Online Manuals RFEM 6 | Multilayer Surfaces

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

2023-10-20

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Overview

Multilayer Surfaces Add-On

Analysis Method

Input

Results

Calculation examples

Stress

Strengths

The following table shows the strengths relevant for the stress analysis of cross-laminated timber.

Symbol	Strength	Graphic
f_m,0,k	flexural strength
f_t,0,k	Tensile Strength
f_t,90,k	Strength for tension perpendicular
f_c,0,k	Compressive Strength
f_c,90,k	Strength for compression perpendicular
f_v,xz,k	Shear strength
f_v,xy,k	Shear strength (failure mechanism 1)
f_v,net,k	Shear strength (failure mechanism 2)
f_v,tor,k	Shear strength (failure mechanism 3)
f_v,yz,k	Rolling Shear Strength

Basic stresses in RFEM are calculated according to the finite element method. The corresponding equations are described in Chapter Stresses of the RFEM manual.

Basic Equation

σ (z) = d_{i} ε (z)

Stress Calculation

Calculation of strain

ε_{t o t} = [\begin{matrix} \begin{matrix} (\frac{\partial u}{\partial x}) + (\frac{\partial φ_{y}}{\partial x}) \\ (\frac{\partial v}{\partial y}) - (\frac{\partial φ_{x}}{\partial y}) \\ \frac{\partial φ_{y}}{\partial y} - \frac{\partial φ_{x}}{\partial x} \end{matrix} \\ \begin{matrix} \frac{\partial w}{\partial x} + φ_{y} \\ \frac{\partial w}{\partial y} - φ_{x} \\ \frac{\partial u}{\partial x} \\ \frac{\partial v}{\partial y} \\ \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \end{matrix} \end{matrix}] \cdot z

Strain Multilayer Surfaces

Bending stresses

The strain is determined according to the equation above as follows:

ε_{x} = (\frac{\partial u}{\partial x}) + (\frac{\partial φ_{y}}{\partial x}) \cdot z

Strain x Multilayer Surfaces

In the case of the pure bending stress without axial force, the calculation of the total strain is simplified by means of the inverse of the total plate stiffness. The following equation shows this for the X-direction.

ε_{x, g} = D 11^{- 1} \cdot m_{x}

Total Strain Bending x

The stress for the entire structure is determined from this total strain. The stress is then calculated on the individual integration points using the local stiffness matrix of each layer.

ε_{x, i} = ε_{x, g} \cdot z_{i}

Strain by Integration Point

σ_{x, i} = d 11 \cdot ε_{x, i}

Stress by Integration Point

The following image shows this method as an example for a three-layer plate. The stress diagram on the left represents the initial stress, and the stress diagram on the right represents the resulting stress in each layer. Since the middle layer has the modulus of elasticity E_y = 0 in this example, there can be no stresses worth mentioning.

Stress Distribution Total (Left) and Local (Right)

Shear Stresses

There are three types of failure modes (CM) to be distinguished for shear under loading in the slab plane.

Failure Mode 1

Failure Mode 1: Shear - Gross Section

Failure Mechanism 2

Failure Mode 2: Shear - Net Section

η = \frac{τ_{x, netto}}{f_{v, n e t t o}} = \frac{\frac{n_{x y} e_{x}}{b_{x} \sum_{i, y} t_{i}}}{f_{v, n e t t o}} \leq 1

η = \frac{τ_{y, netto}}{f_{v, n e t t o}} = \frac{\frac{n_{x y} e_{y}}{b_{y} \sum_{i, x} t_{i}}}{f_{v, n e t t o}} \leq 1

n_xy	Shear force
e_x, e_y	Plank width with gap in the x or y direction
b_x, b_y	Plank width in the x or y direction

Failure Mechanism 2

Failure Mode 3

In principle, this failure mechanism corresponds to a torsion design. It is verified here that the shear stresses due to shearing the intersection surfaces of the boards lying on top of each other do not become too high.

Failure Mode 3: Shear - Adhesive Surfaces

η = \frac{τ_{t o r, x}}{f_{v, t o r}} + \frac{τ_{x} + τ_{x z}}{f_{R}} = \frac{\frac{3 n_{x y}}{e_{x} (n - 1)}}{f_{v, t o r}} + \frac{\frac{\frac{\partial n_{x}}{\partial x}}{n - 1} + τ_{x z}}{f_{R}} \leq 1

η = \frac{τ_{t o r, y}}{f_{v, t o r}} + \frac{τ_{y} + τ_{y z}}{f_{R}} = \frac{\frac{3 n_{x y}}{e_{y} (n - 1)}}{f_{v, t o r}} + \frac{\frac{\frac{\partial n_{y}}{\partial y}}{n - 1} + τ_{y z}}{f_{R}} \leq 1

Failure Mechanism 3

The value e_x or e_y describes the plank width in the x- or y-direction with gap.

Rolling Shear Stresses

The rolling shear stresses are taken into account by the resultants of the shear stresses in the thickness direction x and y according to the following equation.

{τ_{y z}}^{'} = - τ_{x z} \sin β + τ_{y z} \cos β

Rolling Shear Stresses

Thus, for a common orientation of a cross-laminated timber plate of 0°/90° of the individual layers, the shear stress in the x- or y-direction is compared to the rolling shear.

Parent Chapter

Analysis Method