Four Point Bending

1
Four Point Bending
2
Other Types of Bending
Bending by Eccentric Loading
Cantilever Bending
3
Various Boundary Conditions of Beams
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Features of Beam Deformation
5
Neutral Plane and Axis of Symmetry
6
Assumptions for Beam Theory

• Kirchhoff Hypotheses---
• The cross-sections remain
• a straight plane perpendi-
• cular to the mid plane.
• The vertical segments
• are not stretched.

Bernoulli-Euler Beams
7
Deformation of Beams under Pure Bending
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Curvature under Pure Bending
Neutral Axis
Constant Curvature
9
Strain Analysis for Bending
ex d / L -y/r -yk
ex max c/r
d L L (r-y)q rq -yq
ex (-y/c) ex max
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Stress Distribution in Bending
sx (-y/c) sx max (-y/c)sm
Neutral plane should pass through the centroid.
sm Mc/I
11
Stress/Strain Distribution in Beams under Pure
Bending
12
Section Modulus and Bending Stiffness

sm Mc/I sx (-y/c)sm
sx -My/I
Define Section Modulus as S I/c Then sm M/S
Also ex -y/r -yk
My/I Ey/r
k 1/r M/EI (EI Bending Stiffness)
Note d/L P/EA, f/L T/GJ
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Beams with Irregular Cross-sections
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Stress Distribution in Beams with Irregular
Cross-sections
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Asymmetric Bending of Symmetric Beams
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Pure Bending of Asymmetric Beams
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Composite Beams
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Stress Distribution in Composite Beams
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Bending Due to Eccentric Loading