Chapter 5: Deflections of Beams by Double Integration Method

Prepaired BySANJAY KUMAR

Assistant Professor Department of Mechanical Engineering

YMCA University of Science & Technology, Faridabad

DEPARTMENT OF MECHANICAL ENGINEERINGYMCA UNIVERSITY OF SCIENCE & TECHNOLOGY, FARIDABAD

To know application of double integration method to find out slopes and deflections in beams.

To understand application of Macaulay’s method to find out slopes and deflections in beams.

Objectives

To familiarize the importance of computing deformations in beams.

Relation between Curvature, Slope, Deflection, etc. at agiven Section in a Beam

Deflection = y

Slope dy

dx2d y

Moment M =

Shear force F =

Load intensity q =

2

2

d yEI

dx

3

3 dM d y

EIdx dx

4

4 dF d y

EIdx dx

Maximum Moment and Deflections in CantileverSubjected to

Moment at free end qmax = at free end

ymax = at free end

Concentrated load at free end

ML

EI2

2

– ML

EI Concentrated load at free end

qmax = at free end

ymax = at free end

2EI

2

2

WL

EI

3

3

WL

EI

….(Contd..)

UDL over entire span qmax = at free end

ymax = at free end

3

6

wL

EI

4

8

wL

EI

Varying linearly from zero at free end to w unit length at fixed end

qmax = at free end

ymax = at free end

24

3wL

EI

4

30

wL

EI

Maximum Moment and Deflections in Simply SupportedBeam Subjected to

Concentrated load at mid span qmax = at support

Y = at mid span

2

16

– wL

EI

3wLYmax = at mid span

UDL over entire span qmax = at support

Ymax = at mid span

48

wL

EI

3

24

wL

EI

45

384

wL

EI

….. (Contd..) Varying linearly from zero at one end to

w/unit length at other end ymax = 0.006523

at x = 0.5193 from the end

4wL

EI

at x = 0.5193 from the end where intensity of load is zero

Important Points forMacaulay’s Method Constants of integration is written first.

Integration of (x – a) is written as instead

of .

2

2

x a

2

x

axof .

If for any section (x – a) becomes negative, it warns that the term is not applicable.

2 ax

Important Points forMacaulay’s Method (Contd..) In the expression for bending moment, if external moment

M0 appearing at distance ‘a’ should be written as M0 (x – a) to retain warning message.

If uniformly distributed load is not extending up to the end If uniformly distributed load is not extending up to the end of the beam, extend it up to the end and consider the compensating load also.