SLOPE DEFLECTION METHOD

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SLOPE DEFLECTION METHOD
By SUDARSHAN H P
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Introduction
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Determinate Structures
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Indeterminate Structures
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Degree of redundancy
D O R 4 3 1 D O R 4 3 1 D O R
5 3 2 D O R 6 3 3
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Degrees of freedom
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Support moment
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Fixed end moment
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Moment Area Theorems
Theorem 1
When a beam is subjected to external loading, it
under goes deformation. Then the intersection
angle between tangents drawn at any two points on
the elastic curve is given by the area of bending
moment diagram divided by its flexural rigidity.
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Moment Area Theorems
Theorem 2
The vertical distance between any point on the
elastic curve and intersection of a vertical line
through that point and tangent drawn at some
other point on the elastic curve is given by the
moment of area of bending moment diagram between
two points taken about first point divided by
flexural rigidity.
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Fixed end moment due to a point load at the mid
span
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Both moments are negative and hence they produce
hogging bending moment.
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Stiffness coefficients
a) When far end is simply supported
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b) When far end is fixed
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Substituting in (1)
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Fixed end moments due to yielding of support.
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Hence sagging BM
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Fixed end moment for various types of loading
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Assumptions made in slope deflection method

• All joints of the frame are rigid
• 2) Distortions due to axial loads, shear stresses
  being small are neglected.
• 3) When beams or frames are deflected the rigid
  joints are considered to rotate as a whole.

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Sign conventions
Moments All the clockwise moments at the ends
of members are taken as positive. Rotations
Clockwise rotations of a tangent drawn on to an
elastic curve at any joint is taken as
positive. Sinking of support When right support
sinks with respect to left support, the end
moments will be anticlockwise and are taken as
negative.
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Development of Slope Deflection Equation
Span AB after deformation
Effect of loading
Effect of rotation at A
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Effect of rotation at B
Effect of yielding of support B
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Slope Deflection Equations
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EXAMPLES
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Example Analyze the propped cantilever shown by
using slope deflection method. Then draw Bending
moment and shear force diagram.
Solution
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Slope deflection equations
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Boundary condition at B
MBA0
Substituting in equations (1) and (2)
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Free body diagram
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Example Analyze two span continuous beam ABC by
slope deflection method. Then draw Bending moment
Shear force diagram. Take EI constant
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Solution
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Slope deflection equations
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Boundary conditions
i. -MBA-MBC0
MBAMBC0
ii. MCB0
Now
Solving
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Free body diagram
Span AB
Span BC
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BM and SF diagram
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Example Analyze continuous beam ABCD by slope
deflection method and then draw bending moment
diagram. Take EI constant.
Solution
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Slope deflection equations
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Boundary conditions
Solving
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Substituting
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Example Analyse the continuous beam ABCD shown
in figure by slope deflection method. The support
B sinks by 15mm. Take
Solution
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FEM due to yielding of support B
For span AB
For span BC
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Slope deflection equation
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Boundary conditions
Now
Solving
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Final moments
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