CE 203Structural Mechanics Section 3

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CE 203 Structural Mechanics
Week 5
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Bending moment Shearing force diagrams
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What is a beam

• Members that are loaded in a direction
  perpendicular to their longitudinal axis
• Length is more significant than lengths in
  cross-section
• We could have Simply supported, Cantilevered,
  Overhanging or Continuous beams

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Find the internal forces at C.
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Make section at C Indicate internal forces
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Sign Convention
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Draw shear bending moment diagrams
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Draw shear bending moment diagrams
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Load-shear-moment relations

• dV/dx -w(x)
• dM/dx V(x)

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Notes

• No load constant shear linear moment
  variation
• Uniform load linear shear quadratic moment
  variation

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Notes

• Concentrated load causes jump in shear
• Concentrated moment causes jump in moment

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Draw shear force and bending moment diagrams
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Reactions
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SFD
BMD
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• Bending stresses

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Bending of straight beams
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What type of beams are we talking about ?

• Prismatic, straight.
• X-section has an axis of symmetry
• Moment on an axis perpendicular to axis of
  symmetry

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Deformation due to pure bending
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Axis

• x longitudinal axis
• y axis of symmetry in cross section z
  axis of bending
• 
  y
• z

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Assumptions

• No change in length of longitudinal axis but
  becomes a curve
• Cross section remain plane perpendicular to
  longitudinal axis
• Deformation of cross section is neglected

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Deformation of a differential element along the
beam
?s ? ??
?s (?-y) ??
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Longitudinal strain

• ?x (?s ?s)/ ?s
• If ? radius of curvature (can be f(x))
• ?s ? ??
• ?s (?-y) ??
• ?x -y/ ? (1)

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Strain variation along y
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Flextural stresses

• Assume linear elastic material and
• sy , sz much less than sx then
• sx E ?x
• From 1
• sx - E y/ ? (2)

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Resulting stress distribution

• From statics
• Fx ? sx dA
• And
• Mz ? (-y) sx dA

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Location of neutral axis

• But Fx 0
• ? sx dA ? - E y/ ? dA 0
• -E/ ? ? y dA 0
• ? dA 0
• This only true if the z-axis passes by the
  centroid of the cross section.

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Flexture formula

• Mz ? (-y) sx dA ? (-y) (- E y/ ?) dA
• (3) Mz E/ ? ? y2 dA E/ ? Iz

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Flexture formula

• Using equations (2) and (3)
• sx - E y/ ? (2)
• (3) Mz E/ ? ? y2 dA E/ ? Iz
• We come up with the bending stresses
• sx - Mz y / Iz

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Comments about stress distribution

• sx - Mz y / Iz

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• Bending stresses
• Using the flexure formula

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Draw BM SF diagrams Find max tensile stress and
indicate its location Find max compressive
stress and indicate its location Plot stress
distribution in a section through point D.
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What if we measure strain?

• Determine the magnitude of the force P, if the
  strain at point a is measured to be 7x10-5 .Take
  E 30x106psi

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P 460 lb
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Contribution to moment resistance

• What percentage of
• Moment is resisted by
• The shaded area?

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h/2
If a1/2 h only 12.5
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Derivation
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Examples from text

• sx - Mz y / Iz
• Example 6.14 page 299
• Example 6.15 page 301
• Example 6.16 page 303
• Example 6.17 page 304

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