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1) The antisymmetric angle-ply laminate described in Example 7

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1) The antisymmetric angle-ply laminate described in Example 7.6 is subjected to a single uniaxial force per unit length Nx = 50 MPa mm. Determine the resulting stresses associated with the x- and y-axes in each lamina.

2) The symmetric angle-ply laminate described in Example 7.5 is subjected to a single uniaxial force per unit length Nx = 50 MPa mm. Determine the resulting stresses associated with the x- and y-axes in each lamina.

3) Develop a “parallel axis theorem” for the effective laminate stiffnesses A′′ij , B′′ij , and D′′ij associated with the (x′′, z′′) axes, which are parallel to the original (x, z) axes, as shown in Figure 7.22. Express the new A′′ij , B′′ij , and D′′ij in terms of the original Aij, Bij, and Dij, for the (x, z) axes and the distance d between the parallel axes, where z′′ = z + d.

4) A [−60/0/60] laminate and a [0/45/90] laminate both consist of 1.0-mmthick plies having the following properties: E1 = 181 GPa, E2 = 10.3 GPa, G12 = 7.17 GPa, and ν12 = 0.28. Plot the Aij, for both laminates as a function of the orientation to determine which, if any, of the laminates is quasi-isotropic.

5) By expanding the [A] matrix in terms of ply stiffnesses show that a “balanced” cross-ply laminate having equal numbers of 0° and 90° plies is not necessarily quasi-isotropic.

6) A [0/90/0] laminated beam of length L is simply supported at both ends and is loaded by a single concentrated load P at midspan. Find the equation for the maximum flexural deflection of the beam at midspan

7) For the [90/0/90]s E-glass/epoxy beam described in Example 7.1, sketch the distribution of normal and shear stresses through the thickness of the beam. Assume a ply thickness of 0.01 in. (0.254 mm).

8) If the two laminated beams described in Example 7.1 are each 0.6 mm thick and 10 mm wide and the ply strengths are S S L L ( ) ( ) MPa, + − = = 700 S S T T ( ) ( ) . MPa, + − = = 7 0 use the maximum stress criterion to determine the maximum allowable bending moment that each beam can withstand. Compare the maximum allowable bending moments for the two beams.

9) Determine the flexural and Young’s moduli of E-glass/epoxy laminated beams having stacking sequences of [0/90/0]s and [90/0/90]s. The ply moduli are E1 = 5 × 106 psi (34.48 GPa) and E2 = 1.5 × 106 psi (10.34 GPa), and the plies all have the same thickness.

10) Using the Tsai–Hill criterion and the appropriate micromechanics equations, set up the equation for predicting the averaged isotropic shear strength for a randomly oriented short-fiber composite. Your answer should be in terms of the fiber and matrix properties and volume fractions and trigonometric functions of the fiber orientation angle, θ. It is not necessary to solve the equation.

11) Set up the equations for predicting the averaged isotropic shear modulus of a randomly oriented short-fiber composite. Your answer should be in terms of the fiber and matrix properties and volume fractions and trigonometric functions of the fiber orientation angle, θ. It is not necessary to solve the equation

12) Using micromechanics and the Tsai–Hill criterion, set up the equation for the averaged isotropic tensile strength for a randomly oriented short-fiber composite. The equation should be in terms of fiber and matrix properties and volume fractions and the angle θ.

13) Determine the isotropic moduli ­ E and G­ for a composite consisting of randomly oriented T300 CFs in a 934 epoxy matrix if the fibers are long enough to be considered continuous. Use the properties in Table 2.2. Compare the values of ­ E and G­ calculated from the invariant expressions (Equation 6.43) with those calculated from the approximate expressions in Equation 6.44.

14) Express the isotropic moduli ­ E and G­ of a randomly oriented fiber composite in Equation 6.43 in terms of the orthotropic lamina stiffnesses Qij

15) A short-fiber composite is made from boron fibers of length 0.125 in. (3.175 mm) and diameter 0.0056 in. (0.142 mm) randomly oriented in a highmodulus epoxy matrix with a fiber volume fraction of 0.4. Using the fiber and matrix properties in Tables 3.1 and 3.2, respectively, estimate the modulus of elasticity for the composite. Compare the modulus for the randomly oriented short-fiber composite with the longitudinal and transverse moduli of an orthotropically aligned discontinuous fiber lamina of the same material.

16) Determine the Young’s modulus of a randomly oriented fiber composite if the unidirectional form of the composite has an off-axis Young’s modulus that can be described by an equation of the form where θ is the fiber angle in radians and E1 and E2 are the longitudinal and transverse Young’s moduli, respectively, of the unidirectional composite

17) Set up the equations for predicting the off-axis tensile strength of an aligned discontinuous fiber composite based on the maximum strain criterion.

18) If the distribution of the interfacial shear stress, τ, along the discontinuous fiber in Figure 6.4 is described by the linear function shown in Figure 6.8a, find the corresponding expression for the fiber tensile normal stress, σf , and sketch its distribution.

19) Equation 6.5 gives the variation of the fiber normal stress, σf , with the distance along the fiber, x, for the Kelly–Tyson model whose matrix shear stress–shear strain diagram is illustrated in Figure 6.5a. Find the corresponding equation for variation of the fiber normal stress, σf , for the Cox model whose matrix shear stress–shear strain diagram is illustrated in Figure 6.5b, and for which the shear strain varies linearly with the distance x.

20) The interfacial shear stresses, τ, and the fiber normal stresses, σf , acting on a differential element at a distance x from the end of the fiber are shown in Figure 6.4, where the fiber is assumed to be round having a circular cross section of diameter d. For a fiber that has a rectangular cross section of width, a, and depth, b, find the differential equation governing the variation of the stresses τ and σf along the length of the fiber.

21) A unidirectional graphite/epoxy lamina having the properties described in Problem 5.10 is to be designed to have a CTE of zero along a particular axis. Determine the required lamina orientation for such a design.

22) A hybrid unidirectional S-glass/Kevlar/high-modulus (HM) epoxy composite lamina has twice as many S-glass fibers as Kevlar fibers and the total fiber-volume fraction is 0.6. Determine the longitudinal and transverse CTEs for the composite.

23) Develop an analytical model for determination of the CHE, β, for a randomly oriented continuous fiber composite in terms of fiber and matrix properties and volume fractions. Assume that the composite is planar isotropic, and find the β for in-plane hygroscopic expansion.

24) A composite lamina is to be designed to have a specified CTE along a given direction. Outline a procedure to be used in the design.

25) A carbon/epoxy lamina having the properties listed in Problem 5.10 is clamped between two rigid plates as shown in Figure 5.17. If the lamina is heated from 20°C to 120°C, determine the thermal stresses associated with the principal material axes of the lamina.

26) An orthotropic lamina has thermal expansion coefficients α1 = −4.0 (10−6 ) m/m/K and α2 = 79(10−6 ) m/m/K. Determine (a) the angle θ for which the thermal expansion coefficient αy = 0, and (b) the angle θ for which the thermal expansion coefficient αxy has its maximum value.

27) A unidirectional 45° off-axis E-glass/epoxy composite lamina is supported on frictionless rollers between rigid walls as shown in Figure 5.18. The lamina is fixed against displacements in the y-direction, but is free to move in the x-direction. Determine all of the lamina strains associated with the x,y axes if the lamina is heated from 20°C to 120°C. The required properties for E-glass/epoxy are given in Tables 2.2 and 5.3.

28) Samples of unidirectional Kevlar 49/epoxy and S-glass/epoxy composites are subjected to elevated temperatures in an oven and the resulting thermal strains are measured by using strain gages oriented along the 1 and 2

29) A carbon/epoxy lamina is clamped between rigid plates in a mold (Figure  5.17), while curing at a temperature of 125°C. After curing, the lamina/mold assembly (still clamped together) is cooled from 125°C to 25°C. The cooling process occurs in moist air and the lamina absorbs 0.5% of its weight in moisture. The lamina has the following properties:

30) An orthotropic lamina forms one layer of a laminate which is initially at temperature T0. Assuming that the lamina is initially stress free, that the adjacent laminae are rigid, that the properties do not change as a result of the temperature change, and that the lamina picks up no moisture, determine the maximum temperature that the lamina can withstand according to the Tsai–Hill failure criterion.

31) An orthotropic lamina forms one layer of a laminate which is initially at temperature To. Assuming that the lamina is initially stress free, that adjacent lamina are rigid, that the properties do not change as a result of the temperature change, and that the lamina picks up no moisture, determine the maximum temperature that the lamina can withstand according to the maximum stress criterion.