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By J M T Thompson; G W Hunt

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The correct force flow satisfies not only the equilibrium equations, but also the condition that the deformed column has to fit exactly between both fixed supports. Therefore, the actual force flow satisfies not only the equilibrium equations but also the stress-strain (constitutive) equations and the straindisplacement (kinematic) equations. When performing calculations for a statically indeterminate structure we therefore need all three basic relationships: kinematic, constitutive and static, as shown below.

33 34 ENGINEERING MECHANICS. VOLUME 2: STRESSES, DEFORMATIONS, DISPLACEMENTS NBAB = NBBC . 14). • End conditions At a bar end there is always one boundary condition: an end condition. At the bar end, either the displacement u, or the normal force N = EAu is prescribed. If u is prescribed (this happens at a support1), then N is unknown; and vice versa, if N is prescribed (this occurs at a non-supported end), then u is unknown. The prescribed value of N follows from the equilibrium of a small end segment with length x for which x → 0.

2 Bar Subject to Extension the deformation quantity ε (the strain of the fibres in the bar) and the displacement u (of a cross-section in x direction). For a bar segment the change in length “ displacement between the end-sections: “ ”= ” is equal to the difference in u = ε x. The total change in length of the bar is found by summing all contributions ε x of the individual segments over the entire length of the bar: = ε dx. This relationship is the basis for the formulae for calculating the change in length of a bar.

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