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Homework answers / question archive / 1)Express the stress–strain relationships in Equation 2

1)Express the stress–strain relationships in Equation 2


1)Express the stress–strain relationships in Equation 2.37 in terms of offaxis engineering constants such as the moduli of elasticity, shear modulus, Poisson’s ratios, and shear-coupling ratios.

2)An element of a balanced orthotropic carbon/epoxy lamina is under the state of stress shown in Figure 2.20. If the properties of the woven carbon fabric/ epoxy material are E1 = 70 GPa, ν12 = 0.25, and G12 = 5 GPa, determine all the strains along the fiber directions.

3)Describe a series of tensile tests that could be used to measure the four independent engineering constants for an orthotropic lamina without using a pure shear test. Give the necessary equations for the data reduction.

4)A lamina consisting of continuous fibers randomly oriented in the plane of the lamina is said to be “planar isotropic,” and the elastic properties in the plane are isotropic in nature. Find expressions for the lamina stiffnesses for a planar isotropic lamina.

5)A tensile test specimen is cut out along the x direction of the pressure vessel described in Example 2.5. What effective modulus of elasticity would you expect to get during a test of this specimen?

6)Describe the measurements that would have to be taken and the equations that would have to be used to determine G23, ν32, and E2 for a specially orthotropic, transversely isotropic material from a single tensile test.

7)For a specially orthotropic, transversely isotropic material the “plane strain bulk modulus,” K23, is an engineering constant that is defined by the stress conditions σ2 = σ3 = σ and the strain conditions ε1 = 0, ε2 = ε3 = ε. Show that these conditions lead to the stress–strain relationship σ = 2K23 ε, and find the relationship among K23, E1, E2, G23, and ν12

8)Using an example of static equilibrium of an element in pure two-dimensional (2D) shear stress, prove that the shear stresses are symmetric (i.e., prove that σij = σji when i ≠ j)

9)If a specially orthotropic, transversely isotropic material is subjected to a pure shear strain γ23 = ε4, use the contracted notation to determine all of the resulting stresses associated with 1, 2, 3 axes.

10)For the original 4340 steel-reinforced concrete post design of Problem 1.13 and the new IM9 carbon fiber-reinforced concrete post design of Problem 1.16, compare the tensile stress-to-tensile strength ratio for the steel rod with the corresponding stress-to-strength ratio for the carbon fiber bundles.

11)For the original concrete composite post design of Problem 1.13, assume that the steel rods are made of 4340 steel, and that the rods are to be replaced

12)The concrete composite post in Figure 1.47 is 1.2 m long with a 0.3 m × 0.3 m square cross section. The post is reinforced by four vertical steel rods of the same length having a cross-sectional area of As = 0.00125 m2 each, and is loaded by a single vertical load P = 500 kN applied on the rigid cover plate as shown below. The modulus of elasticity for concrete is Ec = 17 GPa, whereas the modulus of elasticity of steel is Es = 200 GPa. Determine the stresses in the steel rods and concrete.

13)The 2000 mm long composite bar shown in Figure 1.46 consists of an aluminum bar having a modulus of elasticity EAl = 70 GPa and length LAl = 500 mm, which is securely fastened to a steel bar having modulus of elasticity ESt = 210 GPa and length LSt = 1500 mm. After the force P is applied, a tensile normal strain of εAL = 1000 × 10−6 is measured in the aluminum bar. Find the tensile normal stress in each bar and the total elongation of the composite bar.

14)Now, we wish to find the stresses and deformations in the composite system of Figure 1.42a, where a solid isotropic bar A is securely bonded inside a hollow isotropic bar B of the same length and both bars are axially loaded by a load P that is transmitted through rigid plates. The free-body diagrams for the bars and one of the rigid plates are shown in Figure 1.42b.

15)Which of the reinforcing fibers listed in Table 1.1 would be best for use in an orbiting space satellite antenna structure that is subjected to relatively low stresses but has very precise dimensional stability requirements? The answer should be based only on the properties given in Table 1.1.

16)A cantilever beam of rectangular cross section and made of 6061T6 aluminum is to be replaced by an IM10 carbon fiber composite beam having the same length L and width b, and it must have the same tip deflection w under the same tip load P. Compare the thicknesses and weights of the two beams, neglecting the contribution of the matrix material in the composite.

17)Compare the total fiber surface area of a group of N small-diameter fibers with that of a single large-diameter fiber having the same length and volume.

18)What research methods might you use to update your knowledge of professional and legislative requirements relating to activity statements?

19)Prepare a memo to the partner making a recommendation as to whether Barnes and Fischer should or should not accept Ocean Manufacturing, Inc. as an audit client. Carefully justify your position in light of the information in the case. Include consideration of reasons both for and against acceptance and be sure to address both financial and nonfinancial issues to justify your recommendation.

20)Maternal/child nurses understand that certain treatments (prophylactic eye treatments, vitamin K injections, phenylketonuria [PKU] tests) are provided for all newborn children. What happens to children born outside an acute care setting (e.g., home births)? What rationales do we have to support these treatments?

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