1. Relation Between BM and Slope
Lecture No-1
J P Supale
Mechanical Engineering Department
SKN SITS LONAVALA
Strength of Materials

2. Deflection And Slope
Deflection– Deflection at any point is vertical distance between the axis of beam before and after loading at that point.
Denoted by ‘y’ or ‘ ‘
Always zero at supports.
Slope– Slope at any point is defined as the angle in radian between the tangent to the elastic curve at that point and the original axis of beam.
Denoted by ‘dy/dx’ or ‘θ’
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3. Deflection And Slope y θ
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4. Boundary Conditions Simply Supported Beam
At supports, the deflection is always zero.
At supports, the slope is always maximum.
Maximum defection occurs at point where slope is zero or vice versa.
θmax y
ymax
y =0
y =0
θ = 0
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5. Boundary Conditions Cantilever Beam
At fixed support, the deflection is always zero.
At fixed support, the slope is always zero.
At free end, the deflection is maximum.
At free end, the slope is maximum.

6. Flexural Rigidity Product of modulus of Elasticity , E and
Moment of Inertia, I
Its Unit is N.m2
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7. Relation between Slope and BM
Let
L = Span of Beam
M = Bending Moment
R = Radius of curvature of beam after bending
Y = Deflection of beam at centre
θ = Slope of beam
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8. Relation between Slope and BMy D
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9. Relation between Slope and BM
From the fig. AC x BC = CE x CD In practice, deflection of beam y is very small and square of y is negligible. 1
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10. Relation between Slope and BM
By using Bending Formula, Put value of R in equation 1,we get
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11. Relation between Slope and BM
From the geometry of fig. Since Sin ΘA is very small quantity Sin ΘA = ΘA
Measured in radian
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12. Differential equations of the deflection curve
and y y y y y
From the Engineering Beam Theory:

13. Differential equations of the deflection curve
Bending Moment
Flexural Stiffness
Curvature
Curvature
Since,
Slope
Deflection
Where C1 and C2 are found using the boundary conditions.

14. Equations of Elastic Curves
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