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analysis angle axial bar tensions bending moments buckling built-in end calculation cantilever carries centre Chapter clockwise collapse load collapse mechanism column compressive configuration continuous beam corresponding displacement critical load cross-section curvature deformation dimensions in metres direction displacement diagram distribution of bending elastic modulus element elongations equilibrium conditions equilibrium equations equilibrium set Example exerted external loads flexural rigidity forces and moments full plastic geometric Hooke's law horizontal load horizontal reaction imaginary unit indeterminate induced integration kilonewtons Fig left-hand end left-hand support linear method move multiplied notation obey pin-jointed framework plastic hinge polygon portal frame positive problem redundant relative displacement rigid foundation rigidity F segment shear force shown in Figure simple simply supported solid mechanics stanchion feet statically determinate structure support reactions take moments uniform beam unit load unknown vector sum vertical deflection vertical reaction yield stress zero