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.