Deflection and Slope of Beam Elastic Curve

Given a set of loading and support conditions for a beam, we will have specific formulae to calculate the deflection and slope at any point on its elastic curve. To ease your engineering analysis, we have prepared this beam elastic curve info sheet, and tabulated the deflection and slope formulae for some commonly encountered cases. The steps to develop those formulae are included as well.

The development of deflection and slope formulae of beam elastic curve is mainly done using double integration method. The function of moment in terms of ordinate, x is first determined using the method of section on beam. Where the moment function is expected to change with location, we may need multiple cut sections, and each of them should have a well-defined limit.

Then, we perform integration on the function with respect to x and obtain the equation for elastic curve slope. Constant C1 is produced in the process. Subsequently, we integrate the slope and obtain the deflection function, as well as producing constant C2. Both constants can be solved using boundary and continuity conditions.

In this worked example, we need to derive the slope and deflection formulae for simply supported beam subjected to trapezoidal loading pattern.

To ease the calculation process, we can apply principle of superposition and convert the loading pattern into the combination of two simpler pattern, namely uniform and triangular load patterns.

Then, we are good to determine the support reaction. Let the ordinate x be the distance from point A. In this case, we only need to calculate the support reaction at A. Then, we can use method of section and determine the internal moment function.

By using double integration, we first get the equations for slope and deflection of elastic curve with two unknowns, C1 and C2. With the aid of boundary conditions, we can solve for these constants.

Finally, we can simplify them and obtain the finalized equations for the slope and deflection of beam elastic curve. The result can be verified by using principle of superposition with readily available formulae.