Energy principles in structural mechanics express the relationships between
stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
es,
strains or
deformation
Deformation can refer to:
* Deformation (engineering), changes in an object's shape or form due to the application of a force or forces.
** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies.
* Defor ...
s,
displacements, material properties, and
external effects 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, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and ...
. 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
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
.
General principles
*
Virtual work principle
**Principle of virtual displacements
**Principle of virtual forces
***
Unit dummy force method
* Modified variational principles
Elastic systems
*
Minimum total potential energy principle
* Principle of stationary total complementary potential energy
*
Castigliano Castigliano is a surname. Notable persons with that name include:
* Carlo Alberto Castigliano (1847–1884), Italian mathematician and physicist
* Caroline Castigliano (born c.1960), British fashion designer
* Eusebio Castigliano (1921–1949), ...
's first theorem (for forces)
Linear elastic systems
*
Castigliano's second theorem (for displacements)
*
Betti's reciprocal theorem
*
Müller-Breslau's principle
The Müller-Breslau principle is a method to determine influence lines. The principle states that the influence lines of an action (force or moment) assumes the scaled form of the deflection displacement.
OR,
This principle states that "ordinate o ...
Applications
* Governing equations by variational principles
* Approximate solution methods
*
Finite element method in structural mechanics The finite element method (FEM) 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 b ...
Bibliography
*Charlton, T.M.; ''Energy Principles in Theory of Structures'', Oxford University Press, 1973.
*Dym, C. L. and I. H. Shames; ''Solid Mechanics: A Variational Approach'', McGraw-Hill, 1973.
*Hu, H. ''Variational Principles of Theory of Elasticity With Applications''; Taylor & Francis, 1984.
*Langhaar, H. L.; ''Energy Methods in Applied Mechanics'', Krieger, 1989.
*Moiseiwitsch, B. L.; ''Variational Principles'', John Wiley and Sons, 1966.
*Mura, T.; ''Variational Methods in Mechanics'', Oxford University Press, 1992.
*
Reddy, J.N.; ''Energy Principles and Variational Methods in Applied Mechanics'', John Wiley, 2002.
*Shames, I. H. and Dym, C. L.; ''Energy and Finite Element Methods in Structural Mechanics'', Taylor & Francis, 1995,
*Tauchert, T.R.; ''Energy Principles in Structural Mechanics'', McGraw-Hill, 1974.
*Washizu, K.; ''Variational Methods in Elasticity and Plasticity'', Pergamon Pr, 1982.
*Wunderlich, W.; ''Mechanics of Structures: Variational and Computational Methods'', CRC, 2002. {{ISBN, 0-8493-0700-7
Structural analysis
Calculus of variations