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Stress in rectangular sections

Flexible type

For a rectangular beam section, a flexible type property may be used.

Bi-directional bending stiffness is to be defined taking into account the different momenta of inertia.

\[ I_{xx}=\frac{ba^3}{12}-\frac{(b-wt_x)(a-2wt_y)^3}{12} \]

and

\[ I_{yy}=\frac{ab^3}{12}-\frac{(a-wt_y)(b-2wt_x)^3}{12} \]

In the post-processing panel, equivalent diameter and wall thickness have to be entered.

These values can be chosen by the user, provided the following procedure is conducted :

  • Post-process the stresses in the equivalent circular beam,

  • Apply a \(SCF= \frac{a}{D}\) on the external bending stress around \(X\) axis;

  • Apply a \(SCF= \frac{b}{D}\) on the external bending stress around \(Y\) axis.

Generic type

With a GENERIC type, a specific area may be defined for the end cap effect and the link between the effective tension and the true tension becomes as such:

\[ T_{true}=T_{eff}+ P_i S_i-P_e S_{endcap} \]

Moreover, the data defining the local section have an impact on the mechanical strain due to the Poisson’s term and so indirectly on the effective tension. Indeed, the total strain ε_axial is given by the positions of the nodes and as mentioned above, the total strain is linked with the mechanical strain by the following relationship:

\[ \epsilon_{axial}=\epsilon_{mechanical}-2\theta \frac{\sigma_p}{E} \]

For generic lines, the pressure stress writes:

\[ \sigma_p=\frac{P_i S_i-P_e S_{endcap}}{A_{steel}} \]

The steel area is deduced from the section local properties as such:

  • Ring stiffened cylinder

\(OD = 2 \times\) (Shell external radius)

\(WT =\) Shell thickness

And the steel area writes: \(A_{steel}=π(OD-WT)WT\)

  • Rectangular stiffened hollow section:

For any point P(x,y) in the local section, stresses (without SCFs) are calculated as such:

\(\sigma_{axial}=\frac{N_{Sd}}{A_{steel}}\) and \(\sigma_{bending}=M_1 \frac{y}{I_x} – M_2 \frac{x}{I_y}\)

For GENERIC type, no corrosion allowance may be defined, it shall be included into SCFs;

\(A_{steel}=A_{sn}=\) Effective section area

\(I_x\) and \(I_y\) : inertia momenta defined in the mechanical properties.

Note

  • To calculate the pressure effect, the solver needs to know which steel area is to be considered. This area is computed with an equivalent external diameter OD and a wall thickness WT translated in the LOG file with keyword *BEAMPRO2.
    In case of a rectangular section, it comes: \(OD=2\sqrt{\frac{hw}{\pi}}\) An equivalent wall thickness \(WT\) is deduced from the effective steel area as such: \(\pi(OD-WT)*WT=A_n\) It comes: \(\Delta=OD^2-4\frac{A_n}{\pi}\) and \(WT=\frac{OD-\sqrt{\Delta}}{2}\) since the lowest value is the most relevant one.
  • This “equivalent” thickness is used to post-process the internal bending stress in the general post-processing. The section local details are only used for code check.
  • It shall be kept in mind that if calculations are performed for a generic line with a given section local property and this property is changed just before regenerating the DSS file, the resulting effective tension may differs between the solver and the interface due to the pressure term. Of course, if the effective steel area is kept unchanged, the same effective tension shall be processed between GUI and the solver.