Transverse load

For plate problems, whether the specially orthotropic laminate has a single layer or multiple layers is essentially immaterial the laminate need only be characterized by 0 2, D22. and Dgg in Equation (5.2). That is, because there is no bending-extension coupling, the force-strain relations, Equation (5.1), are not used in plate analysis for transverse loading causing only bending. However, note that force-strain relations are needed in shell analysis because of the differences between deformation characteristics of plates as opposed to shells. 

The equilibrium differential equations in terms of the force and moment resultants derived in Chapter 4 and the transverse loading p(x,y) are... 

DEFLECTION OF SIMPLY SUPPORTED LAMINATED PLATES UNDER DISTRIBUTED TRANSVERSE LOAD... 

Consider the general class of laminated rectangular plates that are simply supported along edges x = 0, x = a, y = 0, and y = b and subjected to a distributed transverse load, p(x,y). In Figure 5-8. The transverse load can be expanded in a double Fourier sine series ... 

Many different types of transverse loading can easily be represented by Equation (5.25). For example, a uniform load, Pq, is given by... 

Figure 5-8 Simply Supported Laminated Rectangular Plate under Distributed Transverse Load, p(x,y)...

If the transverse loading is represented by the Fourier sine series in Equation (5.25), the solution to this fourth-order partial differential equation and subject to its associated boundary conditions is remarkably simple. As with isotropic plates, the solution can easily be verified to be... 

For a uniform transverse load, the solution is easily shown to be... 

If the transverse load is but one term of the Fourier series, that is. 

Figure 5-13 Deflection of an Antisymmetric Cross-Ply Laminated Plate under Sinusoidal Transverse Load...

Results for a square plate under sinusoidal transverse load with a variable modulus ratio, E1/E2, and a 45° lamination angle are shown in Figure 5-17. There, the effect of bending-extension coupling on deflections is significant for all modulus ratios except those quite close to Ei/E2=1. 

Figure 5-37 Deflection under Uniform Transverse Loading of...

J. M. Whitney, Bending-Extension Coupling in Laminated Plates Under Transverse Loading, Journal of Composite Materials, January 1969, pp. 20-28. 

J. E. Ashton, Clamped Skew Plates of Orthotropic Material Under Transverse Load, in Devebpments in Mechanics, Vol. 5, The Iowa State University Press, Ames, Iowa, 1969, pp. 297-306. 

Pagano studied cylindrical bending of symmetric cross-ply laminated composite plates [6-21]. Each layer is orthotropic and has principal material directions aligned with the plate axes. The plate is infinitely long in the y-direction (see Figure 6-16). When subjected to a transverse load, p(x), that is, p is independent of y, the plate deforms into a cylinder ... 

For a simply supported laminated rectangular plate subjected to the distributed transverse load... 

Whitney solved Equations (6.44) for a square four-layered symmetric cross-ply [0 /90 /90 /0 ] laminated graphite-epoxy plate under the transverse load p = Po sin(7tx/a) sin(jiy/a) [6-30], The material properties are typical of a high-modulus graphite-epoxy ... 

TRANSVERSE LOADS IN-PLANE LOADS EXCITATION FREQUENCIES... 

Note that the shear equation. Equation (D.6), can be substituted in the transverse load equation. Equation (D.4), to get... 

A beam subjected to a simple transverse load (Figure 2-29a) will bend. Furthermore, if the beam is cut (Figure 2-29b) and free body diagrams of the remaining sections are constructed, then a shear force V and a moment M must be applied to the cut ends to maintain static equilibrium. 

For purposes of this specification, stresses in the individual members of a latticed or trussed structure resulting from elastic deformation and rigidity of joints are defined as secondary stresses. These secondary stresses may be taken to be the difference between stresses from an analysis assuming fully rigid joints, with loads applied only at the joints, and stresses from a similar analysis with pinned joints. Stresses arising from eccentric joint connections, or from transverse loading of members between joints, or from applied moments, must be considered primary stresses. 

The shear mode involves the application of a load to a material specimen in such a way that cubic volume elements of the material comprising the specimen become distorted, their volume remaining constant, but with opposite faces sliding sideways with respect to each other. Shear deformation occurs in structural elements subjected to torsional loads and in short beams subjected to transverse loads. 

In many isotropic materials the shear modulus G is high compared to the elastic modulus E, and the shear distortion of a transversely loaded beam is so small that it can be neglected in calculating deflection. In a structural sandwich the core shear modulus G, is usually so much smaller than Ef of the facings that the shear distortion of the core may be large and therefore contribute significantly to the deflection of a transversely loaded beam. The total deflection of a beam is thus composed of two factors the deflection caused by the bending moment alone, and the deflection caused by shear, that is, S = m + Ss, where S = total deflection, Sm = moment deflection, and Ss = shear deflection. 

Under transverse loading, bending moment deflection is proportional to the load and the cube of the span and inversely proportional to the stiffness factor, El. Shear deflection is proportional to the load and span and inversely proportional to shear stiffness factor N, whose value for symmetrical sandwiches is ... 

The microductile/compliant layer concept stems from the early work on composite models containing spherical particles and oriented fibers (Broutman and Agarwal, 1974) in that the stress around the inclusions are functions of the shear modulus and Poisson ratio of the interlayer. A photoelastic study (Marom and Arridge, 1976) has proven that the stress concentration in the radial and transverse directions when subjected to transverse loading was substantially reduced when there was a soft interlayer introduced at the fiber-matrix interface. The soft/ductile interlayer allowed the fiber to distribute the local stresses acting on the fibers more evenly, which, in turn, enhanced the energy absorption capability of the composite (Shelton and Marks, 1988). 

Zhang, W. (1993). Compulation of stress fields in unidirectional n-phase fibrous composite under longitudinal and transverse loads. Compiit. Structures 34, 647-653. 

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