Resultant force and moment

The resultant force F is compensated by the reaction of the thread and, correspondingly, it does not influence the motion of the mass. In general, the total torque T has arbitrary direction, but we will be interested in only its vertical component, that causes a rotation in the horizontal plane. At the same time special measures are taken to remove an influence of motion in the vertical plane. 

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We will proceed in the next section to define the strain and stress variations through the thickness of a laminate. The resultant forces and moments on a laminate will then be obtained in Section 4.2.3 by integrating the stress-strain relations for each layer. Equation (4.6), through the laminate thickness subject to the stress and strain variations determined in Section 4.2.2. 

The resultant forces and moments acting on a laminate are obtained by integration of the stresses in each layer or lamina through the laminate thickness, for example,... 

With the values of Kq and Ks determined from these equations, the value of the resultant force and moment at an anchor, for a given pipe arrangement, may be found. The following equations may be used to solve for the desired resultant forces and moment. 

Example. Find the resultant force and moment for the arrangement shown in Figure 7 -40b. 

The dynamic interaction between two bodies may be described by a resultant force and moment. Thus, when bodies (i - 1) and i interact, whether through a joint or contact, a generalized force, f<, is exerted on body i by body (i - 1) and with a negative sign on body (i - 1). Its form is given in Equation 2.3. This force vector may be resolved in the dual basis defined above ... 

And Npi and Mpt i=3, 4) denote the resultant forces and moments indueed by the electric field E3 applied to the tth bonded piezoelectric patches represented by ... 

The force and moment ia a constrained system can be estimated by the cantilever formula. Leg MB is a cantilever subject to a displacement of and leg CB subject to a displacement Av. Taking leg CB, for example, the task has become the problem of a cantilever beam with length E and displacement of Av. This problem caimot be readily solved, because the end condition at is an unknown quantity. However, it can be conservatively solved by assuming there is no rotation at poiat B. This is equivalent to putting a guide at poiat B, and results ia higher estimate ia force, moment, and stress. The approach is called guided-cantilever method. 

Similarly, is a moment gemnit width as shown In Figure 4-7. However, Nx, etc., and etc T 6ereferred to as forces and moments with the stipulation of per unit width being dropped for convenience. The entire collection of force and moment resultants for an N-layered laminate is depicted in Figures 4-6 and 4-7 and is defined as... 

Again, there is no coupling between bending and extension, so the force and moment resultants are... 

The force and moment resultants take the simplified form... 

The bending-extension coupling stiffnesses, Bjj, vary for different classes of antisymmetric laminates of generally orthotropic laminae, and, in fact, no general representation exists other than in the following force and moment resultants ... 

An antisymmetric cross-ply laminate consists of an even number of orthotropic laminae laid on each other with principal material directions alternating at 0° and 90° to the laminate axes as in the simple example of Figure 4-19. A more complicated example is given in Table 4-4 (where the adjacent layers do not always have the sequence 0°, then 90°, then 0°, etc.). Such laminates do not have A g, Agg, D g, and Dgg, but do have bending-extension coupling. We will show later that the coupling is such that the force and moment resultants are... 

Unsymmetric laminates with multiple specially orthotropic layers can be shown to have the force and moment resultants in Equations (4.59) and (4.60) but with different A22, B22, and D22 korn B.,., and Dll, respectively. That is, there are no shear-extensiorTcoupling terms nor any bend-twist coupling terms, so the solution of problems with this kind of lamination is about as easy as with isotropic layers. 

Unsymmetric laminates with multiple generally orthotropic layers or with multiple anisotropic layers have force and moment resultants no... 

However, in most experiments, the loads are applied, and the resulting deformations are measured, i.e., the deformations are the dependent variables, not the loads. Thus, the expressions for the middle-surface extensional strains and curvatures in terms of the force and moment resultants would be convenient. 

The overall procedure of laminate-strength analysis, which simultaneously results in the laminate load-deformation behavior, is shown schematically in Figure 4-36. There, load is taken to mean both forces and moments similarly, deformations are meant to include both strains and curvatures. The analysis is composed of two different approaches that depend on whether any laminae have failed. 

Actually, only in the restricted case of perfect constraint are the and thermal forces and moments, respectively. However, the force and moment resultants can be rearranged to read... 

For two- and three-layered cross-ply and angle-ply laminates of E-glass-epoxy, Tsai [4-10] tabulates all the stiffnesses, inverse stiffnesses, thermal forces and moments, etc. Results are obtained for various cross-ply ratios and lamination angles, as appropriate, from a short computer program that could be used for other materials. 

Specially orthotropic plates, i.e., plates with multiple specially orthotropic layers that are symmetric about the plate middle surface have, as has already been noted in Section 4.3, force and moment resultants in which there is no bending-extension coupling nor any shear-extension or bend-twist coupling, that is,... 

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

The final level of complexity is to arrange layers of unidirectional laminae to form the laminate composite, as shown in Figure 5.121. The lower surface of the /th lamina is assigned the coordinates hf relative to the midplane, so the thickness of the /th lamina is /i/ — /i/ 1. The mathematics of stress analysis are an extension of those used in the foregoing description, and they include force and moment analysis on the bending of the laminate. The complete development will not be presented here, but only outlined. The interested reader is referred to two excellent descriptions of the results [9,10]. 

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