The ThreeMoment Equation for Continuous （连续的ThreeMoment方程）.pdf

The Three-Moment Equation for Continuous-Beam Analysis CEE 201L. Uncertainty, Design, and Optimization Department of Civil and Environmental Engineering Duke University Henri P. Gavin Spring, 2009 Consider a continuous beam over several supports carrying arbitrary loads, w (x). Using the Moment-Area Theorem, we will analyze two adjoining spans of this beam to find the relationship between the internal moments at each of the supports and the loads applied to the beam. We will label the left, center, and right supports of this two-span segment L, C , and R. The left span has length LL and flexural rigidity EIL ; the right span has length LR and flexural rigidity EIR (see figure (a)). Applying the principle of superposition to this two-span segment, we can separate the moments caused by the applied loads from the internal moments at the supports. The two-span segment can be represented by two simply- supported spans (with zero moment at L, C , and R) carrying the external loads plus two simply-supported spans carrying the internal moments ML , MC , and MR (figures (b), (c), and (d)). The applied loads are illustrated below the beam, so as not to confuse the loads with the moment diagram (shown above the beams). Note that we are being consistent with our sign convention: positive moments create positive curvature in the beam. The internal moments ML , MC , and MR are drawn in the positive directions. The 2 CEE 201L. – Uncertainty, Design, and Optimization – Duke University – Spring 2009 – H.P. Gavin areas under the moment diagrams due to the applied loads on the simply- supported spans (figure (b)) are AL and AR ; x¯L repres