Online Manuals RFEM 6 | Multilayer Surfaces

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Dipl.-Ing. (FH) Bastian Kuhn, M.Sc.

2023-10-19

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Overview

Multilayer Surfaces Add-On

Analysis Method

Input

Results

Calculation examples

Stiffnesses for Multilayer Surfaces

Material Models

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.

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.

The following list shows a selection of the possible combinations:

Isotropic layers (e.g. concrete - steel)
Orthotropic layers (for example, cross-laminated timber)
Isotropic - orthotropic (e.g. steel - GFRP)
Isotropic Plastic - Isotropic (e.g. concrete - steel)
Isotropic Nonlinear Elastic - Orthotropic (e.g. Concrete - Timber)
Isotropic - Orthotropic Plastic (e.g. Concrete - Timber)
Isotropic Damage - Orthotropic (e.g. Concrete - Timber)

Info

For the combination of nonlinear materials, the Nonlinear Material Behavior should be activated.

Stiffnesses for multilayer surfaces without solids

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.

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.

Calculation Process for Multilayer Surfaces

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

stiffness matrix calculation

The material models are based on the following conditions (see Chapter Materials of the RFEM manual):

All stiffness values ≥ 0
Overall stiffness matrix of surface must be positive definite.
Basic equation isotropic:

E = 2 G (1 + ν)

E	Modulus of elasticity
G	Shear modulus
ν	Poisson's ratio

Modulus of Elasticity, Shear Modulus, and Poisson's Ratio

Basic Equation Orthotropic:

\frac{ν_{y x}}{E_{y}} = \frac{ν_{x y}}{E_{x}}

Transversal Strain Relation

Local stiffness matrix of each layer

Isotropic

d_{i}' = [\begin{matrix} d'_{11, i} & d'_{12, i} & 0 \\ d'_{22, i} & 0 \\ s y m . & d'_{33, i} \end{matrix}] = [\begin{matrix} \frac{E_{i}}{1 - ν_{i}^{2}} & \frac{ν_{i} E_{i}}{1 - ν_{i}^{2}} & 0 \\ \frac{E_{i}}{1 - ν_{i}^{2}} & 0 \\ s y m . & G_{i} \end{matrix}] i = 1, . . ., n w h e r e G_{i} = \frac{E_{i}}{2 (1 + ν_{i})}

Stiffness Matrix for Isotropic Layer

Orthotropic

d_{i}' = [\begin{matrix} d'_{11, i} & d'_{12, i} & 0 \\ d'_{22, i} & 0 \\ s y m . & d'_{33, i} \end{matrix}] = [\begin{matrix} \frac{E_{x, i}}{1 - ν_{x y, i}^{2} \frac{E_{y, i}}{E_{x, i}}} & \frac{ν_{x y, i} E_{y, i}}{1 - ν_{x y, i}^{2} \frac{E_{y, i}}{E_{x, i}}} & 0 \\ \frac{E_{y, i}}{1 - ν_{x y, i}^{2} \frac{E_{y, i}}{E_{x, i}}} & 0 \\ s y m . & G_{x y, i} \end{matrix}] i = 1, . . ., n

Stiffness Matrix for Orthotropic Layer

Info

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.

The shear stiffnesses for orthotropic material are as follows:

G_xy	Shear modulus in the panel plane (e.g. 690 N/mm² for C24)
G_xz	Shear modulus in the direction x over the thickness (e.g. 690 N/mm² for C24)
G_yz	Shear modulus in direction y over the thickness (e.g. 690 N/mm² for C24) – also called "rolling shear modulus".

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.

d_{i} = [\begin{matrix} d_{11, i} & d_{12, i} & d_{13, i} \\ d_{22, i} & d_{23, i} \\ s y m . & d_{33, i} \end{matrix}] = T_{3 x 3, i}^{T} d'_{i} T_{3 x 3, i}

Transformation Stiffnesses

T_{3 x 3, i} = [\begin{matrix} c ² & s ² & c s \\ s ² & c ² & - c s \\ - 2 c s & 2 c s & c ² - s ² \end{matrix}] w h e r e c = \cos (β_{i}), s = \sin (β_{i})

Transformation Matrix

Summed element of each layer:

d_{11, i} = c^{4} d'_{11, i} + 2 c ² s ² d'_{12, i} + s^{4} d'_{22, i} + 4 c ² s ² d'_{33, i}

d_{12, i} = c^{2} s^{2} d'_{11, i} + s^{4} d'_{12, i} + c^{4} d'_{12, i} + c ² s ² d'_{22, i} - 4 c ² s ² d'_{33, i}

d_{13, i} = c^{3} s d'_{11, i} + c s^{3} d'_{12, i} - c^{3} s d'_{12, i} - c s^{3} d'_{22, i} - 2 c^{3} s d'_{33, i} + 2 c s^{3} d'_{33, i}

d_{22, i} = s^{4} d'_{11, i} + 2 c ² s ² d'_{12, i} + c^{4} d'_{22, i} + 4 c ² s ² d'_{33, i}

d_{23, i} = c s^{3} d'_{11, i} + c^{3} s d'_{12, i} - c s^{3} d'_{12, i} - c^{3} s d'_{22, i} + 2 c^{3} s d'_{33, i} - 2 c s^{3} d'_{33, i}

d_{33, i} = c^{2} s^{2} d'_{11, i} - 2 c ² s ² d'_{12, i} + c^{2} s^{2} d'_{22, i} + {(c^{2} - s^{2})}^{2} d'_{33, i}

Individual Terms by Layer

Bending and torsional elements [Nm]

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

D_{11} = \sum_{i = 1}^{n} \frac{z_{m a x, i}^{3} - z_{m i n, i}^{3}}{3} d_{11, i}

Bending Stiffness in x

D_{12} = \sum_{i = 1}^{n} \frac{z_{m a x, i}^{3} - z_{m i n, i}^{3}}{3} d_{12, i}

Eccentricity Term (12)

D_{22} = \sum_{i = 1}^{n} \frac{z_{m a x, i}^{3} - z_{m i n, i}^{3}}{3} d_{22, i}

Bending Stiffness in y

D_{13} = \sum_{i = 1}^{n} \frac{z_{m a x, i}^{3} - z_{m i n, i}^{3}}{3} d_{13, i}

Eccentricity Term (13)

D_{23} = \sum_{i = 1}^{n} \frac{z_{m a x, i}^{3} - z_{m i n, i}^{3}}{3} d_{23, i}

Eccentricity Term (23)

D_{33} = \sum_{i = 1}^{n} \frac{z_{m a x, i}^{3} - z_{m i n, i}^{3}}{3} d_{33, i}

Torsional Stiffness

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

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

Eccentricity terms [Nm/m]

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:

D_{16} = \sum_{i = 1}^{n} \frac{z_{m a x, i}^{2} - z_{m i n, i}^{2}}{2} d_{11, i}

Eccentricity Element D16

D_{17} = \sum_{i = 1}^{n} \frac{z_{m a x, i}^{2} - z_{m i n, i}^{2}}{2} d_{12, i}

Eccentricity Element D17

D_{18} = \sum_{i = 1}^{n} \frac{z_{m a x, i}^{2} - z_{m i n, i}^{2}}{2} d_{13, i}

Eccentricity Element D18

D_{27} = \sum_{i = 1}^{n} \frac{z_{m a x, i}^{2} - z_{m i n, i}^{2}}{2} d_{22, i}

Eccentricity Element D27

D_{28} = \sum_{i = 1}^{n} \frac{z_{m a x, i}^{2} - z_{m i n, i}^{2}}{2} d_{23, i}

Eccentricity Element D28

D_{38} = \sum_{i = 1}^{n} \frac{z_{m a x, i}^{2} - z_{m i n, i}^{2}}{2} d_{33, i}

Eccentricity Element D38

Plate Plane [N/m]

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:

D_{66} = \sum_{i = 1}^{n} t_{i} d_{11, i}

Normal Stiffness x D66

D_{67} = \sum_{i = 1}^{n} t_{i} d_{12, i}

Normal Stiffness Eccentric D67

D_{68} = \sum_{i = 1}^{n} t_{i} d_{13, i}

Normal Stiffness Eccentric D68

D_{77} = \sum_{i = 1}^{n} t_{i} d_{22, i}

Normal Stiffness y D77

D_{78} = \sum_{i = 1}^{n} t_{i} d_{23, i}

Normal Stiffness Eccentric D78

D_{88} = \sum_{i = 1}^{n} t_{i} d_{33, i}

Plate Shear D88

Shear in slab plane [N/m]

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.

Shear Deformation (Source: Information Service Timber)

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.

Stiffness Orientation

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."

D_{44, c a l c}^{"} = \frac{1}{\int_{- t / 2}^{t / 2} \frac{G_{22}^{"} (z)}{G_{11}^{"} (z) G_{22}^{"} (z) - {G_{12}^{"}}^{2} (z)} {(\frac{\int_{z}^{t / 2} E_{x}^{"} (\bar{z}) (\bar{z} - z_{0, x}) d \bar{z}}{\int_{- t / 2}^{t / 2} E_{x}^{"} (\bar{z}) {(\bar{z} - z_{0, x})}^{2} d \bar{z}})}^{2} d z}

Shear Stiffness in x-Direction

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.

z_{0, x} = \frac{\int_{- t / 2}^{t / 2} E_{x}^{"} (z) z d z}{\int_{- t / 2}^{t / 2} E_{x}^{"} (z) d z} = \frac{I_{1}}{I_{0}}

Center of Stiffness in x

D_{55, c a l c}^{"} = \frac{1}{\int_{- t / 2}^{t / 2} \frac{G_{11}^{"} (z)}{G_{11}^{"} (z) G_{22}^{"} (z) - {G_{12}^{"}}^{2} (z)} {(\frac{\int_{z}^{t / 2} E_{y}^{"} (\bar{z}) (\bar{z} - z_{0, x}) d \bar{z}}{\int_{- t / 2}^{t / 2} E_{y}^{"} (\bar{z}) {(\bar{z} - z_{0, x})}^{2} d \bar{z}})}^{2} d z}

Shear Rigidity in y-Direction

Center of stiffness in y-direction:

z_{0, x} = \frac{\int_{- t / 2}^{t / 2} E_{y}^{"} (z) z d z}{\int_{- t / 2}^{t / 2} E_{y}^{"} (z) d z} = \frac{I_{1}}{I_{0}}

Center of Stiffness in y-Direction

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

G_{11, i}^{"} = \cos^{2} (φ - ß_{i}) G_{x z, i} + \sin^{2} (φ - ß_{i}) G_{y z, i}

Oriented Shear Stiffness G11

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

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

[\begin{matrix} G_{11} & G_{12} \\ G_{21} & G_{22} \end{matrix}]

Shear Stiffness Matrix

G_{22, i}^{"} = \sin^{2} (φ - ß_{i}) G_{x z, i} + \cos^{2} (φ - ß_{i}) G_{y z, i}

Oriented Shear Stiffness G22

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.

G_{12}^{"} = (G_{x z, i} - G_{y z, i}) \sin (φ - ß_{i}) \cos (φ - ß_{i})

Eccentric Oriented Shear Stiffness G12

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

Calculation of shear stiffness

The shear stiffness is determined in the following steps:

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''.
All stiffnesses are rotated in the oriented direction x'' according to the equations presented above.
The pane stiffness matrix of each position (3 x 3) is transformed from the local coordinate system x', y' to the rotated system x'', y". In addition to calculating the directed-stiffness of each individual layer, this is also done for the moduli of elasticity of each layer.

$E_{x, i}^{"} = d_{12, i}^{"} + \frac{2 d_{12, i}^{"} d_{13, i}^{"} d_{23, i}^{"} - d_{22, i}^{"} {(d_{13, i}^{"})}^{2} - d_{33, i}^{"} {(d_{12, i}^{"})}^{2}}{d_{22, i}^{"} d_{33, i}^{"} - {(d_{23, i}^{"})}^{2}}$

Oriented Modulus of Elasticity in x-Direction

$E_{y, i}^{"} = d_{22, i}^{"} + \frac{2 d_{12, i}^{"} d_{13, i}^{"} d_{23, i}^{"} - d_{11, i}^{"} {(d_{23, i}^{"})}^{2} - d_{33, i}^{"} {(d_{12, i}^{"})}^{2}}{d_{11, i}^{"} d_{33, i}^{"} - {(d_{13, i}^{"})}^{2}}$

Oriented Modulus of Elasticity in y-Direction
The shear stiffness is calculated with the equations (Grashoff integral) described above. The shear stiffness is calculated using the individual parts.

$I = \int_{- h / 2}^{h / 2} d_{11} (z) {(z - z_{0, x})}^{2} d z = \sum_{i = 1}^{n} d_{11, i} \frac{{(z_{m a x, i} - z_{0, x})}^{3} - {(z_{m i n, i} - z_{0, x})}^{3}}{3}$

Second Moment of Area in x-Direction

$χ_{i} = d_{11, i} \frac{{(z_{m a x, i} - z_{0, x})}^{2}}{2} + \sum_{k = i + 1}^{n} d_{11, k} \frac{{(z_{m a x, i} - z_{0, x})}^{2} - {(z_{m i n, i} - z_{0, x})}^{2}}{2}$

Equivalent Stiffness by Layer (in x)

Here, the equations are only shown for the x-direction (D44). The same applies to the y-direction. The equivalent stiffness (Stinter component) is calculated for each layer.
The calculated stiffnesses for the oriented direction of the entire structure are finally calculated back by means of the angular relations and shown as the original stiffnesses D44, D55, and D45 in the stiffness matrix.

$D_{44} = \cos^{2} (φ) D_{44}^{"} + \sin^{2} (φ) D_{55}^{"}$

Recalculated Shear Stiffness D44

$D_{55} = \sin^{2} (φ) D_{44}^{"} + \cos^{2} (φ) D_{55}^{"}$

Recalculated Shear Stiffness D55

$D_{45} = \sin (φ) \cos (φ) (D_{44}^{"} - D_{55}^{"})$

Recalculated Shear Stiffness D45

Increasing shear stiffness

Since geometries with very narrow surface strips are also possible by modeling laminates as a surface, the shear stiffness must be increased accordingly when calculating such problematic geometries.

The following equation shows this for the X-direction

D_{44}^{"} = \max (D_{44, calc}^{"}, \frac{48}{5 l ²} \frac{1}{\frac{1}{\sum_{i = 1}^{n} E_{x, i}^{"} \frac{t_{i}^{3}}{12}} - \frac{1}{\sum_{i = 1}^{n} E_{x, i}^{"} \frac{z_{\max, i}^{3} - z_{\min, i}^{3}}{3}}})

Maximum Shear Stiffness

The length l in the equation above means the shortest length of a box that can be placed over the corresponding geometry.

In another model, which can be downloaded on the right, a narrow surface of 10 cm in width was compared with an identical surface of 20 cm in width.

The shear stiffness of the narrow surface is D44=15,253 kN/m compared to D44=5,970.8 kN/m of the wider surface. As a result, the deformation of the stiffer surface is smaller and the shear load is greater, despite the identical loading.

Stiffnesses for multilayer surfaces with integrated solids

In the future, it will also be possible to define solids together with surfaces in the Multilayer Surfaces add-on. In the case of this type, a surface is also exported to RFEM. Since the generation of stiffnesses and the decomposition of internal forces is more time-consuming, this is explained separately.

References

Cross-Laminated Timber Building - Load-Bearing Mass Timber Elements for Walls, Floors and Roofs - Series 4 Part 6 Part 1. Information Service Holz
Planar surface structures: Basics of modeling and calculation of walls and plates
Jones, R. M. (n.d.). Mechanics of Composite Materials (2nd ed.). Taylor & Francis Inc., Philadelphia.
Arnold, M.: Mechanical Properties of Diagonal Laminated Timber (DLT) with Respect to Point-Supported Mass Timber Slabs .Dissertationin preparation. Chair of Timber and Building Construction, Technical University of Munich, Germanyexpected2023.

Parent Chapter

Analysis Method

Model

445x

10x

Multilayer Models

Shear Stiffness