Chapter 3-part 2

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Load and Stress Analysis

Lecture 5

The McGraw-Hill Companies © 2012

Chapter Outline

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General Three-Dimensional Stress

• All stress elements are actually 3-D.
• Plane stress elements simply have one surface with zero stresses.
• For cases where there is no stress-free surface, the principal stresses are found from the roots of the cubic equation

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Fig. 3−12

General Three-Dimensional Stress

• Always three extreme shear values

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• Maximum Shear Stress is the largest
• Principal stresses are usually ordered such that σ₁ > σ₂ > σ₃, �in which case τ_max = τ_1/3

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Fig. 3−12

Elastic Strain

• Hooke’s law

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• E is Young’s modulus, or modulus of elasticity
• Tension in on direction produces negative strain (contraction) in a perpendicular direction.
• For axial stress in x direction,

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• The constant of proportionality n is Poisson’s ratio
• See Table A-5 for values for common materials.

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Elastic Strain

• For a stress element undergoing σ_x, σ_y, and σ_z, simultaneously,

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Elastic Strain

• Hooke’s law for shear:

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• Shear strain γ is the change in a right angle of a stress element when subjected to pure shear stress.
• G is the shear modulus of elasticity or modulus of rigidity.
• For a linear, isotropic, homogeneous material,

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Uniformly Distributed Stresses

• Uniformly distributed stress distribution is often assumed for pure tension, pure compression, or pure shear.
• For tension and compression,

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• For direct shear (no bending present),

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Normal Stresses for Beams in Bending

• Straight beam in positive bending
• x axis is neutral axis
• xz plane is neutral plane
• Neutral axis is coincident with the centroidal axis of the cross section

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Shigley’s Mechanical Engineering Design

Fig. 3−13

Normal Stresses for Beams in Bending

• Bending stress varies linearly with distance from neutral axis, y

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• I is the second-area moment about the z axis

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Fig. 3−14

Normal Stresses for Beams in Bending

• Maximum bending stress is where y is greatest.

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• c is the magnitude of the greatest y
• Z = I/c is the section modulus

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Assumptions for Normal Bending Stress

• Pure bending (though effects of axial, torsional, and shear loads are often assumed to have minimal effect on bending stress)
• Material is isotropic and homogeneous
• Material obeys Hooke’s law
• Beam is initially straight with constant cross section
• Beam has axis of symmetry in the plane of bending
• Proportions are such that failure is by bending rather than crushing, wrinkling, or sidewise buckling
• Plane cross sections remain plane during bending

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Example 3-5

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Dimensions in mm

Fig. 3−15

Example 3-5

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Example 3-5

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Example 3-5

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Example 3-5

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Two-Plane Bending

• Consider bending in both xy and xz planes
• Cross sections with one or two planes of symmetry only

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• For solid circular cross section, the maximum bending stress is

Shigley’s Mechanical Engineering Design

Example 3-6

Shigley’s Mechanical Engineering Design

Fig. 3−16

Example 3-6

Shigley’s Mechanical Engineering Design

Fig. 3−16

Example 3-6

Shigley’s Mechanical Engineering Design

Example 3-6

Shigley’s Mechanical Engineering Design