Shear Force & Bending Moment Diagrams PDF

## Shear Force & Bending Moment ### Beam - A horizontal member. A load is acting on this. ### Column - A vertical member. Parallel loads are acting through the axis of the member. ### Shear force - The algebraic sum of the vertical forces at any section of the beam to the right or left sides of t...

## Shear Force & Bending Moment ### Beam - A horizontal member. A load is acting on this. ### Column - A vertical member. Parallel loads are acting through the axis of the member. ### Shear force - The algebraic sum of the vertical forces at any section of the beam to the right or left sides of the section. ### Bending moment - The algebraic sum of the moments of all the forces acting to the right or left side of the section. ### Shear force diagram (SFD) - Shows the variation of the shear force along the length of the beam. ### Bending moment diagram (BMD) - Shows the variation of the bending moment along the length of the beam. ### Types of Beam 1. Cantilever beam - A beam which is fixed at one end and free at the other. 2. Simply supported beam - A bream supported or resting freely at the supports at both ends. 3. Over-hanging beam - If an end portion of the beam is extended beyond the support, the beam is known as an over-hanging beam. 4. Fixed beam - The beam at both ends is fixed or built in walls. 5. Continuous beam - A beam which is provided by more than two supports. ### Types of Loads 1. Concentrated or Point Load - A load which is considered to act at a point. It must be distributed over a small area. ### Uniformly Distributed Load (UDL) - A uniformly distributed load is one which is spread over a beam in such a manner that the rate of loading is uniform along the length. ### Uniformly Varying Loads (UVL) - A uniformly varying load is one which is spread over a beam in such a manner that the rate of loading varies from point to point along the beam. ### Sign Convention - **Shear force:** - If the resultant force to the left of the section is upwards, or to the right of the section is downwards, the shear force at the section is considered `positive`. - If the resultant force to the left of the section is downwards, or to the right of the section is upwards, the shear force at the section is considered `negative`. - **Bending moment:** - If the bending moment at the section tends to bend the beam to a curvature having concavity at the top, it is considered `positive` (sagging moment). - If the bending moment at the section tends to bend the beam to a curvature having concavity at the bottom, it is considered `negative` (hugging moment). ### Important Points for drawing shear force and bending moment diagrams 1. Consider the left (or) right portion of the section. 2. Add the forces (including reactions) normal to the beam on one of the portions. 3. The right portion of the section is chosen. 4. The shear force and bending moment are plotted as follows: - `Positive` values are plotted above the base line. - `Negative` values are plotted below the base line. ### Shear Force & Bending Moment Diagrams of a cantilever beam with a point load at the free end - **SF Diagram:** - The shear force at any section is constant and equal to the point load (W) and acting downwards. This results in a straight line diagram. - At the fixed end (A), shear force is `zero`. - At the free end (B), shear force is `W`. - **BMD Diagram:** - The bending moment at any section is proportional to `x` (distance of the section from the free end). - At the fixed end (A), BM is `WL`. - At the free end (B), BM is `zero`. ### Shear Force & Bending Moment Diagrams of a cantilever beam with a uniformly distributed load - **SFD Diagram:** - The shear force at any section `x` will be equal to the resultant force acting on the right portion of the section. - The shear force follows a straight line law. - At the fixed end (A), shear force is `zero`. - At the free end (B), shear force is `WL`. - **BMD Diagram**: - The bending moment at any section is given by `M_x = -(w.x).(x/2)`. - The bending moment varies according to a parabolic law. - At the free end (B), BM is `zero`. - At the fixed end (A), BM is `WL^2/2`. ### Shear Force & Bending Moment Diagrams for the cantilever beam shown on page 14 - **SFD Diagram:** - At D, SF is `800N`. - SF remains constant between D & C. - At C, SF is `800N+500N=1300N`. - SF remains constant between C & B. - At B, SF is `800N+ 500N +300N = 1600N`. - SF remains constant between B & A. - At A, SF is `1600N`. - **BMD Diagram:** - At D, BM is `zero`. - From D to C, BM varies according to a straight line law. - At C, BM is `-6400N`. - From C to B, BM varies according to a straight line law. - At B, BM is `-1550N`. - From B to A, BM varies according to a straight line law. - At A, BM is `-2350N`.