Summary

The displacement method of finite element stress analysis is a powerful tool for the solution of complex boundary value problems. If the element properties are derived in accordance with energy principles (a consistent formulation), then the numerical solution converges monotonically to the exact solution as the element size decreases.

The displacement method of finite element stress analysis can be used to calculate the linear and nonlinear response of structures subjected to static and dynamic loads. Computer codes featuring the full range of capabilities of the method are readily available.

The method is summarized below for the limiting case of linear elastic (small displacements and small displacement gradients) response of structures subjected to static loads. Details of the complete range of application of the method can be found in numerous technical references including Reference 1 through Reference 4.

Static Analysis

The governing equations for the linear elastic response of an idealized structure subjected to static loads are:

(1)

where		=	the system stiffness matrix;
		=	the prescribed system nodal load vector; and
		=	the developed system nodal displacement vector.

The system stiffness matrix is obtained by assembling the element stiffness matrices consistent with the compatibility requirements between the system or global displacements and the element or local displacements.

The state of stress of the idealized structure is obtained by compiling the results for the states of stress of the individual elements. The state of stress of an individual element is obtained by solving Equation 1 for the system displacements, identifying the element displacements, solving for the element strains, and then solving for the element stresses.

The compatibility equations between the system nodal displacements and the element nodal displacements are:

(2)

where		=	the compatibility matrix for element i; and
		=	the nodal displacement vector for element i.

The results for the strains are:

(3)

where		=	the strain-displacement matrix for element i; and
		=	the strain vector for element i.

The results for the stresses are:

(4)

where		=	the elastic constants matrix; and
		=	the stress vector for element i.

Technical References

1. Rubinstein, M.F., Matrix Computer Analysis of Structures, Prentice-Hall, Inc., 1966.
2. Bathe, K.J., and Wilson, E.L., Numerical Methods in Finite Element Analysis, Prentice-Hall, Inc., 1976.
3. Zienkiewicz, O.C., The Finite Element Method, Third Edition, McGraw-Hill Book Company (UK) Limited, 1977.
4. Cook, R.D., Malkus, D.S., and Plesha, M.E., Concepts and Applications of Finite Element Analysis, Third Edition, John Wiley & Sons, 1989.