Type "Cross Laminated Timber"
The type "Cross Laminated Timber" is used for 2D structures made out of timber in which the stiffness is calculated based on the composition of the different cross laminated timber layers. The formula used for calculating the stiffness values is obtained from the publication Cross-Laminated Timber Structural Design, pro:Holz - November 2014.
It is important to emphasize that within the solver settings the Mindlin bending theory should be used in combination with Cross-laminated orthotropy.
Mathematical description of orthotropy model
Flexural orthotropy
Flexural stiffness in direction xp [MNm/m] (1)
Flexural stiffness in direction yp [MNm/m] (2)
Transverse contraction xp — yp [MNm/m] (3)
Torsion stiffness related to (x,y)p [MNm/m] (4)
Shear flexural stiffness in direction xp [MN/m] (5)
Shear flexural stiffness in direction yp [MN/m] (6)
Membrane orthotropy
In the Wall model the membrane stiffnesses (4 parameters) play the role of the flexural ones in the Plate model. The most general Shell model defines both parameter sets simultaneously (in total 10 orthotropy parameters):
Normal membrane stiffness in direction xp [MN/m] (7)
Normal membrane stiffness in direction yp [MN/m] (8)
Transversal contraction xp—yp [MN/m] (9)
Shear membrane stiffness related to (x,y)p [MN/m] (10)
Symbols in formulae (1) – (10):
| b | Unit width of the CLT-panel (=1meter). |
| d | Total thickness of the CLT-panel. |
| E0,mean | Mean value of modulus of elasticity (Young's Modulus) along the grain of the timber laminate. |
| G0,mean | Mean value of shear modulus along the grain of the timber laminate. |
| I0,net,x | Second moment of area along the grain for the x-direction. |
| I0,net,y | Second moment of area along the grain for the y-direction. |
| νx,y | Coefficient of transverse expansion, a value of 0 is a good approximation coefficient. |
| νy,x | Coefficient of transverse expansion, a value of 0 is a good approximation coefficient. |
| ktors | Torsion reduction coefficient, a value of 0.65 is a good approximation coefficient. |
| kshear | Shear stiffness coefficient, a value of 0.75 is a good approximation coefficient. |
| k0,z | Shear correction coefficient (input range 1.20<=k0,z<=6.67) for the x-direction |
| k90,z | Shear correction coefficient (input range 1.20<=k0,z<=6.67) for the y-direction |
| A0,net,x | Net area of the CLT-panel in the x-direction. |
| A0,net,y | Net area of the CLT-panel in the y-direction. |
| Agross | Gross area of the CLT-panel (Agross=b*d) |
Torsion reduction coefficient ktors
The torsion reduction coefficient which is used for determining the torsion stiffness D33. According to the research done by Silly, 2010 a value of 0.80 is recommended for CLT-panels without any cracks. Considering cracks a value of 0.65 is recommended. The coefficient can also be calculated according to ÖNORM B 1995-1-1 (NA.K.2) in which the following equation is used:
With:
dmax: Thickness of the thickest individual layer.
| 3 layers | 5 layers | 7+ layers | |
| pD | 0.89 | 0.67 | 0.55 |
| qD | 1.33 | 1.26 | 1.23 |
Shear stiffness coefficient kshear
The shear stiffness coefficient which is used for determining the shear membrane stiffness d33. According to Silly (2010) when stressing a plate, the shear stiffness of cross-laminated timber must be reduced compared to homogeneous materials. A value of 0.75 is a good approximation for the shear stiffness coefficient.
The coefficient can also be calculated according to ÖNORM B 1995-1-1 (NA.K.1) in which the following equation is used:
With:
, with ax and ay the individual lamination widths for the given direction.
dmax: Thickness of the thickest individual layer.
| 3 layers | 5, 7+ layers | |
| pS | 0.53 | 0.43 |
| qS | 1.21 | |
Shear correction coefficients k0,z and k90,z
The shear correction coefficients k0,z and k90,z are used to properly take into account shear deformations within the Reissner-Mindlin model. These shear correction coefficients have been investigated by Jöbstl for CLT-elements in Schickhofer et al., 2010. Within that paper multiple CL-panels with constant thickness were tested and the outcome is summarized by means of shear correction factors kappa:
In tabular form the following recommended values for kappa are [Cross-Laminated Timber Structural Design Volume 2, pro:Holz, 2018]:
Keep in mind that the above table describes the kappa-values which is the shear correction factor, while the orthotropy dialog expects the shear correction coefficients as input which is 1/kappa.
The shear correction coefficient can also be calculated automatically, in the background the CLT-panel is taken as a cross-section with 1m width. Afterwards its shear area Az is calculated per direction. Finally the shear correction factor kappa is obtained as the ratio between the shear area and the total area from which the shear correction coefficient can be derived as 1/kappa.