Bending Moment Experiment

Objective

The objective of this experiment is to compare the theoretical internal moment with the measured bending moment for a beam under various loads.

Introduction and Background

Theory

Definition of a Beam

Members that are slender and support loadings that are applied perpendicular to their longitudinal axis are called beams. Beams are important structural and mechanical elements in engineering. Beams are in general, long straight bars having a constant cross-sectional area, often classified as to how they are supported. For example, a simply supported beam is pinned at one end and roller-supported at the other, a cantilevered beam is fixed at one end and free at the other, and an overhanging beam has one or both of its ends freely extended over the supports.

Figure 1: Types of beams

Types of Internal Loading

The design of a structural member, such as a beam, requires an investigation of the forces acting within the member which is necessary to balance the force acting externally on it. There are generally four types of internal loading that can be resisted by a structural member:

Figure 2: Types of Loadings

A. Normal Force, N

This force acts along the member’s longitudinal axis and passes through the centroid or geometric centre of the cross-sectional area. It acts perpendicular to the area and is developed whenever the external loads tend to push or pull on the two segments of the body.

B. Shear Force, V

If the external force is applied perpendicular to the axis of a member, it causes an internal stress contribution acting tangent to the member’s cross section. The resultant of this stress distribution is called the �shear force’. The shear force is developed when the external loads tend to cause the two segments of the body to slide over one another.

C. Bending Moment, M

When external moment is applied perpendicular to the axis of a member, the internal distribution of stress is directed perpendicular to the member’s cross-sectional area and varies linearly from a �neutral’ axis passing the member’s centroid. The resultant of this stress distribution is called the �bending moment’. The bending moment is caused by the external loads that tend to bend the body about an axis lying within the plane of the area.

D. Torsional moment or Torque, T

An external torque tends to twist a circular member about its longitudinal axis. It cause an internal distribution of stress that varies linearly when measured in a radial direction. The resultant of this stress distribution is called the �torque’ or �torsional moment’.

Bending Moment

When applied loads act along a beam, an internal bending moment which varies from point to point along the axis of the beam is developed. A bending moment is an internal force that is induced in a restrained structural element when external forces are applied. Failure by bending will occur when loading is sufficient to induce a bending stress greater than the yield stress of the material. Bending stress increases proportionally with bending moment. It is possible that failure by shear will occur before this, although while there is a strong relationship between bending moments and shear forces, the mechanics of failure are different.

A bending moment may be defined as "the sum of turning forces about that section of all external forces acting to one side of that section". The forces on either side of the section must be equal in order to counter-act each other and maintain a state of equilibrium. For systems allowed to rotate, then the equivalent force would be referred to as torque.

Moments are calculated by multiplying the external vector forces (loads or reactions) by the vector distance at which they are applied. When analysing an entire element, it is sensible to calculate moments at both ends of the element, at the beginning, centre and end of any uniformly distributed loads, and directly underneath any point loads. Of course any "pin-joints" within a structure allow free rotation, and so zero moment occurs at these points as there is no way of transmitting turning forces from one side to the other.

If clockwise bending moments are taken as negative, then a negative bending moment within an element will cause "sagging", (e.g. a closet rod �sagging’ under the weight of clothes on clothes hangers), and a clockwise moment will cause "hogging". 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.

Shearing Force and Bending Moment Diagrams

In order to properly design a beam it is necessary to first determine the maximum shear and moment in the beam, by expressing V and M as functions of the arbitrary position x along the beam’s axis. These shear and moments functions can therefore be plotted and represented by graphs called shearing force and bending moment diagrams. In these diagrams, the abscissa (horizontal axis) indicates the position of the section along the beam, and the ordinate (vertical axis) represent the values of the shearing force and bending moment at any section along the length of the beam. From these graphs, the maximum values of V and M can therefore be obtained. In addition, since these diagrams provide detailed information about the variation of the shear force and bending moment along the beam’s axis, they are often used by engineers as a guide to decide where to place reinforcement materials within the beam or how to proportion the size of the beam at various points along it length.

Critical values within the beam are most commonly annotated using a bending moment diagram, where negative moments are plotted to scale above