The three-moment equation gives us the relation between the moments between any three points in a beam and their relative vertical distances or deviations. This method is widely used in finding the reactions in a continuous beam.
Consider three points on the beam loaded as shown.
From proportions between similar triangles:
→ equation (1)
Substitute t1/2 and t3/2 to equation (1)
Multiply both sides by 6
Combine similar terms and rearrange
If E is constant this equation becomes,
If E and I are constant then,
For the application of three-moment equation to continuous beam, points 1, 2, and 3 are usually unsettling supports, thus h1 and h3 are zero. With E and I constants, the equation will reduce to
Factors for the three-moment equation
The table below list the value of and for different types of loading.
|Type of Loading|
|Concentrated load anywhere on the span.
|Concentrated load at the midspan.
|Uniform load over the entire span.
|Increasing triangular load on the whole span.
|Decreasing triangular load on the whole span.
|Isosceles triangular load over the entire span.
|Moment load at any point on the span.
|General uniform loading.