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The three moment equation expresses the relation between bending moments at three successive supports of a continuous beam, subject to a loading on a two adjacent span with or without settlement of the supports.
Clapeyron's equation, which uses the final force only, may be puzzling at first, but is nevertheless true because it includes a corrective factor of one half. Another theorem, the theorem of three moments used in bridge engineering is also sometimes called Clapeyron's theorem.
In a three-point bend test, a fatigue crack is created at the tip of the notch by cyclic loading. The length of the crack is measured. The specimen is then loaded monotonically. A plot of the load versus the crack opening displacement is used to determine the load at which the crack starts growing.
Load and moment boundary conditions involve higher derivatives of and represent momentum flux. Flux boundary conditions are also called Neumann boundary conditions. As an example consider a cantilever beam that is built-in at one end and free at the other as shown in the adjacent figure. At the built-in end of the beam there cannot be any ...
where , are the coordinates of a point on the cross section at which the stress is to be determined as shown to the right, and are the bending moments about the y and z centroid axes, and are the second moments of area (distinct from moments of inertia) about the y and z axes, and is the product of moments of area. Using this equation it is ...
Then each fixed joint is sequentially released and the fixed-end moments (which by the time of release are not in equilibrium) are distributed to adjacent members until equilibrium is achieved. The moment distribution method in mathematical terms can be demonstrated as the process of solving a set of simultaneous equations by means of iteration.
Fig. 3 - Beam under 3 point bending. For a rectangular sample under a load in a three-point bending setup (Fig. 3), starting with the classical form of maximum bending stress: = M is the moment in the beam; c is the maximum distance from the neutral axis to the outermost fiber in the bending plane
For a 3-point test of a rectangular beam behaving as an isotropic linear material, where w and h are the width and height of the beam, I is the second moment of area of the beam's cross-section, L is the distance between the two outer supports, and d is the deflection due to the load F applied at the middle of the beam, the flexural modulus: [1]