Strain-displacement relations

If the laminate is subjected to uniform axial extension on the ends X = constant, then all stresses are independent of x. The stress-displacement relations are obtained by substituting the strain-displacement relations, Equation (4.162), in the stress-strain relations. Equation (4.161). Next, the stress-displacement relations can be integrated under the condition that all stresses are functions of y and z only to obtain, after imposing symmetry and antisymmetry conditions, the form of the displacement field for the present problem ... 

Then, integration of the strain-displacement relation. Equation (2.2), with respect to z (with w assumed to be independent of z) yields... 

The fundamental equations treated in structural analyses are the mechanical equilibrium, strain-displacement relation, and stress-strain relation. The equilibrium equations in an elementary volume can be expressed ... 

Equation (3.12) is the governing partial differential equation for two-dimensional elasticity. Any function that satisfies this fourth-order partial differential equation will satisfy all of the eight equations of elasticity namely, the equilibrium equations, Hooke s law, and the strain-displacement relations. 

A state of plane stress exists with respect to x-y plane when Ox, T yz and r z are zero, and a, Oy and x y are functions of x and y only. The equilibrium equations are given by (A.17) for the x and y directions. In this case, however, cXx is zero, but is nonzero. The strain displacement relations are the same as Eq. (A.17) with the additional relationship... 

If these features are incorporated into the model, the strain-displacement relations take the form... 

One way to proceed with analysis of the circular buckle is to adopt the von Karman plate theory for small strain and moderate rotation. In this case, the strain-displacement relations are... 

Since one of the boundary conditions in Eki. 9.46 is now a displacement boundary condition, we also require the expressions for the displacements in terms of the constants A and C. These are found first by using the stress-strain-displacement relations and then by integrating the strain components to determine the displacements. The stress-strain-displacement equations for the condition of plane strain in cylindrical coordinates are,... 

Since the thin walls of the prismatic beams under discussion resemble the cylindrical thin shells of Section 6.2.2, the associated formulation of strains may be adopted. Thereby, also the respective assumptions are inherited. A comparison of Remarks 6.4 and 7.6 reveals that the ratio of thickness and radius of curvature is additionally confined and Remark 6.5 indicates a hnear strain displacement relation. These shell strains are given by Eqs. (6.12) and (6.15). The insertion of Eqs. (7.26) and (7.24) leads to rather complicated expressions. Through tedious manipulations with the aid of Eqs. (7.17), (7.22), and (7.23), a remarkably compact formulation may be found. For the sake of correlation to the beam displacements and rotations again in anticipation of the upcoming considerations of torsional warping, the warping function 0 s) is employed and also appears in the abbreviation... 

Employing the strain-displacement relations Eq. 2 and stress-strain relations Eq. 4, the nonvanishing stress components are obtained as... 

The strain vector e in the finite element is also associated with the nodal displacement vector a via the strain-displacement relation matrix B of the element as follows ... 

Using the stress-strain and strain-displacement relations for linear elastic deformation and using the variational principle, one may write [8] ... 

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