Stress at a Point and

Change in Failure Mode

Mark Walter and Jerome P. CaseEMA 307-004April 6, 2005

Combined Loading

Outline

• Stress at a Point – Normal and Shear Equations

• Experimental Stress Measurements– Rosette Strain Gauges

• Description of Lab

• Failure Theories– Rankine and Tresca

Combined Loading Case-Walter April 6, 20052

• Normal and Shear Equations– Sum forces and moments: normal, shear, bending,

torsion

Stress at a Point

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• Only Torsion– Assume no deflection

at point #1

Stress at a Point

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• Only bending – Tubular beam– No torsion

Stress at a Point

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Stress Measurements

• Rosette Strain Gauges– 3 elements: 0°, 45 °, 90°

from direction of force– Normal strain in any

other direction found by equation

– Micro-strain

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Stress Measurements

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• 2-D Hook’s Law for Plane Stress and Strain

• Rearrange strain equation to get shear strain

x

E ( )x

y

1 2y

E ( )y

x

1 2

x

x

y

Ey

y

x

E

xy

2 45

x

y

Mohr’s Circle• German engineer Otto Mohr (1835-1918)• Used to derive formulas relating to

transformation of plane stress and strain

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Mohr’s Circle (Stress)

• Normal stress (sigma) is x-axis• Shear stress (tau) is y-axis• Use experimental data and

equations • Find normal stress in x and y

directions and shear stress in xy plane

• Plot point (sigma x, - tau xy)• Plot point (sigma y, + tau xy)• Connect points with a straight line

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Mohr’s Circle (Stress)• Straight line is diameter of circle• Center is at intersection of line and sigma axis• Draw circle• Where circle intersects sigma axis are minimum and maximum normal stresses• Maximum shear stress is circle’s radius• Principle planes (orientation where max

normal stress occurs) can be calculated• Max shear stress planes 45 degrees to

principle planes

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Mohr’s Circle (Strain)• Normal strain (epsilon) is x-axis• One half of the shear strain (gamma) is the y-axis• Points now (epsilon x, -gamma xy/ 2) and (epsilon

y, gamma xy/ 2)• Max shear strain equals 2*radius• Find epsilon max and min just like sigma max and

min• For principle angle calculations use gamma (not

gamma/2)• Max shear strain planes are 45 degrees to max normal strain planes

11Combined Loading

Failure Theories

• For combined loading

• Rankine’s Theory (maximum normal stress theory)

• Tresca Theory (maximum shear stress theory)

• Theories can be seen using Mohr’s circle

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Failure Theories• Tensile specimen with circumferential groove can cause brittle

behavior in ductile material• Caused by change in max values of shear or normal stress• Groove produces triaxial state of tensile stress• Necking and deformation much smaller than on ungrooved specimen

due to reduced shear stress• Brittle failure now predicted by max normal stress criterion

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Description of Lab Tests

• Part one• 10, 20, 30, 40, and 50 pounds placed on hook

#1 to demonstrate pure bending• Next, a single load of 50 pounds placed on hook

#2 to demonstrate pure bending with torsion

Description of Lab Tests

• 3 element rosette strain gage attached at top of pipe at position A

• Using Wheatstone bridge strain is measured at 0, 45, and 90 degrees from the axis of the tube

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Description of Lab Tests

• Part two

• Compare loads and stresses

of grooved vs. ungrooved specimen

• Compare yield point of bar with clamp vs. without clamp

• Tests illustrate how combined loads affect failure type

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Conclusion• Physics• Examples• Combined

Loading means very complicated

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Gravity and wind loads subject post to

compression, bending and torsion.

“Nothing is as simple as the example problems.”

References(no date.) Lesson 9 – Combined Loading [Lab Manual]. ME/EMA 307:

Mechanics of Materials Lab. Retrieved April 2, 2005 from https://uwmad.courses.wisconsin.edu/d2l/

Jenkins, M. (Sept. 2002). PDF Chapter 12, Pressure Vessels: Combined Loading. Mechanics of Materials Laboratory Notes. University of Washington. Retrieved April 4, 2005 from http://courses.washington.edu/mengr354/jenkins/notes/chap12.pdf

Beer, F., Johnston, E., DeWolf J. (2001). Mechanics of Materials. McGraw-Hill. New York.

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