shear force and bending moment diagram

Presented by Sr no. Name Enroll.no. Roll no. 1 Avaiya Milan V. 130030106007 7 2 Dabhi Harikrushan n. 130030106020 18 4 Dabhi Mehul 130030106021 19

Subject : mechanics of solid (2130003)

Introduction Structural Members are usually classified according to the types of loads that they support (axially loaded bar, etc.) Now we will begin to look at beams, which are structural members subjected to lateral loads. The first beams we will investigate are called planar structures, because they lie in a single plane.

Introduction If all loads act in that same plane, and if all deflections act in that plane, then the plane is called the Plane of Bending. First we will look at shear forces and bending moments in beams. Once we know these, we can find the stresses, strains and deflections of a beam

Types of Beams Beams are usually described by the way they are supported. Simply supported beam – pin support at one end and a roller support at the other Cantilever beam – fixed at one end and free at the other. Beam with an Overhang – simply supported beam that projects beyond the support (similar to a cantilever). Draw pictures

Review: Types of Loads Concentrated Loads Couple Distributed Loads Uniformly distributed Linearly varying Ty

Review: Types of Loads Concentrated Loads Couple Distributed Loads Uniformly distributed Linearly varying Ty

Review: Reactions Simple Beam FBD Cantilever Beam FBD Beam with Overhang FBD

Shear and Bending Moment in Beams If we have a beam that is loaded by a system of forces all in the y direction. The beam is classified as a simple beam (pin support at one end and a roller support at the other) The beam is in equilibrium with the application of these forces and its reactions.

Shear & Moment Sign Convention The signs associated with the shear force and bending moment are defined in a different manner than the signs associated with forces and moments in static equilibrium. The Shear Force is positive if it tends to rotate the beam section clockwise with respect to a point inside the beam section. The Bending Moment is positive if it tends to bend the beam section concave facing upward. (Or if it tends to put the top of the beam into compression and the bottom of the beam into tension.) + shear + moment

Shear & Moment Sign Convention The figure shows the shear force V and Bending Moment M acting in positive directions. Notice there is a possibility for confusion with sign notation. When summing forces, the direction of V is in the negative y-direction, but is positive shear. We will deal with this confusion by always selecting the V and M directions as shown in the diagram. This will simplify the sign conventions.

Simple Beam Looking at figure c and mentally summing torque about the center of the beam, we see that the horizontal x-forces cause a net toque - which we call the internal bending moment, M. This is the cause of the internal bending moment (torque) inside a loaded beam.

Shear & Moment Diagrams These are a quite useful way of visualizing how the shear force and bending moments vary through out the beam. We have completed our first Shear Force/Bending Moment Problem. We have determined the expressions for the shear forces and bending moments in the beam, and have made accompanying shear force and bending moment diagrams

Concentrated Load Moment Diagram The equations for moment are plotted below the Shear diagram of the beam The Bending Moment diagram The bending moment in the left side increases linearly from zero at the support to P(ab/L) at the concentrated load x=a In the right side, the bending moment is again a linear function of x, varying from P(ab/L) at x=a to zero at the support x=L. The maximum bending moment is therefore P(ab/L), which occurs at the concentrated load. In this example a=b=L/2

Uniform Load Next look at a simple beam with a uniformly distributed load of constant intensity w. Find reactions, because the beam and its loading is symmetric, the reactions are equal to wL/2 The slope of the shear diagram at each point equals the negative distributed load intensity at each point

Shear and moment diagram Axial load diagram Torque diagram Both of these diagrams show the internal forces acting on the members. Similarly, the shear and moment diagrams show the internal shear and moment acting on the members

Shear Diagram M V F x Lecture 1 Sign convention: V= -9kN

Shear Diagram F M=-9x Sign convention: M= -9x kNm X=0: M= 0 V = -9 kN F x M=-9x Sign convention: M= -9x kNm X=0: M= 0 X=3: M=-27kNm

Distributed Load For calculation purposes, distributed load can be represented as a single load acting on the center point of the distributed area. Total force = area of distributed load (W : height and L: length) Point of action: center point of the area

Example

Example

External Forces Draw the shear and moment diagram Absolute Bending Stress Mmax = 2.25kNm

Thank you…!!!