The element integration point strains and stresses are computed by combining equations Equation 2–1 and Equation 2–44 to get:
(2–59) |
(2–60) |
where:
{ε^{el}} = strains that cause stresses (output as EPEL) |
[B] = strain-displacement matrix evaluated at integration point |
{u} = nodal displacement vector |
{ε^{th}} = thermal strain vector |
{σ} = stress vector (output as S) |
[D] = elasticity matrix |
Nodal and centroidal stresses are computed from the integration point stresses as described in Nodal and Centroidal Data Evaluation.
Surface stress output may be requested on “free” faces of 2-D and 3-D elements. “Free” means not connected to other elements as well as not having any imposed displacements or nodal forces normal to the surface. The following steps are executed at each surface Gauss point to evaluate the surface stresses. The integration points used are the same as for an applied pressure to that surface.
Compute the in-plane strains of the surface at an integration point using:
(2–61) |
Hence, , and are known. The prime (') represents the surface coordinate system, with z being normal to the surface.
A each point, set:
(2–62) |
(2–63) |
(2–64) |
where P is the applied pressure. Equation 2–63 and Equation 2–64 are valid, as the surface for which stresses are computed is presumed to be a free surface.
At each point, use the six material property equations represented by:
(2–65) |
to compute the remaining strain and stress components ( , , , , and ).
Repeat and average the results across all integration points.
For elastic shell elements, the forces and moments per unit length (using shell nomenclature) are computed as:
(2–66) |
(2–67) |
(2–68) |
(2–69) |
(2–70) |
(2–71) |
(2–72) |
(2–73) |
where:
T_{x}, T_{y}, T_{xy} = in-plane forces per unit length (output as TX, TY, and TXY) |
M_{x}, M_{y}, M_{xy} = bending moments per unit length (output as MX, MY, and MXY) |
N_{x}, N_{y} = transverse shear forces per unit length (output as NX and NY) |
t = thickness at midpoint of element, computed normal to center plane |
σ_{x}, etc. = direct stress (output as SX, etc.) |
σ_{xy}, etc. = shear stress (output as SXY, etc.) |
For shell elements with linearly elastic material, Equation 2–66 to Equation 2–73 reduce to:
(2–74) |
(2–75) |
(2–76) |
(2–77) |
(2–78) |
(2–79) |
(2–80) |
(2–81) |
For shell elements with nonlinear materials, Equation 2–66 to Equation 2–73 are numerically integrated.
It should be noted that the shell nomenclature and the nodal moment conventions are in apparent conflict with each other. For example, a cantilever beam located along the x axis and consisting of shell elements in the x-y plane that deforms in the z direction under a pure bending load with coupled nodes at the free end, has the following relationship:
(2–82) |
where:
b = width of beam |
F_{MY} = nodal moment applied to the free end (input as VALUE on F command with Lab = MY (not MX)) |
The shape functions of the shell element result in constant transverse strains and stresses through the thickness. Some shell elements adjust these values so that they will peak at the mid-surface with 3/2 of the constant value and be zero at both surfaces, as noted in the element discussions in Element Library.
The through-thickness stress (σ_{z}) is set equal to the negative of the applied pressure at the surfaces of the shell elements, and linearly interpolated in between.