Stiffnesses transformation

Tsai and Pagano [2-7] ingeniously recast the stiffness transformation equations to enable ready understanding of the consequences of rotating a lamina in a laminate. By use of various trigonometric identities between sin and cos to powers and sin and cos of multiples of the angle, the transformed reduced stiffnesses. Equation (2.85), can be written as... 

The topic of invariant transformed reduced stiffnesses of orthotropic and anisotropic laminae was introduced in Section 2.7. There, the rearrangement of stiffness transformation equations by Tsai and Pagano [7-16 and 7-17] was shown to be quite advantageous. In particular, certain invariant components of the lamina stiffnesses become apparent and are heipful in determining how the iamina stiffnesses change with transformation to non-principal material directions that are essential for a laminate. 

Ki, = load or stiffness transformation factor Km = mass transformation factor... 

Step 7. The pushover curves were idealized by means of a trilinear force-displacement relationship, also taking into account negative post-capping stiffness. Transformation of the... 

For the situation where the loading is applied off the fibre axis, then the above approach involving the Plate Constitutive Equations can be used but it is necessary to use the transformed stiffness matrix terms Q. 

Not all of the strength and stiffness advantages of fiber-reinforced composite materials can be transformed directly into structural advantages. Prominent among the reasons for this statement is the fact that the joints for members made of composite materials are typically more bulky than those for metal parts. These relative inefficiencies are being studied because they obviously affect the cost trade-offs for application of composite materials. Other limitations will be discussed subsequently. 

Note that the transformed reduced stiffness matrix Qy has terms in all nine positions in contrast to the presence of zeros in the reduced stiffness matrix Qy. However, there are still only four independent material constants because the lamina is orthotropic. In the general case with body coordinates x and y, there is coupling between shear strain and normal stresses and between shear stress and normal strains, i.e., shear-extension coupling exists. Thus, in body coordinates, even an orthotropic lamina appears to be anisotropic. However, because such a lamina does have orthotropic characteristics in principal material coordinates, it is called a generally orthotropic lamina because it can be represented by the stress-strain relations in Equation (2.84). That is, a generally orthotropic lamina is an orthotropic lamina whose principai material axes are not aligned with the natural body axes. 

Accordingly, we have supposedly found the shear modulus G.,2. However, a relationship such as Equation (2.107) does not exist for strengths because strengths do not transform like stiffnesses. Thus, this experiment cannot be relied upon to determine S, the ultimate shear stress (shear strength), because a pure shear deformation mode has not been excited with accompanying failure in shear. Accordingly, other approaches to obtain S must be used. 

Thus, the Tsai-Wu tensor failure criterion is obviously of more general character than the Tsai-Hill or Hoffman failure criteria. Specific advantages of the Tsai-Wu failure criterion include (1) invariance under rotation or redefinition of coordinates (2) transformation via known tensor-transformation laws (so data interpretation is eased) and (3) symmetry properties similar to those of the stiffnesses and compliances. Accordingly, the mathematical operations with this tensor failure criterion are well-known and relatively straightforward. 

In Section 2.2, the stress-strain relations (generalized Hooke s law) for anisotropic and orthotropic as well as isotropic materials are discussed. These relations have two commonly accepted manners of expression compliances and stiffnesses as coefficients (elastic constants) of the stress-strain relations. The most attractive form of the stress-strain relations for orthotropic materials involves the engineering constants described in Section 2.3. The engineering constants are particularly helpful in describing composite material behavior because they are defined by the use of very obvious and simple physical measurements. Restrictions in the form of bounds are derived for the elastic constants in Section 2.4. These restrictions are useful in understanding the unusual behavior of composite materials relative to conventional isotropic materials. Attention is focused in Section 2.5 on stress-strain relations for an orthotropic material under plane stress conditions, the most common use of a composite lamina. These stress-strain relations are transformed in Section 2.6 to coordinate systems that are not aligned with the principal material... 

The Ajj are the sum of the product of the individual laminae Qy and the laminae thjcknesses. Thus, the only ways to obtain a zero individual Ay are for all Qy to be zero or for some Qy to be negative and some positive so that their products with their respective thicknesses sum torero. From the expressions for Uie transformed larnina stiffnesses, Qy, in Equation (2.80), apparently Qii, Q- 2, Q22 66 positive-definite... 

Angle-ply laminates have more complicated stiffness matrices than cross-ply laminates because nontrivial coordinate transformations are involved. However, the behavior of simple angle-ply laminates (only one angle, i.e., a) will be shown to be simpler than that of cross-ply laminates because no knee results in the load-deformation diagram under uniaxial loading. Other than the preceding two differences, analysis of angle-ply laminates is conceptually the same as that of cross-ply laminates. 

The example considered to illustrate the strength-analysis procedure is a three-layered laminate with a [4-15°/-15°/+15°] stacking sequence [4-10]. The laminae are the same E-glass-epoxy as in the cross-ply laminate example with thickness. 005 in (.1270 mm), so that the total laminate thickness is. 015 in (.381 mm). In laminate coordinates, the transformed reduced stiffnesses are... 

The essence of Ashton s contribution is that he identified the skew plate stiffnesses as being a transformation of the symmetric angle-ply stiffnesses or, more generally, the anisotropic bending stiffnesses, that is,... 

The stiffnesses in Table 7-6 are not transformed stiffnesses in analogy to Table 2-2. That is, the x-axis of the laminate is fixed relative to the 0, of each lamina. However, the transformed stiffnesses can be obtained by rotating the entire laminate through angle < ), that is, by substituting (0 - < )) for 0 in Equation (7.29). For example. 

A final result of interest is the integral of the area under the transformed stiffness versus angle of rotation curve from <)) = 0 to <)) = 2k, that is, one complete revolution of the laminate ... 

Figure 7-54 Transformed Laminate Stiffnesses (After Tsai and Pagano [7-17])...

We strongly suggest the use of the reduced sensitivity whenever we are dealing with differential equation models. Even if the system of differential equations is non-stiff at the optimum (when k=k ), when the parameters are far from their optimal values, the equations may become stiff temporarily for a few iterations of the Gauss-Newton method. Furthermore, since this transformation also results in better conditioning of the normal equations, we propose its use at all times. This transformation has been implemented in the program for ODE systems provided with this book. 

The behavior of cristobalite PON has been studied as a function of pressure. No in situ evidence for pressure-induced amorphization was noticed. Whereas cristobalite Si02 displays four crystalline phases up to 50 GPa (195), PON remains in a cristobalite phase (193, 196). By using Raman spectroscopy and synchrotron X-ray diffraction, Kingma et al. (193, 197) observe a displacive transformation below 20 GPa to a high-pressure cristobalite-related structure, which then remains stable to at least 70 GPa. The high value of the calculated bulk modulus (71 GPa) (196) is indicative of the remarkable stiffness of the phase. 

Note that Table 6.1 does not include a case for three point loads. In lieu of a derivation of the needed values, the stiffness and transformation factors for uniform loading will be used as an approximation. 

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