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 E4.35 Graph of ζ as a function of LengthtoWidth ratio .3. Zeta LengthtoWidth 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 LengthtoWidth 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 Offaxis Shear Strength Maximum Stress 34 MPa 7.44 MPa 7.44 MPa Maximum Strain: 34 MPa MPa MPa TsaiHill: 6.4 MPa 6.4 MPa 6.4 MPa TsaiWu (Mises Hencky criterion) 39 MPa MPa MPa.36 The maximum stress failure theory gives the mode of failure. The TsaiWu failure theory is a unified theory and gives no indication of the failure mode..37 The TsaiWu 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 TsaiWu (MisesHencky 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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