SLOPE AND DEFLECTION CLASS NOTES FOR MECHANICAL ENGINEERING

SLOPE AND DEFLECTION

CLASS NOTES FOR MECHANICAL

ENGINEERING

 

SLOPE

  It is angular shift at any point of the

beam between no load condition and

loaded beam. Its value is different at

different points on the length of the beam.

It is represented by dy/dx or θ. Its units are

radians. There is a maximum limit for slope

for any loaded beam.

Deflection

 It is the vertical shift of a point on the

beam between no load condition and

loaded beam. Its value is different at

different points on the length of the beam.

It is represented by y or 𝜹. Its units are

mm. There is a  limit for maximum

deflection for any loaded beam.

METHODS TO FIND SLOPE AND DEFLECTION

  1. Double Integration Method: It is valid for finding slope and deflection for one load at a time. Thus it is time consuming.
  2. Macaulay’s Method: Uses SQUARE BRACKETS. It is applicable for any number and any types of loads.
  3. Superposition Method
  4. Moment Area Method Or Graphical Method : Uses Two Mohr’s Theorems

MOHR’S FIRST THEOREM

  For finding Slope.
Statement says that the difference of slopes between any two points on the loaded beam.
It is equal to the area of the BMD between those two points divided by Bending Stiffness EI.
θB –θA=AREA OF BMD BETWEEN A AND B/EI

MOHR’S SECOND THEOREM

 For finding deflection.
 Statement says that the difference of deflections between any two points on the loaded beam.
Difference is equal to Moment  of the BMD area between those two points divided by Bending Stiffness EI.
              This moment is to be about the first point say A.
               δBδA=YB–yA =(AREA OF BMD BETWEEN A AND B) x XB/EI

             5.Strain Energy Method

DIFFERENT DIFFERENTIAL EQUATIONS OR THE BENT BEAM OR DEFLECTED BEAM
Exact differential equation of the deflected beam
(EI d2y/dx2)/[1+(dy/dx)2] =± Mx
Approximate Differential equation of the bent beam:
Neglecting (dy/dx)2 since dy/dx is small.
we get approximate Differential equation of the bent beam.
The use of this approximate differential equation introduces maximum error of 4 % in maximum deflection. It is in comparison to value found by exact differential equation. It is also much simpler to use.
(EI d2y/dx2) =±Mx
 

SLOPE AND DEFLECTIONS FOR DIFFERENT LOADED BEAMS

 

Sr. No.

TYPE OF

LOADED BEAM

MAXIMUM DEFLECTION AND ITS LOCATION ON THE BEAM

MAXIMUM SLOPE  AND

ITS LOCATION ON THE BEAM

1.
SIMPLY SUPPORTED BEAM WITH A CONCENTRATED LOAD AT THE CENTER
ymax  = (1/48)WL3/EI
It is at the center.
WL2/16EI and it
is at the
ends of the beam.
2.
SIMPLY SUPPORTED BEAM WITH A UDL OVER THE ENTIRE SPAN
ymax  = (5/384)WL3/EI,
W =w L
It is at the center.
WL2/24EI,    W =w L
and it is at the ends
of the beam.
3.
CANTILEVER BEAM WITH A CONCENTRATED LOAD AT THE FREE END
ymax  = WL3/3EI
It is at the free end.
WL2/2EI and it is
at the free end
4.
CANTILEVER BEAM WITH A UDL OVER THE ENTIRE SPAN
ymax  = WL3/8EI,    W =w L
It is at the free end.
WL2/6EI,
where W =w L
and it is at the
free end
5.
SIMPLY SUPPORTED BEAM WITH A CONCENTRATED LOAD NOT AT THE CCENTER,  (length ‘a’ > length ‘b’
(i)                Deflection under the load will be
yW = Wa2b2/3EIL.
(ii)              Maximum deflection
ymax= [Wb(L2—b2)3/2]/9
and it will be
at x= [(L2 –b2)/3]0.5 from the center towards left portion ‘a’ . It is irrespective of the load placed anywhere on the beam span.
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https://www.mesubjects.net/wp-admin/post.php?post=4290&action=edit        MCQ Bending Stresses Beams

https://www.mesubjects.net/wp-admin/post.php?post=4363&action=edit             MCQ slope Deflection