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user5

1 Years ago Sep 05 Sun 2021 at 01:12 PM

ANISOTROPIC MATERIALS

The positive definition of stresses, in reference to an orthogonal 1-2-3 system, is shown in Figure

All stresses are by definition in units of force-per-unit-area. In matrix notation, the six independent stresses can be defined by:

From equilibrium,               

The six corresponding engineering strains are:

The most general form of the three dimensional strain-stress relationship for linear structural materials subjected to both mechanical stresses and temperature change can be written in the following matrix form:

E  modulus of elasticity, ν Poisson’s ratio and α coefficient of thermal expansion.

Or, in symbolic matrix form:

The C matrix is known as the compliance matrix and can be considered to be the most fundamental definition of the material properties because all terms can be evaluated directly from simple laboratory experiments. Each column of the C matrix represents the strains caused by the application of a unit stress. The temperature increase  is in reference to the temperature at zero stress. The a matrix indicates the strains caused by a unit temperature increase.

USE OF MATERIAL PROPERTIES WITHIN COMPUTER PROGRAMS

Most of the modern computer programs for finite element analysis require that the stresses be expressed in terms of the strains and temperature change.

Therefore, an equation of the following form is required within the program:

in which  . Therefore, the zero-strain thermal stresses are defined by:

    (1.7).

The numerical inversion of the 6 x 6 C matrix for complex anisotropic materials is performed within the computer program. Therefore, it is not necessary to calculate the E matrix in analytical form as indicated in many classical books on solid mechanics. In addition, the initial thermal stresses are numerically evaluated within the computer program. Consequently, for the most general anisotropic material, the basic computer input data will be twenty-one elastic constants, plus six coefficients of thermal expansion.

Initial stresses, in addition to thermal stresses, may exist for many different types of structural systems. These initial stresses may be the result of the fabrication or construction history of the structure. If these initial stresses are known, they may be added directly to Equation (1.7).

ISOTROPIC MATERIALS

An isotropic material has equal properties in all directions and is the most commonly used approximation to predict the behavior of linear elastic materials.

For isotropic materials, Equation (1.3) is of the following form:

It appears that the compliance matrix has three independent material constants. It can easily be shown that the application of a pure shear stress should result in pure tension and compression strains on the element if it is rotated 45 degrees.

Using this restriction, it can be shown that:

Therefore, for isotropic materials only Young's modulus E and Poisson's ratio ν need to be defined. Most computer programs use Equation (1.10) to calculate the shear modulus if it is not specified.

Reference

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

user5

1 Years ago Sep 05 Sun 2021 at 02:26 PM

GENERAL SOLUTION OF STRUCTURAL SYSTEMS

The starting point is always joint equilibrium.

Or,

where A is a load-force transformation matrix and is a function of the geometry of the structure only.

From the element force-deformation equation,


  , the joint equilibrium equation can be written as

From the compatibility equation,  

joint equilibrium can be written in terms of joint displacements as

(The element deformation-displacement transformation matrix, B, is a function of the geometry of the structure. Of greater significance, however, is the fact that the matrix B is the transpose of the matrix A defined by the joint equilibrium.)

Therefore, the general joint equilibrium can be written as:

The global stiffness matrix K is given by one of the following matrix equations:

It is of interest to note that the equations of equilibrium or the equations of compatibility can be used to calculate the global stiffness matrix K.

The standard approach is to solve Equation  for the joint displacements and then calculate the member forces from:

 

It should be noted that within a computer program, the sparse matrices A, B, k and K are never formed because of their large storage requirements. The symmetric global stiffness matrix K is formed and solved in condensed form.


(The C matrix is known as the compliance matrix and can be considered to be the most fundamental definition of the material properties because all terms can be evaluated directly from simple laboratory experiments. Each column of the C matrix represents the strains caused by the application of a unit stress).

Reference

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

user5

1 Years ago Sep 05 Sun 2021 at 02:44 PM

user5

1 Years ago Sep 05 Sun 2021 at 07:12 PM

user5

1 Years ago Sep 05 Sun 2021 at 07:12 PM

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