Analysis of Beams in Bending (5.1-5.3)

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Analysis of Beams in Bending (5.1-5.3)

• MAE 314 Solid Mechanics
• Yun Jing

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Bending Moment Along a Beam

• In this chapter, we will learn how to find the
  bending moment M along the beam.
• M is not necessarily constant sometimes M is a
  function of x.
• We will also solve for the shear force V(x),
  which will be used in Chapter 6.
• As before, we need a new FBD every time the
  loading changes.

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3
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Review of Beam Supports

• 3 equilibrium equations S FY 0, S FX 0, S M
  0
• Ignore the horizontal (x-direction) components,
  because these are axial loading.

R1
R3
R1
R1
R3
M1
R2
R2
R2
R1
R1
R3
M1
M2
M1
R1
R3
R3
R4
R2
R2
R4
R2
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Sign Convention

• Recall the applied loading results in both a
  bending moment M and a shear force V.

Positive shear and bending moment
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Example Problem

• Draw the shear and bending moment diagrams for
  the beam and
• loading shown, and determine the maximum absolute
  value (a) of the
• shear and (b) of the bending moment.

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Relations Between F, V, and M

• For beams with more complicated loading, it is
  helpful to develop a relationship between load,
  shear and bending moment.
• Sum forces in the vertical direction.

or
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Relations Between F, V, and M

• Sum moment about C.
• Neglect (?x)2 term since it is much smaller
  than?x term.

or
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Example Problem

• Determine (a) the equations of the shear and
  bending moment curves
• for the beam and loading shown, and (b) the
  maximum absolute value
• of the bending moment in the beam.

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Example Problem

• Draw the shear and bending-moment diagrams for
  the beam and loading shown.

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Design of Beams in Bending (5.4)

• MAE 314 Solid Mechanics
• Yun Jing

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Design of Beams for Bending

• Recall the largest normal stress in the beam
  subject to bending occurs at the surface and can
  be defined as
• A safe design requires the maximum stress is no
  more than the allowable stress (smax sall), so

where
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Procedure for Design

• Determine the value for sall.
• Draw shear and moment diagrams.
• From the diagrams, determine the maximum absolute
  bending moment.
• Determine the minimum allowable value Smin.
• Use Smin to determine best cross section
  dimensions.
• Timber beam Smin bh2/6
• Rolled-steel beam Use Appendix C in textbook

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Example Problem

• For the beam and loading shown, design the cross
  section of the beam,
• knowing that the grade of timber used has an
  allowable normal stress of
• 12 MPa.

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Example Problem

• Knowing that the allowable stress for the steel
  used is 160 MPa, select
• the most economical S-shape beam to support the
  loading shown.

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Example Problem

• A beam is to be made of steel that has an
  allowable bending stress of 170MPa. Select an
  appropriate W shape that will carry the loading
  shown in the figure below