200 Equation (4.179) has been termed the “reduced” or “apparent” flexural stiffness. 4.12 References 1. 2. 3. 4. 5. 6. 7. 8. Gere, J. and Timoshenko, S. (1984) Mechanics of Materials, Second Edition, PWS- Kent Publishing Company, Boston, MA, Appendix G. Warburton, G. (1968) “The Vibration of Rectangular Plates,” Proceedings of the Institute of Mechanical Engineers, Institute of Mechanical Engineers, London, U.K., 371. Young, D. and Felgar, R.F. Jr., “Tables of Characteristic Functions Representing Normal Modes of Vibration for a Beam,” The University of Texas Engineering Research Series Report No. 44, July 1, 1949. Felgar, R.P. Jr. (1950) “Formulas for Integrals Containing Characteristic Functions of a Vibrating Beam,” The University of Texas Bureau of Engineering Research Circular No. 14. Vinson, J.R. and Dee, A.T. (1998) Use of Asymmetric Sandwich Construction to Minimize Bending Stresses, Sandwich Constructions 4, ed. K A. Olson, Vol. 1, EMAS Publishers, Ltd. UK, pp 391-402. Whitney, J.M. (1969) The Effect of Transverse Shear Deformation on the Bending of Laminated Plates, Journal of Composite Materials, Vol. 3, pp. 534-547. Timoshenko, S.P. and Gere, J.M. (1961) Theory of Elastic Stability, McGraw-Hill Book Co., Inc., 2 nd Edition. Bleich, H.H. (1952) Buckling of Metal Structures, McGraw-Hill Book Co., Inc. 4.13 Problems 4.1. The governing differential equation for a beam composed of a composite material in which and is where b is the beam width, is the flexural stiffness in the x -direction, and q(x) is the lateral load per unit length. If the beam is of length L, and the beam is simply supported at x = 0, clamped at x = L, and the lateral load is a constant determine explicitly the expression w(x ) . Given a beam made of T300/5208 graphite epoxy with the following mechanical properties at 4.2. 70º F 201 assume Consider the beam to be made of an all layup, of 30 laminae, each thick, such that the total beam thickness is The beam is one inch wide and twelve inches long If the beam is simply supported at each end, and subjected to a uniform lateral load of 10 lbs./in. of length, what is the maximum deflection? What is its maximum stress? What is its fundamental natural frequency in bending? If the beam were subjected to a compressive end load what would the critical buckling load be? (a) (b) (c) (d) 4.3. 4.4 . 4.5. 4.6. 4.7. Consider a beam of length width of the material of Problem 4.2. If the beam is simply supported at each end, what minimum beam thickness, h, is necessary to insure that the beam is not overstressed when it is subjected to a uniform lateral load of For a beam of the material of Problem 4.2, with and what is the fundamental natural frequency in cycles per second (Hz) if the beam is simply supported at each end? Neglect transverse shear deformation. For the beam of Problem 4.4 above, what is the critical buckling load, neglecting transverse shear deformation? For a clamped-clamped composite beam of uniform cross-section but mid-plane asymmetric subjected to mechanical and hygrothermal loads, determine explicit expressions for the lateral (transverse) deflection, w ( x ), and the in-plane stresses. For a composite beam simply supported at each end, made of a laminate of i.e., [0/90/90/0] whose ply thickness is 0.01 inches, subjected to a uniform lateral load determine the following stress profiles of (a) (b) The material properties are: 202 4.8 . Consider the following simply supported beam subjected to the loading shown, where the beam is mid-plane asymmetric So from the equilibrium equations, prove that 4.9. Consider a composite material beam, with constant flexural stiffness, simply supported at each end, and subjected to a lateral load per unit length given by 4.10. Solving the governing differential equation and satisfying the boundary conditions, determine the location and the magnitude of the maximum displacement. Consider a composite material beam, with constant flexural stiffness, simply supported at each end, and subjected to a lateral load per unit length given by Solving the governing differential equation and satisfying the boundary conditions, determine the location and the magnitude of the maximum displacement. For a beam of the material below where and shown in Figure 4.1, what is the fundamental natural frequency in cycles per second (Hz) if the beam is simply supported on each end. Neglect transverse shear deformation. The following material properties are given for a unidirectional, 4 ply laminate. 4.11. 4.12. 4.13. 4.14. For the same beam as in Problem 4.11 above, what is the critical compressive buckling load, neglecting transverse shear deformation. A beam is made of unidirectional Spectra 900 fibers in a Metton matrix. The properties are: The beam is made by injection molding, hence is uniform in construction. What is the longitudinal stiffness in the fiber direction? If the beam is thick, wide, long what is its fundamental natural frequency if it is simply supported at each end? What is its critical buckling load when subjected to an axial compressive load? If subjected to a unit uniform lateral pressure of 1 psi, what is the maximum deflection? (a) (b) (c) (d) Consider a composite beam of T300/5208 graphite/epoxy, composed of 4 unidirectional laminae each of thickness whose properties are given in Problem 4.2. The beam is wide and long, simply supported at the end x = 0 and clamped at the end x = L. A later load of q = 10 lbs./in. of length is imposed. (a) (b) What is the maximum stress (which occurs at the clamped end, incidentally)? Is the beam overstressed? 203 204 4.15. 4.16. 4.17. (c) If so, how many plys are needed for an adequate design so that the maximum stress is not greater than the strength of the material? What is the fundamental natural frequency of the beam of Problem 4.14 if the weight density is 0.06 lbs./cubic inch, and the gravitational constant is 386 in./second squared? For a clamped-clamped beam, the axial buckling load is four times that of a beam simply supported at each end. For the beam of Problem 4.14, clamped at each end, what is the axial buckling load? A cantilever beam of length L, composed of a graphite/epoxy composite material T300/5208 with the stacking sequence shown is subjected to a uniform mechanical load For the beam shown plot the stress distribution at the section of maximum moment. Compare the shape of this stress distribution with that of an isotropic beam. Calculate the bending and shear stresses at x = L/ 2 for a simply supported [0/90/90/0], composite beam composed of the following materials, whose properties are given in Appendix 2: • E-glass/Epoxy • Graphite/Epoxy (T300/5208) for the mechanical loads shown. Each ply is thick and the beam width is (a) Uniform transverse load 4.18. 205 (b) Triangular transverse load 4.19. A load P is supported by three vertical bars as shown below. The horizontal bar remains horizontal during deformation. Consider the middle bar to be made from a composite material consisting of an aluminum matrix with boron fibers continuous and aligned parallel to the load. Assume a 50% volume fraction of fibers to matrix and a structurally contiguous bond between fibers and matrix. (a) Find the fiber and matrix stresses for the middle bar (b) If all bars were made of aluminum, what would the middle load stress be? Compare to the composite bar. 4.20. Consider a simply supported laminated beam subjected to a single concentrated load as shown 206 For this beam find: (a) (b) (c) (d) The displacements Forces and Moments Strains If the beam consists of a laminate, find the stress distribution at x - a. Consider the beam to be made of a Kevlar-epoxy composite. Use properties from Appendix 2. Each ply thickness is 0.010 inches. Find the stress distribution in the cantilever beam shown below at the section of maximum moment. The beam is made of T300/5208 Graphite-epoxy with the following properties 4.21. Compare the shape of the stress distribution with the case of an isotropic beam. Consider a beam simply supported at each end under a constant (uniform) load as shown below. Find the stresses and plot the stress distribution at the mid-span L /2 and at L/4. 4.22. Kevlar/Epoxy Composite [0/+45/-45/90/90/-45/+45/0], 207 Stacking sequence with Consider a simply supported beam under a transverse ramp loading. Find the lateral deflection and in-plane displacement. The governing differential equations can be reduced to 4.24. A graphite-epoxy structure with fiber orientation 30° from the x-axis is loaded by a 4.23 208 triangular loading as shown below. Assume for this problem that it is a beam. Obtain the expressions for the in-plane displacement normal displacement w and the rotation Find the maximum deflection w and identify the shear correction. Assume no axial loading, no hygrothermal effects, and mid-plane symmetry. Assume the displacement field is: Assume Make sure to clearly write down the governing equations and all the boundary conditions. 4.25. Consider a simply supported beam with two equal concentrated loads symmetrically placed as shown below, i.e., four point bending. For a composite laminated stacking sequence of [0/90/0/90] with each ply of thickness and properties given by 209 Find the maximum stresses in the beam for each ply. For a clamped-clamped beam, the axial buckling load is 4 times that of a beam simply supported at each end. Consider a composite beam made of T300/5208 (see Problem 4.2) graphite/epoxy composed of 4 unidirectional laminae each of thickness the beam being wide and in length, find the axial buckling load. Find the variation in the flexural stress for the beam shown below and compare the solution with that of a homogeneous/isotropic beam. Use the material properties given in Problem 4.17. 4.26. 4.27. Find the variation in the flexural stress for the laminated beam cross-section shown below, subjected to a bending moment M . 4.28. Take each ply as thick and consider two composite materials, with the properties indicated below. [...]... displacement shell, composed of composed of two composite materials, subjected to an and du(0)/dx = 0 at one edge Each of the eight plys are and the two materials are T300/52 08 graphite/epoxy and E-glass/epoxy with the following properties: T300/52 08 E-glass/epoxy 0.30 0.021 0.23 0.0941 0.70 0.72 Using the methods of Chapter 2, the components of the stiffness matrices are: T300/52 08 E-glass/epoxy 0 0... where These equilibrium equations are independent of the material system The circumferential terms can also be written in terms of ds, the arc distance where The quantities and are functions of the surface shear stresses on the outer and inner surfaces of the composite shell wall, and is the laterally distributed load per unit area, positive in the positive direction 2 18 For the case where the transverse... 216 where all of the terms have been explained in the previous chapters In addition, there is one other assumption known as Love’s First Approximation, which is consistent with the neglect of transverse shear deformation: It is true that the accurate analysis of shells of composite materials should include transverse shear deformation because of the fact that the modulus of elasticity in the fiber direction... structure, with , the particular solution can be written as 227 5.3.6 BEST FORM OF SOLUTION UTILIZING THE BENDING BOUNDARY LAYER Instead of using A, B, C and E in Equation (5. 38) as the constants of integration, one employs and as the integration constants The particular advantage of using Equation (5.54) rather than Equation (5. 38) is easily seen It can be shown that each term of the homogeneous solution... solving the problem (b) Write the governing differential equations (c) Find the deflection w(x) (d) Find the deflection at the center of the beam (e) Discuss the effects of the end supports 4.36 Plot the first five vibration mode shapes for a beam simply supported at each end with a stacking sequence and a ply thickness of Assume the material is E-Glass/Epoxy 4.37 Consider a composite beam made of T300/52 08. .. T300/52 08 Graphite/epoxy (see Problem 4.2), 213 and composed of 4 unidirectional lamina each of thickness The beam is wide, long, carries a uniform lateral load of q = 10 lbs./in of length, has a weight density of and the gravitational constant is Find the fundamental natural frequency 4. 38 Starting from general equations for the free-free vibrations of a composite beam shown below, obtain the frequency... that of a shell of one lamina only Subsequently, generalizations are made to the configurations In the current case, the stress-strain relations and the strain-displacement relations, utilizing the displacement assumptions of Equation (5.1) are; for the case of one ply The in-plane stiffnesses and the flexural stiffnesses are given as follows: Remember that if the fibers are in the axial direction then... beyond the scope of this book One recent textbook dealing with shells of composite materials as well as isotropic materials is that of Reference [1] Even to derive the governing differential equations for a shell of general curvature from first principles require several lectures in topology Then, to complicate shell theory all of the material complexities associated with laminated composite materials. .. positive directions of the displacements u, v, and w are shown, as well as the positive directions of the coordinates x and The remaining coordinate is the circumferential coordinate The positive value of all stress resultants and stress couples are shown in Figure 5.2 below In the classical shell theory discussed in this section, all of the assumptions used in classical plate theory of Chapters 2 and... direction in which the middle surface is curved On the other hand in a spherical shell there is curvature in both directions Such mundane shells as a front fender of a car and an eggshell are examples of shells with double curvature Shell theory is greatly complicated, compared to beam and plate theory, because of this curvature The treatment of shell theory in its proper detail is the subject of a graduate . analysis of shells of composite materials should include transverse shear deformation because of the fact that the modulus of elasticity in the fiber direction is a fiber dependent property, while the. the strength of the material? What is the fundamental natural frequency of the beam of Problem 4.14 if the weight density is 0.06 lbs./cubic inch, and the gravitational constant is 386 in./second. inches. Find the stress distribution in the cantilever beam shown below at the section of maximum moment. The beam is made of T300/52 08 Graphite-epoxy with the following properties 4.21. Compare the shape