## 2.1. Structural Fundamentals

The following topics concerning structural fundamentals are available:

### 2.1.1. Stress-Strain Relationships

This section discusses material relationships for linear materials. Nonlinear materials are discussed in Structures with Material Nonlinearities. The stress is related to the strains by: (2–1)

where:

 {σ} = stress vector = (output as S) [D] = elasticity or elastic stiffness matrix or stress-strain matrix (defined in Equation 2–14 through Equation 2–19) or inverse defined in Equation 2–4 or, for a few anisotropic elements, defined by full matrix definition (input with TB,ANEL.) {εel} = {ε} - {εth} = elastic strain vector (output as EPEL) {ε} = total strain vector = {εth} = thermal strain vector (defined in Equation 2–3) (output as EPTH)

Note:  {εel} (output as EPEL) are the strains that cause stresses.

The shear strains (εxy, εyz, and εxz) are the engineering shear strains, which are twice the tensor shear strains. The ε notation is commonly used for tensor shear strains, but is used here as engineering shear strains for simplicity of output.

A related quantity used in POST1 labeled “component total strain” (output as EPTO) is described in Structures with Material Nonlinearities.

The stress vector is shown in the figure below. The sign convention for direct stresses and strains used throughout the ANSYS program is that tension is positive and compression is negative. For shears, positive is when the two applicable positive axes rotate toward each other.

Figure 2.1:  Stress Vector Definition Equation 2–1 may also be inverted to: (2–2)

For the 3-D case, the thermal strain vector is: (2–3)

where: = secant coefficient of thermal expansion in the x direction (see Temperature-Dependent Coefficient of Thermal Expansion) ΔT = T - Tref T = current temperature at the point in question Tref = reference (strain-free) temperature (input on TREF command or as REFT on MP command)

The flexibility or compliance matrix, [D]-1 is: (2–4)

where typical terms are:

 Ex = Young's modulus in the x direction (input as EX on MP command) νxy = major Poisson's ratio (input as PRXY on MP command) νyx = minor Poisson's ratio (input as NUXY on MP command) Gxy = shear modulus in the xy plane (input as GXY on MP command)

The difference between νxy and νyx is described below.

Also, the [D]-1 matrix is presumed to be symmetric, so that: (2–5) (2–6) (2–7)

Because of the above three relationships, νxy, νyz, νxz, νyx, νzy, and νzx are not independent quantities and therefore the user should input either νxy, νyz, and νxz (input as PRXY, PRYZ, and PRXZ), or νyx, νzy, and νzx (input as NUXY, NUYZ, and NUXZ). The use of Poisson's ratios for orthotropic materials sometimes causes confusion, so that care should be taken in their use. Assuming that Ex is larger than Ey, νxy (PRXY) is larger than νyx (NUXY). Hence, νxy is commonly referred to as the “major Poisson's ratio”, because it is larger than νyx, which is commonly referred to as the “minor” Poisson's ratio. For orthotropic materials, the user needs to inquire of the source of the material property data as to which type of input is appropriate. In practice, orthotropic material data are most often supplied in the major (PR-notation) form. For isotropic materials (Ex = Ey = Ez and νxy = νyz = νxz), so it makes no difference which type of input is used.

Expanding Equation 2–2 with Equation 2–3 through Equation 2–7 and writing out the six equations explicitly, (2–8) (2–9) (2–10) (2–11) (2–12) (2–13)

where typical terms are:

 εx = direct strain in the x direction σx = direct stress in the x direction εxy = shear strain in the x-y plane σxy = shear stress on the x-y plane

Alternatively, Equation 2–1 may be expanded by first inverting Equation 2–4 and then combining that result with Equation 2–3 and Equation 2–5 through Equation 2–7 to give six explicit equations: (2–14) (2–15) (2–16) (2–17) (2–18) (2–19)

where: (2–20)

If the shear moduli Gxy, Gyz, and Gxz are not input for isotropic materials, they are computed as: (2–21)

For orthotropic materials, the user needs to inquire of the source of the material property data as to the correct values of the shear moduli, as there are no defaults provided by the program.

The [D] matrix must be positive definite. The program checks each material property as used by each active element type to ensure that [D] is indeed positive definite. Positive definite matrices are defined in Positive Definite Matrices. In the case of temperature dependent material properties, the evaluation is done at the uniform temperature (input as BFUNIF,TEMP) for the first load step. The material is always positive definite if the material is isotropic or if νxy, νyz, and νxz are all zero. When using the major Poisson's ratios (PRXY, PRYZ, PRXZ), h as defined in Equation 2–20 must be positive for the material to be positive definite.

### 2.1.2. Orthotropic Material Transformation for Axisymmetric Models

The transformation of material property data from the R-θ-Z cylindrical system to the x-y-z system used for the input requires special care. The conversion of the Young's moduli is fairly direct, whereas the correct method of conversion of the Poisson's ratios is not obvious. Consider first how the Young's moduli transform from the global cylindrical system to the global Cartesian as used by the axisymmetric elements for a disc:

Figure 2.2:  Material Coordinate Systems Thus, ER Ex, Eθ Ez, EZ Ey. Starting with the global Cartesian system, the input for x-y-z coordinates gives the following stress-strain matrix for the non-shear terms (from Equation 2–4): (2–22)

Rearranging so that the R-θ-Z axes match the x-y-z axes (i.e., x R, y Z, z θ): (2–23)

If one coordinate system uses the major Poisson's ratios, and the other uses the minor Poisson's ratio, an additional adjustment will need to be made.

Comparing Equation 2–22 and Equation 2–23 gives: (2–24) (2–25) (2–26) (2–27) (2–28) (2–29)

This assumes that: νxy, νyz, νxz and νRZ, ν, ν are all major Poisson's ratios (i.e., Ex EY Ez and ER EZ Eθ).

If this is not the case (e.g., Eθ > EZ): (2–30)

### 2.1.3. Temperature-Dependent Coefficient of Thermal Expansion

Considering a typical component, the thermal strain from Equation 2–3 is: (2–31)

where:

 α se(T) = temperature-dependent secant coefficient of thermal expansion (SCTE)

α se(T) is input in one of three ways:

1. Input α se(T) directly (input as ALPX, ALPY, or ALPZ on MP command)

2. Computed using Equation 2–34 from α in(T), the instantaneous coefficients of thermal expansion (input as CTEX, CTEY, or CTEZ on MP command)

3. Computed using Equation 2–32 from εith(T), the input thermal strains (input as THSX, THSY, or THSZ on MP command)

α se(T) is computed from εith(T) by rearranging Equation 2–31: (2–32)

Equation 2–32 assumes that when T = Tref, εith = 0. If this is not the case, the εith data is shifted automatically by a constant so that it is true. α se at Tref is calculated based on the slopes from the adjacent user-defined data points. Hence, if the slopes of εith above and below Tref are not identical, a step change in α se at Tref will be computed.

εth(T) (thermal strain) is related to α in(T) by: (2–33)

Combining this with equation Equation 2–32, (2–34)

No adjustment is needed for α in(T) as α se(T) is defined to be α in(T) when T = Tref.

As seen above, α se(T) is dependent on what was used for Tref. If α se(T) was defined using Tref as one value but then the thermal strain was zero at another value, an adjustment needs to be made (using the MPAMOD command). Consider: (2–35) (2–36)

Equation 2–35 and Equation 2–36 represent the thermal strain at a temperature T for two different starting points, To and Tref. Now let To be the temperature about which the data has been generated (definition temperature), and Tref be the temperature at which all strains are zero (reference temperature). Thus, is the supplied data, and is what is needed as program input.

The right-hand side of Equation 2–35 may be expanded as: (2–37)

also, (2–38)

or (2–39)

Combining Equation 2–35 through Equation 2–38, (2–40)

Thus, Equation 2–40 must be accounted for when making an adjustment for the definition temperature being different from the strain-free temperature. This adjustment may be made (using the MPAMOD command).

Note that:

 Equation 2–40 is nonlinear. Segments that were straight before may be no longer straight. Hence, extra temperatures may need to be specified initially (using the MPTEMP command). If Tref = To, Equation 2–40 is trivial. If T = Tref, Equation 2–40 is undefined.

The values of T as used here are the temperatures used to define α se (input via the MPTEMP command). Thus, when using the α se adjustment procedure, avoid defining a T value to be the same as T = Tref (to a tolerance of one degree). If a T value is the same as Tref, the value of α se remains unchanged.