The Behavior of Structures Composed of Composite Materials Part 6 pptx

139 Proceeding as before for the mid-plane symmetric rectangular plate of Section 3.3, the resulting three coupled equations using classical plate theory, i.e. no transverse shear deformation, have the following form: Because of the bending-stretching coupling not only are lateral displacements, w ( x,y ) , induced but in-plane displacements, and as well; hence, three coupled equations (3.165) through (3.167). 3.23 Governing Equations for a Composite Material Plate With Bending-Twisting Coupling Looking at Equation (2.66), the moment curvature relations for a rectangular mid- plane symmetric plate with bending-twisting coupling are: 140 Of course if transverse shear deformation is ignored, i.e. classical theory then the curvatures are given by (3.23), and the moment curvature relations become: Substituting these into (3.19), provides the following governing differential equation. Comparing (3.170) with (3.29), it is seen that due to the presence of the and bending-twisting coupling terms, odd numbered derivatives appear in the governing differential equation. That precludes the use of both the Navier approach of Section 3.5, and the use of the Levy approach of Section 3.7 in obtaining solutions for plate with bending-twisting coupling. With these complications one may want to obtain solutions using the Theorem of Minimum Potential Energy discussed in Chapter 6 below. 3.24 Concluding Remarks It appears that there is no end in trying to more adequately describe mathematically the behavior of composite materials utilized in structural components. Unfortunately, the more sophisticated one gets in such descriptions the more difficult the mathematics becomes, as is evidenced in the increasing difficulty observed as one progresses through the sections of Chapter 3. One additional complication that is important in some composite material structures is that the stiffness (and other properties) are different in tension than they are in compression. This occurs because (1) sometimes the tensile and compressive mechanical properties of both fiber and matrix materials, differ and (2) sometimes it occurs because the matrix material is very weak compared to the fiber (that is such that the fibers buckle in compression under a small load so that for 141 the composite the stiffness in compression differs markedly than the stiffness in tension. Hence, one can idealize a little and say that one has one set of elastic properties in tension and another set of elastic properties in compression. Bert [25] has termed this a bimodular material, typical of some composites, certainly typical of aramid (Kevlar) fibers in a rubber matrix as used in tires, and also typical of certain biological tissues modeled in biomedical engineering. In this context and all of the complications that result therefrom are too difficult to treat in this text for first year students trying to learn the fundamentals of composite materials. Lastly, time dependent effects in the stresses, deformation and strains of composite materials are becoming more important design considerations. Viscoelasticity and creep are respected disciplines about which entire books have been written. These effects have been deemed important in some composite material structures. Crossman, Flaggs, Vinson and Wilson have all commented thereupon. Wilson and Vinson [26] have shown that the effects of viscoelasticity on the buckling resistance of polymer matrix composite material plates is very significant. Similarly, the effect of viscoelasticity on the natural vibration frequencies will also be significant. Many of these effects have been included in a survey article by Reddy [27] who has focused primarily on plates composed of composite materials. 3.2 5 References 1. 2. 3. 4. 5. 6. 7. 8. 9. Vinson, J.R. (1974) Structural Mechanics: The Behavior of Plates and Shells, Wiley-Interscience, John Wiley and Sons, New York. Vinson, J.R. and Chou, T.W. (1975) Composite Materials and Their Use in Structures, Applied Science Publishers, London. Levy, M. (1899) Sur L’equilibrie Elastique d’une Plaque Rectangulaire, Compt Rend 129, pp. 535-539. Timoshenko, S. and Woinowsky-Krieger, S. (1959) Theory of Plates and Shells, McGraw-Hill Book Co. Inc., edition, New York. Vinson, J.R. (1961) New Techniques of Solutions for Problems in Orthotropic Plates, Ph.D. Dissertation, University of Pennsylvania. Vinson, J.R. and Brull, M.A. (1962) New Techniques of Solutions of Problems in Orthotropic Plates, Transactions of the Fourth United Stated Congress of Applied Mechanics, Vol. 2, pp. 817-825. Vinson, J.R. (1989) The Behavior of Thin Walled Structures: Beams, Plates and Shells, Kluwer Academic Publishers, Dordrecht, The Netherlands. Whitney, J.M. (1987) Structural Analysis of Laminated Anisotropic Plates, Technomic Publishing Co. Inc., Lancaster, Pa. Vinson, J.R. (1999) The Behavior of Sandwich Structures of Isotropic and Composite Materials, Technomic Publishing Co. Inc., Lancaster, Pa. 142 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. Dobyns, A.L., (1981) The Analysis of Simply-Supported Orthotropic Plates Subjected to Static and Dynamic Loads, AIAA Journal, May, pp. 642-650. Leissa, A.W. (1973) The Free Vibration of Rectangular Plates, Journal of Sound and Vibration, Vol. 31, No. 3, pp. 257-293. Nashif, A.D., Jones, D.I.G. and Henderson, J. (1985) Vibration Damping, Wiley Interscience. Inman, D.J. (1989) Vibration with Control Measurement and Stability, Prentice Hall, Englewood Cliffs, New Jersey. Warburton, G. The Vibration of Rectangular Plates, Proceedings of the Institute of Mechanical Engineers, 1968 (1954), pg. 371-384. Young, D. and Felgar, R., Jr. (1944) Tables of Characteristic Functions Representing Normal Modes of Vibration of a Beam, The University of Texas Publication Number 4913. Felgar, R., Jr. (1950) Formulas for Integral Containing Characteristic Functions of a Vibrating Beam, Bureau of Engineering Research, The University of Texas Publication. Moh, J-S and Hwu, C. (1997) Optimization for Buckling of Composite Sandwich Plates, AIAA Journal, Vol. 35, pp. 863-868. Kerr, A.D. (1964) Elastic and Viscoelastic Foundation Models, Journal of Applied Mechanics, Vol. 31, pp. 491-498. Paliwal, D.N. and Ghosh, S.K. (1944) Stability of Orthotropic Plates on a Kerr Foundation, AIAA Journal, Vol. 38, pp. 1993-1997. Zenkert, D. (1995) An Introduction to Sandwich Construction, EMAS Publications, West Midlands, UK. Sierakowski, R.L. and Mukhopadhyay, A.K. (1990) On Sandwich Beams With Laminate Facings and Honeycomb - Cores Subjected to Hygrothermal Loads: Part I – Analysis, Journal of Composite Materials, Vol. 24, No. 4, pp. 382-400. Sierakowski, R.L. and Mukhopadhyay, A.K. (1990) On Sandwich Beams With Laminate Facings and Honeycomb – Cores Subjected to Hygrothermal Loads: Part II – Application, Journal of Composite Materials, Vol. 24, No. 4, pp. 401- 418. Sierakowski, R.L., Mukhopadhyay, A.K., and Yu, Y.Y. (1994) On Sandwich Beams With Laminate Facings and Honeycomb – Cores Subjected to Hygrothermal and Mechanical Loads: Part III – Timoshenko Beam Theory, Journal of Composite Materials, Vol. 28, No. 11, pp. 1057-1075. Sierakowski, R.L. and Mukhopadhyay, A.K. (2000) On Thermoelastic and Hygrothermal Response of Sandwich Beams With Laminate Facings and Honeycomb – Cores: Part IV – A Dynamic Theory, Journal of Composite Materials, Vol. 34, pp. 174-199. Bert, C.W., Reddy, J.N. Reddy, V.S. and Chao, W.C. (1981) Analysis of Thick Rectangular Plates Laminated of Bimodulus Composite Materials, AIAA Journal, Vol. 19, No. 10, October, pp. 1342-1349. Wilson, D.W. and Vinson, J.R. (1984) Viscoelastic Analysis of Laminated Plate Buckling, AIAA Journal, Vol. 22, No. 7, July, pp. 982-988. Reddy, J.N. (1982) Survey of Recent Research in the Analysis of Composite Plates, Composite Technology Review, Fall. 3.26 Problems and Exercises 3.1. Find the critical buckling load, in lbs./in. for a plate simply supported on all four edges made of a material whose flexural stiffness properties are given as follows and whose thickness is 1 inch. (a) If a = 30 inches and b = 20 inches. (b) If a = 50 inches and b=12 inches. 3.2. Find the fundamental natural frequency in Hz (cps) for each of the plates of Problem 3.1, if the mass density of the material is 3.3. The following material properties are given for a unidirectional, 4 ply laminate, h = 0.020” the mass density (corresponding to Consider a plate made of the above material with dimensions a = 20”, b = 30”, h = 0.020”. For the first perturbation method of Section 3.8 determine and Is a proper value to use this perturbation technique? 3.4. Consider the plate of problem 3.3. If it is simply supported on all four edges, what is its fundamental natural frequency in cycles per seconds neglecting transverse shear deformation? 143 144 3.5. For a box beam whose dimensions are b = 4”, h = 2”, L = 20”, composed of T300/934 graphite/epoxy, whose properties are given in Appendix 2, determine the extensional stiffness, EA; the flexural stiffness, EI, and the torsional stiffness, GJ, if the box beam is made of a 4 laminae, unidirectional composite, with a lamina thickness of 0.0055”, all fibers being in the length direction. 3.6. Consider a composite material plate of dimensions of thickness h, composed of an E Glass/epoxy, which is modeled as being simply supported on all four edges. It is part of a structural system, which is subjected to a hydraulic load as shown below. The load is where is the weight density of the water. (a) To utilize the Navier approach determine which is given by (b) At what value of x will the maximum deflection occur? (c) At what value of x will the maximum stress occur? 3.7. Consider a square plate in which a = b = 20” made of a unidirectional Kevlar/epoxy composite, whose properties are: 145 (a) Determine the flexural stiffness matrix [ D ] . (b) In the first perturbation technique of Section 3.8, calculate and (c) Can this perturbation technique be used for this problem? (d) What is the total weight of this plate? (e) If this plate is simply supported on all four edges at what location (i.e., x = ? and y = ?) will the maximum deflection occur? (f) For the plate in e. above at what location will the maximum bending-stress occur? 3.8. Consider a plate composed of aluminum, an isotropic material of modulus of elasticity E, shear modulus G, and Poisson’s ratio The plate is of thickness h. Analogous to the stiffness matrix of Equation (2.66) determine the values of and for this construction. 3.9. Consider a plate measuring 16” x 16” in planform of [ 0°, 90°, 90°, 0° ], of total thickness 0.022”. The [ D ] matrix for this construction is If the plate is subjected to an in-plane compressive load in the direction, what is the critical buckling load per inch of the edge distance, using classical plate theory? 3.10. What is the fundamental natural frequency of the plate of Problem 3.9 in Hz (i.e. cycles per second), using classical plate theory? The weight density of the composite is 3.11. In designing a test facility to demonstrate the buckling of the plate of Problem 3.9, what load cell capacity (force capability) is needed to attain the loads necessary to buckle the plate? 3.12. (a) The plate of Problems 3.9 through 3.11 above will be used in an environment in which it will be exposed to a sinusoidal frequency of 6 Hz. Is it likely there will be a vibration problem requiring detailed study? Why? (b) What about 12 Hz? Why? 3.13. Could the first perturbation solution technique of Section 3.8 be used to obtain solution for the plate of Problem 3.9 subjected to a static lateral load, p ( x,y )? 3.14. Consider a rectangular plate of composite materials which is part of a space vehicle structure. Its dimensions are 10” x 10”. It is composed of Kevlar 49/epoxy 146 (properties are given in Appendix 2 of the text). It is composed of six laminae, unidirectional (all 0°), with ply thickness The density of the material is The plate is simply supported on all four edges. (a) (b) (c) (d) What are the flexural stiffnesses and Because the panel is part of a large space vehicle structure, care must be taken to identify all natural frequencies in the 0-1.5 Hz. Range. What, if any natural frequencies fall in this range? If the plate is subjected to a uniform in-plane compressive load in the x- direction, what is the critical buckling load, Will the plate buckle before it is overstressed or will it be overstressed before it buckles? 3.15. For a plate or panel, what are the four ways in which it may fail or become subjected to a condition which may terminate its usefulness? 3.16. A panel simply supported on all four edges, measuring a = 30”, b = 10”, composed of T300-5208 graphite epoxy, composed of laminae with the following properties: In the October 1986 issue of the AIAA Journal, M.P. Nemeth discuses the conditions in which one can ignore and in determining the buckling load for a composite plate. He defines: 147 If both of these ratios are less than 0.18, one can use Equation (3.149) to determine the buckling load within 2% of the correct value for a plate simply supported on all four edges. If either of the ratios is greater than 0.18 one must replace the left hand side of Equation (3.146) with the left hand side of Equation (3.170), which negates the use of the Navier and Levy methods being used, thus complicating the solution. For a four ply panel with stacking sequence of [+45°, – 45°, – 45°, + 45°], determine and to see if the simpler solution can be used. Determine the fundamental natural frequency in Hertz (cycles per second) for the panel of Problem 3.16 made of four plys, unidirectionally oriented (all 0º plys). Determine the critical buckling load, for the same panel as in Problem 3.16. For the panel of Problems 3.16 and 3.17, could the perturbation method of Section 3.8 be used to solve for deflections and stresses, i.e., is Consider a plate of dimensions a = 18” and b = 12”, composed of a laminated composite material whose lamina properties are: 3.17. 3.18. 3.19. 3.20. The stacking sequence of the plate is [ 0°, 90°, 90°, 0° ] in which each lamina is The plate is simply supported on each edge. What are and for this plate? For the plate of Problem 3.20, at what values of x and y will the maximum deflection occur if the plate is subjected to a uniform lateral load (a constant)? For the plate of Problems 3.20 and 3.21 above at which values of x and y would maximum ply stresses occur? For the plate of Problem 3.20, calculate the critical buckling load per unit width, 3.21. 3.22. 3.23. and 148 if the plate is subjected to a uniform compressive load in the x direction. What is the fundamental natural vibration frequency in Hz for the plate of Problem 3.20. Assume a weight density for the composite to be Suppose the plate of Problem 3.24 were designed to be subjected to a continuing harmonic forcing function at: (a) 38 to 48 Hz (b) 10 Hz. Would there be a problem structurally with this due to dynamic effects? Why? Consider a Kevlar 49/epoxy composite, whose properties are given in Appendix 2 of the text, and whose weight density is A plate whose stacking sequence is [0°, 90°, 90°, 0° ] is fabricated wherein each ply is 0.0055” thick. The plate is in planform dimensions, and is simply supported on all four edges. Determine and Could the perturbation solution technique of Section 3.8 be used to solve problems for the plate of Problem 3.26? If the plate of Problem 3.26 is subjected to an in-plane compressive load in the x- direction only, what is the critical buckling load per inch of edge distance, using classical plate theory? What is the fundamental natural frequency of the plate of Problem 3.26 in Hz., using classical plate theory? If the fundamental natural frequency were calculated including the effects of transverse shear deformation, would that frequency be higher, lower or equal to the frequency calculated in Problem 3.29 above? Consider a Kevlar 49/3501-6 epoxy composite with the following properties: For a unidirectional composite of thickness 0.1 inches, calculate 3.24. 3.25. 3.26. 3.27. 3.28. 3.29. 3.30. 3.31. [...]... for the mid-plane symmetric beam can be written as: Once the solution of w(x) is found, the bending stresses in each of the laminae are found by Obviously if both in-plane and lateral loads occur simultaneously, then the stresses are found by the sum of Equations (4.19) and (4. 26) 160 Thus the theory for the classical beam, i.e., no transverse shear deformation, with composite materials or isotropic materials, ... “stretch” the structure The term “column” is used when the structure shown is subjected to compressive forces in the xdirection which reduces the length of the structure, resulting in the compressive stresses, and/or the elastic instability (buckling) of the structure, a topic which will be discussed 1 56 later in this chapter Combination of these loads may occur, such as when the first and third types of. .. that the maximum value occurs at x = 0 and L, and is 162 To tie in the mechanics of single layer isotropic classical beams, with the mechanics of a symmetric laminated composite beam or rectangular cross-section, the following analyses are made Suppose the beam were of a one layer, isotropic material then the maximum stress would be at the top and bottom of the beam at each end, that is, and the traditional... all four edges that is 6 inches wide and 15 inches long made up of the unidirectional four ply graphite epoxy described in (a) above of Problem 3.32, what is the critical buckling load, if the compressive load is applied parallel to the longer direction of the plate? 3.34 If the plate of Problem 3.33 were subjected only to a uniform lateral load, where would the maximum value of the tensile stress be... the maximum stress can be calculated, and compared to the allowable stress properties for the material For a beam of a given composite material, the calculation is not so simple Having found through Equation (4.32), Equation (4.14) must be utilized to find the maximum curvature Then, and only then, can the maximum stress be calculated for each lamina through the use of Equations (4. 26) and (4.34) Then,... 1 or Section 2 The equations to solve involve solving an 8 × 8 set of algebraic equations and lengthy manipulations The final constants in this example are: The resulting displacement equations are: The location and magnitude of the maximum deflection will occur in either Section 1 or 2 depending upon the extent of the load, i.e., where is located Then by taking second derivatives of the deflections... 2 The boundary conditions at the ends can be easily written as follows: Four other boundary conditions must be determined, and they describe the compatibilities that must exist at the junction of beam parts 1 and 2, namely 166 Using the above, it is seen that there are eight equations to determine the eight boundary constants and in Equations (4.22) through (4.25), where the second subscript is used... overstressed before it could buckle? (d) What is the total weight of this plate? (e) What thickness would this plate have to be to have the buckling stress equal to the compressive strength of the composite material? CHAPTER 4 Beams, Columns and Rods of Composite Materials 4.1 Development of Classical Beam Theory A beam, column or rod is a long thin structural component of width b, height h and length L, where... there is no lateral distributed load, q(x), so only the homogeneous solution of Equation (4.25) is used for each of the two sections, or For this class of problems, again there are eight boundary conditions; four to be determined to satisfy the boundary conditions at the ends of the beam, and four to properly match conditions at the load P At the ends, the boundary conditions such as simple support, clamped,... now for some solutions 4.2 Some Composite Beam Solutions From the theory for a composite beam-rod-column developed in Section 4.1, solutions to all such problems can be found directly For the bending of a beam, the equations (4.21) through (4.25) result in four constants of integration, which are used to satisfy the boundary conditions at each end of the beam Discussions of classical boundary conditions . is composed of Kevlar 49/epoxy 1 46 (properties are given in Appendix 2 of the text). It is composed of six laminae, unidirectional (all 0°), with ply thickness The density of the material is The. edges. If either of the ratios is greater than 0.18 one must replace the left hand side of Equation (3.1 46) with the left hand side of Equation (3.170), which negates the use of the Navier and. classical plate theory? 3.10. What is the fundamental natural frequency of the plate of Problem 3.9 in Hz (i.e. cycles per second), using classical plate theory? The weight density of the composite