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a-line ae(l/aT ae/ac ae/aT ah8t application assumed basic solution basic variables beam boundary conditions calculus of variations components compression members concentrated members constant corresponding curvilinear coordinates cycloids defined dual constraints dual problem end load equal Equations equilibrium 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 Mr=Me 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 rhumb lines satisfy Section shown slack variables solved straight lines strain energy strain field stress resultants sufficient conditions system of given tension members theorem TxTy valid values variations virtual deformation virtual displacement virtual strain Vmin volume of material Warren truss written yield Young's modulus zero