Bending Deformation, Strain and Stress in Beams

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Bending Deformation, Strain and Stress in Beams
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6.2 Bending Deformation and Strain

• Key Points
• Bending moment causes beam to deform.
• X longitudinal axis
• Y axis of symmetry
• Neutral surface does not undergo a change in
  length

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• Key Points
• Internal bending moment causes beam to deform.
• For this case, top fibers in compression, bottom
  in tension.

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• Key Points
• Neutral surface no change in length.
• All cross-sections remain plane and perpendicular
  to longitudinal axis.
• Neglect deformation of cros section within its
  own plane.

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Radius of curvature
Length decreases
Length constant
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Says normal strain is linear Maximum at outer
surface (where y c)
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6.2 Bending Stress The Flexure Formula
What about Stress????
Recall from section 6.1
Therefore, it follows that
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Sum moments about cut
This is the moment of inertia, I
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The Flexure Formula
Internal bending moment, lb-in
Max bending stress, psi
Distance from NA to outer fiber, in
Moment of inertia, in4
Or in general
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Examples

• Find maximum moment
• Find area properties, I and c
• Calculate stress

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WHERE IS BENDING STRESS MAXIMUM???

• Answer
• Outer surface (furthest away from Neutral Axis)
• Value of x along length where moment is maximum!!

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• Example The T-shape beam is subjected to the
  loading below.
• Draw shear and moment diagram. Identify location
  and magnitude of Mmax.
• Determine location and magnitude of maximum
  bending stress and draw stress profile. Is the
  beam safe if the material is aluminum w/ sy 15
  ksi?
• What is the largest internal moment the beam can
  resist if sallow 2 ksi?

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45K
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Statics Example 1 - Pliers
Given Typical household pliers as shown. Find
Force applied to wire and force in pin that
connects the two parts of the pliers.
Do this for homework. See solution Link
Side what is the shear stress in pin and bending
stress in handle? SofM
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Statics Example 2 Crane Structure
Given Crane structure as shown. Find Forces and
FBDs for cables A-B and A-E, boom DEF and post
AFC.
Do this for homework. See solution Link
Side what is the normal stress in cables
(average normal only) and normal stress in boom
and post (combined loading)? SofM
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Example 4 Determine the resultant internal
loadings acting on the cross sections located
through points D and E of the frame. (1-114)