Methods of analysis of steel structures

1. INTRODUCTION

Checking the strength of cross sections, the stability of structural members or section components, and possibly fatigue requires that the internal force distribution within the structure is known beforehand. From this, the stress distribution within any cross section may be deduced as required. The words 'internal forces' (also termed 'member forces') are used generally and refer to axial forces, shear forces, bending moments, torque moments, etc.

The internal forces in a statically determinate structure can be obtained using statics only. In a statically indeterminate structure, they cannot be found from the equations of static equilibrium alone; a knowledge of some geometric conditions under load is additionally required. It is important, at this stage, to recognise this fundamental difference between statically determinate and indeterminate (hyperstatic) structures. The internal forces in a structure may be determined using either an elastic or a plastic global analysis. While elastic global analysis may be used in all cases, plastic global analysis can be used only where both the member cross sections and the steel material satisfy special requirements.

The internal forces may be determined using different approaches depending on whether the effects of the deformations in the structure can or cannot be disregarded. In first order theory, the computations are carried out by referring only to the initial geometry of the structure; in this case, the deformations are so small that the resulting displacements do not significantly affect the geometry of the structure, and, hence, do not significantly change the forces in the members. Second order theory takes into account the influence of the deformation of the structure and, therefore, reference must be made to the deflected geometry under load. First order theory may, for instance, be used for the global analysis in cases where the structure is appropriately braced, is prevented from sway, or when the design methods make indirect allowances for second-order effects. Second order theory may be used for the global analysis in all cases without any restrictions.

When first order theory can be used, the behaviour of a structure made with a material obeying Hooke's law is itself linear; the displacements - translation or rotation of any section - vary linearly with the applied forces. That is, any increment in displacement is proportional to the force causing it. Under such conditions, stresses, strains, member forces, and displacements due to different actions can be added using the principle of superposition. This principle indeed states that the displacements (internal forces) due to a number of loads acting simultaneously is equal to the sum of the displacements (internal forces) due to each load acting separately. This does not apply if the stress-strain relationship of the material is nonlinear or if the structure (even if it is made of a material obeying Hooke's law) behaves non-linearly because of changes in the geometry caused by the applied loads.

The principle of superposition, when it can be used, is especially useful when determining the most severe condition in each individual member of a statically indeterminate structure; the interaction between different parts of the structure makes it difficult to identify the exact loading which produces the critical condition for design.

In practice, elastic global analysis is generally used to study the serviceability performance of a structure, i.e. limit states beyond which specified service criteria are no longer met. Plastic global analysis is particularly useful when investigating states associated with an actual collapse of the structure and to assess the actual ultimate resistance, i.e. ultimate limit states.