Questions and Answers of Principles Of Composite

The isochronous stress-strain curves for an epoxy material at different times are shown in Figure 8.53. This material is used as the matrix in a unidirectional E-glass-epoxy composite having a fiber
For the nonsymmetrical laminated beam of Problem 29 in Chapter 7, determine the effects of \(B_{i j}\)-type coupling on the flexural vibration frequencies.Data From Problem 29 Of Chapter 7:The
If the complex moduli of the \(0^{\circ}\) and \(90^{\circ}\) plies in a laminated beam are the same as those described in Example 8.12, but the laminate has a stacking sequence of \([90 / 0 /
Derive Equation 10.2.\(\frac{1}{G}=\frac{\left(1+v_t\right)}{E_t}+\frac{\left(1+v_c\right)}{E_c} \tag{10.2}\)
A \([0 / 90]_{\mathrm{S}}\) laminate is subjected to a single bending moment per unit length, \(M_{x}\). If the laminate is unconstrained, so that bending along both the \(x\) and the \(y\)
The laminate described in problem 9 is subjected to a single bending moment per unit length, \(M_{x}\) and the two edges on which \(M_{x}\) acts are fixed so that bending along the \(x\) direction
The non-symmetrically laminated beam shown in Figure 7.71 consists of a substrate material having Young's modulus \(E_{\mathrm{S}}\) and a coating material having Young's modulus \(E_{\mathrm{C}}\),
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
A flywheel for energy storage is modeled as a rotating thin-walled cylindrical ring \((tthe kinetic energy stored per unit mass of a ring made from 4340 steel with that of a ring made from IM-7
Derive Equation 2.27.\(\begin{aligned}& Q_{11}=\frac{S_{22}}{S_{11} S_{22}-S_{12}^2}=\frac{E_1}{1-v_{12} v_{21}} \\& Q_{12}=-\frac{S_{12}}{S_{11} S_{22}-S_{12}^2}=\frac{v_{12} E_2}{1-v_{12}
Derive Equation 3.45.\(v_{12}=v_{\mathrm{f} 12} v_{\mathrm{f}}+v_{\mathrm{m}} v_{\mathrm{m}} \tag{3.45}\)
Derive Equation 3.47.\(\frac{1}{G_{12}}=\frac{v_{\mathrm{f}}}{G_{\mathrm{f} 12}}+\frac{v_{\mathrm{m}}}{G_{\mathrm{m}}} \tag{3.47}\)
Derive Equation 3.59.\(G\left[\frac{\partial^2 w}{\partial x^2}+\frac{\partial^2 w}{\partial y^2}\right]=0 \tag{3.59}\)
Derive Equation 3.66.\(\frac{1}{E_2}=\frac{1}{v_{\mathrm{f}}+\eta_2 v_{\mathrm{m}}}\left[\frac{v_{\mathrm{f}}}{E_{\mathrm{f}}}+\frac{\eta_2 v_{\mathrm{m}}}{E_{\mathrm{m}}}\right] \tag{3.66}\)
A unidirectional IM-9 carbon fiber/Hexply 8551-7 epoxy composite is to be designed to replace a 6061-T6 aluminum alloy rod which is to be loaded in longitudinal tension.(a) What fiber volume fraction
How would the answer to Problem 9 in Chapter 1 change if the flywheel ring is made of IM-7/8552 carbon/epoxy composite with fibers oriented in the circumferential direction? Assume a fiber volume
A unidirectional continuous fiber composite is to be made from T300 graphite fibers in a high-modulus (HM) epoxy matrix, and the composite is to have a longitudinal coefficient of thermal expansion
A hybrid unidirectional E-glass/T-300 carbon/IMHS epoxy composite is to be designed to have an overall longitudinal thermal expansion coefficient of zero in order to insure the best possible thermal
Derive Equations 5.26 and 5.30 .\(\alpha_1=\frac{E_{\mathrm{f} 1} \alpha_{\mathrm{f} 1} v_{\mathrm{f}}+E_{\mathrm{m} 1} \alpha_{\mathrm{m} 1} v_{\mathrm{m}}}{E_1}=\frac{E_{\mathrm{f} 1}
Express the isotropic moduli \(\tilde{E}\) and \(\tilde{G}\) of a randomly oriented fiber composite in Equations 6.43 in terms of the orthotropic lamina stiffnesses \(Q_{i j}\).\(\begin{aligned}&
Determine the isotropic moduli \(\tilde{E}\) and \(\tilde{G}\) for a composite consisting of randomly oriented T300 CFs in a 934 epoxy matrix if the fibers are long enough to be considered
The off-axis Young's modulus for a particular unidirectional fiber-reinforced orthotropic composite lamina is given by\[ E_{x}=E_{1}-\left(E_{1}-E_{2}\right) \sin \theta \]where \(\theta\) is the
A laminated \([0 / 90 / 0 / 90]_{s}\) carbon/epoxy beam is \(1 \mathrm{~mm}\) thick, \(20 \mathrm{~mm}\) wide, and has \(0.125-\mathrm{mm}\)-thick plies. The lamina properties area. Determine the
The laminated beam shown in Figure 7.9 is made up of two outer plies of material " \(\mathrm{A}\) " having Young's modulus \(E_{\mathrm{A}}\), two inner plies of material " \(\mathrm{B}\) " having
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
A thin-walled composite tube having an outside diameter of \(48 \mathrm{~mm}\) is made by securely bonding an aluminum tube inside a steel tube, as shown in Figure 7.10. Determine the maximum
Determine the stiffness matrix for a \([+45 /-45]\) antisymmetric laminate consisting of 0.25-mm-thick unidirectional AS/3501 carbon/epoxy plies.
Show that for symmetric laminates the coupling stiffnesses, \(B_{i j}\), must all be equal to zero.
By expanding the \([A]\) matrix in terms of ply stiffnesses show that a "balanced" cross-ply laminate having equal numbers of \(0^{\circ}\) and \(90^{\circ}\) plies is not necessarily quasi-isotropic.
A [-60/0/60] laminate and a [0/45/90] laminate both consist of 1.0-mmthick plies having the following properties: \(E_{1}=181 \mathrm{GPa}, E_{2}=10.3 \mathrm{GPa}\), \(G_{12}=7.17 \mathrm{GPa}\),
Develop a "parallel axis theorem" for the effective laminate stiffnesses \(A_{i j}^{\prime \prime}\), \(B_{i j}^{\prime \prime}\), and \(D_{i j}^{\prime \prime}\) associated with the
The [+45/-45] laminate described in Problem 7.6 is subjected to a uniaxial force per unit length \(N_{x}=30 \mathrm{MPa} \mathrm{mm}\). Find the resulting stresses and strains in each ply along the
A \([90 / 0 / 90]_{\mathrm{s}}\) laminate is fabricated from laminae consisting of isotropic fibers \(\left(E_{\mathrm{f}}=220 \mathrm{GPa}, v_{\mathrm{f}}=0.25\right)\) embedded in an isotropic
A symmetric [0/90/0] laminate is \(0.75 \mathrm{~mm}\) thick and its full compliance matrix is given below:where the units of the matrix areDetermine the following effective engineering constants for
The laminate described in Problem 7.15 has laminae that are \(0.25 \mathrm{~mm}\) thick and the stiffness matrix associated with the \(0^{\circ}\) lamina is given byIf a single bending moment per
A four ply symmetric \([0 / 90]_{\mathrm{s}}\) laminate has the following ply properties; ply thickness \(=0.25 \mathrm{~mm}\);\[E_{1}=129 \mathrm{GPa}, \quad E_{2}=12.8 \mathrm{GPa}, \quad
An antisymmetric angle-ply \([+\theta /-\theta]\) laminate is to be made of carbon/ epoxy and designed to have a laminate CTE, \(\alpha_{x}\) as close to zero as possible. Determine the ply
Repeat Problem 7.18 for a Kevlar \(^{\circledR} /\) epoxy composite having lamina properties as follows:Problem 7.18An antisymmetric angle-ply \([+\theta /-\theta]\) laminate is to be made of carbon/
The sensing element in many thermostats is a bimetallic strip (Figure 7.45), which is a nonsymmetric laminate consisting of two plies made from different metals. If the strip is subjected to a
The distribution of the in-plane shear stress, \(\tau_{x y \prime}\) along the \(y\) direction at a particular distance \(z\) from the middle surface of a uniaxially loaded laminate is idealized, as
A \([0 / 90 / 0]_{\mathrm{s}}\) laminate consisting of AS/3501 carbon/epoxy laminae is subjected to uniaxial loading along the \(x\)-direction. Use the maximum strain criterion to find the loads
Solve Example 7.15 using the Tsai-Hill criterion instead of the maximum stress criterion. That is, determine the largest torque \(T\) that can be transmitted by the shaft without failure, according
Solve Example 7.15 using the maximum strain criterion instead of the maximum stress criterion. See Problem 7.23, for additional information.Problem 7.23Solve Example 7.15 using the Tsai-Hill
Solve Example 7.15 using the maximum strain criterion instead of the maximum stress criterion. See Problem 7.23, for additional information.Problem 7.23Solve Example 7.15 using the Tsai-Hill
Prove that for the specially orthotropic plate shown in Figure 7.48 under the loading described by Equation 7.144, the solution given by Equation 7.146 satisfies the differential Equation 7.139 and
Find expressions for the moments \(M_{x^{\prime}} M_{y^{\prime}}\) and \(M_{x y}\) and the stresses \(\left(\sigma_{x}\right)_{k^{\prime}}\left(\sigma_{y}\right)_{k^{\prime}}\) and \(\left(\tau_{x
Derive the differential equation and the boundary conditions governing the small transverse deflections of a simply supported, rectangular, symmetric angle-ply laminate that is subjected to
Derive the coupled differential equations and the boundary conditions governing the small transverse deflections of a simply supported, rectangular, antisymmetric angle-ply laminate that is subjected
A simply supported, specially orthotropic plate is subjected to an in-plane compressive load per unit length \(N_{x}\) and an in-plane tensile load per unit length \(N_{y}=-0.5 N_{x}\) as shown in
Derive the differential equation and the boundary conditions governing the buckling of a simply supported, rectangular, symmetric angle-ply laminate that is subjected to a uniaxial in-plane load,
The plate in Figure 7.70 has edge dimensions \(a\) and \(b\) and is made from a \([90 / 0 / 90]_{s}\) symmetric cross-ply laminate. The plate is simply supported on all edges and is subjected to a
A filament-wound composite drive shaft for a helicopter transmits a torque \(T\) that generates shear loading of the shaft material, as shown in Figure 7.71. The shaft is to be designed as a hollow
Using the carpet plots of Figure 7.58, select the percentages of \(0^{\circ}, \pm 45^{\circ}\), and \(90^{\circ}\) plies that are needed in a \(\left[0_{i} / \pm 45_{i} / 90_{k}\right]\) laminate if
A thin-walled cylindrical pressure vessel has mean diameter \(d=18\) in. and wall thickness \(h=0.25\) in. The vessel is made of filament-wound unidirectional composite material with all fibers
For a linear viscoelastic material, the creep response under a constant stress is followed by a "recovery response" after the stress is removed at some time, \(t_{0}\). Using the Boltzmann
In general, the creep compliances, \(S_{i j}(t)\) and the relaxation moduli, \(C_{i j}(t)\) are not related by a simple inverse relationship. Show that only when \(t \rightarrow 0\) and when \(t
The shear creep compliance, \(S_{66}(t)\) for a unidirectional viscoelastic composite is given by \(S_{66}(t)=\gamma_{12}(t) / \tau_{12}\), where \(\gamma_{12}(t)\) is the time-dependent shear creep
The time-dependent axial stress, \(\sigma_{x}(t)\) and the time-dependent circumferential stress, \(\sigma_{y}(t)\) in the wall of the filament-wound, thin-walled composite pressure vessel shown in
A linear viscoelastic, orthotropic lamina has principal creep compliances given in contracted notation by\[S_{i j}(t)=E_{i j}=F_{i j} t, \quad i, j=1,2, \ldots, 6\]where \(E_{i j}\) and \(F_{i j}\)
Derive the equations for the stress-strain relationship, the creep compliance, and the relaxation modulus for the Kelvin-Voigt model.
Derive the equations for the stress-strain relationship, the creep compliance, and the relaxation modulus for the Zener model.
Derive Equation 8.45. C(t) = ko+ " E=1 kie-/ (8.45)
Derive Equation 8.47. S(t)= 1 ko + i=1 ki [1-et/pi] (8.47)
The shear relaxation modulus, \(G_{12}(t)\), for an orthotropic lamina is idealized, as shown in Figure 8.30. Find the corresponding equations for the shear storage modulus,
For the Maxwell model in Figure 8.8, express the storage modulus, \(E^{\prime}(\omega)\), the loss modulus, \(E^{\prime \prime}(\omega)\), and the loss factor, \(\eta(\omega)\), in terms of the
The composite pressure vessel in Problem 8.6 is subjected to an internal pressure that varies sinusoidally with time according to the relationship \(p(t)=P_{0}\) \(\sin t\), and the principal complex
The polymer matrix material in a linear viscoelastic, unidirectional composite material has a relaxation modulus that can be characterized by the Maxwell model in Figure 8.8. The fibers are assumed
The matrix material in a linear viscoelastic, unidirectional composite material is to be modeled by using a Maxwell model having parameters \(k_{\mathrm{m}}\) and \(\mu_{m}\), whereas the fiber is to
The dynamic mechanical behavior of an isotropic polymer matrix material may be characterized by two independent complex moduli such as the complex extensional modulus, \(E^{*}(\omega)\) and the
A drive shaft in the shape of a hollow tube and made of a linear viscoelastic angle-ply laminate is subjected to a torque, \(T\), as shown in Figure 8.32. Develop an analytical model for predicting
Part of the required input to the viscoelastic option in some finite element codes is a table showing the time-dependent, isotropic shear modulus \(G(t)\) at different times \(t\). Explain how you
For the Maxwell model shown in Figure 8.8, it can be shown that the complex modulus is given by\[E^{*}(\omega)=E^{\prime}(\omega)[1+i \eta(\omega)]\]where the frequency-dependent storage modulus is
Longitudinal vibration of an isotropic, particle-reinforced composite bar may be modeled by using the 1D wave equation (Equation 8.99) if the material is linear elastic. Derive the equation of motion
Find the separation of variables solution for the longitudinal displacement, \(u(x, t)\), of the equation derived in Problem 8.21. Leave the answer in terms of constants, which must be determined
Derive the equation of motion for free transverse vibration of a simply supported, specially orthotropic plate that is subjected to in-plane loads per unit length \(N_{x}\) and \(N_{y}\) as shown in
For the plate described in Problem 8.23, find the equations for the plate natural frequencies and determine the effects of positive (tensile) and negative (compressive) in-plane loads \(N_{x}\) and
If the plate described in Problem 8.23 is clamped on all edges, investigate solutions of the formDoes this solution satisfy the boundary conditions? Can it be used to find the natural frequencies?
A randomly oriented, short-fiber-reinforced composite plate having a central through-thickness crack of length \(2 a\) is subjected to a uniaxial stress \(\sigma\), as shown in Figure 9.1. If the
The thin-walled tubular shaft shown in Figure 9.8 is made of a randomly oriented, short-fiber-reinforced metal matrix composite. The shaft has a longitudinal through-thickness crack of length \(2 a\)
a. Determine the allowable torque, \(T\), if the crack length for the shaft in Figure 9.8 is \(2 a=10 \mathrm{~mm}\). Use the same dimensions and fracture toughness values that were given in Problem
The tube shown in Figure 9.8 is subjected to an internal pressure, \(p=5 \mathrm{MPa}\), instead of a torque. Neglecting the stress along the longitudinal axis of the tube, and assuming that the mode
As in Problem 9.4, assume that the tube in Figure 9.8 is subjected only to an internal pressure and neglect the longitudinal stress.a. Determine the allowable internal pressure, \(p\), if the crack
A 3-mm-thick composite specimen is tested, as shown in Figure 9.4a, and the compliance, \(s=u / P\), as a function of the half-crack length, \(a\), is shown in Figure 9.9. In a separate test, the
Use the Whitney-Nuismer average stress criterion to estimate the allowable internal pressure for Problem 9.5 if the unnotched tensile strength of the material is \(\sigma_{0}=1500 \mathrm{MPa}\) and
Repeat Problem 9.8 using the Whitney-Nuismer point stress criterion and the parameter \(d_{0}=1 \mathrm{~mm}\).Problem 9.8Use the Whitney-Nuismer average stress criterion to estimate the allowable
The \(920-\mathrm{mm}\)-diameter, \(1.6-\mathrm{mm}\)-thick spherical pressure vessel in Figure 9.27 is a filament wound quasi-isotropic composite laminate with a single 50-mm-diameter entrance hole.
A laminated plate consisting of the [90/0/90] \(]_{\mathrm{s}} \mathrm{AS} / 3501\) laminate described in Example 7.13 has a central hole, as shown in Figure 9.28. The plate is loaded uniaxially
A unidirectional [0] composite beam of longitudinal modulus \(E_{1}\), thickness \(b\), and depth \(h\) has a crack of length \(a\) and is loaded by the equal and opposite forces \(P\), as shown in
Solve Example 10.1 for the case of a bundle of \(N=10,000\) fibers instead of a single fiber. EXAMPLE 10.1 A carbon fiber having a diameter d = 0.0003 in. is to be tested in tension, according to
In order to determine the tensile Young's modulus and tensile strength of carbon fibers, a tensile test of a resin-impregnated carbon/epoxy yarn having a fiber volume fraction of 0.6 is conducted. A
Derive Equation 10.2. S11(t)= = E1(t) 01 (10.26)
The results of longitudinal, transverse, and \(45^{\circ}\) off-axis tensile tests on samples from an orthotropic lamina are shown in Figure 10.35. Based on these results, find numerical values for
The in-plane shear modulus, \(G_{12}\), of a carbon/epoxy lamina is to be measured by using the rail shear test shown in Figure 10.36. The test is conducted on a 10 in. \(\times 10 \mathrm{in} .
The \(45^{\circ}\) off-axis test shown in Figure 10.37 is conducted on a 10 in. \(\times 1\) in. \(\times 0.1\) in. \((254 \mathrm{~mm} \times 25.4 \mathrm{~mm} \times 2.54 \mathrm{~mm})\)
A \(45^{\circ}\) off-axis specimen cut from an AS/3501 carbon/epoxy lamina is subjected to a tensile test. The specimen is \(3 \mathrm{~mm}\) thick and \(25 \mathrm{~mm}\) wide, and a tensile load of
A \(45^{\circ}\) off-axis rail shear test specimen of an orthotropic lamina is shown in Figure 10.38. Attached to the specimen are three strain gages that measure the normal strains
Describe the measurements that must be taken and the equations that must be used to determine the shear creep compliance, \(S_{66}(t)\), of a unidirectional viscoelastic lamina by using a rail shear
Extensional vibration experiments are conducted on longitudinal, transverse, and \(45^{\circ}\) off-axis unidirectional composite specimens, and the complex moduli results for a particular vibration
Using the results from Problem 10.10, derive the equations for both parts of the off-axis complex modulus, \(E_{x}^{*}=E_{x}^{\prime}\left(1+i \eta_{x}\right)\), for an arbitrary angle \(\theta\);
Describe an experiment, and give the necessary equations for measurement of the complex flexural modulus, \(E_{\mathrm{fx}}^{*}\), of a symmetric laminated beam.

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