1. Deflection of Beams Chapter 12
Stress Analysis -MDP N161
Deflection
of Beams
Chapter 12
MDP: N161- Stress Analysis Fall 2009 1

2. MDP: N161- Stress Analysis - Fall 2009
Deflection of beam is a design limit. It is affected by the material properties, cross-section properties and the position of the supports.
MDP: N161- Stress Analysis Fall 2009

3. MDP: N161- Stress Analysis - Fall 2009
Elastic Curve
The elastic curve is the deflection diagram of the longitudinal axis that passes through the centroid of each cross-sectional area
MDP: N161- Stress Analysis Fall 2009

4. Internal Moment and Deflection
MDP: N161- Stress Analysis Fall 2009

5. MDP: N161- Stress Analysis - Fall 2009

6. MDP: N161- Stress Analysis - Fall 2009v
Elastic curve can be expressed as
V = f(x) v
Elastic curve can be expressed as
V = f(x)
MDP: N161- Stress Analysis Fall 2009

7. MDP: N161- Stress Analysis - Fall 2009

8. MDP: N161- Stress Analysis - Fall 2009
Since x = E x
and x = -Mz y/ Iz
Then 1 ρ = M EI
MDP: N161- Stress Analysis Fall 2009

9. MDP: N161- Stress Analysis - Fall 2009
From Calculus and after approximation for sake of application to engineering structures which specify small deformation, the relation between the deflection and the curvature is: 1 ρ
d2v
dx2 ≈ M EI =
d2v
dx2
Since M EI 1 ρ =
MDP: N161- Stress Analysis Fall 2009

10. Slope and Deflection by IntegrationM EI =
d2v
dx2
To get the elastic curve v (x) for a beam of constant cross-sectional area and made of the same material, you have to know the variation of the moment M with x and then double integrate the above equation
MDP: N161- Stress Analysis Fall 2009

11. MDP: N161- Stress Analysis - Fall 2009
The integrations produces two constants, which could be evaluated from the known values of deflection v or the slope =dv/dx at certain sections
MDP: N161- Stress Analysis Fall 2009

12. Boundary Conditions at supports
At roller or pin supports; the deflection v =0
MDP: N161- Stress Analysis Fall 2009

13. MDP: N161- Stress Analysis - Fall 2009
For fixed support v =0
and the slope  = dv/dx =0
MDP: N161- Stress Analysis Fall 2009

14. MDP: N161- Stress Analysis - Fall 2009
Example 8-1
MDP: N161- Stress Analysis Fall 2009

15. MDP: N161- Stress Analysis - Fall 2009
Solution
MDP: N161- Stress Analysis Fall 2009

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18. MDP: N161- Stress Analysis - Fall 2009