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Besides deflection, the beam equation describes forces and moments and can thus be used to describe stresses. For this reason, the Euler–Bernoulli beam equation is widely used in engineering, especially civil and mechanical, to determine the strength (as well as deflection) of beams under bending.
A beam is a structural element that primarily resists loads applied laterally across the beam's axis (an element designed to carry a load pushing parallel to its axis would be a strut or column). Its mode of deflection is primarily by bending, as loads produce reaction forces at the beam's support points and internal bending moments, shear ...
Shear and Bending moment diagram for a simply supported beam with a concentrated load at mid-span. Shear force and bending moment diagrams are analytical tools used in conjunction with structural analysis to help perform structural design by determining the value of shear forces and bending moments at a given point of a structural element such as a beam.
If clockwise bending moments are taken as negative, then a negative bending moment within an element will cause "hogging", and a positive moment will cause "sagging". It is therefore clear that a point of zero bending moment within a beam is a point of contraflexure—that is, the point of transition from hogging to sagging or vice versa.
The cantilever method is an approximate method for calculating shear forces and moments developed in beams and columns of a frame or structure due to lateral loads. The applied lateral loads typically include wind loads and earthquake loads, which must be taken into consideration while designing buildings.
The deflection downward positive. (Downward settlement positive) Let ABC is a continuous beam with support at A,B, and C. Then moment at A,B, and C are M1, M2, and M3, respectively. Let A' B' and C' be the final positions of the beam ABC due to support settlements. Figure 04-Deflection Curve of a Continuous Beam Under Settlement
Joint B is released once again to induce moment distribution and to achieve equilibrium.* Steps 5 - 10: Joints are released and fixed again until every joint has unbalanced moments of size zero or neglectably small in required precision. Arithmetically summing all moments in each respective columns gives the final moment values.
Using the beam sign convention and cutting the beam at B, we can deduce the figure shown. Part (e) of the figure shows the influence line for the bending moment at point B. Again making a cut through the beam at point B and using the beam sign convention, we can deduce the figure shown.