Axial Loads

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Bending in Beams I II
Structural Engineering Laboratory 560.208 Spring
2000
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Overview

• Shear force and bending moment diagrams
• Normal strains and stresses
• Shear stresses
• Deflections
• Lab sequence will cover
• Cantilever
• Simply-supported beam

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Shear Force/Bending Moment Diagrams Important
Ideas

• Loads
• point/concentrated
• distributed
• uniform
• linearly varying
• Supports
• simple support roller support fixed support

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Characteristics

• Shear force Internal member forces
• Bending moment Internal moments
• Member must resist both
• Recall Slope of bending moment diagram
  ...
• Later...
• slope and displacement along beam related as well

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Normal Strains

• Derived from geometric arguments
• curvature
• compatibility of displacements
• plane sections remain plane
• plane sections remain normal to neutral axis
• Note Transverse strains exist also

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Normal Stresses

• From Hookes law
• As expected introduces
• material parameter E (Youngs modulus)
• geometric parameter, I (2nd moment of area)

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Shear Stresses

• Shear stress at location in section
• V Shear at section I 2nd moment of area
• Q 1st moment of area about point where stress
  required
• b Width of section

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Deflections of Beams

• Differential equation for deflection
• load
• shear
• moment-curvature
• slope
• deflection
• Obtaining deflections
• integration
• moment-area

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Curvature

• From calculus
• Relate to spanwise location, x, deflection, v
• Exact
• Small deflections

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Moment-Curvature

• Recall
• Small deflections

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Beam Differential Relations

• Using load, shear, moment relations

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Laboratory Exercise

• You will be provided cantilever/SS beams
• Load with test rig
• central load on simple beam
• mid-span load on 2-span continuous beam
• central load on cantilever
• central load on propped cantilever
• Record (3 increments of load) in each case
• strain gauge outputs
• deflection (dial gauge) and
• load cell outputs.
• Compute applied load, deflection and stresses

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Summary Report Requirements

• Outline equipment used and procedures
• Compute expected moments/shears/stresses in
  members using load beam dimensions
• Compute stresses from measurements and compare
• Plot deflection vs. load and compare with
  expected
• Make sure all group members names are indicated
  and signed off.