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Shearing and Bearing Stress 1

Shearing and Bearing

Stress (1.6-1.8, 1.12)MAE 314 – Solid MechanicsXiaoning Jiang

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Shearing and Bearing Stress 2

What is Shearing Stress?

We learned about normalstress ( σ) , which actsperpendicular to thecross-section.

Shear stress ( τ ) acts

tangential to the surfaceof a material element.

Normal stress resultsin a volume change.

Shear stress resultsin a shape change.

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Shearing and Bearing Stress 3

Where Do Shearing Stresses Occur?

Shearing stresses are commonly found in bolts, pins, andrivets.

Free Body Diagram of Bolt

Bolt is in “single” shear

Force P results in shearing stress

Force F results in bearing stress (will discuss later)

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Shearing and Bearing Stress 4

We o not assume τ is uniform over the cross-section,because this is not the case.

τ is the average shear stress.

The maximum value of τ may be considerably greaterthan τ ave , which is important for design purposes.

Shear Stress Defined

A

F

A

P ave

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Shearing and Bearing Stress 5

Double Shear

Free Body Diagram of Bolt Free Body Diagram of Centerof Bolt

A

F

A

F

A

P

ave 22

Bolt is in “double” shear

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Shearing and Bearing Stress 6

Bearing Stress

Bearing stress is a normal stress , not a shearing stress.

Thus,

where

Ab = projected area where bearing pressure is appliedP = bearing force

td P

A P

b

b

Single shear case

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Shearing and Bearing Stress 7

Equilibrium of Shear Stresses Consider an infinitesimal element of material. Apply a single shearstress, τ 1.

Total shear force on surface is ( τ 1)bc .

For equilibrium in the y-direction, applyτ 1 on (-) surface.

For moment equilibrium about the z-axis,apply τ 2 on top and bottom surfaces.

Moment equilibrium equation about z-axis:

Thus, a shear stress must be balanced by three other stresses for theelement to be in equilibrium.

1

2

2

1

bacabc )()( 21

21

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Shearing and Bearing Stress 8

Equilibrium of Shear Stresses

1

2

2

1

Face Direction ShearStress

+ + +

+ - -

- - +

- + -

What does this tell us?Shear stresses on opposite (parallel) faces of an element are equal inmagnitude and opposite in direction.Shear stress on adjacent (perpendicular) faces of an element are equal inmagnitude and both point towards or away from each other .

Sign convention for shear stressesPositive face – normal is in (+) x, y, or z directionNegative face - normal is in (-) x, y, or z direction

+

++

+

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Shearing and Bearing Stress 9

σx = stress in x-direction applied in the plane normal to x-axisσ y = stress in y-direction applied in the plane normal to y-axisσ z = stress in z-direction applied in the plane normal to z-axisτ xy = stress in y-direction applied in the plane normal to x-axisτ xz = stress in z-direction applied in the plane normal to x-axisτ zy = stress in y-direction applied in the plane normal to z-axisAnd so on…

Define General State of Stress y

z x

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Shearing and Bearing Stress 10

There are 9 components of stress:σx, σ y, σ z, τ xy, τ xz, τ yx, τ yz, τ zx, τ zy

As shown previously, in order to maintain equilibrium:τ xy= τ yx, τ xz = τ zx, τ yz = τ zy

There are only 6 independent stress components.

Define General State of Stress y

z x

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Shearing and Bearing Stress 11

Example Problem

A load P = 10 kips is applied to a rod supported as shownby a plate with a 0.6 in. diameter hole. Determine the shearstress in the rod and the plate.

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Shearing and Bearing Stress 12

Example Problem

Link AB is used to support the end of a horizontal beam. If link AB issubject to a 10 kips compressive force determine the normal andbearing stress in the link and the shear stress in each of the pins.

ind

int

inb

1

41

2

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Shearing and Bearing Stress 13

Allowable Dimension or Load

Given dimension allowable load/stress: lower load/stress value to beconsidered for a safe design

Given load/stress allowable dimension: larger dimension (cross-section)value to be considered

Example:1. A structure design uses wood and steel rods with the same diameters.

The allowable stress of wood and steel are 70 MPa and 140 MPa,respectively. Consider a load of 1000 N for these rods, what should bethe designed diameter?

2. Given diameter and allowable stresses maximum load?

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14

Key Concepts

• Shear stress

• Bearing stress

•

Components of stress• Allowable stress

Next reading: 1.11-1.13