Types of Structural Analysis

1 Classification According to the Effects of Time Variation

Structural analysis can be classified according to the variation of actions and results with time as follows:

1. Static analysis: actions are assumed not to vary with time (static loading), and inertial and damping effects are neglected. Because the actions are constant with time, the responses are also constant. The displacement of any point in the structure is therefore constant, and its velocity and acceleration are zero;

2. Dynamic analysis: actions are assumed to vary with time (dynamic loading). Because the actions vary with time (for example, those arising from earthquakes, wind, vehicle traffic, or explosions near the structure), the responses also vary. The displacement of a point in the structure is not constant with time, and its velocity and acceleration are therefore nonzero. In this type of analysis, inertial effects must be considered. Realistic analyses must also consider energy dissipation (damping). Even after a time-varying action ceases, the structure may continue to vibrate;

3. Pseudo-static or quasi-static analysis: actions are assumed to vary very slowly with time (quasi-static loading). Inertial and damping effects can therefore be neglected, and a series of static analyses is performed, one for each loading configuration. This approach is widely used in bridge studies, where a vehicle is assumed to move very slowly across the structure and a static analysis is performed for each vehicle position. Because the actions vary with time, the responses also vary. The displacement of a point changes with time, but so slowly that its velocity and acceleration can be neglected. This approach underlies the static use of influence lines, but does not, by itself, represent the dynamic effects of vehicle–structure interaction.

Another analysis of great importance in the study of structural vibrations is modal analysis. Its purpose is to determine the natural frequencies and free-vibration modes of a linearized structural model. It is not, in itself, a dynamic response analysis: it does not independently provide the response to a time-varying load, but characterizes vibratory behavior and can form the basis of dynamic analysis procedures.

2 Classification According to Linearity

Structural analysis can be classified according to linearity (proportionality) between actions and analysis results as follows:

1. Linear analysis: proportionality is assumed between cause (external actions) and effect (structural responses), allowing the principle of superposition to be applied. Two requirements are important for this proportionality. First, the structural material must exhibit linear elastic behavior: stresses are assumed proportional to strains (roughly speaking, the stress–strain diagram must be a straight line). Second, structural strains and displacements must be small (infinitesimal), so that equilibrium equations can be written using the initial, undeformed geometry;

2. Nonlinear analysis: there is no proportionality between cause (external actions) and effect (structural responses). This lack of proportionality has two main origins. The first is nonlinear material behavior, with a curved or piecewise-linear stress–strain diagram, known as material or physical nonlinearity. The second is the occurrence of large strains and/or displacements, known as geometric nonlinearity. Geometric effects can be significant even when material strains are small. Another source of nonlinearity is a change in contact, support, or connection conditions, such as gaps opening or supports lifting off (for example, a support may resist compression but not tension because contact is lost).

The effects of time-varying actions and the linearity of the action–response relationships can be considered independently: a structural analysis may be static linear, static nonlinear, dynamic linear, or dynamic nonlinear.

It is also important to note that fully accounting for local or global buckling requires a nonlinear analysis or, at least, a critical buckling-load analysis.

A critical buckling-load analysis determines a factor corresponding to the maximum multiplier of the applied loading at which buckling occurs. It also identifies possible buckling modes: the possible deformed configurations (shapes) that the structure can assume when buckling instability occurs.

Geometric imperfections and residual stresses can alter the response and require more comprehensive analyses. The assessment of local buckling also depends on whether the model represents the relevant local modes, which is not always possible with member elements.

3 Methods of Linear Static Analysis of Reticulated Structures

There is a very wide variety of structural analysis methods, particularly when considering the different types of structures (framed, surface, solid, and mixed) and analyses (static, dynamic, modal, critical-load, and others).

In broad terms, quantities of interest in structural analysis can be classified into two groups:

1. Kinematic variables – this group describes structural motion and includes displacements and their time derivatives (velocities and accelerations), rotations, curvatures, elongations, strains, and shear distortions. In broad terms, the name “displacements” can be used to represent this group;

2. Static variables – this group describes the forces in the structure and includes forces, internal forces and moments, stresses, and tractions. In broad terms, the name “forces” can be used to represent this group.

Structural analysis combines three sets of relationships: equilibrium equations (relationships between static variables only), displacement compatibility conditions (relationships between kinematic variables only), and material constitutive relationships (generally relating static variables, such as stresses, to kinematic variables, such as strains). Structural analysis methods differ in the choice of primary unknowns and in how these relationships are organized.

Because there are two groups of variables in the analysis of reticulated structures, structural analysis methods can fundamentally be classified into two types:

1. Force method – adopts redundant forces or moments as the primary unknowns. It starts from a statically determinate reference system and imposes compatibility conditions to restore the constraints of the original structure;

2. Displacement method – adopts independent nodal displacements and rotations as the primary unknowns. It expresses internal forces and moments in terms of these unknowns and imposes equilibrium at the free degrees of freedom.

This distinction does not cover every possible formulation, but provides a foundation for understanding classical procedures and their computational implementation.