DERIVATION OF BEAM EQUILIBRIUM EQUATIONS

Therefore, unless axial loads are introduced along the beam, the axial force is constant. 

the lateral loading causes a change in the shear force from point to point along the beam. 

Note that the shear equation. Equation (D.6), can be substituted in the transverse load equation. Equation (D.4), to get 

a fourth-order differential equation such as Equation (D.11) has four boundary conditions which are the second and third of the conditions in Equation (D.8) at each end of the beam. The first boundary condition in Equation (D.8) applies to the axial force equilibrium equation, Equation (D.2), or its equivalent in terms of displacement (u). 

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