SLS 13: Principal forces and stresses in a plate

Description

Geometry	Plate section:	$h=200mm$
Material:	C25/30
Loads	Load group “300”:	Axial load along line BC of 300kN/m
Load group “50”:	Distributed load of 50 kN/m²
Internal forces in point A	Load group “50+300”:	$M_{xx}=35.085\text{kNm}$ $M_{zz}=223.700\text{kNm}$ $M_{xz}=5.866\text{kN}$ $N_{xx}=-19.146\text{kN}$ $N_{zz}=-303.884 \text{kN}$ $N_{xz}=-14.841 \text{kN}$ $V_{x}=-23.621 \text{kN}$ $V_{z}=163.424 \text{kN}$

Independent reference results

Mesh points and interpolation

Diamonds calculates the internal forces/stresses at each vertex and in the centre of each triangle side of a mesh triangle. The colour gradient of the results is created by interpolation.
To avoid having to worry about interpolation here as well, we specifically use internal forces that occur in mesh nodes.

You can easily retrieve the internal forces in the mesh nodes via the results table . Or if you add a point in the geometry, like we did for point A, that becomes a meshnode by default.

Principal internal forces

The principal moment (max) $M_1$ :

$M_1=\frac{1}{2}\left( M_{xx} + M_{zz} + \sqrt{\left( M_{xx}-M_{zz} \right)^2 + 4 \cdot M_{xz}^2} \right)=223.882 \text{kNm}$

The principal moment (min) $M_2$ :

$M_2=\frac{1}{2}\left( M_{xx} + M_{zz} - \sqrt{\left( M_{xx}-M_{zz} \right)^2 + 4 \cdot M_{xz}^2} \right)=34.903\text{kNm}$

The angle of the principal moment $\alpha_{bending}$ . This result can only be verified graphically in Diamonds.

$\alpha_{bending}=\frac{1}{2}\left( atan\left( \frac{2 \cdot M_{xz}}{M_{xx} - M_{zz}} \right) \right)=1.780^{\circ}$

The principal force (max) $N_1$ :

$N_1=\frac{1}{2}\left( N_{xx} + N_{zz} + \sqrt{\left( N_{xx}-N_{zz} \right)^2 + 4 \cdot N_{xz}^2} \right)=-18.375 \text{kNm}$

The principal force (min) $N_2$ :

$N_2=\frac{1}{2}\left( N_{xx} + N_{zz} - \sqrt{\left( N_{xx}-N_{zz} \right)^2 + 4 \cdot N_{xz}^2} \right)=-304.655 \text{kNm}$

The angle of the principal force $\alpha_{membrane}$ . This result can only be verified graphically in Diamonds.

$\alpha_{membrane}=\frac{1}{2}\left( atan\left( \frac{2 \cdot N_{xz}}{N_{xx} - N_{zz}} \right) \right)=-2.976^{\circ}$

$M_1, M_2, N_1$ and $N_2$ can be requested in the Data table for the combination “300+50”. Mesh point 5 corresponds to the position of point A. $\alpha_{bending}$ and $\alpha_{membrane}$ can only be verified graphically.

Axial stresses due to axial force

$\sigma_{N_{xx}}=\frac{N_{xx}}{b \cdot h}=-0.096 \text{MPa}$

$\sigma_{N_{zz}}=\frac{N_{zz}}{b \cdot h}=-1.519 \text{MPa}$

$\sigma_{N_{xz}}=\frac{N_{xz}}{b \cdot h}=0.07 \text{MPa}$

$\sigma_{N_{xx}$ and $\sigma_{N_{zz}$ can be requested in the Data table when you look at the load group “300”. The results for $\sigma_{N_{xz}$ cannot be consulted in Diamonds.

Bending stresses due to the bending moment

$\sigma_{M_{xx}}=\frac{6\cdot M_{xx}}{b \cdot h^2}=5.263\text{MPa}$

$\sigma_{M_{zz}}=\frac{6\cdot M_{zz}}{b \cdot h^2}=33.555\text{MPa}$

$\sigma_{M_{xz}}=\frac{6\cdot M_{xz}}{b \cdot h^2}=0.880 \text{MPa}$

$\sigma_{M_{xx}$ and $\sigma_{M_{zz}}$ can be requested in the Data table when you look at the load group “50”. The results for $\sigma_{M_{xz}}$ cannot be consulted in Diamonds.

Combined stresses

$\sigma_{xx,i}=\sigma_{N_{xx}}+\sigma_{M_{xx}}}=5.167\text{MPa}$

$\sigma_{xx,s}=\sigma_{N_{xx}}-\sigma_{M_{xx}}=-5.358\text{MPa}$

$\sigma_{zz,i}=\sigma_{N_{zz}}+\sigma_{M_{zz}}=32.036\text{MPa}$

$\sigma_{zz,s}=\sigma_{N_{zz}}-\sigma_{M_{zz}}=-35.074\text{MPa}$

$\sigma_{xz,i}=\sigma_{N_{xz}}+\sigma_{M_{xz}}=0.806\text{MPa}$

$\sigma_{xz,s}=\sigma_{N_{xz}}-\sigma_{M_{xz}}=-0.954\text{MPa}$

$\sigma_{xx,i}, \sigma_{xx,s}, \sigma_{zz,i}$ and $\sigma_{zz,s}$ can be requested in the Data table when you look at the load group “300+50”. The results for $\sigma_{xz,i}$ and $\sigma_{xz,i}$ cannot be consulted in Diamonds.

Principal stresses

$\sigma_{1,s} &=\frac{1}{2} \cdot \left( \sigma_{xx,s}+ \sigma_{zz,s} + \sqrt{\left(\sigma_{xx,s}+\sigma_{zz,s} \right)^2+ 4 \cdot \sigma_{xz,s} }\right= 32.060\text{MPa}$

$\sigma_{2,s} &=\frac{1}{2} \cdot \left( \sigma_{xx,s}+ \sigma_{zz,s} - \sqrt{\left(\sigma_{xx,s}+\sigma_{zz,s} \right)^2+ 4 \cdot \sigma_{xz,s} }\right)= 5.143 \text{MPa}$

$\sigma_{1,i} &=\frac{1}{2} \cdot \left( \sigma_{xx,i}+ \sigma_{zz,i} + \sqrt{\left(\sigma_{xx,i}+\sigma_{zz,i} \right)^2+ 4 \cdot \sigma_{xz,i} }\right)= -5.328 \text{MPa}$

$\sigma_{2,i} &=\frac{1}{2} \cdot \left( \sigma_{xx,i}+ \sigma_{zz,i} - \sqrt{\left(\sigma_{xx,i}+\sigma_{zz,i} \right)^2+ 4 \cdot \sigma_{xz,i} }\right)= --35.105\text{MPa}$

$\sigma_{1,s} , \sigma_{1,i}, \sigma_{2,s}$ and $\sigma_{2,i}$ can be requested in the Data table when you look at the load group “300+50”.

Effective stresses

$\sigma_y=0 \text{MPa}$

$\tau_x=\frac{3 \cdot V_x}{2 \cdot b \cdot h} = -0.177 \text{MPa}$

$\tau_z=\frac{3 \cdot V_z}{2 \cdot b \cdot h} = 1.226 \text{MPa}\[tau_y=\sigma_{N_{xz}}+\sigma_{M_{xz}}=0.806 \text{MPa}$

$\sigma_{eff,s}=\sqrt{\frac{1}{2} \cdot \left[ \left( \sigma_{xx,s}-\sigma_{y} \right)^2+ \left(\sigma_{y}- \sigma_{zz,s} \right)^2+ \left( \sigma_{zz,s}- \sigma_{xx,s} \right)^2 \right] + 3 \cdot \left(\tau_x^2+\tau_y^2+\tau_z^2 \right)} = 29.900 \text{MPa}$

$\sigma_{eff,i}=\sqrt{\frac{1}{2} \cdot \left[ \left( \sigma_{xx,i}-\sigma_{y} \right)^2+ \left(\sigma_{y}- \sigma_{zz,i} \right)^2+ \left( \sigma_{zz,i}- \sigma_{xx,i} \right)^2 \right] + 3 \cdot \left(\tau_x^2+\tau_y^2+\tau_z^2 \right)} = 32.826 \text{MPa}$

$\sigma_{eff,s}$ and $\sigma_{eff,i}$ can be requested in the Data table when you look at the load group “300+50”. The results for $\tau_x, \tau_xz, \tau_y$ cannot be consulted in Diamonds.

Diamonds results and comparison

	Independant reference	Diamonds	Difference
$\sigma_{1,s}$	32.060 MPa	32.060 MPa	0%
$\sigma_{1,i}$	-5.328 MPa	-5.328 MPa	0%
$\sigma_{2,s}$	5.143 MPa	5.143 MPa	0%
$\sigma_{2,i}$	-35.105 MPa	-35.105 MPa	0%
$\sigma_{xx,s}$	5.167 MPa	5.167 MPa	0%
$\sigma_{xx,i}$	-5.358 MPa	-5.358 MPa	0%
$\sigma_{zz,s}$	32.036 MPa	32.036 MPa	0%
$\sigma_{zz,i}$	-35.074 MPa	-35.074 MPa	0%
$\sigma_{eff,s}$	29.900 MPa	29.857 MPa	≈0%
$\sigma_{eff,i}$	32.829 MPa	32.799 MPa	≈0%