# Essay on Flexural Stresses

Submitted By drickymagosi
Words: 1205
Pages: 5

FLEXUAL STRESSES IN BENDING

OBJECTIVE

To investigate the validity of the simple bending equation for slender members.

THEORY

Stress-strain relationship;

[pic]Ɛx= σx/E from ɛx=y/R

Ɛx=[pic]/E=y/R or [pic][pic]x/y=E/R {6.7}

Thus bending stress is also distributed in a linear manner over the cross-section, being zero where y=0, that is at the neutral plane and being maximum tension and compression at the two outer surface where y (I) maximum.

EQUILIBRIUM FORCES

Consider an element, dA, at a distance y from some arbitrary location of the neutral surface.

The force on the element in the x-direction is

(fx = σx dA

Therefore the total longitudinal force on the cross-section is

F(x) =∫σx dA

Where A is the total area of the section

i) Since there is no external axial force in pure bending, the internal force resultant must be zero; therefore

F(x) =∫σx dA=0

Using equation {6.7} above to substitute for [pic]x

F(x) =∫σx dA=0

Since E/R is not zero, the integral must be zero, as this is the first moment of area about the neutral axis. (The first moment of area of a section about its centroid is zero).

The moment of the axial force about the neutral surface is yd f(x).

Therefore the total internal resisting moment is

This must balance the external applied moment M, so that for equilibrium

∫AyσxdA=M {6.8}

Now ∫AydA is the second moment of area of the cross-section about the neutral axis and will be denoted by I. Thus

M=EI/R or I/R=M/EI {6.9}

USING EQUATION {6.7}

M/I=σx/y and hence σx= My /I which relates the stress to the moment and the geometry of the beam.

Combining equation {6.7} and {6.9} gives the fundamentals relationship between bending stress, moment and geometry.

M/I=σ/y=E/R=Eɛ/y

Figure 1[pic]

M=Bending moment

I=Second moment of the cross-sectional area with respect to the N.A I=bd3/12

[pic]=Stress

Y=Distance from the neutral axis

E=Young modulus

R=Radius of curvature of neutral axis

σ=stress

ɛ=Strain

EQUIPMENT

▪ Steel and aluminum Slender strip

▪ Electrical strain gauges

▪ Digital strain recorder

▪ Dial gauges

PROCEDURE

CASE 1

□ The strip was simply supported on the two reaction beam supports

□ Strain gauge points were marked on each of the beams

□ A number of points on the beam were marked for application of loads and for dial gauges

□ The gauges were initialized to zeros

□ Strain gauges were connected to the bridge recorder and brought to null point under no loads

□ The strain readings were also recorded at each stage

□ The recordings were tabulated.

CASE 2

□ the beam was fixed to a clamp to form a cantilever

□ The strain gauge points were marked on each of the beams

□ Number of points on the beam were marked as reasonably close as possible, some earmarked for application of loads

□ The strain gauge points and the gauges were initialized to zero

□ Strain gauge were connected to the bridge recorder and brought to null point under load