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a-line ae(l/aT ae/ac ah8t application assumed basic solution basic variables beam boundary conditions calculated calculus of variations components compression members concentrated members constant corresponding curvilinear coordinate cycloids defined dual constraints dual problem end load equal Equations equilibrium example feasible solution field of Fig follows force F given forces Green's theorem Hencky implies kinematic Lagrangian multipliers layout least volume limit linear programming maximum stiffness Michell minimum solution nodes non-basic variables non-negative obtained optimum design optimum elastic design optimum framework optimum solution optimum structure orthogonal plastic design plate present problem primal problem principal strains regions satisfy Section shown simplex method slack variables solved straight lines strain energy strain field strains ae/aT stress resultants structure of least sufficient conditions system of given tension members theorem TxTy valid values virtual deformation virtual displacement virtual strain Vmin volume of material Warren truss written yield Young's modulus zero