Online Manuals RFEM 6 | Multilayer Surfaces

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003636

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

2023-10-20

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Overview

Multilayer Surfaces Add-on

Theory

Input

Results

Calculation Examples

Stresses

Strengths

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

Symbol	Strength	Graphic
f_m,0,k	Flexural strength	Bending Stiffness
f_t,0,k	Tensile strength	Tensile Strength
f_t,90,k	Tensile strength perpendicular	Tensile Strength Perpendicular
f_c,0,k	Compressive strength	Compressive Strength
f_c,90,k	Compressive strength perpendicular	Perpendicular Compressive Strength
f_v,xz,k	Shear strength	Shear Strength
f_v,xy,k	Shear strength (Failure Mode 1)	Shear Strength (Failure Mode 1)
f_v,net,k	Shear strength (Failure Mode 2)	Shear Strength (Failure Mode 2)
f_v,tor,k	Shear strength (Failure Mode 3)	Shear Strength (Failure Mode 3)
f_v,yz,k	Rolling shear strength	Rolling Shear Strength

The basic stresses are calculated in RFEM using the finite element method. The corresponding equations are presented in the chapter Stresses of the RFEM manual.

Basic Equation

σ (z) = d_{i} ε (z)

Stress Calculation

Strain Calculation

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 above equation 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 pure bending stress without axial force, the calculation of the total strain is simplified by using the inverse of the total panel stiffness. This is displayed for the X-direction in the following equation.

ε_{x, g} = D 11^{- 1} \cdot m_{x}

Total Strain Bending x

The stress for a complete structure is determined from this total strain. The stress at the individual integration points is then calculated 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 process using a three-layer panel as an example. The stress distribution on the left represents the initial stress, while the stress distribution on the right represents the resulting stress in each layer. Since the middle layer in this example has a modulus of elasticity E_y = 0, there can be no significant stress there.

Stress Distribution Total (Left) and Local (Right)

Shear Stresses

Three types of failure modes (FM) can be distinguished for shear stress in the panel plane.

Failure Mode 1

Failure Mode 1: Shear – Gross Section

Failure Mode 2

Failure Mode 2: Net Section Shear

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	Brettbreite in x- oder y-Richtung

Failure Mode 2

Failure Mode 3

In principle, this failure mode corresponds to torsion design. It is verified here that the shear stresses caused by the shearing of the intersection surfaces of the overlapping boards do not become too great.

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 Mode 3

The value e_x or e_y refers to the board width in the x or y direction with the joint.

Rolling Shear Stresses

The rolling shear stresses are taken into account using the resultant of the shear stresses in the thickness direction x and y via the following equation.

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

Rolling Shear Stresses

With a typical orientation of a cross-laminated timber panel of 0° / 90° for the individual layers, the shear stress in the x and y direction is therefore compared with the rolling shear.

Parent Chapter

Theory