Transverse deflection

Then, upon substitution of the derivative of the transverse deflection, v, fory, Equation (3.95), in the work expression, Equation (3.107),... 

Obvious and sometimes drastic simplifications occur when the laminate is symmetric about the middle surface (By = 0), specially orthotropic (all the terms with 16 and 26 subscripts vanisn in addition to the By), homogeneous ( = 0 and Djj = Ayr/12), or isotropic. In all those cases. Equations (5.6) and (5.7) are coupled to each other, but uncoupled from Equation (5.8). That is. Equation (5.8) contains derivatives of the transverse displacement w only, and Equations (5.6) and (5.7) contain both u and V but not w. Accordingly, only Equation (5.8) must be solved to determine the transverse deflections of a plate with the aforementioned... 

A specially orthotropic laminate has either a single layer of a specially orthotropic material or multiple specially orthotropic layers that are symmetrically arranged about the laminate middle surface. In both cases, the laminate stiffnesses consist solely of A, A 2> 22> 66> 11> D 2, D22, and Dgg. That is, neither shear-extension or bend-twist coupling nor bending-extension coupling exists. Thus, for plate problems, the transverse deflections are described by only one differential equation of equilibrium ... 

Symmetric angle-ply laminates were described in Section 4.3.2 and found to be characterized by a full matrix of extensional stiffnesses as well as bending stiffnesses (but of course no bending-extension coupling stiffnesses because of middle-surface symmetry). The new facet of this type of laminate as opposed to specially orthotropic laminates is the appearance of the bend-twist coupling stiffnesses D. g and D2g (the shear-extension coupling stiffnesses A. g and A2g do not affect the transverse deflection w when the laminate is symmetric). The governing differential equation of equilibrium is... 

Figure 5-10 Transverse Deflection Coefficient versus Principal Stiffness Ratio (After Ashton [5-11 ])...

Hyer shovys the conditions under vyhich the cylindrical shape must exist and when the saddle shape must exist [6-38], He approximates the transverse deflection of an unsymmetric cross-ply laminate as... 

Aitemativeiy, the beam end couid have compiete rotational restraint and no transverse displacement, i.e., clamped. However, a third boundary condition exists in Rgure D-3 just as in Figure D-2. That is, an axial condition on displacement or force must exist in addition to the conditions usually thought of as comprising a clamped-end condition. Note that the block-like device at the end of the beam prevents rotation and transverse deflection. A similar device will be used later for plates. Whether all of the three boundary conditions can actually be enforced depends on the order of the differential equation set when (necessarily approximate) force-strain and moment-curvature relations are substituted in Equations (D.2), (D.4), and (D.7). 

A mathematical treatment will be made for a still simpler dimer configuration shown in Fig. 56, which illustrates possible motions of specific parts of this structure due to thermal excitation. Figure 56a refers to a pure transverse deflection of the left water molecule perpendicular to an equilibrium (horizontal) position of an H-bonded molecule. The right molecule is assumed... 

We see from Fig. 57b that if the transverse deflection c is small, the potential (20) has also a flat bottom similar to that calculated for the potential (443) at a small angular deflection p. As shown in Fig. 57b at q > 0.25, the function (443) rapidly increases. In view of Fig. 58d and Table XXI the estimated mean frequency (v) of transverse vibrations is about order of magnitude less than a mean rotational frequency. This result roughly justifies our neglect of the translational motion in derivations of the formulas for rotational motion. 

Hu Y, Gao Y, Singamaneni S, Tsukruk VV, Wang ZL (2009) Converse piezoelectric effect induced transverse deflection of a free-standing ZnO microbelt. Nano Lett 9 2661-2665... 

Formally, the differential equation relating the transverse deflection of a simple beam of width b , thickness h is given by ... 

For URPs, the emphasis is somewhat different. Due to their relatively low stiffness, component deformations under load may be much higher than for metals and the design criteria in step (b) are often defined in terms of maximum acceptable deflections. Thus, for example, a metal panel subjected to a transverse load may be limited by the stresses leading to yield and to a permanent dent. Whereas a URPs panel may be limited by a maximum acceptable transverse deflection even though the panel may recover without permanent damage upon removal of the loads. Even when the design is limited by material failure it is usual to specify the materials criterion in terms of a critical failure strain rather than a failure stress. Thus, it is evident that strain and deformation play a much more important role for URP than they do for metals. As a consequence, step (a) is usually required to provide a full stress/strain/ deformation analysis and, because of the viscoelastic nature of plastics, this can pose a more difficult problem than for metals. 

Schmidt et alT used a pair of high speed cameras to record the dynamic deformation, showing shape and strain details of a fabric upon ballistic impact The information obtained is used to validate the FE model in LS-DYNA and to quantily the transverse deflection. Nurick used light rays emitted from a silicon photovoltaic... 

K. Rubin, M.S. Lubell, A proposed study of photon statistics in fluorescence through high resolution measurements of the transverse deflection of an atomic beam, in Laser Cooled and Trapped Atoms. NBS Special Publ., No. 653 (1983), p. 119... 

With large transverse deflections, contacts between adjacent convolutions occur and the whole bellows must be modeled. The shape of the beUows is asymmetric, i.e., the overall shape of the deformed stmcture possesses 180° rotational symmetry about the midpoint of the neutral axis. Three-dimensional shell elements (SHELL43) are thus necessary (Fig. 42.9c). An element size of 12 X 12 mm yields 11,760 elements and 11,979 nodes to mesh only one half a bellows. The computational requirements for this are large. Increases in element size result in an increase in axial stiffness. 

Figure 42.10 illustrates the relative transverse deflection of the present model as element numbers are increased. When the sum of elements exceeds between 500-650 a near asymptote is reached. A maximum element size of 125 mm was therefore chosen using 720 elements with 803 nodes to model the bellows (Fig. 42.9c). 

FIGURE 42.10 Total element numbers versus relative transverse deflection (stiffness). 

The plate undergoes transverse deflection and inplane deformation. Thus, for the transverse deflection the equation of motion employing the Hnearized second order theory can be written as... 

The v y = v y(x ) transverse deflection with respect to y axis must satisfy the following quasi-static (transverse inertia forces with respect to y axis are ignored) boundary value problem... 

For the finite element discretization beam finite elements are used, with two degrees of freedom at each node the transversal deflection u>i and the rotation ijfi. They are gathered to form the degrees of freedom vector X = wi After assembling the mass and stiffness matrices for all elements, we obtain the equation of motion in the form ... 

Figure 7.3 (a) Axial compression (b) axial elongation (c) transverse deflection (d) angular deflection... 

Transverse deflection the relative displacement of the two ends of an expansion joint perpendicular to its longitudinal axis is called transverse deflection. 

FIGURE 21. Transverse deflection versus distance along the length of a bonded cantilever beam. 

For a uniform, slender cantilever beam of length L, total mass m and elastic bending stiffness El, the small amplitude transverse deflection Wb x,t) of the midplane is governed by the partial differential equation... 

The assumption which underlies the derivation of (2.79), that the radial curvature is uniform everywhere in the film—substrate system, is an essential feature of the deformation in the linear range. In the nonlinear deformation range, on the other hand, there is no basis for expecting the curvature to be uniform. The finite element method of numerical analysis can be used to determine the deformed shape of the substrate midplane in the nonlinear range without a priori assumptions on the distribution of curvature. The deformation is constrained to be consistent with the Kirchhoff hypothesis but it is otherwise general. In particular, transverse deflections which are large compared to the substrate thickness hs are accommodated. 

The problem was analyzed in the following way. The approximate deflection was computed for values of mismatch strain which spanned a range of practical interest. An eighth order polynomial in r was fit to the computed transverse deflection w r) for each value of and the polynomial was differentiated to determine the radial curvature k(v) = w"(r)/(l + w (r ) /. The overall dependence of local normalized radial curvature R.(r) = R K(r)/4hs on mismatch strain and radial position was then represented by means of a contour plot of R on the plane of r/R versus Cm- The result is shown in Figure 2.20 for the case when hg/hf = 100 and R/hs = 100 (Freund 2000). The approximation (r) = w" r) leads to results which are nearly indistinguishable from those shown. 

In the discussion in Section 2.6.1 of substrate curvature in the range of behavior where the deformation is not axially symmetric, it was assumed a priori that the coordinate axes coincided with the axes of principal curvature. This assumption was incorporated in writing the transverse deflection w x, y) in the form given in (2.82). On the other hand, in determining curvature from measurements in this range of behavior, the directions of principal curvature are not known in advance. How is data to be interpreted in order to extract complete curvature information under the circumstances An approximation based on measurements obtained with the CGS method, as discussed in Section 2.3.4, is briefly considered here on the basis of a graphical construction known commonly as Mohr s circle. 

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