University of Illinois, Urbana Champaign
TAM 251
TAM 251 Worksheet 12
Objectives:
• Determine resultant forces for a three-dimensional loading scenario
• Construct Mohr’s circle and determine principal stresses
• Use the Tresca yield criterion to predict the failure stress
Introduction:
A lug wrench is firmly a!xed to a lug nut. A torque T is require
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TAM 251 Worksheet 12
Objectives:
• Determine resultant forces for a three-dimensional loading scenario
• Construct Mohr’s circle and determine principal stresses
• Use the Tresca yield criterion to predict the failure stress
Introduction:
A lug wrench is firmly a!xed to a lug nut. A torque T is required to loosen the nut. You decide
to step on the end of the wrench in order to turn it. Assume that the force provided by your foot
can be modeled as a concentrated force with magnitude F. We also assume that the force vector
lies within the vertically-oriented yz-plane and makes an angle ✓ with the y-axis. Note that the lug
wrench’s coordinate system has its origin at the center of the wrench, the nut is located at position
x = !L and the force is applied at position y = !L.
Sidharth Asur
John Cantieri
Serenity Day
Mehul Shukla
a) Determine the magnitude of the force F required to tighten the nut for an arbitrary angle ✓.
b) The force F determined in part (a) is applied to the lug wrench. Calculate the resultant forces
and moments acting on the cross section B, which is at location x = !L. Dimension values are
Parameters
L 250 mm Length
✓ 60 deg Force angle
F ? N Applied force
T 40 Nm Required nut torque
do 20 mm Outer diameter
di 16 mm Inner diameter
.9914m Negative b/c they are
Mz = Fcoso ☐ L = -23.09 Nm Resultant moments
Mx =
Fsino • L = -39.9914m
c) Determine the stress state of a di↵erential element at point K, which is located on cross section
B from part (b), as shown below. The tube has outer diameter do and inner diameter di. Recall
the first area moment of a semi-circle: Q = 12 1 d3.
= ÷¥.oii" = 43.12
d) Use Mohr’s circle to determine the principal stresses for the stress element at point K. Sketch
the principal stresses on a properly oriented stress element.
e) Determine the maximum shear stress at point K. Sketch this stress (and any other relevant
stresses) on a properly oriented stress element.
4
0-1=19 MPa
0-2=-105.4 MPa
Txy = 44.766
§
"
" " """
0×1 = 0
Jang = -43.12
Op
• • >
⑤+
• Op : Etan" /¥o*) • 66°
-0-2=-105.4 Gang = -43.12 01--19
0s = 22°
• ✗
✓+I
Imax = 0
,
-Oaug = 19 - C-43.12) = 62.12hPa
f) There are many theories which predict the failure of materials. The Tresca yield criterion, a.k.a.
the maximum shear stress theory, predicts that materials will fail when their maximum shear stress
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