Transformed reduced stiffnesses

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. 

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 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 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. 

Here, the array [g]/ is the transformed reduced stiffness matrix, and each gy term can be related to the gy terms of the reduced stiffness matrix, [g], through the angle, 6. 

Coupling terms of laminate stiffness matrix Bending terms of laminate stiffness matrix Longitudinal Young s modulus of the lamina Transverse Young s modulus of the lamina In-plane shear modulus of the lamina Out-of-plane shear modulus of lamina (in the 1-3 plane) Out-of-plane shear modulus of lamina (in 2-3 plane) Moment stress resultants per unit width Force stress resultants per unit width Laminate reduced stiffness terms Transformed reduced stiffness terms... 

In a last step, the reduced stiffness and mass matrices of all components are assembled with the non-reduced residual structure, to form the reduced stiffness and reduced mass matrix of the complete structural system. These can then be used to perform finite element analyses (e.g. an eigenfrequency or frequency response function analysis) on the global structure. Data recovery for each superelement is performed by expanding the solution at the attachment points, using the same transformation matrices that were used to perform the original reduction on the superelement. 

The yarn composite is thus modeled by concentrically piling up each lamina. By considering the transformed reduced stiffness and the area of each lamina, a new... 

The matrices [5] and [Q] appearing in Equations 8.42 and 8.43 are called the reduced compliance and stiffness matrices, while [5 ] and Q in Equations 8.54 and 8.55 are known as the transformed reduced compliance and stillness matrices. In general, the lamina mechanical properties, from which compliance and stiffness can be calculated, are determined experimentally in principal material directions and provided to the designer in a material specification sheet by the manufacturer. Thus, a method is needed for transforming stress-strain relations from off-axis to the principal material coordinate systan. 

The purpose of introducing the concepts developed in Equation 8.56 through Equation 8.63 has been to provide the tools needed to present a straightforward derivation of the transformed reduced compliance and stiffness matrices [5] and [Q based on matrix algebra. The derivation makes use of the following sequence of operations to obtain stress-strain relations in the reference (x, y) coordinate system ... 

The transformed reduced compliance and stiffness matrices [5] and [G1 relate off-axis stress and strain in an orthotropic lamina. Since these matrices are fully populated, the material responds to off-axis stresses as though it was fully anisotropic. Some consequences of the anisotropic nature of a unidirectional lamina are discussed in the following. 

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. 

Crystallinity plays a large role in the physical behavior of polymers. The amorphous regions play perhaps an even greater role. Some amorphous polymers such as polymethyl methacrylate (PMMA) are stiff, hard plastics at room temperature, whereas polymers such as polybutadiene are soft and flexible at room temperature. If PMMA is healed to lOS C, it will soften, and its modulus wiU be reduced by orders of magnitude. If polybutadiene is cooled at to —73 C, it will become stiff and hard. The temperature at which this hard-to-soft transformation takes place is called the glass transition temperature T. ... 

Having obtained the lamina reduced stiffness terms Qy for each lamina, calculate the transformed lamina reduced stiffness terms ifor a given angle of orientation... 

Since WRA transform stiff concrete mixes into more plastic mixes at a given waterxement ratio, they can also be used to improve the pumpability. Concrete mixes with WRA were reported to have increased pumpability even with reduced water and cement contents for a specified workability [23]. 

Since Ci and 8 are symmetric tensors, each of them has 6 independent components, the fourth order stiffness tensor Cyia contains at most 36 independent constants, such that it can be displayed as a 6 x 6 matrix of components using contracted notation, Cto), where m,n — 1,2,3,4,5,6. There is a unique correspondence between of and Cijki- The index m is related to ij, and n is related to Id, as shown in Table B.l. For instance, Cu22 = C12, C1323 = C54- Since C = C the number of independent constants is generally 21. For orthotropic materials, the the number of independent constants further reduces to 9. When the fourth order tensor is transformed to the principal axes, all Qj — 0, except for Cn, C22, C33, C12. 13. 23. 44, 55, and... 

Using this transformation, the component stiffness and mass matrix are reduced to form the reduced superelement matrices (Craig Bampton, 1968 MSC Software Corporation, 2001) ... 

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