Fixed End Moments and Their Derivation

Fixed end moment (FEM) is useful when we want to solve for a statically indeterminate beam.

For both propped-cantilevered and fixed-end beams, we can straightaway calculate their fixed end moments and reduce it to a statically determinate problem, based on the loading condition.

When dealing with continuous beam or statically indeterminate frame, determining fixed end moment is an essential step to solve the problem using displacement method, namely slope-deflection equation or moment distribution. With the presence of fixed end moment formula table, we can avoid spending too much time in deriving and calculating the fixed end moment for a specific case.

The derivation of fixed end moment is done by implementing force method. For propped-cantilevered beam, it is sufficient to make one support reaction redundant. For fixed-end beam, we can make two reactions redundant, while neglecting the horizontal reaction. To ease the derivation process, we often choose the resisting moment at support to be our redundant reaction.

Now, let's try and derive the fixed end moment formulae for a fixed end beam subjected to trapezoidal loading pattern.

We can use force method to solve for this problem. First, we simplify the problem by converting the loading pattern into the summation of uniform and triangular load patterns.

Next, we implement aii = L/3EI and aij = L/6EI, considering the beam as a simply supported beam subjected to couple moment at the support.

Then, we need the elastic curve slope formula for simply supported beam subjected to trapezoidal load to proceed. This can be adopted directly from our previous worked example.
The elastic curve slope of simply supported beam due to the actual loading needs to be calculated for both supports, namely at x=0 and x=L.

At this point, we can substitute all the values into the compatibility equations developed using force method. The actual rotation at fixed end beam supports should be taken as zero.

From here, we obtain two simultaneous equations. By solving them, we determine the fixed end moment formulae for our problem. Our result can be verified by using principle of superposition with readily available fixed end moment formulae.