RF-CONCRETE Surfaces – Online Manual Version 5

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RF-CONCRETE Surfaces – Online Manual Version 5

2.7.8 Material Stiffness Matrix D

Material Stiffness Matrix D

Bending stiffness – plates and shells

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

$D_{d, d} = I_{0, d} \cdot \frac{E}{(1 - ν^{2})} mit d = {1, 2}$

$D_{d, d} = I_{d} \cdot \frac{E}{(1 - ν^{2})} mit d = {1, 2}$

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

$D_{1, 2} = D_{2, 1} = ν \cdot \sqrt{(D_{1, 1} \cdot D_{2, 2})}$

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.

Torsional stiffness – plates and shells

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

$D_{3, 3} = \frac{1 - ν}{2} \cdot \sqrt{(D_{1, 1,} \cdot D_{2, 2})}$

Shear stiffness – plates and shells

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:

$D_{3 + d, 3 + d} = \frac{5}{6} \cdot G \cdot h mit d = {1, 2}$

Membrane stiffness – shells

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

$D_{5 + d, 5 + d} = E \cdot \frac{A_{d}}{(1 - ν^{2})} mit d = {1, 2}$

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

$D_{6, 7} = D_{7, 6} = ν \cdot \sqrt{(D_{6, 6} \cdot D_{7, 7})}$

The shear stiffness component is:

$D_{8, 8} = G \cdot h$

Eccentricity – shells

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

$D_{d, 6} = D_{6, d} = D_{5 + d, 5 + d} \cdot e_{d} mit d = {1, 2}$

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

$D_{1, 7} = D_{7, 1} = \frac{ν}{2} \cdot (e_{ϕ_{1}} + e_{ϕ_{2}}) \cdot \sqrt{(D_{6, 6} \cdot D_{7, 7})}$

The eccentricity components for torsion are calculated as follows:

$D_{3, 8} = D_{8, 3} = \frac{1}{2} \cdot G \cdot h \cdot (e_{ϕ_{1}} + e_{ϕ_{2}})$

Quick Overview of this Section

Material Stiffness Matrix D

Quick Overview of this Section

Material Stiffness Matrix D

Content

2.7.8 Material Stiffness Matrix D

Material Stiffness Matrix D

Quick Overview of this Section

Quick Overview of this Section