Coupling stiffness matrix

The stifftiess matrix associated with support degrees of freedom, and the coupling stiffness matrix between... 

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. 

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... 

The result of this analysis is a series of coupled algebraic equations in the unknown nodal displacements. For the linear problem we have considered here, these equations are themselves linear (though they are not in general) and our problem reduces to little more than diagonalization of the relevant stiffness matrix which, as seen above, can be computed as products of material parameters and geometric factors. 

The final algebraic equations are set up by assembling the element contributions the stiffness matrix and the coupling matrix for each desired process or coupling, respectively. Additionally, the boundary conditions have to be incorporated. Consequently, the resulting equations are strongly or weakly coupled with each other and it has to be decided whether they can be solved iteratively in a partitioned scheme or in one step in a directly coupled system of equations, see Eq. (12),... 

In-plane membrane stiffnesses of a laminate (units force/length) Membrane/bending coupling stiffnesses of a laminate (units force) Measure of crack density in matrix with cracks Inverse of D matrix... 

Entries of inverse of laminate stiffness matrix relating to mem-brane/bending coupling... 

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... 

A laminate in which the stacking sequence below the midplane is a mirror image of the sequence above the midplane is called a symmetric laminate. For symmetric laminates the coupling terms of its stiffness matrix. By, vanish. 

It is seen that [A] is the extensional stiffness matrix, [D] is the flexural stiffness matrix and [B] is the bending-stretching coupling matrix. 

The stiffness matrix associated with support DOFs, and the coupling submatrix between structural DOFs and support DOFs (from Problem 9.24).are... 

Another approach for approximate component reduction is to update the non-reduced substructure matrices and recalculate the reduced matrices, in each iteration step of a global optimisation procedure on the reduced model, while the static and dynamic superelement transformation matrices are kept constant. This leads to a large reduction in calculation time, as on the superelement level only the system matrix assembly has to be performed, and no time-consuming matrix inversions or substructure eigenvalue analyses has to be performed. Mathematically, the approximative method comes down to the recalculation of the matrices in equations (17)-(23). However, care should be taken when calculating the reduced stiffness matrix, as coupling terms [Ktq] similar to the mass coupling terms [Mtq arise due to the fact that the deterministic static transformation matrix [Gof] does not necessarily equal the term —[Aoo] [f of] with updated system matrices. 

Coupled Pile Foundation Stiffness Matrix In this method, a quasi-dynamic analysis for the pile group is conducted by applying loading (either as forces or displacements) at the interface node between the superstructure and foundation model using linearized properties for the soils. Linearized properties for a single pile can be achieved by assuming secant foundation stiffness... 

The simplified method assumes coupling only between shear and overturning moment for the single pile. The general form of a single pile stiffness matrix can take the form ... 

Bridge Drilled Non-linear Soil Equivalent Coupled Foundation Shaft System Spring Model Cantilever Stiffness Matrix... 

Coupled Foundation Stiffness Matrix This approach is usually employed in regions of low to moderate seismicity. As illustrated in Fig. 16, a 6 X 6 can be determined for a single shaft at the ground line. As indicated in section Coupled Pile Foundation Stiffness Matrix, ... 

The effect of the specific values of the B j can be readily calculated for some simple laminates and can be calculated without significant difficulty for many more complex laminates. The influence of bending-extension coupling can be evaluated by use of the reduced bending stiffness approximation suggested by Ashton [7-20]. If you examine the matrix manipulations for the inversion of the force-strain-curvature and moment-strain-curvature relations (see Section 4.4), you will find a definition that relates to the reduced bending stiffness approximation. You will find that you could use as the bending stiffness of the entire structure,... 

A chain of N equally spaced atoms with masses m coupled by N- springs of stiffness ki obeying Hooke s law is described by the A-dimensional matrix system (for details see for example [5]) ... 

Recent experiments by Dr. Bar-Cohen el al. ha e shown that ultrasonic oblique insonification can be used to characterize thermal damage to composites [156]. Using an inversion technique based on a micromechanical model, the reflected ultrasonic signals arc analyzed to determine the overall laminate stiffness constant before and after loading. Another technique developed by the NASA to encompass the limitation of pulse-echo ultrasonic and photomicroscopic methods is diffuse-field acoustoultrasonic coupled vibration damping [157]. Both NASA techniques are complementary and arc used to assess microstructural damage accumulation in ceramic matrix composites. 

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