Variational formulation of the B.V.P. in terms of displacements is. This principle is particularly useful in analyzing large deflection

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1 4-8 CATIGLIANO' THEOREM OF LEAT WORK. Principle of stationar potential energ: Variational formulation of the B.V.P. in terms of displacements is achieved b the principle. This principle is particularl useful in analing large deflection or large strain problems because it does not require use of the compatibilit conditions. However, it requires inclusion of higher order terms in the strain-displacement law, not included in the following derivation. When approimate methods are used, use of the principle provides reasonable approimations for the displacements but not stresses. For better approimation of stresses, Castigliano's theorem of least work becomes the guiding principle. Castigliano s theorem of least work: Principle of stationar complementar energ: Uses variation of the stresses. This is also sometimes called the principle of stationar complementar enrg. Consider the entire bod occuping region R. Let the surface have outward unit normal vector (l,m,n). Let σ i be the stress field that satisfied the differential equations of equilibrium. Then the variations of the stresses δσ i also satisf these equations. Lecture #22 101

2 Complementar Energ: = 0dV (4-58) V 0 = complementar energ densit in terms of stress components. is generall written in terms of forces. First variation of complementar energ densit: δ = δ0dv ; where δ0 = εiδσi V ubstituting for ε i and using divergence theorem, we obtain δ = u δpd = (uδp + vδp + wδp )d (4-59) Let the boundar surface be divided into two parts: 1 on which the stress vector is given, i.e., δp i = 0; and 2 on which the displacement vector is given, i.e., δu i = 0. Now Eq. (4-59) ma be epressed as: Among all states of stress that satisf the differential equations of equilibrium and boundar conditions on 1 that which represents the actual equilibrium state provides a stationar value to Ψ; i.e., δψ = 0, Ψ = 2 ( up + vp wp ) d = u p d + If 0 is a quadratic function of the stress components (i.e., Hookean or linearl elastic bod, AUMPTION) in which the second-degree terms are a positive definite quadratic form (i.e., 0 is a strictl conve function), then the stationar principle becomes a minimum principle: 2 Lecture #22 102

3 Among all the states of stress of a Hookean bod which satisf the differential equations of equilibrium and boundar conditions on 1, that which represents the actual equilibrium state provides an absolute minimum to Ψ. This principle gives compatibilit equation in terms of stresses and the associated boundar conditions. AUMPTION. The above principle is valid even if there are thermal strains; also the strains need not be measured from the unstressed state; displacements are assumed to be small, however. If all the boundar conditions are stress tpe, the region 2 vanishes, and Ψ =. Castigliano s theorem is epressed as dg = 0. dg = 0 also gives the compatibilit equation in terms of stresses since these equations are not dependent on B.C.; variations of stresses are restricted to class of functions that satisf equilibrium. It is also possible to represent the general solution of the equilibrium equation b means of the stress function. Then dg = 0 provides the compatibilit equation for the stress function. An eample of this is the plane stress problem: σ = τ = τ = 0. General solution of equilibrium equation in terms of Air s stress function: σ = F, σ = F, τ = - F For isotropic Hookean material with θ = 0, U 0 = 0. For a rectangular plate with thickness h: Lecture #22 103

4 h = U = [ F + F 2νFF + 2(1 + ν )F]dd 2E The Euler equation for this problem is 2 2 F = REINER' VARIATIONAL THEOREM OF ELATICITY Reissner s variational theorem of elasticit simultaneousl provides the stress displacement relations, the equilibrium equations, and the boundar conditions. AUMPTION: mall displacement approimations are assumed here, although the theorem can be etended to finite deformations. Let the boundar be divided into two parts: tress specified part 2 : p = p, p = p, p = p Displacement specified part 1 : u = u,v = v,w = w A composite functional J is defined: 0 is regarded as a function of the stress components: J = R u ρ(uf [ σ 1 + σ + vf [(u - u)p v + σ + wf w + (v - v)p ) 0 ]dv (w - w)p (up ) d + vp + wp ) d (4-61) In the small displacement theor of elasticit, the equilibrium state of the bod is such that δj = 0 for arbitrar variations of u, v,,σ,,τ. The identical vanishing of δj ensures the satisfaction of the differential equations of equilibrium, the stress-displacement Lecture #22 104

5 relations, and the boundar conditions. It is a mied variational principle; δj = 0 formulates the elasticit problem completel. This provides an opportunit to use Rit's method where both stresses and displacements are approimated. Lecture #22 105