The loads on a Section Cut are determined by examining the loads developed in the shell and beam elements making up the cross-section. For each Design Case, the loads in the finite elements are assembled as the scaled sum of the loads in the sub-cases. The multipliers and limit/ultimate factors are used as the coefficients in this linear super positioning.
The loads of interest in the shell elements are:
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Sectional Forces - \(N_{xx}\) and \(N_{yy}\)
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Sectional Moment - \(M_{xx}\)
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Out-of-Plane Shear Load - \(Q_{x}\)
For beams, the loads are:
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Axial Force - \(F_x\)
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Shear Forces - \(V_y\) and \(V_z\)
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Torque - \(T_x\)
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Bending Moments - \(M_y\) and \(M_z\)
HyperX Beam Force and Moment Diagram
Coordinate System and Loads on Shell and Beam Element Cross-Sections
\(\begin{align} \text{Axial Force } (F_N)=\sum_{i=1}^{n_{shells}}N_{xx,i}l_i+\sum_{i=1}^{n_{beams}}F_{x,i} \end{align}\)
\(\begin{align} \text{Shear H } (V_H)=\sum_{i=1}^{n_{shells}}\left(N_{xy,i} \cos \theta_i-Q_x \sin \theta_i\right)l_i+\sum_{i=1}^{n_{beams}}V_y \cos \theta_i-V_z\sin\theta_i \end{align}\)
\(\begin{align} \text{Shear V } (V_V)=\sum_{i=1}^{n_{shells}}\left(N_{xy,i} \sin \theta_i+Q_x \cos \theta_i\right)l_i+\sum_{i=1}^{n_{beams}}V_y \sin \theta_i+V_z\cos\theta_i \end{align}\)
\(\begin{align} \text{Torque } (T) &= \sum_{i=1}^{n_{shells}}\left(N_{xy,i}\left[\Delta h_{CN,i}\sin\theta_i - \Delta \nu_{CN,i} \cos\theta_i\right] + Q_x\left[\Delta \nu_{CN,i}\sin\theta_i+\Delta h_{CN,i}\cos\theta_i\right]\right)l_i \\ &+\sum_{i=1}^{n_{beams}}T_x + V_y \left[\Delta h_{CN,i}\sin\theta_i-\Delta \nu_{CN,i}\cos\theta_i\right] \\ &+ V_z \left[\Delta \nu_{CN,i}\sin \theta_i+\Delta h_{CN,i}\cos\theta_i\right]\end{align}\)
\(\begin{align} \text{Moment H } (M_H) &= \sum_{i=1}^{n_{shells}}\left(N_{xx,i} \Delta \nu_{CN,i}+M_x \cos\theta_i\right)l_i \\ &+\sum_{i=1}^{n_{beams}}\left(F_x \Delta \nu_{CN,i}+M_y \cos \theta_i - M_z \sin \theta_i\right)\end{align}\)
\(\begin{align} \text{Moment V } (M_V) &= \sum_{i=1}^{n_{shells}} -\left(N_{xx,i} \Delta h_{CN,i}-M_x \sin\theta_i\right)l_i \\ &-\sum_{i=1}^{n_{beams}}\left(F_x \Delta h_{CN,i}-M_y \sin\theta_i - M_z \cos\theta_i\right)\end{align}\)
Section Cut loads are converted to beam forces as shown below in the figure below.
Beam loads can be converted to panel loads if the desired design is a panel property.
\(N_x=\frac{P_{axial}}{s} \text{ } N_{xy}=\frac{V_2}{s} \text{ } M_x = \frac{M_1}{s} \text{ } Q_x = \frac{V_1}{s}\)
HyperX will perform this conversion automatically when assigning the beam loads created from a Section Cut to a panel design. For Standard Sizing, \(s\) is the spacing from a stiffened panel concept. For HyperX Solver, \(s\) is the panel y-span.
The beam loads are calculated about the neutral axis of the Section Cut. For stiffened panel concepts, HyperX assumes a reference plane at the mid-plane of the top facesheet, thus requiring a transformation of loads. To do this, a Virtual Moment is calculated so that the curvature caused by the membrane loading being applied at the panel’s reference plane is zero. This Virtual Moment is then superimposed onto the actual applied moment. In other words, the resulting curvature \(\kappa_x\) will only be caused by the applied moment \(M_x\) and not from the shift in reference plane.
For HyperX Solver, it is also assumed that there will be no anticlastic bending in the panel so moments \(M_y\) and \(M_{xy}\) are calculated and applied so that \(\kappa_y = \kappa_{xy} = 0\). For Standard Sizing, anticlastic bending is allowed.
To verify the above, a User General Panel Load can be created where the loads are set to be equal to the values given by the equation at the top of this section. A fixed boundary condition can be applied to \(M_y\) and \(M_{xy}\) of the user load, as shown below in the figure below. The moment reference plane is also set relative to the zero-curvature plane.