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How to Find Fixed End Moments of Trapezoidal Loadings

You can find the reaction forces and moments of force on a fixed beam through the equations of statics, which derive from Newton's second law of motion. Furthermore, through analysis, you can transform a trapezoidal load into a force that acts at a specific point. The magnitude and position of that force depend on the shape and the magnitude of the loading applied to the beam. However, with careful algebra and some knowledge of geometry, you can find the moment a fixed beam provides to counteract a trapezoidal load.

Things You'll Need

Centroid table

Instructions

- 1
  Break the trapezoidal load down into a rectangular load and one or two triangular loads. Remember that each loading can be replaced with a vector that acts at a specific point, even if they overlap. That same principle also lets you add and subtract loadings together.
- 2
  Find the area of each distributed load. The area of a rectangle is its base times its height, while the area of a triangle is one-half its base times height. Make sure you note the units attached to the height and width of each figure. The area of each loading will be the magnitude of the force you replace them with.
- 3
  Find the centroid of each load using a centroid table. The centroid of a figure is the point at which one-half its area lies on each side. The force you'll replace each loading with will act at the centroid of each load.
- 4
  Replace each loading with the forces calculated in Step 2 at the point calculated in Step 3.
- 5
  Sum all forces in the X and Y directions and set them equal to zero. Sum the moments of force about any point and also set them equal to zero. Represent any unknown forces or moments with a variable. Do not neglect the reaction forces and moment the fixed end of the beam provides.
- 6
  Solve the system of equations found in Step 5. One of the unknowns you find is the reaction moment the fixed end provides.

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