Bending Stress Case Study

Bending Stress Case Study

A fundamental equation for the bending of beams is:

where

s stress due to bending at distance y from the neutral axis
M =bending moment
I = second moment of area of section of beam about its neutral axis
E = modulus of elasticity
R =radius of curvature

Section modulus Z = I/ ymax
The second moments of area of the beam sections most commonly met with are (about the central axis XX)

(a) Solid rectangle

 (b) Symmetrical hollow rectangle or I-section 

(c) Solid rod

(c) Tube

The neutral axis of any section, where bending produces no strain and therefore no stress, always passes through the centroid of the section. For the symmetrical sections listed above this means that for vertical loading the neutral axis is the horizontal axis of symmetry.

STRESSES IN BEAMS DUE TO BENDING

TYPES OF BEAMS A beam is a structure, which is loaded transversely (sideways). The loads may be point loads or uniformly distributed loads (udl). The diagrams show the way that point loads and uniform loads are illustrated.

Transverse loading causes bending and bending is a very severe form of stressing a structure. The bent beam goes into tension (stretched) on one side and compression on the other.

RADIUS OF CURVATURE
Normally the beam does not bend into a circular arc. However, what ever shape the beam takes under the sideways loads; it will basically form a curve on an x – y graph. In maths, the radius of curvature at any point on a graph is the radius of a circle that just touches the graph and has the same tangent at that point

RELATIONSHIP BETWEEN STRAIN AND RADIUS OF CURVATURE
The length of AB AB = R θ
There is a layer of material distance y from the neutral axis and this is stretched because it must become longer.
The radius of the layer is R + y. The length of this layer is denoted by the line DC. DC = (R + y)θ This layer is strained and strain (ε) is defined as ε= change in length/original length

The modulus of Elasticity (E) relates direct stress (σ) and direct strain (ε) for an elastic material and the definition is as follows.

It follows that stress and strain vary along the length of the beam depending on the radius of curvature.
RELATIONSHIP BETWEEN STRESS AND BENDING MOMENT
The centroid in this case is on the neutral axis. The areas above and below the neutral axis are equal. Half the force is a compressive force pushing into the diagram, and half is tensile pulling out. They are equal and opposite so it follows that F = 0 which is sensible since cross sections along the length of a beam obviously are held I equilibrium.

The diagram indicates that the two forces produce a turning moment about the neutral axis because half the section is in tension and half in compression. This moment must be produced by the external forces acting on the beam making it bend in the first place.
This indicates that the stress in a beam depends on the bending moment and so the maximum stress will occur where the bending moment is a maximum along the length of the beam. It also indicates that stress is related to distance y from the neutral axis so it varies from zero to a maximum at the top or bottom of the section. One edge of the beam will be in maximum tension and the other in maximum compression. In beam problems, we must be able to deduce the position of greatest bending moment along the length.

NEUTRAL AXIS
When bending alone occurs in a member such as a beam, the neutral axis passes through the centroid. The stress varies from top to bottom over the structure from a maximum tensile on one edge to a maximum compressive on the other. The stress distribution is typically as shown. The stress is zero on the neutral axis and this passes through the centroid.

When a compressive stress is added to the bending stress, the stress everywhere is decreased by σB and the neutral axis moves away from the centroid towards the tensile edge as shown in figure . It is quite possible for the neutral axis to move beyond the edge.