Deflection of Beams and Shafts (Chapter 12)

[Sections to be covered: 12.1, 12.2, 12.3, 12.5, 12.7, 12.9, 12.4 and 12.8]

Elastic curve: The deflection diagram (that assists in visualizing the deformed shape) of the longitudinal axis (that passes through the centroid of each cross-sectional area) of the beams is called the elastic curve.

1

Longitudinal axis of the beam

Assumptions(i) Cross-sections remain plane and perpendicular to the longitudinal axis of the beam, during elastic deformation. Sections rotate compared to the vertical undeformed plane, but do not distort out-of-plane

2

P1

No distortion of the plane occurs due to purebending

900

Section remains in a plane but gets rotatedDue to shearThe plane getsdistorted out-of-plane

(ii) Longitudinal axis, which lies within the neutral surface, does not experience and change in length.

(iii) The beam bends along the longitudinal axis of the beam, in a + ve or - ve manner. Perpendicular to this longitudinal axis, the beam obeys the position effect and deforms in a contrary manner. At locations where the tensile stresses occur, the transverse section contracts, and at locations where the compressive stresses occur, the transverse section expands.

3

b

d

900

db

d

Bending is contrary tolongitudinal bending

This shows that + ve moment causes a - ve stress on the + ve side of the axis.

When s tends to zero, the curve AC and straight line AC tend to be the same

4

From Eqn. 1

s

O

x

vs

A

C

B

Since ds = d,

Eqn. (8) becomes,

5

Summing up vertical forces, i.e.,

6

xx

x

yw(x)

z

w(x)

M M+MV

V+dV

x

O

B

Cross-sectionof beam

7

When w(x) is a continuous function, equation (16) can be used to solve the problem of deflection of beams. When w(x) is not continuous additional procedures have to be developed to handle the problem.

Macaulay or Singularity Functions

Take the moment at a section, where all the loads acting on the beam are included in the

b.m. equation.

are called singularity functions

….., 2, 1, 0, -1, -2, ….

We can write the shear and load equation by,

8

X

X

b

a

c

x

A B

R2R1

P

Mo

wo /u. l.

Differentiation

Integration

When the index of the function is less than zero, it does not exist.

i.e., do not exist

When the index of the zero or greater than zero, it exists at and beyond the parameter

within the function.

i.e.,

Method of superposition

As long as the deformations in a structure are elastic, and the deformations are small, the deflection due to a series of loads, acting on the structure, can be obtained by summing up the response of the structure to each of those loadings. (a)

9

l1 a

P2 P1

= P1

W /u. l.

(b)

With the condition that

10

+

P2

l1 a

w /u. l.

+

L

w /u. l.

A B

RB

L

w /u. l.

A

v1B

=

+

RB

V2B

(c)

Vertical deflection at c,

= vc cantilever+B a

Horizontal deformation at c

= - vhB

11

a

CB

A

b

P

negligible

B

vc cantilever

CP

B

vhB

A

B

B

Deformation of Given Structures

12

4 (=L)

L1=3

C

B

A

20 lb/in (=w0 lb/in)

CB

A

CB

A

20 lb/in (=w0 lb/in)

=

+

80 lb

80 lb

13

C

B

A

+

80 lb

80 lb (=P1)

B

C

C

B

A

B

C

20 lb/in (=w0 lb/in)

=

14

A

BCB

L2

80 lb 80 lb80 lb

A

B80 lb

80 lb 80 lb

A

BCB

A

B

BRB = w0L2

RB

MB

MBRB

MB

w0 /u. l

12.87 :

15

L/2

L/2

C

A

B

L/2

16

L/2

A

PP

P

M=(P)(L/2)

T = PL/2

Angle of twist = B

= (TL)/(GJ)

BA

B

A

17

A

B

C

RB

L L

P

At B, Compatibility condition

18

RB

2L

B CA

RB

L L

w / u. l

+

w / u. l

RB

At B, (Compatibility condition)

12.117

19

A

D C

B

L/2 L/2P

P

RC = RB

RC = RB

RB = RC

At B,

20

L/2 L/2

RB

h

RC = RB

RC = RB

L

L/2

(p)(L/2)p

RB

Moment-Area Method

It is a semigraphical procedure for finding the slope and deflection at specific points on

the elastic curve of the beam.

21

dxx

L

A B

wx

B/A = Angular (or tangential deviation of angles) deviation between A & B

22

B/A

A/B

Elastic curve

Elastic curve

x

Mx

dx

x

M/EIdiagram

This can be stated as: “The angular deviation between B and A is equal to the area under

the M/EI diagram between the points A and B”.

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BS T

Elastic curve

A B

tB/A

+veB

A

td

d

dx

M/EI

Enlarge section between S and T on elastic curve

Tangential deviation of point B on elastic curve from the tangent drawn from A to meet

the vertical line through B (at B)

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x xO

tA/B

S TBA d

td xd

O

Elastic curve

Tangent from B

ST

d

= Vertical deviation of the tangent at a point A on the

elastic curve with respect to the tangent drawn from B

tA/B = Moment of the area under the M/EI diagram between A and

B, computed about point A.

When tA/B is +ve, point A on the elastic curve is above the point O, the intersection of the

vertical line from A to the tangent drawn from B.

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+ve

- veM/EI diagram

- ve

tD/CC

Tangent from CElastic curve

When tD/C is – ve, point D on the elastic curve is below point O, the intersection point of

the vertical line from D to the tangent drawn from C.

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