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

∫Adfx=∫AyσxdA

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

□ Loads were gradually applied and slowly increased, recording the dial gauge reading at each stage

□ 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

□ Loads were gradually applied and slowly increased recording the dial gauge readings at each stage

□ The strain readings were also noted at each stage

□ The recorded readings were tabulated and the curvatures of the beam were evaluated by finite differences of recorded deflections.