Limit and shakedown analyses are powerful methods in designing pressure vessel components and other engineering structures. With the development of computer technology the use of finite element analysis as an alternative tool for engineering structure design has become ever more increasing. In this thesis the finite element method utilises the novel elastic compensation method to carry out the limit and shakedown analyses on variety engineering structures: nozzles in pressure vessels, plates with a central hole and frame structures. There are two main objectives of the present study. One of them is to conduct limit and shakedown analyses on a series of thick cylinders, nozzle/sphere intersections under internal pressure, plates with a hole and frame structures under multiple loading conditions using the initially developed elastic compensation method based on 2-D solid element models. A comparision of the lower, upper bound limit loads and shakedown solutions is made with the result s available in literature or with new elasto-plastic analyses. The results obtained using the elastic compensation method were found to be of useful accuracy. Another is to further develop the elastic compensation method using generalised yield criteria. Then the procedure is implemented to beam and shell finite elements to calculated limit loads for beam and shell structures. A number of 2-D and 3-D frames were examined using a general yield surface. The obtained results were compared with that of theoretical plastic analysis and with the results available in literature and were found to be in good agreement. Parametric studies of nozzle/sphere intersections and nozzle/cylinder intersections under internal pressure were carried out using Ilyushin's and Ivanov's generalised yield criteria. The results calculated were compared with the solutions obtained using the initially elastic compensation method and with the solutions available in literature and were found to also be in good agreement. The newly developed elastic compensation procedure using generalised yield criteria was found to be more economic and useful in engineering design. From this study, some new design methods based on limit and shakedown loads are proposed for nozzle/sphere intersections and for other engineering structures. The newly developed elastic compensation procedure using generalised yield criteria is highly recommended in structural design for a quick limit load estimation.