RF-CONCRETE Surfaces Version 5

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RF-CONCRETE Surfaces Version 5

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2.7.8 Material Stiffness Matrix D

Material Stiffness Matrix D

Bending stiffness – plates and shells

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

Dd,d = I0,d · E1 - ν2          mit d = {1,2} 

Dd,d = Id · E1 - ν2            mit d = {1,2} 

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

D1,2 = D2,1 = ν · D1,1 · D2,2 

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

D3,3 = 1 - ν2 · D1,1, · D2,2  

Shear stiffness – plates and shells

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

D3+d,3+d = 56 · G · h          mit d = {1,2} 

Membrane stiffness – shells

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

D5+d,5+d = E · Ad1 - ν2          mit d = {1,2} 

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

D6,7 = D7,6 = ν · D6,6 · D7,7 

The shear stiffness component is:

D8,8 = G · h 

Eccentricity – shells

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

Dd,6 = D6,d = D5+d,5+d · ed          mit d = {1,2} 

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

D1,7 = D7,1 = ν2 · eϕ1 + eϕ2 · D6,6 · D7,7 

The eccentricity components for torsion are calculated as follows:

D3,8 = D8,3 = 12 · G · h ·eϕ1 + eϕ2 

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