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aluminum arch architect architectural axis barrel bars behavior bending action bending stresses boundary bridge building built cable action carry loads circular columns compressive stresses considered cross section curvature curved cylinder dead load deflection deformation depth develop direction displacements dome efficient elastic elastic limit elastic modulus ends equilibrium feet fibers flanges folded plates frame funicular grid hinged horizontal hyperbolic paraboloid increase l-beam length lever arm ln order longitudinal maximum membrane action membrane stresses midspan parabola parallel plane plastic flow post and lintel pounds per square prestressed principal curvature principal stress ratio rectangular reinforced concrete reinforced-concrete resistance ribs right angles rigid roof rotation saddle surface shape shear stresses sides simply supported beam slabs Space frames span square inch steel stiff stiffeners strength structural elements structural materials structural systems surface tensile stresses thickness thin shells thrust tie-rod transverse truss twist vertical loads weight wind