Energy principles in structural mechanics

Encyclopedia

**Energy principles in structural mechanics**express the relationships between stress

Stress (physics)

In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

es, strains

Strain (materials science)

In continuum mechanics, the infinitesimal strain theory, sometimes called small deformation theory, small displacement theory, or small displacement-gradient theory, deals with infinitesimal deformations of a continuum body...

or deformation

Deformation (mechanics)

Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particles of the body...

s, displacement

Displacement (vector)

A displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P...

s, material properties, and external effects

Structural load

Structural loads or actions are forces, deformations or accelerations applied to a structure or its components.Loads cause stresses, deformations and displacements in structures. Assessment of their effects is carried out by the methods of structural analysis...

in the form of energy or work done by internal and external forces. Since energy is a scalar quantity, these relationships provide convenient and alternative means for formulating the governing equations of deformable bodies in solid mechanics

Solid mechanics

Solid mechanics is the branch of mechanics, physics, and mathematics that concerns the behavior of solid matter under external actions . It is part of a broader study known as continuum mechanics. One of the most common practical applications of solid mechanics is the Euler-Bernoulli beam equation...

. They can also be used for obtaining approximate solutions of fairly complex systems, bypassing the difficult task of solving the set of governing partial differential equations.

## General principles

- Virtual workVirtual workVirtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. Historically, virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies, but they have also been developed for the...

principle- Principle of virtual displacements
- Principle of virtual forces
- Unit dummy force methodUnit dummy force methodThe Unit dummy force method provides a convenient means for computing displacements in structural systems. It is applicable for both linear and non-linear material behaviours as well as for systems subject to environmental effects, and hence more general than Castigliano's second theorem.-Discrete...

- Unit dummy force method

- Modified variational principles

## Elastic systems

- Minimum total potential energy principleMinimum total potential energy principleThe principle of minimum total potential energy is a fundamental concept used in physics, chemistry, biology, and engineering. It asserts that a structure or body shall deform or displace to a position that minimizes the total potential energy, with the lost potential energy being dissipated as heat...
- Principle of stationary total complementary potential energy
- CastiglianoCarlo Alberto CastiglianoCarlo Alberto Castigliano was an Italian mathematician and physicist known for Castigliano's method for determining displacements in a linear-elastic system based on the partial derivatives of strain energy....

's first theorem (for forces)

## Linear elastic systems

- Castigliano's second theoremCastigliano's methodCastigliano's method, named for Carlo Alberto Castigliano, is a method for determining the displacements of a linear-elastic system based on the partial derivatives of the strain energy...

(for displacements) - Betti's reciprocal theorem
- Müller-Breslau principleMüller-Breslau principleThe Müller-Breslau principle is a method to determine influence lines. The principle states that the influence lines of an action assumes the scaled form of the deflection that the structure displays after removing the restraint on the point where the action is evaluated and applying a point that...

## Applications

- Governing equations by variational principles
- Approximate solution methods
- Finite element method in structural mechanicsFinite element method in structural mechanicsThe Finite element method is a powerful technique originally developed for numerical solution of complex problems in structural mechanics, and it remains the method of choice for complex systems. In the FEM, the structural system is modeled by a set of appropriate finite elements interconnected at...