72 kN

With reference to §9.7, the element stiffness matrix with respect to global coordinates is given by cos2 a sin a cos a

— sin a cos a sin2 a ■ sin a cos a — sin2 a symmetric cos2 a sin a cos a sin2 a J

Evaluating the stiffness matrix for each element:

Element a

(4/3),

cosa(lJ) =

-0.6,

sin<x(a)

= 0.8

" 0.36

0 0

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