The Response Characteristic of Carbon-Reinforced Composite Plates under Transverse Loading Joshua Keegan Faculty Sponsor: Olanrewaju Aluko Department of Computer Science, Engineering and Physics University of Michigan-Flint Abstract Fiber-reinforced laminated plate behavior is governed by laminate shape, laminate size, boundary conditions, fiber orientation, and material properties and these influence deflections, force resultants, moment resultants, and stresses on the plate under loading. In this analysis, two graphite reinforced symmetrically laminated 0⁄90 cross-ply plates are analytically utilized to document the effects of boundary conditions on the response of fiber-reinforced structures. One plate was examined under simple support conditions with fixed horizontal motion, and the other under built-in conditions. Both support conditions showed no in-plane loading. The built-in plate was found to have much less out-of-plane deflection than the simply and are the same for both cases, though supported plate. Furthermore, the variations of the magnitude for both moment resultants was greater in the simply supported plate. The built-in plate also experienced less stress in both the x and y directions. 1.0 Introduction In order to examine the response characteristic of fiber-reinforced laminated plates, it is imperative to first examine the theories governing fiber-reinforced structures in general. The study of laminated plates is quite extensive, and subsequently an overview of the linear response of laminated plates will be the primary focus. The principles of the classical laminated plate theory are imperative to analysis. These principles are used by establishing equations that will dictate the response of the plate, and, based on specification of the conditions at the exterior and interior of the plate, these equations are used to predict the plate's response. There are two methods that may be used to derive governing conditions: through energy and variational principles, and through the Newtonian approach [1]. The Newtonian approach was used in the present analysis, to derive the differential equations of equilibrium by summing forces and moments of the laminates. The classical laminated plate theory generates three equilibrium equations which govern the response of a laminated plate [1]. These equations are valid for any rectangular plate, regardless of material or type of laminate. The equations are: 0 0 (1a) (1b) 0 (1c)
When examining a structure’s response to loading, it is necessary to express the equilibrium equations in terms of displacements [1]. This can be done by expressing stress resultants in terms of reference surface strains and curvatures, and then by expressing the surface strains and
2
curvatures in terms of surface displacements. The matrix relation between the stress resultants and the reference surface deformations are: (2)
where the laminate stiffness matrix is defined as: ∑ ∑
(3a) (3b) (3c) (4a)
,
∑ and the reference strains and curvatures are defined as: , ,
, ,
,
,
,
(4b) (4c) (4d)
,
,
Boundary conditions are extremely important in the examination of laminated plates. For the governing equations to be effective, boundary conditions must be satisfied along each edge of the plate. This results in four boundary conditions, which are used to simplify the extensive equilibrium equations. Furthermore, the specific class of the laminate can be used to simplify the equations further, resulting in a more manageable calculation. The most common simplifications result from symmetric laminates, symmetric balanced laminates, symmetric cross-ply laminates, and isotropic plates [1]. After expressions for the displacements are generated and simplified, the stresses and strains of the plates can be determined. This can be accomplished via kinematics for classical plates. Kinematics describes how a
