# Chapter 2 - Macromechanical Analysis of a Lamina. Exercise Set. 2.1 The number of independent elastic constants in three dimensions are: 2.

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1 Chapter - Macromechanical Analysis of a Lamina Exercise Set. The number of independent elastic constants in three dimensions are: Anisotropic Monoclinic 3 Orthotropic 9 Transversely Orthotropic 5 Isotropic C Msi S Msi a) σ 8.69 σ 6.73 σ 3 τ kpa τ 3 τ 9 b) Compliance matrix -

2 S GPa c) E GPa E GPa E GPa ν ν ν G 4 GPa G 3 GPa d) G 3.5 GPa W Pa. S GPa Q GPa ε ε γ µm m

3 . Modifying Equation (.7) for an isotropic lamina under plane stress - G Inverting the compliance matrix gives the reduced stiffness matrix - E E. ν ν ν S Q E ν E. E ν ν E E ν E ν G.3 Compliance matrix for a two dimensional orthotropic material per Equations (.87) - ν E E S ν E E Matrix inversion yields - G. E ν. ν ν. E ν. ν Q ν. E ν. ν. E ν. ν G Compliance matrix for three dimensional orthotropic material per Equation (.7) -

4 E ν E ν 3 E ν E E ν 3 E S ν 3 E 3 ν 3 E 3 E 3 G 3 G 3 G Matrix inversion yields - ν. 3 ν 3 C. E ν. 3 ν. 3 ν ν.. ν ν 3 ν.. 3 ν 3 ν ν. 3 ν 3 ν 3 C 66 G Proving Q C and Q 66 C E Msi, E.99 Msi ν.6 G.66 Msi.6 σ 9.9 σ 5.98 MPa τ ε ε γ µin in.8

5 Q GPa S GPa.9 S S S S S S S 6 S 6 S 66 S 66 The values of c and s are interchanged for and 9 laminas. The values of S and S are interchanged also since the local axes for the lamina are rotated 9.. A) ε x ε y γ xy µm m σ 9.8 σ 6.98 MPa τ 6.34 ε ε γ µm m c) Principal normal stresses produced by applied global stresses - σ max σ x σ y σ x σ y τ xy σ max 6.6 MPa

6 σ min σ x σ y σ x σ y Orientation of maximum principal stress -. τ θ. xy. pσ atan 8. σ x σ y π Principal normal strains - τ xy σ min.63 MPa θ pσ ε max ε x ε y ε x ε y γ xy ε max 339. µm m ε min ε x ε y ε x ε y γ xy ε min 6.64 µm m Orientation of maximum principal strain - γ θ. xy. pε atan 8. ε x ε y π θ pε 39.3 d) Maximum shear produced by applied global stresses - τ max σ x σ y Orientation of maximum shear - τ xy σ θ. x σ y. sτ atan 8.. τ xy π τ max 3.6 MPa θ sτ 9. Maximum shear strain - γ. max ε x ε y γ xy Orientation of maximum shear strain - ε θ. x ε y. sγ atan 8. γ xy π γ max µm m θ sγ 5.7

7 . θ E x.7 Msi ν xy.363 m x.853 E y Msi m y G xy.6 Msi.3 The maximum value of G xy occurs at θ 45 G xy θ 3 The minimum value of G xy occurs at θ G xy θ Displayed graphically GPa. GPa 8 Engineering Shear Modulus (GPa) Ply Angle ( ).4 a) Graphite/Epoxy lamina b) Aluminum plate ε a.6 %

8 ε b.36 %.5 Given a) Elastic modulus in x direction per Equation (.) - G. GPa b) Elastic modulus - E x 3.4 GPa.6 E x ( 3. ) 4.73 Msi. ν b) Only can be determined, G and ν cannot be determined individually. G E.7 Yes, for some values of θ, E x < E if E E < E and G <. E ν E.8 Yes, for some values of θ, E x > E if E G >. ν.9 Basing calculations on a Boron/Epoxy lamina The value of ν xy is maximum for a 6 lamina. Displayed graphically

9 .3 Given Tabulation: LWR ζ E x (Msi).54E E E E-4.35 Graph of ζ as a function of Length-to-Width ratio -.3. Zeta Length-to-Width Ratio (L/W) Graph of Engineering modulus for Finite LWR as a function of ζ -

10 3.5 Young's modulus for Finite L/W (Msi) 3.5 E x Length-to-Width Ratio (L/W).3 G.488 GPa b) From 35 ply data, one can find S and S for the lamina. However, for Equations (.a) and (.b), there are four unknowns: S, S, S, and S 66. Therefore, one cannot find S 66 nor, in turn, the shear modulus, G. Although the same is true in the case of the 45 ply, no manipulation of Equations (.a) and (.b) will allow S 66 to be expressed in terms of S and S for the 35 ply..33 Elastic properties of Boron/Epoxy from Table. - U.7 Msi U 3.5 Msi U Msi U Msi V 3.46 Msi V.695 Msi V 3.4 Msi V 4.9 Msi

11 .35 Given Failure Criterion Magnitude of Maximum Positive Shear Stress Magnitude of Maximum Negative Shear Stress Off-axis Shear Strength Maximum Stress 34 MPa 7.44 MPa 7.44 MPa Maximum Strain: 34 MPa MPa MPa Tsai-Hill: 6.4 MPa 6.4 MPa 6.4 MPa Tsai-Wu (Mises- Hencky criterion) 39 MPa MPa MPa.36 The maximum stress failure theory gives the mode of failure. The Tsai-Wu failure theory is a unified theory and gives no indication of the failure mode..37 The Tsai-Wu failure theory agrees closely with experimentally obtained results. The difference between the maximum stress failure theory and experimental results are quite pronounced..38 Given Maximum Strain: σ Mε MPa Tsai-Wu (Mises-Hencky criterion): σ TWMH MPa The maximum biaxial stress that can be applied to a 6 lamina of Graphite/Epoxy is conservatively estimated at MPa..4 Given the strength parameters for an isotropic material σ σ. 6.5 τ... σ σ < σ T.4 Given the strength parameters for a unidirectional Boron/Epoxy system - Since H < H. H the stability criterion is satisfied..4 The units for the coefficient of thermal expansion in the USCS system are in/in/ F. In the SI system the units for the coefficient of thermal expansion are m/m/ C..43 ε C ε C γ C µm m

12 T _Offset 54.3 C.44 α x α y α xy µin in F.45 The units for the coefficient of moisture expansion in the USCS system are in/in/lbm/lbm. In the SI system the units for the coefficient of moisture expansion are m/m/kg/kg..46 β x β y β xy m m kg kg

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