Bending

Shear Forces & Bending MomentsShear & Moment Diagrams

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 otherCantilever 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).

Review: Types of Loads

Concentrated LoadsCoupleDistributed Loads– Uniformly distributed– Linearly varying

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 and Bending Moments

To determine the internal effects of the applied loads we imagine a cutting plane to isolate either the left side or right side of the beam.In order for the isolated section of the beam to be in equilibrium a force (V) and couple (M) are required at the cut.– V is termed Shear force– M is termed Bending Moment

Shear and Bending Moments

The shear V and Moment M are the force and couple applied to the left part of the beam by the right side of the beam to maintain equilibrium.Equilibrium of both sides is required because the entire beam is in equilibrium.The section can be made anywhere along the length of the beam.Shear and Moment are a function of the distance from the origin.

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

Simple Beam

A simply supported loaded beamAn exaggerated view of the way the bending caused by the load is shown

Simple Beam

If the beam were cut and we looked at the left side section

These forces develop since, as the beam bends, the top region of the beam is put into compression and the bottom region of the beam is put into tension.

Simple Beam

As a result there are internal horizontal (x-direction) forces acting in the beam; however for every positive x-force, there is an equal and opposite negative x-force. Thus the net horizontal (x-direction) internal force in the beam section is zero. However, even though the actual x-forces cancel each other, the torque produced by these x-forces is not zero.

Concentrated Load Shear Diagram

The Shear Force diagram– Shear force at the end is equal to the

reaction– Remains constant until the point of load P– Shear force decreases abruptly by an

amount equal to P– In the right hand part, the shear force is

constant but equal to the reaction at B.