X.C.1 Real Results

A summary of the Results that can be read is given in TableX.C.1 to X.C.10.

In the following remarks about the information given in Tables X.C.1 to X.C.10, one assumes that the international unit system is used.

1.

“Beam Forces” and “Beam Moments” are assumed in FeResPost to be tensorial Results expressed in N or Nm respectively. However several components of the tensor are systematically nil. The non-zero components are:

$$F=\left(\begin{array}{ccc}\hfill F_{xx}\hfill & \hfill F_{xy}\hfill & \hfill F_{xz}\hfill \\ \hfill F_{xy}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill F_{xz}\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right),M=\left(\begin{array}{ccc}\hfill M_{xx}\hfill & \hfill M_{xy}\hfill & \hfill M_{xz}\hfill \\ \hfill M_{xy}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill M_{xz}\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

There is an approximation for the moments above because the torsional component is calculated wrt cross-section shear centre, and bending components are calculated wrt cross-section centre of inertia.

One assumes that the beam forces are calculated from the Cauchy stress tensor components as follows:

$$\left\{F\right\}=\left(\begin{array}{c}\hfill F_{xx}\hfill \\ \hfill F_{xy}\hfill \\ \hfill F_{xz}\hfill \end{array}\right)={\int \; <!--nolimits-->}_{\mathrm{\Omega }}\left(\begin{array}{c}\hfill {\sigma }_{xx}\hfill \\ \hfill {\tau }_{xy}\hfill \\ \hfill {\tau }_{xz}\hfill \end{array}\right)\text{d}\mathrm{\Omega }.$$

In which $\mathrm{\Omega }$ is the surface defined by a cross-section through the beam orthogonal to beam longitudinal axis. (One presumes here that the beam longitudinal direction corresponds to “$x$” axis.) Similarly, one assumes that the bending moments tensor is calculated from the Cauchy stress tensor components as follows:

$$\left\{M\right\}=\left(\begin{array}{c}\hfill M_{xx}\hfill \\ \hfill M_{xy}\hfill \\ \hfill M_{xz}\hfill \end{array}\right)={\int \; <!--nolimits-->}_{\mathrm{\Omega }}\left(\begin{array}{c}\hfill \left(y-y_C\right){\tau }_{xz}-\left(z-z_C\right){\tau }_{xy}\hfill \\ \hfill \left(z-z_0\right){\sigma }_{xx}\hfill \\ \hfill -\left(y-y_0\right){\sigma }_{xx}\hfill \end{array}\right)\text{d}\mathrm{\Omega }.$$

In which $y$ and $z$ are the components of coordinates in $\mathrm{\Omega }$ section, $y_0$ and $z_0$ correspond to the coordinates of center of gravity of the section, and $y_C$ and $z_C$ correspond to the shear center coordinates. Note also that “Beam Forces” and “Beam Moments” are Results that correspond to most 1D elements. (Bars, beams, rods, bushing elements...). However spring elements do not produce “Beam Forces” and “Beam Moments”.

These conventions ensure that beam forces and moments behave like real order 2 tensors when transformation of coordinates systems are performed. The vectorial forces and moments at the two extremities are easily obtained. Vectors

$$\left\{F\right\}=\left(\begin{array}{c}\hfill F_{xx}\hfill \\ \hfill F_{xy}\hfill \\ \hfill F_{xz}\hfill \end{array}\right)\text{ and }\left\{M\right\}=\left(\begin{array}{c}\hfill M_{xx}\hfill \\ \hfill M_{xy}\hfill \\ \hfill M_{xz}\hfill \end{array}\right)$$

correspond to the forces and moments that must be applied on $+x$ side of the beam. On the $-x$ side of the beam the components of these vectors must be multiplied by -1.

2.

“Beam Deformations” is a tensorial Result corresponding to the difference of displacements of grids B and A of the beam element. The tensor is expressed in element axes. The “Beam Velocities” Result is the time derivative of the “Beam Deformations”.

3.

Spring forces are scalar. The units depend on the connected components: one has N for displacements and Nm for rotations. (Of course, it is also possible to define springs connecting translational and rotational degrees of freedom, but it is generally an error.)

4.

“Shell Forces” and “Shell Moments” are tensorial Results expressed in N/m or N respectively. These Results contain all the force and moment tensors produced by 2D elements. The non-zero components are:

$$F=\left(\begin{array}{ccc}\hfill F_{xx}\hfill & \hfill F_{xy}\hfill & \hfill Q_{xz}\hfill \\ \hfill F_{xy}\hfill & \hfill F_{yy}\hfill & \hfill Q_{yz}\hfill \\ \hfill Q_{xz}\hfill & \hfill Q_{yz}\hfill & \hfill 0\hfill \end{array}\right),M=\left(\begin{array}{ccc}\hfill M_{xx}\hfill & \hfill M_{xy}\hfill & \hfill 0\hfill \\ \hfill M_{xy}\hfill & \hfill M_{yy}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

(Symbol $Q$ has been used for the out-of-plane shear force.)

One assumes that the shell in-plane forces are calculated from the Cauchy stress tensor components as follows:

$$\begin{array}{llll}\hfill & F_{xx}={\int \; <!--nolimits-->}_{-h∕2}^{h∕2}{\sigma }_{xx}\text{d}z,& \hfill & \\ \hfill & F_{xy}={\int \; <!--nolimits-->}_{-h∕2}^{h∕2}{\sigma }_{xy}\text{d}z,& \hfill & \\ \hfill & ...& \hfill & \end{array}$$

Similarly, one assumes that the bending moments tensor is calculated using the distribution through the thickness of the Cauchy stress tensor components as follows:

$$\begin{array}{llll}\hfill & M_{xx}={\int \; <!--nolimits-->}_{-h∕2}^{h∕2}z{\sigma }_{xx}\text{d}z,& \hfill & \\ \hfill & M_{xy}={\int \; <!--nolimits-->}_{-h∕2}^{h∕2}z{\sigma }_{xy}\text{d}z,& \hfill & \\ \hfill & ...& \hfill & \end{array}$$

This is a usual convention for shell elements. For example, this is generally the convention used to present the theory of classical laminate analysis. When a component of the bending tensor is positive, the corresponding component is positive on the upper surface of the shell, and negative on the other face.

5.

The shear components of strain tensors Results stored in FeResPost are the ${ϵ}_{ij}={\gamma }_{ij}∕2$.

6.

The shell curvature tensor is defined as follows:

$$\left\{\kappa \right\}=\left(\begin{array}{c}\hfill {\kappa }_{xx}\hfill \\ \hfill {\kappa }_{yy}\hfill \\ \hfill {\kappa }_{xy}\hfill \end{array}\right)=\frac{12}{h^3}{\int \; <!--nolimits-->}_{-h∕2}^{h∕2}z\left(\begin{array}{c}\hfill {ϵ}_{xx}\hfill \\ \hfill {ϵ}_{yy}\hfill \\ \hfill {ϵ}_{xy}\hfill \end{array}\right)\text{d}z=\left(\begin{array}{c}\hfill w_{,xx}^0\hfill \\ \hfill w_{,yy}^0\hfill \\ \hfill w_{,xy}^0\hfill \end{array}\right).$$

Here again, a positive curvature means that the corresponding component of strain tensor is in tension on the upper face, and in compression on the lower face of the shell.

7.

Beam stresses and strains are always scalar Results corresponding either to the axial component, or to the norm of the shear components. Depending on the type of element, the axial stress may be calculated from the axial or bending loads, or to involve both contributions.

8.

Gap elements produce various results. Results are vectorial or tensorial and:

• Gap forces results are stored in “Beam Forces” tensorial Result as indicated in remark 1. The value of the axial component is multiplied by “-1.0”, because it is a compression component.

• “Beam Deformations” and “Beam Velocities” are tensorial Results containing the relative displacements or velocities of end nodes B and A in element coordinate system.

• “Gap Slips” is identical to “Beam Deformations” except that the axial component is set to “0.0”.

9.

Bushing elements produce Stress and Strain tensors obtained by multiplying the Beam Forces and Beam Moments by specified constants. These constants are by default set to 1. Therefore, stresses and strains have often values identical to the forces and moments. Note that the meaning of modifications of coordinate systems for bush stresses and strains have may be discussed. This is particularly so for stresses and strains corresponding to moments.

10.

Composite Results have non-linear dependence on the primary unknowns (displacements). Therefore, composite Results obtained by linear combination of elementary Results are false. This remark applies to all non-linear Results, plastification Results...

11.

Actually “Maximum Strain” and “Maximum Stress” composite Results are not tensorial because each component is a separate scalar Result. So no modification of coordinate system can be done for these Results.

The corresponding “CompMax” scalar Results are obtained by selecting the maximum failure index value among the six components.