6. Loads#
The different load types available for element POU_D_TG are:
Types or options |
|
CHAR_MECA_FR1D1D |
load distributed by real values |
CHAR_MECA_FF1D1D |
load distributed by function |
CHAR_MECA_PESA_R |
loading due to gravity |
CHAR_MECA_TEMP_R |
“thermal” loading |
CHAR_MECA_EPSI_R |
loading by imposing deformation (such as thermal stratification) |
Loads are calculated in the same way as for items without warpage [R3.08.01]. So there’s nothing particular about element POU_D_TG. The other load types described in [R3.08.01] are not available for this item.
As far as warping is concerned, it is possible to give boundary conditions involving the degree of freedom \(\mathit{GRX}\) (which makes it possible to model the hampered twist: \(\mathrm{GRX}=0\)), but on the other hand, nothing is planned to affect a loading of the bi-moment type, whose physical interpretation is difficult to establish.
With regard to the connection between elements, the transmission of warping is an open question as indicated by the reference [bib]: the continuity of the variable \(\mathit{GRX}\) from one element to another (on which the warping directly depends) in fact depends on the technology of the connection between the various beams (welding in the axis, in which case the warping can be transmitted in full, connection by gusset,…).
For an assembled structure such as a truss, it seems more reasonable to assume that twisting is impeded, so that there is zero warping at the ends. To identify the influence of this hypothesis, we can refer to test SSLL102 (angle section beam) whose C and D models use the element POU_D_TG, with free twist for C modeling, and hampered twist for D modeling [V3.01.102B].
It can be seen that for flexural loading, the difference in displacement is small (2.5%), but for torsional loading, for this section, a non-zero lateral displacement (overflow) is obtained whose value differs significantly depending on the hypothesis taken:
\({u}_{z}=2\text{.}2{\text{10}}^{\text{-}5}\) for free twisting and \({u}_{z}=2\text{.}\text{62}{\text{10}}^{\text{-}5}\) for awkward twisting.
Likewise, the rotation varies greatly:
\({\theta }_{x}=\mathrm{3,}\text{79}{\text{10}}^{\text{-}4}\) for free twisting and \({\theta }_{x}=\mathrm{6,}\text{39}{\text{10}}^{\text{-}4}\) for awkward twisting (GRX is zero at the ends).
6.1. Distributed loads, options: CHAR_MECA_FR1D1D and CHAR_MECA_FF1D1D#
Loads are given under the FORCE_POUTRE keyword, either by real values in AFFE_CHAR_MECA (option CHAR_MECA_FR1D1D), or by functions in AFFE_CHAR_MECA_F (option CHAR_MECA_FF1D1D). The load is only given by distributed forces, not by distributed moments.
The second member associated with distributed tension-compression loading is:
\(\begin{array}{}\left[\begin{array}{c}{f}_{1}\\ {f}_{2}\end{array}\right]\end{array}\) with \({f}_{1}={\int }_{0}^{1}{f}_{\text{ext}}(x)(1-\frac{x}{L})\text{dx}\)
\({f}_{2}={\int }_{0}^{1}{f}_{\text{ext}}(x)\frac{x}{L}\text{dx}\)
For a loading that is constant or linearly varying, we obtain:
\(\begin{array}{}{F}_{{x}_{1}}=L(\frac{{n}_{1}}{3}+\frac{{n}_{2}}{6})\text{,}\\ {F}_{{x}_{2}}=L(\frac{{n}_{1}}{6}+\frac{{n}_{2}}{3})\text{.}\end{array}\)
\({n}_{1}\) and \({n}_{2}\) are the components of axial loading at points 1 and 2 from user data placed in the local coordinate system.
If \({t}_{{y}_{1}}\text{,}{t}_{\mathrm{y2}}\text{,}{t}_{\mathrm{z1}}\) and \({t}_{\mathrm{z2}}\) are those of the sharp effort, we have:
\(\begin{array}{cccc}{F}_{{y}_{1}}=L(\frac{7{t}_{{y}_{1}}}{\text{20}}+\frac{3{t}_{{y}_{2}}}{\text{20}})& & {M}_{{z}_{1}}={L}^{2}(\frac{{t}_{{y}_{1}}}{\text{20}}+\frac{{t}_{{y}_{2}}}{\text{30}})& \\ {F}_{{y}_{2}}=L(\frac{3{t}_{{y}_{1}}}{\text{20}}+\frac{7{t}_{{y}_{2}}}{\text{20}})& & {M}_{{z}_{2}}=-{L}^{2}(\frac{{t}_{{y}_{1}}}{\text{30}}+\frac{{t}_{{y}_{2}}}{\text{20}})\text{,}& \\ {F}_{{z}_{1}}=L(\frac{7{t}_{{z}_{1}}}{\text{20}}+\frac{3{t}_{{z}_{2}}}{\text{20}})\text{,}& & {M}_{{y}_{1}}=-{L}^{2}(\frac{{t}_{{z}_{1}}}{\text{20}}+\frac{{t}_{{z}_{2}}}{\text{30}})\text{,}& \\ {F}_{{z}_{2}}=L(\frac{3{t}_{{z}_{1}}}{\text{20}}+\frac{7{t}_{{z}_{2}}}{\text{20}})\text{,}& & {M}_{{y}_{2}}={L}^{2}(\frac{{t}_{{z}_{1}}}{\text{30}}+\frac{{t}_{{z}_{2}}}{\text{20}})\text{.}& \end{array}\)
6.2. Gravity loading, option: CHAR_MECA_PESA_R#
The force of gravity is given by the acceleration module \(g\) and a standard vector \(n\) indicating the direction of loading.
Remarks (simplifying hypothesis) :
The shape functions used for this calculation are those of the Euler-Bernoulli model.
The approach is similar to that used for distributed forces, provided that you first transform the load vector due to gravity into the coordinate system local to the element. In the local coordinate system of the beam, we obtain:
\(\begin{array}{}{F}_{{x}_{i}}={\int }_{o}^{L}{\xi }_{i}\rho Sg\cdot x\text{dx}\\ ({\xi }_{1}=1-\frac{x}{L}\text{,}{\xi }_{2}=\frac{x}{L})\end{array}\) from where: \(\begin{array}{}{F}_{{x}_{1}}=\rho g\cdot xL(\frac{S}{3}+\frac{S}{6})\text{}\text{au point 1,}\\ {F}_{{x}_{2}}=\rho g\cdot xL(\frac{S}{6}+\frac{S}{3})\text{}\text{au point 2}\end{array}\)
Flexion in the plane \((\mathrm{Gxz})\) :
\(\begin{array}{}{F}_{{z}_{1}}=\rho g\cdot zL(\frac{7S}{\text{20}}+\frac{3S}{\text{20}})\\ {M}_{{y}_{1}}=-\rho g\cdot z\text{}{L}^{2}(\frac{S}{\text{20}}+\frac{S}{\text{30}})\\ {F}_{{z}_{2}}=\rho g\cdot zL(\frac{3S}{\text{20}}+\frac{7S}{\text{20}})\\ {M}_{{y}_{2}}=\rho g\cdot z{L}^{2}(\frac{S}{\text{30}}+\frac{S}{\text{20}})\end{array}\)
Flexion in the plane \((\mathrm{Gxy})\) :
\(\begin{array}{}{F}_{{y}_{1}}=\rho g\cdot yL(\frac{7S}{\text{20}}+\frac{3S}{\text{20}})\\ {M}_{{z}_{1}}=\rho g\cdot y\text{}{L}^{2}(\frac{S}{\text{20}}+\frac{S}{\text{30}})\\ {F}_{{y}_{2}}=\rho g\cdot yL(\frac{3S}{\text{20}}+\frac{7S}{\text{20}})\\ {M}_{{z}_{2}}=-\rho g\cdot y{L}^{2}(\frac{S}{\text{30}}+\frac{S}{\text{20}})\end{array}\)
6.3. Thermal loading, option: CHAR_MECA_TEMP_R#
To obtain this load, it is necessary to calculate the axial displacements induced by the temperature difference \(T-{T}_{\text{référence}}\):
\(\begin{array}{}{u}_{1}=-L\alpha (T-{T}_{\text{référence}})\\ {u}_{2}=L\alpha (T-{T}_{\text{référence}})\end{array}\)
(\(\alpha\): thermal expansion coefficient)
Then, we simply calculate the forces induced by \(F=\text{K u}\).
Since \(K\) is the stiffness matrix local to the element, we must then change the coordinate system to obtain the values of the load components in the global coordinate system.
6.4. Loading by imposed deformation, option: CHAR_MECA_EPSI_R#
As for elements POU_D_T, the load is calculated from a state of deformation (this option was developed to take into account thermal stratification in pipes). The model only takes into account work in tension - compression and in pure bending (no shear force, no torsional moment).
The deformation is given by the user using the PRE_EPSI keyword in AFFE_CHAR_MECA. By giving ourselves \(\frac{\partial u}{\partial x}\text{,}\frac{\partial {\theta }_{y}}{\partial x}\) and \(\frac{\partial {\theta }_{z}}{\partial x}\) on the beam, we get the second elementary member associated with this load:
at node 1:
\(\begin{array}{cccc}& {F}_{{x}_{1}}& =& E{S}_{1}\frac{\partial u}{\partial x}\text{,}\\ & {M}_{{y}_{1}}& =& E{I}_{{y}_{1}}\frac{\partial {\theta }_{y}}{\partial x}\text{,}\\ & {M}_{{z}_{1}}& =& E{I}_{{z}_{1}}\frac{\partial {\theta }_{z}}{\partial x}\text{,}\end{array}\)
at node 2:
\(\begin{array}{cccc}& {F}_{{x}_{2}}& =& E{S}_{2}\frac{\partial u}{\partial x}\text{,}\\ & {M}_{{y}_{2}}& =& E{I}_{{y}_{2}}\frac{\partial {\theta }_{y}}{\partial x}\text{,}\\ & {M}_{{z}_{2}}& =& E{I}_{{z}_{2}}\frac{\partial {\theta }_{z}}{\partial x}\end{array}\)