Stiffnesses for multilayer surfaces

Material Models

1. The material models are the basis for composing multilayer surfaces to obtain an effective surface stiffness. The Multilayer Surfaces Add-on allows you to freely combine the material models in the RFEM 6 program. The basis of the material models is described in Chapters Materials and Nonlinear Material Behavior of the RFEM manual.

1. A selection of the possible combinations of the material models is created in the "Multilayer Models" model (see the right column), which you can download for further study of the combinations.

1. The following list shows a selection of the possible combinations:
2. * Isotropic layers (e.g. concrete - steel)
3. * Orthotropic layers (e.g. cross-laminated timber)
4. * Isotropic - orthotropic (e.g. steel - GFRP)
5. * Isotropic Plastic - Isotropic (e.g. Concrete - Steel)
6. * Isotropic Nonlinear Elastic - Orthotropic (e.g. Concrete - Timber)
7. * Isotropic - Orthotropic Plastic (e.g. Concrete - Timber)
8. * Isotropic Damage - Orthotropic (e.g. Concrete - Timber)

1. #banner.text@For the combination of nonlinear materials, the Nonlinear Material Behavior should be activated.

1. == Stiffnesses for Multilayer Surfaces Without Solids ==

1. The simpler calculation option in the Multilayer Surfaces add-on is to define different surface layers in the thickness type 'Layers' without solids. However, you can also freely combine the material models here.

1. Once the layers are defined, the Multilayer Surfaces add-on creates a global stiffness matrix of the surface. In RFEM, the internal forces and deformations are calculated for this surface. In the respective design add-on, such as Timber Design or Stress-Strain Analysis, these internal forces are then divided into the existing layers. Usually, the internal forces are displayed as three integration points per position.

1. 

   Calculation Process for Multilayer Surfaces

1. This article explains how to calculate the stiffness matrix for isotropic and orthotropic materials.

1. === Calculation of stiffness matrix ===

1. The material models are based on the following conditions (see Chapter Materials of the RFEM manual):
2. * All stiffness values ≥ 0
3. * Overall stiffness matrix of surface must be positive definite.
4. * Basic equation isotropic:

1. $E=2G\left(1+\nu \right)$

   E	modulus of elasticity
   G	shear modulus
   ν	Poisson's ratio

   Modulus of Elasticity, Shear Modulus, and Poisson's Ratio
2. * Basic orthotropic equation:

1. $\frac{{\nu }_{yx}}{E_y}=\frac{{\nu }_{xy}}{E_x}$

   Transversal Strain Relation

1. ==== Local Stiffness Matrix of Each Layer ====

1. * Isotropic
2. $d_i\text{'}=\left[\begin{array}{ccc}d{\text{'}}_{11,i}& d{\text{'}}_{12,i}& 0\\ & d{\text{'}}_{22,i}& 0\\ sym.& & d{\text{'}}_{33,i}\end{array},),=,\left[\begin{array}{ccc}\frac{E_i}{1-{\nu }_i^2}& \frac{{\nu }_i{E}_i}{1-{\nu }_i^2}& 0\\ & \frac{E_i}{1-{\nu }_i^2}& 0\\ sym.& & G_i\end{array},), ,i,=,1,,,.,.,.,,,n, ,w,h,e,r,e, ,G_i,=,\frac{E_i}{2\left(1+{\nu }_i\right)}\right]\right]$

   Stiffness Matrix for Isotropic Layer

1. * Orthotropic
2. $d_i\text{'}=\left[\begin{array}{ccc}d{\text{'}}_{11,i}& d{\text{'}}_{12,i}& 0\\ & d{\text{'}}_{22,i}& 0\\ sym.& & d{\text{'}}_{33,i}\end{array},),=,\left[\begin{array}{ccc}\frac{E_{x,i}}{1-{\nu }_{xy,i}^2{\displaystyle \frac{E_{y,i}}{E_{x,i}}}}& \frac{{\nu }_{xy,i}{E}_{y,i}}{1-{\nu }_{xy,i}^2{\displaystyle \frac{E_{y,i}}{E_{x,i}}}}& 0\\ & \frac{E_{y,i}}{1-{\nu }_{xy,i}^2{\displaystyle \frac{E_{y,i}}{E_{x,i}}}}& 0\\ sym.& & G_{xy,i}\end{array},), ,i,=,1,,,.,.,.,,,n\right]\right]$

   Stiffness Matrix for Orthotropic Layer

1. #banner.text@For orthotropic material, the shear modulus in the pane plane (G_xy ) is defined by means of the material values whereas for isotropic material it is determined from the modulus of elasticity and the transversal strain. Therefore, the Poisson's ratio with the "location-cause" principle is important for orthotropic material.

1. The shear stiffnesses for orthotropic material are as follows:

1. #table.uni#
2. width=10%|width=90%
3. G_xy |Shear modulus in the wall plane (e.g. 690 N/mm² for C24)
4. G_xz | Shear modulus in direction x over the thickness (e.g. 690 N/mm² for C24)
5. G_yz | Shear modulus in direction y over the thickness (e.g. 690 N/mm² for C24) – also called "rolling shear modulus".

1. Furthermore, orthotropic material has the special feature that directional stiffnesses can be defined in a surface. In the default case, the local orientation of the surface or layer in the x-direction corresponds to the stiffness in the x-direction. However, since this can be freely defined by means of the angle β in the 'Layers' thickness type, it is necessary to transform the stiffnesses accordingly.

1. $d_i=\left[\begin{array}{ccc}{d}_{11,i}& d_{12,i}& d_{13,i}\\ & d_{22,i}& d_{23,i}\\ sym.& & d_{33,i}\end{array},),=,T_{3x3,i}^T,d,{\text{'}}_i,T_{3x3,i}\right]$

   Transformation Stiffnesses

1. $T_{3x3,i}=\left[\begin{array}{ccc}c²& s²& cs\\ s²& c²& -cs\\ -2cs& 2cs& c²-s²\end{array},), ,w,h,e,r,e, ,c,=,\cos ,(,{\beta }_i,),,, ,s,=,\sin ,(,{\beta }_i,)\right]$

   Transformation Matrix

1. Summed element of each layer:
2. $d_{11,i}=c^{4}d{\text{'}}_{11,i}+2c²s²d{\text{'}}_{12,i}+s^{4}d{\text{'}}_{22,i}+4c²s²d{\text{'}}_{33,i}$
   
   $d_{12,i}=c^2{s}^{2}d{\text{'}}_{11,i}+s^{4}d{\text{'}}_{12,i}+c^{4}d{\text{'}}_{12,i}+c²s²d{\text{'}}_{22,i}-4c²s²d{\text{'}}_{33,i}$
   
   $d_{13,i}=c^{3}sd{\text{'}}_{11,i}+c{s}^{3}d{\text{'}}_{12,i}-c^{3}sd{\text{'}}_{12,i}-c{s}^{3}d{\text{'}}_{22,i}-2c^{3}sd{\text{'}}_{33,i}+2c{s}^{3}d{\text{'}}_{33,i}$
   
   $d_{22,i}=s^{4}d{\text{'}}_{11,i}+2c²s²d{\text{'}}_{12,i}+c^{4}d{\text{'}}_{22,i}+4c²s²d{\text{'}}_{33,i}$
   
   $d_{23,i}=c{s}^{3}d{\text{'}}_{11,i}+c^{3}sd{\text{'}}_{12,i}-c{s}^{3}d{\text{'}}_{12,i}-c^{3}sd{\text{'}}_{22,i}+2c^{3}sd{\text{'}}_{33,i}-2c{s}^{3}d{\text{'}}_{33,i}$
   
   $d_{33,i}=c^2{s}^{2}d{\text{'}}_{11,i}-2c²s²d{\text{'}}_{12,i}+c^2{s}^{2}d{\text{'}}_{22,i}+{\left(}^{c^2-s^2}d{\text{'}}_{33,i}$

   Individual Terms by Layer

1. === Bending and torsional elements [Nm] ===

1. The matrix elements for bending and torsion are given in the equations below.

1. $D_{11}=\sum _{i=1}^n\frac{z_{max,i}^3-z_{min,i}^3}{3}{d}_{11,i}$

   Bending Stiffness in x

1. $D_{12}=\sum _{i=1}^n\frac{z_{max,i}^3-z_{min,i}^3}{3}{d}_{12,i}$

   Eccentricity Term (12)

1. $D_{22}=\sum _{i=1}^n\frac{z_{max,i}^3-z_{min,i}^3}{3}{d}_{22,i}$

   Bending Stiffness in y

1. $D_{13}=\sum _{i=1}^n\frac{z_{max,i}^3-z_{min,i}^3}{3}{d}_{13,i}$

   Eccentricity Term (13)

1. $D_{23}=\sum _{i=1}^n\frac{z_{max,i}^3-z_{min,i}^3}{3}{d}_{23,i}$

   Eccentricity Term (23)

1. $D_{33}=\sum _{i=1}^n\frac{z_{max,i}^3-z_{min,i}^3}{3}{d}_{33,i}$

   Torsional Stiffness

1. If there is only one layer of the thickness type 'Layers', the calculation is based on the parameters described in the RFEM manual]].

1. For the shear (Element D44/55) different equations apply in the thickness type 'Layers'. They are described in the Shear in Slab Plane section.

1. === Eccentricity Terms [Nm/m] ===

1. For asymmetrical plates, eccentricity terms arise. An asymmetrical surface can be, for example, in a fire resistance design due to one-sided charring of a cross‑laminated timber plate. The matrix elements are as follows:

1. $D_{16}=\sum _{i=1}^n\frac{z_{max,i}^2-z_{min,i}^2}{2}{d}_{11,i}$

   Eccentricity Element D16

1. $D_{17}=\sum _{i=1}^n\frac{z_{max,i}^2-z_{min,i}^2}{2}{d}_{12,i}$

   Eccentricity Element D17

1. $D_{18}=\sum _{i=1}^n\frac{z_{max,i}^2-z_{min,i}^2}{2}{d}_{13,i}$

   Eccentricity Element D18

1. $D_{27}=\sum _{i=1}^n\frac{z_{max,i}^2-z_{min,i}^2}{2}{d}_{22,i}$

   Eccentricity Element D27

1. $D_{28}=\sum _{i=1}^n\frac{z_{max,i}^2-z_{min,i}^2}{2}{d}_{23,i}$

   Eccentricity Element D28

1. $D_{38}=\sum _{i=1}^n\frac{z_{max,i}^2-z_{min,i}^2}{2}{d}_{33,i}$

   Eccentricity Element D38

1. === Layer Plane [N/m] ===

1. In the "Pane Wall" plane, the normal stiffnesses are represented in the plane of the glass pane. The shear force in the pane is calculated using element D88. The matrix elements are as follows:

1. $D_{66}=\sum _{i=1}^n{t}_i{d}_{11,i}$

   Normal Stiffness x D66

1. $D_{67}=\sum _{i=1}^n{t}_i{d}_{12,i}$

   Normal Stiffness Eccentric D67

1. $D_{68}=\sum _{i=1}^n{t}_i{d}_{13,i}$

   Normal Stiffness Eccentric D68

1. $D_{77}=\sum _{i=1}^n{t}_i{d}_{22,i}$

   Normal Stiffness y D77

1. $D_{78}=\sum _{i=1}^n{t}_i{d}_{23,i}$

   Normal Stiffness Eccentric D78

1. $D_{88}=\sum _{i=1}^n{t}_i{d}_{33,i}$

   Plate Shear D88

1. === Shear in slab plane [N/m] ===

1. To determine the shear stiffness for an orthotropic material, you have to rotate the stiffnesses according to their orientation to the local surface axis. This has to be done for each layer of the thickness type 'Layers'. In a simple layer structure with a 0° orientation of the cover layer and a 90° orientation of the underlying layer, there is a high shear stiffness, which has to be considered accordingly for the multilayer model. The following image (Source [1]) shows this using an example of a cross-laminated timber plate.

1. 

   Shear Deformation (Source: Information Service Timber)

1. In the laminate theory, the shear stiffness of a layered structure is calculated by transforming all bending and shear components in the respective directions of each layer. You can find more information about this in the literature listed below.

1. 

   Stiffness Orientation

1. By using the transformation of the stiffness shown in the figure, the stiffnesses are added up. This summation is also known as the "Grashoff integral."

1. $D_{44,calc}^{"}=\frac{1}{{\int }_{-t/2}^{t/2}{\displaystyle \frac{{\displaystyle G_{22}^{"}\left(z\right)}}{{\displaystyle G_{11}^{"}\left(z\right)G_{22}^{"}\left(z\right)-{G_{12}^{"}}^2\left(z\right)}}}{\left(}^{{\displaystyle \frac{{\int }_z^{t/2}{E}_x^{"}\left(\overline{z}\right)(\overline{z}-z_{0,x})d\overline{z}}{{\int }_{-t/2}^{t/2}{E}_x^{"}\left(\overline{z}\right){(\overline{z}-z_{0,x})}^{2}d\overline{z}}}}dz}$

   Shear Stiffness in x-Direction

1. In order to calculate the stiffness in the x- and y-directions, a centroid of stiffness is calculated for each structure of a multilayer surface.

1. $z_{0,x}=\frac{{\int }_{-t/2}^{t/2}{E}_x^{"}\left(\overline{z}\right)\overline{z}d\overline{z}}{{\int }_{-t/2}^{t/2}{E}_x^{"}\left(\overline{z}\right)d\overline{z}}=\frac{I_1}{I_0}$

   Center of Stiffness in x

1. $D_{55,calc}^{"}=\frac{1}{{\int }_{-t/2}^{t/2}{\displaystyle \frac{G_{11}^{"}\left(z\right)}{G_{11}^{"}\left(z\right)G_{22}^{"}\left(z\right)-{G_{12}^{"}}^2\left(z\right)}}{\left(}^{{\displaystyle \frac{{\int }_z^{t/2}{E}_y^{"}\left(\overline{z}\right)(\overline{z}-z_{0,x})d\overline{z}}{{\int }_{-t/2}^{t/2}{E}_y^{"}\left(\overline{z}\right){(\overline{z}-z_{0,x})}^{2}d\overline{z}}}}dz}$

   Shear Rigidity in y-Direction

1. Center of stiffness in y-direction:
2. $z_{0,x}=\frac{{\int }_{-t/2}^{t/2}{E}_y^{"}\left(\overline{z}\right)\overline{z}d\overline{z}}{{\int }_{-t/2}^{t/2}{E}_y^{"}\left(\overline{z}\right)d\overline{z}}=\frac{I_1}{I_0}$

   Center of Stiffness in y-Direction

1. In order to determine the orientation per position in the calculation of the shear stiffness, the stiffnesses are determined according to the following equations.

1. $G_{11,i}^{"}={\cos }^2(\phi -{ß}_i)G_{xz,i}+{\sin }^2(\phi -{ß}_i)G_{yz,i}$

   Oriented Shear Stiffness G11

1. G stands for the shear stiffness of the layers in order to avoid mistakes for the elements of the stiffness matrix (D).

1. The shear stiffness of each layer can also be displayed in matrix form as follows:

1. $\left[\begin{array}{cc}{G}_{11}& G_{12}\\ G_{21}& G_{22}\end{array},)\right]$

   Shear Stiffness Matrix

1. $G_{22,i}^{"}={\sin }^2(\phi -{ß}_i)G_{xz,i}+{\cos }^2(\phi -{ß}_i)G_{yz,i}$

   Oriented Shear Stiffness G22

1. The eccentric shear stiffness in the following equation would always be zero and thus irrelevant for the symmetrical structure of cross-laminated timber (0°/90°/0°) mentioned above. In the case of diagonally glued cross-laminated timber DLT ( Diagonal Laminated Timber ), for example, this eccentricity element is not zero and therefore plays an important role.

1. $G_{12}^{"}=(G_{xz,i}-G_{yz,i})\sin (\phi -{ß}_i)\cos (\phi -{ß}_i)$

   Eccentric Oriented Shear Stiffness G12

1. Further information can be found in [4] and in this YouTube video.

1. ==== Calculation of shear stiffness ====

1. The shear stiffness is determined in the following steps:

1. #First, the angle of maximum stiffness is determined. The angle φ shows the modification of the local coordinate system x of the surface with respect to the oriented direction x ''.
2. #All stiffnesses are rotated in the oriented direction x according to the equations presented above .
3. #The pane stiffness matrix of each layer (3 x 3) is transformed from the local coordinate system x', y' to the rotated system x'', y". In addition to calculating the directed shear stiffness of each individual layer, this is also done for the moduli of elasticity of each layer.
   $E_{x,i}^{"}=d_{12,i}^{"}+\frac{2d_{12,i}^{"}{d}_{13,i}^{"}{d}_{23,i}^{"}-d_{22,i}^{"}{\left(d_{13,i}^{"}\right)}^2-d_{33,i}^{"}{\left(d_{12,i}^{"}\right)}^2}{d_{22,i}^{"}{d}_{33,i}^{"}-{\left(d_{23,i}^{"}\right)}^2}$

   Oriented Modulus of Elasticity in x-Direction
   $E_{y,i}^{"}=d_{22,i}^{"}+\frac{2d_{12,i}^{"}{d}_{13,i}^{"}{d}_{23,i}^{"}-d_{11,i}^{"}{\left(d_{23,i}^{"}\right)}^2-d_{33,i}^{"}{\left(d_{12,i}^{"}\right)}^2}{d_{11,i}^{"}{d}_{33,i}^{"}-{\left(d_{13,i}^{"}\right)}^2}$

   Oriented Modulus of Elasticity in y-Direction
4. #The shear stiffness is calculated with the equations (Grashoff integral) described above. The shear stiffness is calculated using the individual parts.
   $I={\int }_{-h/2}^{h/2}{d}_{11}\left(z\right){(z-z_{0,x})}^{2}dz=\sum _{i=1}^n{d}_{11,i}\frac{{(z_{max,i}-z_{0,x})}^3-{(z_{min,i}-z_{0,x})}^3}{3}$

   Second Moment of Area in x-Direction
   ${\chi }_i=d_{11,i}\frac{{(z_{max,i}-z_{0,x})}^2}{2}+\sum _{k=i+1}^n{d}_{11,k}\frac{{(z_{max,i}-z_{0,x})}^2-{(z_{min,i}-z_{0,x})}^2}{2}$

   Equivalent Stiffness by Layer (in x)
   Here, the equations are only displayed for the x-direction (D44). The same applies to the y-direction. The equivalent stiffness (Stinter component) is calculated for each layer.
5. #Finally, the calculated stiffnesses for the oriented direction of the entire structure are recalculated using the angular relations and shown as the original stiffnesses D44, D55, and D45 in the stiffness matrix.
   $D_{44}={\cos }^2\left(\phi \right)D_{44}^{"}+{\sin }^2\left(\phi \right)D_{55}^{"}$

   Recalculated Shear Stiffness D44
   $D_{55}={\sin }^2\left(\phi \right)D_{44}^{"}+{\cos }^2\left(\phi \right)D_{55}^{"}$

   Recalculated Shear Stiffness D55
   $D_{45}=\sin \left(\phi \right)\cos \left(\phi \right)(D_{44}^{"}-D_{55}^{"})$

   Recalculated Shear Stiffness D45