004771

0001-01-01

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Overview

1 Introduction

2 Theoretical Background

2.7.10.9 Stiffness matrix of the material

Stiffness matrix of the material

Bending stiffness

$D_{11} = \frac{I_{0, ϕ_{1}} \cdot E}{1 - ν^{2}} = \frac{3.322 \cdot 10^{- 4} \cdot 11 \cdot 10^{9}}{1 - 0 . 0328^{2}} = 3656.74 kNm$

$D_{12} = D_{2.1} = 0.0328 \cdot \sqrt{(3.656 \cdot 10^{6} \cdot 7.344 \cdot 10^{6})} = 170.58 kNm$

Torsional stiffness

$D_{33} = \frac{1 - ν}{2} \cdot \sqrt{D_{1.1} \cdot D_{2.2}} = \frac{1 - 0.0328}{2} \cdot \sqrt{(3.656 \cdot 10^{6} \cdot 7.344 \cdot 10^{6})} = 2505.84 kNm$

Shear stiffness

$D_{44} = D_{55} = \frac{5}{6} \cdot G \cdot h = \frac{5}{6} \cdot 11.8 \cdot 10^{9} \cdot 0.2 = 1 966 670 kN/m$

Membrane stiffness

$D_{66} = \frac{E \cdot A_{ϕ_{1}}}{1 - ν^{2}} = \frac{11 \cdot 10^{9} \cdot 0.096}{1 - 0 . 0328^{2}} = 1 055 590 kN/m$

$D_{67} = D_{76} \cdot ν \cdot \sqrt{(D_{66} \cdot D_{77})} = 0.0328 \cdot \sqrt{(1 055.59 \cdot 10^{6} \cdot 2505.84 \cdot 10^{6})} = 50 210.6 kN/m$

$D_{88} = G \cdot h = 11.8 \cdot 10^{9} \cdot 0.2 = 2 360 000 kN/m$

Eccentricity

$D_{16} = D_{61} = D_{61} \cdot e_{ϕ_{1}} = 1 055.590 \cdot 10^{9} \cdot 0.0393 = 41 499.2 kNm/m$

$D_{27} = D_{72} = D_{77} \cdot e_{ϕ_{1}} = 0$

$D_{17} = D_{71} = \frac{ν}{2} \cdot (e_{ϕ_{1}} + e_{ϕ_{2}}) \cdot \sqrt{D_{66} \cdot D_{77}} = = \frac{0.0328}{2} \cdot (0.393 + 0) \cdot \sqrt{1 055.59 \cdot 10^{6} \cdot 2505.84 \cdot 10^{6}} = 987.0 kNm / m$

$D_{38} = D_{83} = \frac{1}{2} \cdot G \cdot h \cdot (e_{ϕ_{1}} + e_{ϕ_{2}}) = \frac{1}{2} \cdot 11 \cdot 10^{9} \cdot 0.2 \cdot (0.393 + 0) = 46 390.2 kNm / m$

Image 2.125 Stiffness matrix of the material

Parent section

2.7.10 Example