2.7.8 Material Stiffness Matrix D

Material Stiffness Matrix D

The bending stiffnesses in the reinforcement directions φ are determined as follows:

$D_{d,d} = I_{0,d} · \frac{E}{\left(1 - {\nu }^2\right)} \text{where} d =\hspace{0.17em}\left\{1,2\right\}$

$D_{d,d}\hspace{0.17em}=\hspace{0.17em}{I}_d · \frac{E}{\left(1 - {\nu }^2\right)} \text{where} d =\hspace{0.17em}\left\{1,2\right\}$

The non-diagonal component of the material stiffness matrix is calculated identically for plates and shells:

$D_{1,2} = D_{2,1} = \nu · \sqrt{\left(D_{1,1} · D_{2,2}\right)}$

For shells, the differences in the bending stiffnesses due to the moments of inertia are compensated via the eccentricity components in the material stiffness matrix.

The stiffness matrix elements for torsion are calculated as follows for plates and shells:

$D_{3,3} = \frac{1 - \nu }{2} · \sqrt{\left(D_{1,1,} · D_{2,2}\right)}$

The stiffness matrix elements for shear are not reduced for the deformation analysis. They are calculated from the shear modulus G of the ideal cross-section and the cross-section height h. The following applies for shells and plates:

$D_{3+d,3+d} = \frac{5}{6} · G · h \text{where} d\hspace{0.17em}=\hspace{0.17em}\left\{1,2\right\}$

The membrane stiffnesses in the reinforcement directions φ are determined as follows:

$D_{5+d,5+d} =\hspace{0.17em}E · \frac{A_d}{\left(1 - {\nu }^2\right)} \text{where} d =\hspace{0.17em}\left\{1,2\right\}$

The non-diagonal component of the material stiffness matrix is calculated from:

$D_{6,7} = D_{7,6} = \nu · \sqrt{\left(D_{6,6} · D_{7,7}\right)}$

The shear stiffness component is:

$D_{8,8} = G · h$

The stiffness matrix elements for the eccentricity of the centroid (ideal cross-section) in the reinforcement direction φ are calculated as follows:

$D_{d,6} = D_{6,d} = D_{5+d,5+d} · e_d \text{where} d\hspace{0.17em}=\hspace{0.17em}\left\{1,2\right\}$

The non-diagonal component of the material stiffness matrix is determined from:

$D_{1,7} = D_{7,1} = \frac{\nu }{2} · \left(e_{{\varphi }_1} + e_{{\varphi }_2}\right) · \sqrt{\left(D_{6,6} · D_{7,7}\right)}$

The eccentricity components for torsion are calculated as follows:

$D_{3,8} = D_{8,3} = \frac{1}{2} · G · h ·\left(e_{{\varphi }_1} + e_{{\varphi }_2}\right)$