The Behavior of Structures Composed of Composite Materials Part 4 potx

79 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. Sloan, J.G. (1979) The Behavior of Rectangular Composite Material Plates Under Lateral and Hygrothermal Loads, MMAE Thesis, University of Delaware (also AFOSR-TR-78-1477, July 1978). Tsai, S.W. and Pagano, N.J. (1968) Invariant Properties of Composite Materials, Tsai, S.W. et al., eds., Composite Materials Workshop, Technomic Publishing Co., Inc., Lancaster, PA, pp. 233-253. Reissner, E. (1950) On a Variational Theorem in Elasticity, J. Math. Phys., 29, pp. 90. Mindlin, R.D. (1951) Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic Elastic Plates, Journal of Applied Mechanics, pg. 73. Lee, C.K. and Moon, F.C. (1990) Modal Sensors/Actuators, Journal of Applied Mechanics, Vol. 57, pp. 434-441. Yu., Y.Y. (1977) Dynamics for Large Deflection of a Sandwich Plate With Thin Piezoelectric Face Layers, Ed. G.S. Simitses, Analysis and Design Issues for Modern Aerospace Vehicles, ASME AD-Vol. 55, pp. 285-292. Leibowitz, M. and Vinson, J.R. (1991) Intelligent Composites: Design and Analysis of Composite Material Structures Involving Piezoelectric Material Layers. Part A-Basic Formulation, Center for Composite Materials Technical Report 91-54, University of Delaware, November. Larson, P.H. (1993) The Use of Piezoelectric Materials in Creating Adaptive Shell Structures, Ph.D. Dissertation, Mechanical Engineering, University of Delaware. Newill, J.F. (1996) Composite Sandwich Structures Incorporating Piezoelectric Materials, Ph.D. Dissertation, Mechanical Engineering, University of Delaware. Hilton, H.H., Yi, S. and Vinson, J.R. (1998) Probabilistic Structural Integrity of Piezoelectric Viscoelastic Composite Structures, Proceedings of the 39th AIAA/ASME/ASCE/AHS/ASC SDM Conference , April. 2.12 Problems 1. Consider a laminate composed of boron-epoxy with the following properties: If the laminate is a cross–ply with [0°/90°/90°/0°], with each ply being 0.25 mm (0.11") thick, and if the laminate is loaded in tension in the x direction (i.e., the 0° direction): (a) (b) (c) What percentage of the load is carried by the 0 plies? The 90° plies? If the strength of the 0° plies is MPA(198,000 psi), and the strength of the 90° plies is 44.8 MPA (6,500psi) which plies will fail first? What is the maximum load, that the laminate can carry at incipient failure? What stress exists in the remaining two plies, at the failure load of the other two others? 80 2. 3. 4. 5. 6. (d) If the structure can tolerate failure of two plies, what is the maximum load, that the other two plies can withstand to failure? A laminate is composed of graphite epoxy (GY70/339) with the following properties: MPA MPA MPA (0.6×106 psi) and Determine the elements of the A, B and D matrices for a two-ply laminate [+45°/–45°], where each ply is 0.15 mm (0.006") thick. Consider a square panel composed of one ply with the fibers in the directions as shown in Figure 2.12. Which of the orientations above would be the stiffest for the loads given in Figure 2.13? For a panel consisting of boron-epoxy with the properties of Problem 1 above, and a stacking sequence of [0 0 /+45°/-45 0 /0°] , and a ply thickness of 0.14 mm (0.005"), determine the elements of the elements of the A, B and D matrices. The properties of graphite fibers and a polyimide matrix are as follows: (a) Find the modulus of elasticity in the fiber direction, of a laminate of graphite- polymide composite with 60% fiber volume ratio. (b) Find the Poisson’s ratio, ? (c) Find the modulus of elasticity normal to the fiber direction, (d) What is the Poisson’s ratio, ? Consider a laminate composed of GY70/339 graphite epoxy whose properties are given above in Problem 2. For a lamina thickness of 0.127 mm (0.005"), calculate the elements of the A, B and D matrices for the following: (a) [0°, 0°, 0°, 0°] (unidirectional); (b) [0°, 90°, 90°, 0°] (across-ply); (c) i.e. [+45°/-45°/-45°/+45°] (an angle-ply); (d) (a quosi-isotropic, 8 plies); (e) Compare the various stiffness quantities for the four laminates above. 7. Consider a laminate composed of GY70/339 graphite epoxy whose properties are given above in Problem 2. For a lamina thickness of 0.127 mm (0.005") cited in Problem 6, calculate the elements of the [A], [B] and [D] matrices for the following laminates: (a) (b) (c) [+45/-45/-45/+45] [+45/-45/+45/-45] 81 8. 9. 10. 11. 12. 13. (d) Compare the forms of the A, B and D matrices between laminate type. What type of coupling would you expect in the (B) matrix for (a) and (b) below: (a) 0º/90° laminate (b) laminate Given a composite laminate composed of continuous fiber laminate laminae of High Strength Graphite/Epoxy with properties of Table 2.2, if the laminate architecture is [0°, 90°, 90°, 0°], determine if and each ply thickness is 0.006". Consider a plate composed of a 0.01" thick steel plate joined perfectly to an aluminum plate, 0.01" thick. Using the properties of Table 2.2 calculate if the Poisson’s Ratio of each material is Consider a unidirectional composite composed of a polyimide matrix and graphite fibers with properties given in Problem 5 above. In the fiber direction, what volume fraction is required to have a composite stiffness of psi to match an aluminum stiffness. A laminate is composed of ultra high modulus graphite epoxy with properties given in Table 2.2 below. Determine the elements of the [A], [B] and [D] matrices for a two ply laminate [+45°/-45°], where each ply is 0.006 " thick. For the material 0.31. A laminate is composed of boron-epoxy with the properties of Problem 2.1 and a stacking sequence of [0/+45°/–45°/0°], and a ply thickness of 0.006". Determine the elements of the A, B and D matrices. 82 Consider a composite laminae made up of continuous Boron fibers imbedded in an epoxy matrix. The volume fraction of the Boron fibers in the composite is 40%. Assuming that the modulus of elasticity of the Boron fiber is psi and the epoxy is psi, find: (a) The Young’s moduli of the composite in the 1 and 2 direction. (b) Consider an identical second lamina to be glued to the first so that the fibers of the second lamina are parallel to the 2 direction. Assuming the thickness of each lamina to be 0.1" and neglecting Poisson’s Ratio, what are the new moduli in the 1 and 2 directions. 14. 15. The properties of graphite fibers and a polyimide matrix are as follows: GRAPHITE POLYIMIDE Finder the modulus of elasticity in the fiber direction, of a lamina of graphite – polyimide composite with 70% fiber volume ratio. Find the Poisson’s Ratio, Finder the modulus of elasticity normal to the fiber direction, What is the Poisson’s Ratio, Compare these properties with those obtained for the same material system but with in problem 2.5. In a given composite, the coefficient of thermal expansion for the epoxy and the graphite fibers are in/in/°F and in/in/°F respectively. For space application where no thermal distortion can be tolerated what volume fractions of each component are required to make zero expansion and contraction in the fiber direction for an all 0° construction? (Hint: Use the Rule of Mixtures). Find the A, B and D matrices for the following composite: 50% volume Fraction Boron- Epoxy Composite (a) (b) (c) (d) (e) 16. 17. Stacking Sequence (each lamina is 0.0125" thick) Three composite plates are under uniform transverse loading. All the conditions, such as materials, boundary conditions and geometry, etc. are the same except the 18. 83 19 20. 21. 22. 23. 24. stacking sequence as shown below. Without using any calculation, indicate which plate will have maximum deflection and will have minimum deflection. Consider a Kevlar 49/epoxy composite, whose properties are given on p303 in the text. A plate whose stacking sequence is [0,90,90,0] (i.e. a cross ply laminate) is fabricated wherein each thickness is 0.0055 inches (a) Determine the A, B, and D matrix component. (b) What if any are the couplings in this cross-ply construction that are decided below Equation (2.62)? (c) If only in-plane loads are applied, is the plate stiffer in the x direction or y direction, or are they the same? (d) If only plate bending is considered, is the plate stiffer in the x direction, the y direction, or are they equally stiff? Given the following fiber and matrix properties for HM-S/epoxy composite components: Epoxy HM-S/Graphite Determine each of the following properties for a unidirectional composite: and for the fiber volume fractions of 30%, 60%. Which properties increase linearly with volume fraction? Which do not increase linearly with volume fraction? Given a cross-ply construction of four lamina of the same composite material system oriented as 0°, 90°, 90°, 0°, each lamina being equally thick, which elements of the [A], [B] and [D] matrices of Equation (2.66) will be equal to zero. Given an angle-ply construction of five plies of the same composite material oriented as each of equal thickness, which elements of the [A], [B] and [D] matrices of Equation (2.66) will be equal to zero. Determine the elements of the matrix analogous to the of Equation (2.10) through (2.12) for orthotropic materials. (Hint: start with Equation (2.17) and solve for the A laminate is composed of graphite epoxy (GY70/339) with the following properties: . 84 (a) Determine the elements of the [A], [B], and [D] matrices for a two-ply laminate [+45/-45], where each ply is 0.15mm. (0.006 inches) thick. (b) What couplings exist as discussed below Equation (2.66) for this laminate? For a panel consisting of Boron-Epoxy with the properties and a stacking sequence of [0°, +45°, -45°, 0°], and a ply thickness of 0.006 inches, determine the elements of the A, B and D matrices. What would the elements be if the ply thickness were 0.0055 inches? Determine how the A, B, D matrices are populated for the following two stacking sequences and The subscript QS mean symmetric Q times where Q=2, 3, The composite material is orthotropic and has the properties with each lamina having thickness In the [A], [B], and [D] matrices, place an x or an O for each element, where an x shows that the component is non zero, and O shows that the component is zero. What type of couplings, as discussed below Equation (2.66) would you expect in the B matix for (a) and (b) below: (that is, identify the non-zero terms) (a) 0°/90° laminate (b) Find the [A], [B] and [D] for the following laminates. 25. 27. 26. 28. Given: i)=0.3. In problem 2.28 which laminate is stiffest and which is the least stiff for (a) In-plane loads in the 0° direction. (b) In-plane loads in the 90° direction. (c) Bending in the 0° direction. (d) Bending in the 90° direction. Consider a laminate composed of GY 70/339 graphite/epoxy with the following properties, 29. 30. 85 Using the laminate thickness as 0.127mm (0.005 inches) calculate the elements of the [A], [B], and [D] matrices for the following laminates. (a) (b) (b) (c) Compare the forms of the [A], [B] and [D] matrices between laminate types. A composite material has stiffness matrix as follows, 31. Determine the state of stress if the strains are given by, Consider the stress acting on an element of a composite material to be as shown below. The material axes 1,2 are angle with respect to the geometry loading axes for the element. Taking the material properties as noted below, find the m-plane displacements u(x,y), v(x,y). 33. A 50% boron-epoxy orthotropic material is subjected to combined stress as shown below. 32. 86 Find the stress on the material element for a 45° rotation about the z-axis in a positive sense. If the strain components in the non-rotated system are given by: Find the corresponding strains in the rotated system. Comment on the corresponding stresses and strains in the rotated system. (a) (b) CHAPTER 3 PLATES AND PANELS OF COMPOSITE MATERIALS 3.1 Introduction In Chapter 2, the constitutive equations were developed in detail, describing the relationships between integrated stress resultants integrated stress couples in-plane mid-surface strains and the curvatures as seen in Equation (2.66). These will be utilized with the strain- displacement relations of Equations (2.48) and (2.50) and the equilibrium equations to be developed in Section 3.2 to develop structural theories for thin walled bodies, the configuration in which composite materials are most generally employed. Plates and panels are discussed in this chapter. Beams, rods and columns are discussed in Chapter 4. Shells will be the subject of Chapter 5. The use of energy methods for solving structures problems is discussed in Chapter 6. However to study any of the structural equations it is necessary to first develop suitable equilibrium equations. 3.2 Plate Equilibrium Equations The integrated stress resultants ( N ) , shear resultants (Q ) and stress couples ( M ), with appropriate subscripts, are defined by Equation (2.54), and their positive directions are shown in Figure 2.13, for a rectangular plate, defined as a body of length a in the x - direction, width b in the y -direction, and thickness h in the z -direction, where h<< b, h<< a, i.e. a thin plate. In mathematically modeling solid materials, including the laminates of Chapter 2, a continuum theory is generally employed. In doing so, a representative material point within the elastic solid or lamina is selected as being macroscopically typical of all material points in the body or lamina. The material point is assumed to be infinitely smaller than any dimension of the structure containing it, but infinitely larger than the size of the molecular lattice spacing of the structured material comprising it. Moreover, the material point is given a convenient shape; and in a Cartesian reference frame that convenient shape is a small cube of dimensions d x, d y, and d z as shown in Figure 3.1 below. 88 This cubic material point of dimension d x, d y and d z is termed a control element. The positive values of all stresses acting on each surface of the control element are shown in Figure 3.1, along with how they vary from one surface to another, using the positive sign convention consistent with most scientific literature, and consistent with Figure 2.1. Details of the nomenclature can be found in any text on solid mechanics, including Vinson [1,2], In addition to the surface stresses acting on the control element shown in Figure 3.1, body force components and can also act on the body. These body force components such as gravitational, magnetic or centrifugal forces are proportional to the control element volume, i.e., its mass. A force balance can now be made in the x, y and z directions resulting in three equations of equilibrium. For instance, a force balance in the x-direction would yield Canceling terms and dividing the remaining terms by the volume results in the following Similarly, equilibrium in the y and z directions yields: [...]... solution of isotropic plate problems [4] Aware of all of the above, and based upon the fact that the solution of the second case of the Levy solution of (3.52) has the same form as that of the isotropic case, Vinson showed that the cases of (3.51) and (3.53) can be dealt with as perturbations about the solution of the same plates composed of isotropic materials [5,6] Consider the governing equation for the. .. if the x-edges are clamped or free Then whatever the relevant form of the boundary conditions on x = 0 and x = a, the total and hence w(x,y) is known from Equation (3 .44 ) Then, for a composite- material laminated plate, one must calculate the curvatures, as was done is the previous section for the Navier approach: Knowing these, one can calculate the bending stresses in each of the k laminae through the. .. 105 Otherwise, one must seek another particular solution In any case, one must then add the relevant homogeneous to the particular to satisfy any set of boundary conditions on the x-edges of the plate For example, suppose the x = 0 edge is simply supported, then from Equation (3.30) the boundary conditions are However, when w(x,y) has the form of Equation (3 .44 ) this then implies that: where primes... the first term on the right, clearly the moments of all the interlaminar stresses between plies cancel each other out, and the only non-zero terms are the moments of the applied surface shear stresses hence that term becomes Using the former expression, the equation of equilibrium of moments in the xdirection is Similarly in the y-direction the moment equilibrium equation is where all the terms are defined... For the same material there is little difference between the maximum value of the stress for both the unidirectional and cross-ply composites at similar plate locations, however, the stresses differ significantly 3 The stress at a fixed location for the graphite/epoxy laminate is much smaller relative to the value (10%), compared to that in the E glass/epoxy laminate where is 33% of the value of at the. .. Finally, depending upon whether the panel is a laminate or a single layer, the maximum stresses are determined through calculating the curvatures at the locations of maximum stress couples Knowing these, one can calculate the stresses in each of the k laminae through the following, since in this case of a lateral loading only, there is no in-plane response, i.e and are zero The number of terms necessary to... desired accuracy depends upon the particular load p(x, y ) , the aspect ratio of the plate (b/a), and the material system of which the plate is fabricated 98 3.6 Navier Solution for a Uniformly Loaded Simply Supported Plate – An Example Problem The case of a uniformly loaded, simply supported plate is solved by means of the Navier series solution of Section 3.5, for two composite materials systems: unidirectional... notation, these three equations can be consolidated to the following: i, j = x, y, z These three equations comprise the equilibrium equations for a three dimensional elastic body However, for beam, plate and shell theory, whether involving composite materials or not, one must integrate the stresses across the thickness of the thin walled structures to obtain solutions Recalling the definitions of Chapter... The above technique can be very useful However, even if then the composite may fall within another range where for In that case, the plate behaves as a plate in the x-direction, but because it behaves as a membrane in the y-direction, with the following simpler governing differential equation, with the solution in the form of the following for a plate simply supported on the y edges where Even if the. .. h(y) are specified The form of Equation (3 .44 ) requires that the h(y) portion of the load also be expanded in terms of a half range sine series, such as where 103 Substituting Equations (3 .44 ) through (3 .46 ) into (3.29) and observing that the equation exists only if it is true term by term, it is seen that, after dividing by and the trigonometric function: where Note that Equation (3 .48 ) was derived without . 79 36. 37. 38. 39. 40 . 41 . 42 . 43 . 44 . 45 . Sloan, J.G. (1979) The Behavior of Rectangular Composite Material Plates Under Lateral and Hygrothermal Loads, MMAE Thesis, University of Delaware (also AFOSR-TR-78- 147 7,. thickness of 0.127 mm (0.005") cited in Problem 6, calculate the elements of the [A], [B] and [D] matrices for the following laminates: (a) (b) (c) [ +45 / -45 / -45 / +45 ] [ +45 / -45 / +45 / -45 ] 81 8. 9. 10. 11. 12. 13. (d) Compare. 0° direction): (a) (b) (c) What percentage of the load is carried by the 0 plies? The 90° plies? If the strength of the 0° plies is MPA(198,000 psi), and the strength of the 90° plies is 44 .8 MPA (6,500psi) which