STATIONARY ENERGY PRINCIPLE

It is a fact for linear systems that the internal strain energy must equal the external work performed on the structure. For a single degree-of-freedom system, we can use this principle to solve for the displacement. However, for a multi degree-of-freedom system, a different approach is required. The energy plots, shown in Figure, illustrate that a new energy function  can be defined.

 

It is apparent that the solution at the point of minimum potential energy is where the internal energy equals the external energy. Therefore, the major advantage of the use of the potential energy function is that the solution must satisfy the following equation for all displacement degrees-of-freedom  :

The energy function written in matrix form is:

The resultant load vector R associated with the four types of loading is:

Therefore, the node equilibrium equation for all types of structural systems can be written as the following matrix equation:

 

The only approximation involved in the development of this equation is the assumption of the displacement patterns within each element. If the same displacement approximation is used to calculate the kinetic energy, the resulting mass matrix is termed a consistent mass matrix.

Another important fact concerning compatible displacement-based finite elements is that they converge from below, to the exact solution, as the mesh is refined. Therefore, the displacements and stresses tend to be lower than the exact values. From a practical structural engineering viewpoint, this can produce very dangerous results. To minimize this problem, the structural engineer must check statics and conduct parameter studies using different meshes.

Reference

Three-Dimensional Static and Dynamic Analysis of Structures A Physical Approach With Emphasis on Earthquake Engineering Edward L. Wilson

EXTERNAL WORK

The external work  performed by a system of concentrated node, or joint, loads  is:

Within each element " i ", the external work  performed by the body forces because of gravitational loads is:

Application of the displacement assumptions given by Equation (1),

                                (1)

N The shape functions used to calculate the strain energy within the element

Integration over the volume of the element, and summation over all elements produces the following equation for the energy because of body forces:

The external work  performed because of element surface stresses (tractions) , for a typical surface "j" is of the form:

Application of the displacement assumptions given by Equation (1), integration over the surface of the element, and summation over all surface elements produces the following equation for the energy because of surface tractions:

Therefore, the total external work performed on any system of structural elements is:

Reference

Three-Dimensional Static and Dynamic Analysis of Structures A Physical Approach With Emphasis on Earthquake Engineering Edward L. Wilson

STRAIN ENERGY

The strain energy stored in an element (i) within a general structural system is the area under the stress-strain diagram integrated over the volume of the element. For linear systems, the stress-strain matrix   , including initial thermal stresses   , can be written in matrix form as:

The column matrices  and  are the stresses and strain respectively.

Therefore, the strain energy within one element is given by:

Within each element, an approximation can be made on the displacements. Or:

N The shape functions used to calculate the strain energy within the element

Hence, after the application of the strain-displacement equations, the element strains can be expressed in terms of nodal displacements. Or:

The column matrix  contains all of the node, or joint, displacements of the complete structural system.

The element deformation-displacement transformation matrix  , is a function of the geometry of the structure.

After integration over the volume of the element, the strain energy, in terms of the global node displacements, can be written as:

Therefore, the element stiffness matrix is by definition:

And the element thermal force matrix is:

The total internal strain energy is the sum of the element strain energies. Or:

The global stiffness matrix  is the sum of the element stiffness matrices   .

Or:

The summation of element stiffness matrices to form the global stiffness matrix

is termed the direct stiffness method. The global thermal load vector  is the

sum of the element thermal load matrices:

Reference

Three-Dimensional Static and Dynamic Analysis of Structures A Physical Approach With Emphasis on Earthquake Engineering Edward L. Wilson

POTENTIAL ENERGY AND KINETIC ENERGY

One of the most fundamental forms of energy is the position of a mass within a gravitational field near the earth's surface. The gravitational potential energy  is defined as the constant weight  moved against a constant gravitational field of distance  . Or:

A mass that is moving with velocity  has kinetic energy given by the following equation:

One of the most common examples that illustrates the physical significance of both the potential and kinetic energy is the behavior of a pendulum.

If the mass of the pendulum has an initial position of  , the kinetic energy is zero and the potential energy is  . When  equals zero, the potential energy is zero; therefore, from conservation of energy, the kinetic energy is:

Reference

Three-Dimensional Static and Dynamic Analysis of Structures A Physical Approach With Emphasis on Earthquake Engineering Edward L. Wilson

VIRTUAL AND REAL WORK

The principles of virtual work are very simple and are clear statements of conservation of energy. The principles apply to structures that are in equilibrium in a real displaced position  when subjected to loading  . The corresponding real internal deformations and internal forces are  and  respectively.

The principle of virtual forces states if a set of infinitesimal external forces,  , in equilibrium with a set of infinitesimal internal forces  that exist before the application of the real loads and displacements, the external virtual work is equal to the internal virtual work.

If only one joint displacement  is to be calculated, only one external virtual load exists,  . For this case, the equation is the same as the unit load method.

It is apparent for nonlinear analysis that the principle of virtual forces cannot be used, because the linear relationship between  and  may not hold after the application of the real loads and displacements.

The principle of virtual displacements states if a set of infinitesimal external displacements, , consistent with a set of internal virtual displacements,  , and boundary conditions are applied after the application of the real loads and displacements, the external virtual work is equal to the internal virtual work.

It is important to note that the principle of virtual displacements does apply to the solution of nonlinear systems because the virtual displacements are applied to real forces in the deformed structure.

In the case of finite element analysis of continuous solids, the virtual work principles are applied at the level of stresses and strains; therefore, integration over the volume of the element is required to calculate the virtual work terms.

For linear analysis, it is apparent that the real external work, or energy, is given by:

The real internal work, or strain energy, is given by:

Reference

Three-Dimensional Static and Dynamic Analysis of Structures A Physical Approach With Emphasis on Earthquake Engineering Edward L. Wilson

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