Bending of Simply Supported Isotropic and Composite Laminate Plates

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1 Bending of Simply Supported Isotropic and Composite Laminate Plates Ernesto Gutierrez-Miravete 1 Isotropic Plates Consider simply a supported rectangular plate of isotropic material (length a, width b, thickness t, - with t << a, b -, elastic modulus E, Poisson s ratio ν) under a uniform pressure q acting normal to the surface of the plate. Let the x axis be aligned with the length of the plate, the y axis with its width and the z axis with its thickness with z = located in the middle of the plate thickness. For isotropic plates, the (engineering) stress-strain relationships are given by Q 11 Q 1 Q 1 Q 11 Q 66 ǫ x ǫ y ǫ xy where Q is the reduced stiffness matrix with components Q 66 = Q 11 = Q 1 = E 1 ν νe 1 ν E (1 + ν) = G The Kirchoff-Love hypothesis is commonly used to simplify the analysis of plates. If the thickness of a plate is much smaller that its length and its width plate straight normals to the plane of the plate remain approximately straight, of constant length and normal to the plane of the plate when the plate deforms. This is equivalent to neglecting the shearing strains in planes perpendicular to the middle surface (z = z = ) as well as the normal strain normal to the plate (ǫ z = ). Under the hypothesis, the strain components in the 1

2 plate are given in terms of the strains and curvatures along the middle surface of the plate, respectively ǫ x, ǫ y, ǫ xy, κ x, κ y, κ xy, i.e. ǫ x ǫ y ǫ xy ǫ x κ ǫ x y + z κ ǫ y xy κ xy So that the stress-strain relationships become Q 11 Q 1 Q 1 Q 11 Q 66 κ x κ y ǫ x ǫ y + z ǫ xy κ xy The mid plane strains and curvatures are given respectively by ǫ x ǫ y ǫ xy u / x v / y u / y + v / x and κ x w / x κ y w / y κ xy w / x y and where u, v, w = w are the components of displacement at the middle surface of the plate. The normal and shear forces acting on the plate are given by N x N y N xy t t dz The bending and twisting moments acting on the plate are given by M x M y M xy t t zdz Combination and rearrangement of the above yields finally N x N y N xy Et 1 ν νet 1 ν νet 1 ν Et 1 ν 1 ν Et 1 ν ǫ x ǫ y ǫ xy

3 and M x M y M xy Et 3 1(1 ν ) νet 3 1(1 ν ) νet 3 1(1 ν ) Et 3 1(1 ν ) 1 ν Et 3 1(1 ν ) κ x κ y κ xy Under the Kirchoff-Love hypothesis, commonly used when analyzing the deformation of plates and shells, and assuming small deflections (linear elasticity), it can be shown that the condition of mechanical equilibrium assuming that the edges of the plate are free to move in the plane of the plate has the form M x x + M y y M xy x y = q Combining with the above yields the governing equation in terms of the deflection of the mid-surface of the plate: where D 4 w x 4 + D 4 w x y + D 4 w y 4 = q D = Et 3 1(1 ν ) is the flexural rigidity (bending stiffness) of the plate. This is a bi-harmonic equation that must be solved subject to specific boundary conditions. For simply supported edges, deflections and normal moments are zero along the edges, i.e. for x = and x = a. w =, w =, w x = w y = for y = and y = b. An exact solution of this problem was obtained by Navier in 18 and it is w = 16q π 6 D with m = 1, 3, 5,... and n = 1, 3, 5,... m=1 n=1 sin( mπx a )sin(nπy b ) mn( m a + n b ) 3

4 The bending and twisting moments are given by M x = D( w x + ν w y ) M y = D( w y + ν w x ) M xy = D(1 ν) w x y Finally, the elastic strain energy of the plate V is given by V = D a b ( w x + w y ) (1 ν)[ w x w y ( w x y ) ]dxdy Composite Laminate Plates Composite plates are produced by stacking thin sheets of fiber reinforced polymer called plies. Consider simply a supported rectangular ply (ply thickness h, ply length a, width b) consisting of unidirectionally aligned reinforcing fibers embedded in a polymer matrix, (elastic moduli E 1 (longitudinal, parallel to the fibers), E (transverse to the fibers), shear modulus G 1, longitudinal Poisson ratio ν 1, transverse Poisson ratios ν 1, ν 3 ) under a uniform load q acting normal to the surface of the ply. If a ply is aligned with the fiber direction coinciding with the x axis and the transverse direction with the y axis, the material is called specially orthotropic and the stress-strain relations are Q 11 Q 1 Q 1 Q Q 66 where components of the reduced stiffness matrix are κ x κ y ǫ x ǫ y + z ǫ xy κ xy Q 11 = E 1 1 ν 1 ν 1 Q 1 = ν 1E 1 ν 1 ν 1 Q = E 1 ν 1 ν 1 4

5 Q 66 = G 1 An important design feature of composite plies is that the longitudinal and transverse directions of any individual ply can be oriented at any angle θ with respect the a x y axis system used as reference. Hence for a ply where the fiber axis is oriented at an angle θ with respect to the reference x axis the stress-strain relationships are instead given by Q 11 Q 1 Q 16 Q 1 Q Q 6 Q 16 Q6 Q66 where Q is the transformed reduced stiffness matrix with components given by Q 11 = Q 11 cos 4 θ + (Q 1 + Q 66 )sin θ cos θ + Q sin 4 θ ǫ x ǫ y ǫ xy Q 1 = (Q 11 + Q 4Q 66 )sin θ cos θ + Q 1 (sin 4 θ + cos 4 θ) Q = Q 11 sin 4 θ + (Q 1 + Q 66 )sin θ cos θ + Q cos 4 θ Q 16 = (Q 11 Q 1 Q 66 )sin θ cos 3 θ + (Q 1 Q + Q 66 )sin 3 θ cos θ Q 6 = (Q 11 Q 1 Q 66 )sin 3 θ cosθ + (Q 1 Q + Q 66 )sinθ cos 3 θ Q 66 = (Q 11 + Q Q 1 Q 66 )sin θ cos θ + Q 66 (sin 4 θ + cos 4 θ) Another very important design feature of composites is that plies can be stacked to form thicker sections called laminates. Moreover, the stacking pattern can be selected to optimize the properties of the laminate. Consider a composite laminate (thickness t) formed by stacking N composite plies each with thickness h k, k = 1,, 3,..., N such that t = N h k, following a suitably selected stacking pattern. The normal and shear forces acting on the laminate are given by N x N y N xy t t dz = N zk z k 1 dz And the bending and twisting moments acting on the laminate are given by t N zk zdz = zdz M x M y M xy t 5 z k 1

6 and Performing the indicated integrations and rearranging yields N x A 11 A 1 A 16 ǫ x B 11 B 1 B 16 N y A 1 A A 6 ǫ y + B 1 B B 6 N xy A 16 A 6 A 66 B 16 B 6 B 66 M x M y M xy ǫ xy B 11 B 1 B 16 ǫ x B 1 B B 6 ǫ y B 16 B 6 B 66 ǫ xy + κ x κ y κ xy D 11 D 1 D 16 κ x D 1 D D 6 κ y D 16 D 6 D 66 κ xy where A, B and D are, respectively, the extensional, coupling and bending stiffnesses of the laminate with components given by N A ij = ( Q ij ) k (z k z k 1 ) B ij = 1 N ( Q ij ) k (zk zk 1) D ij = 1 N ( 3 Q ij ) k (zk 3 zk 1) 3 In practice, for the computation of all the A ij, B ij, D ij, the mid-plane of the laminate is selected as the origin of the z axis (z = = z (N+1)/ ) and the locations z, z 1, z,..., z N (except z (N+1)/ ) representing the boundaries of each ply. For instance for a three ply laminate formed with plies of thickness (in mm) h 1 =.1, h =., h 3 =.1, one has z =., z 1 =.1, z =., z 3 =.1, z 4 =.. Again, under the Kirchoff-Love and small deflection hypotheses, the equation governing the deflection w(x, y) of the plate in this case is 4 w D 11 x + 4D 4 w 4 16 x 3 y + (D 4 w 1 + D 66 ) x y + 4D 4 w 6 x y + D 4 w 3 y = q 4 where D 11, D 16, D 1, D 66, D, D 6 are the bending stiffnesses of the composite plate. The boundary conditions are, for x = and x = a. w =, w =, w D 11 x + D w 1 y + D 16 w D 1 x + D w y + D 6 6 w x y = w x y =

7 for y = and y = b. The elastic strain energy of the plate V is given by V = 1 a b [D 11 ( w x ) + D 1 w x w y + D ( w y ) + 4D 66 ( w x y ) + w w 4D 16 x x y + 4D w w 6 y x y ]dxdy A specially kind of laminate is obtained when specially orthotropic plies are symmetrically arranged about the laminate middle surface. A laminate without shear or twist coupling nor bending-extension coupling is obtained (i.e. B ij = and D 16 = D 6 = ). The deflection equation in this case becomes 4 w D 11 x + (D 4 w D 66 ) x y + D 4 w y = q 4 An exact solution of this problem can be obtained by an approach similar to the one used by Navier in 18 for the isotropic plate and it is w = 16q π 6 m=1 n=1 ( 1 mn )sin(mπx a )sin(nπy b ) D 11 ( m a )4 + (D 1 + D 66 )( m a ) ( n b ) + D ( n b )4 with m = 1, 3, 5,... and n = 1, 3, 5,... However, for many laminates, the stiffness components B ij, D 16 and D 6 may be non-zero, an exact solution cannot be obtained and approximation methods are required. 3 The Ritz Method The Ritz method is a technique discovered by Ritz to determine approximate solutions to the partial differential equations encountered in plate theory. The method is an application of the principle of minimum potential energy. The total potential energy of a loaded plate consists of the internal elastic strain energy of the plate V minus the potential energy of the external forces. In the case considered here, the potential energy associated with the external forces is given by Ω = So that the total potential energy E is a B E = V Ω qw dxdy 7

8 The Ritz method is then implemented by assuming that the deflection can be expressed as a linear combination of simple orthogonal functions (basis functions) satisfying the specified boundary conditions. For the situation at hand, a suitable expression is then w (x, y) = I J i=1 j=1 c k sin( iπx a )sin(jπy b ) where the c k are unknown coefficients (k = i + (j 1)I = 1,, 3,..., K where K = I J) and the numbers I and J will determine the accuracy of the approximation. The assumed expression for w is then substituted into V and Ω and the integrations performed. The result of this is a more or less complicated looking equation involving all the unknown coefficients. The principle of minimum potential energy is implemented by differentiating the total potential energy with respect to each coefficient and equation the result to zero, i.e. E = E =... = E = c 1 c c K This operation produces a system of K simultaneous linear algebraic equations whence the values of the unknown coefficients can be determined using standard methods. Substitution of the obtained values into the assumed equation for w yields the desired approximate solution. 4 The Finite Element Method The finite element method can be regarded as a generalization of the Ritz method. First, the computational domain is subdivided into a collection of contiguous, non-overlapping sub-domains connected at nodes. As in the Ritz method, one writes down an expression for the approximated quantity as a linear combination of the basis functions and then minimizes the total potential energy of the system with respect to a set of unknown coefficients. The unknown coefficients in the case of the finite element method turn out to be the nodal values of the dependent variable whose approximation is being sought. The basis functions selected for the finite element approximation are simple functions of compact support. Perhaps the best known example are the linear roof functions with value of one at a node, zero at all next neighbor nodes and varying linearly in between according to a simple first order Lagrange interpolating polynomial. In implementing the method one proceeds as with the classic Ritz approach and minimizes the total potential energy. The result is a set of linear algebraic equations for the unknown coefficients. Most importantly, since the basis functions are of compact support, the resulting system is sparse. Solution of the system produces the nodal values and substitution into the assumed expression for the approximated quantity yields the finite element approximation. The accuracy of the approximation can be increased either by increasing the number of elements in the subdivision or by increasing the order of the interpolating polynomial or both. 8

9 5 References 1.- S.P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, nd ed. McGraw-Hill, New York, L.P. Kollar and G.S. Springer, Mechanics of Composite Structures, Cambridge U.P. Cambridge, 3. 9

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