Appendix A: Basic Relations of Elasticity
Isotropic Material
Hooke Law 3D (Principal Stress and Strain)
Hooke Law for 2D Plan Stress
;
;
Hooke Law for 2D Plane Strain
;
;
E, | E,G | E,B | G, | G, B | B, | ||
---|---|---|---|---|---|---|---|
E | E | E | E | 2(1+v)G | 3(1-2v)B | ||
G | G | G | |||||
B=K | B | B | B | ||||
Orthotropic Material
General 3D Orthotropic Case
- Three Young's modulus in orthotropic directions 1, 2 and 3: , ,
- Three shear modulus in planes 12, 13 and 23: , ,
- Three Poisson ratio's satisfying the relations:
; ;
;
2D In-plane Orthotropic Material
- Orthotropic plane 1-2, isotropic plane 2-3
- Orthotropy coefficients in the plane 1-2:
- Isotropy coefficients in plane 2-3:
- Five independent coefficients
;
Stiffness Matrix of Beam Element
Terms of the stiffness matrix:
For a rectangle cross-section: