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

Double Integration Method: It is valid for finding slope and deflection for one load at a time. Thus it is time consuming.

Macaulay’s Method: Uses SQUARE BRACKETS. It is applicable for any number and any types of loads.

Superposition Method

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.
θ_{B} –θ_{A}=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.
δ_{B} —δ_{A}=Y_{B}–y_{A =}(AREA OF BMD BETWEEN A AND B) x X_{B}/EI
5.Strain Energy Method
DIFFERENT DIFFERENTIAL EQUATIONS OR THE BENT BEAM OR DEFLECTED BEAM
Exact differential equation of the deflected beam
(EI d^{2}y/dx^{2})/[1+(dy/dx)^{2}] =± M_{x}
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 d^{2}y/dx^{2}) =±M_{x}
SLOPE AND DEFLECTIONS FOR DIFFERENT LOADED BEAMS
Sr. No. 
TYPE OFLOADED BEAM 
MAXIMUM DEFLECTION AND ITS LOCATION ON THE BEAM 
MAXIMUM SLOPE ANDITS LOCATION ON THE BEAM 
1. 
SIMPLY SUPPORTED BEAM WITH A CONCENTRATED LOAD AT THE CENTER 
y_{max} = (1/48)WL^{3}/EIIt is at the center. 
WL^{2}/16EI and itis at theends of the beam. 
2. 
SIMPLY SUPPORTED BEAM WITH A UDL OVER THE ENTIRE SPAN 
y_{max} = (5/384)WL^{3}/EI,W =w LIt is at the center. 
WL^{2}/24EI, W =w Land it is at the endsof the beam. 
3. 
CANTILEVER BEAM WITH A CONCENTRATED LOAD AT THE FREE END 
y_{max} = WL^{3}/3EIIt is at the free end. 
WL^{2}/2EI and it isat the free end 
4. 
CANTILEVER BEAM WITH A UDL OVER THE ENTIRE SPAN 
y_{max} = WL^{3}/8EI, W =w LIt is at the free end. 
WL^{2}/6EI,where W =w Land it is at thefree end 
5. 
SIMPLY SUPPORTED BEAM WITH A CONCENTRATED LOAD NOT AT THE CCENTER, (length ‘a’ > length ‘b’ 
(i) Deflection under the load will bey_{W} = Wa^{2}b^{2}/3EIL.(ii) Maximum deflectiony_{max}= [Wb(L2—b^{2})^{3/2}]/9and it will beat x= [(L^{2} –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/wpadmin/post.php?post=4290&action=edit MCQ Bending Stresses Beams
https://www.mesubjects.net/wpadmin/post.php?post=4363&action=edit MCQ slope Deflection