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PH0101 UNIT 1 LECTURE 3 1

PH0101 UNIT 1 LECTURE 3

Bending of Beams

Bending moment of a BeamUniform Bending (Theory and Experiment)

Worked Problem

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Bending of Beams

A beam is a rod or bar of uniform crosssection(circular or rectangular) whose length is verymuch greater than its thickness as shown inFigure

A beam is considered to be made up of a largenumber of thin plane layers called surfacesplaced one above the other.

AB

MN

DC

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Consider a beam to bebent into an arc of acircle by the applicationof an external couple

as shown Figure Taking the longitudinal

section ABCD of thebent beam ,the layers

in the upper half areelongated while thosein the lower half arecompressed.

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In the middle there is a layer (MN) which is notelongated or compressed due to bending of the

beam.

This layer is called theneutral surfaceand the

line (MN) at which the neutral layer intersects the

plane of bending is called the neutral axis.

It is obvious that the length of the filamentincreases or decreases in proportion to its distanceaway from the neutral axis MN.

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The layers below MN are compressed and thoseabove MN are elongated and there will be such

pairs of layers one above MN and one below

MN experiencing same forces of elongation and

compression due to bending and each pairforms a couple.

The resultant of the moments of all these

internal couples are called the in ternal bend ing

momentand in the equilibrium condition, this is

equal to the external bending moment.

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Bending Moment of a Beam

Consider the section PBCP, the extended filaments

lying above the neutral axis MN are in state oftension and exert an inward pull on the filamentadjacent to them towards the fixed end of the beam.

M

N

C

B

P

D

AP

Load

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In the same way the shortened filaments lying below theneutral axis MNare in a state of compression and exertan outward push on the filaments adjacent to themtowards the loaded end of the beam.

As a result tensile and compressive stresses develop inthe upper and lower halves of the beam respectively andform a couple which opposes to bending of the beam.

The moment of this couple is called the moment of the

resistance.

When the beam is in equilibrium position the bendingmoment and restoring moment or moment of resistanceshould be equal.

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To find an expression for

the moment of therestoring couple considera fiber DCat a distance rfrom the neutral axis MNas shown in Figure

Let the radius ofcurvature be Rand bethe angle subtended by itat the centre of curvature.In unstrained position ofthe beam, the length ofthe fiber DC= MN = R.

In the strained positionthe length of the fibre,

C

A

M

B

D

N

DC= (R + r)

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Strain in the fiber DC, =

or

lengthOriginal

lengththeinChange

R

r

R

RrR

Strain

)(

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i.e., strain is proportional to the distance from the

neutral axis.

Let the area of the fiber beAand its neutral axis

be at a distance r from neutral axis of the beam

and the strain produced be r/R. We haveStress = Yx Strain = Y r / R

Hence, force on the areaA

F = Y(r/R) xA

Therefore the moment of this force about MN

= Y(r/R) xAx r= Y Ar2/ R

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As the moment of the forces acting on both the upper andlower halves of the section are in the same direction, thetotal moment of the forces acting on the filaments due tostraining

gIR

YarR

Y

R

raY

22

where Igis the geometrical moment of inertia and

is equal to AK2

, A being the total area of thesection and Kbeing the radius of gyration of the

beam

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gIRY

gIR

Y

:. moment of the forces

In equilibrium, bending moment of the beam is equal and

opposite to the moment of bending couple due to the loadon one end.

:. Bending moment of the beam =

The quantity YIg (=YAK2) is called the flexural rigidity of the

beam. Flexural r igid i ty is defined as the bending moment

required to produce a unit radius of curvature.

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Uniform Bending The beam is loaded

uniformly on its both ends,

the bent beam forms an

arc of a circle. An elevation

in the beam is produced.

This bending is calleduni form bending.

Consider a beam (or bar)

AB arranged horizontally

on two knife edges C

and D symmetrically so

that AC = BD = a ,as

shown in Figure

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The beam is loaded with equal weights W and Wat the ends A and B.

The reactions on the knife edges at C and D areequal to W and W acting vertical upwards.

The external bending moment on the part AF ofthe beam is

= W x AFW x CF = W (AFCF)= W x AC = W x a

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R

YIWa

g

Internal bending moment =where

Y - Youngsmodulus of the material of the bar

Ig - Geometrical moment of inertia of the cross-

section of beamR - Radius of curvature of the bar at F

In the equilibrium position,

external bending moment = internal bending

moment

R

YIg

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Since for a given value of W, the values of a, Y and

Igare constants, R is constant so that the beam isbent uniformly into an arc of a circle of radius R.

CD = l and yis the elevation of the midpoint E of the

beam so that y= EF

D

F

C

O

y

R

l/2 E

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EF (2REF) = (CE)2

y(2Ry) =

y2R = (y2is negligible)

y=

2

l

R8

2

l

4

2

l

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or

Wa =

or

g2 I8 Yy

2

8R1

ly

yIalWY

g8

2

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If the beam is of rectangular cross-section,

, where b is the breath and d is the

thickness of beam. If M is the mass, the

corresponding weight W = Mg

12

3bd

Ig

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Hence

ybd

aMglY

3

2

2

3

from which Y the Youngs modulus of thematerial of the bar is determined.

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Exper iment

A rectangular beam (or bar) AB of uniformsection

is supported horizontally on two knife edges A

and B as shown in Figure

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Two weight hangers of equal masses aresuspended from the ends of the beam.

A pin is arranged vertically at the mid-pointof the beam.

A microscope is focused on the tip of the pin.

Initial reading of the microscope in thevertical scale is noted.

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Equal weights are added to both hangerssimultaneously and the reading of the

microscope in the vertical scale in noted.

The experiment is repeated for decreasing

order of magnitude of the equal masses.

The observations are then tabulated and the

mean elevation (y) at the mid point of the bar

is determined.

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Load in Kg

Microscope reading elevation Meanelevation(y)

for

a load of M.

Load

increasing

Load

decreasingMean

WW+50 gms

W+100 gms

W+150 gms

W+200 gms

W+250 gms

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The length of the bar between the knifeedges lis measured.

The distance of one of the weight hangersfrom the nearest knife edge p is measured.

The breadth (b) and thickness (d) of the barare measured using vernier calipers andscrew gauge.

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The youngs modulus of the material of the

beam is determined by the relation

Nm-2

ybd

lagMY

3

2

2

3

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