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Unformatted text preview: Midterm Exam ME572 (Prof. Pan) Name
3:40PM  5:10 PM, October 23, 2006 A formula sheet is allowed. Box your answers and include units. 1. We consider a tensile bar under uniaxial cyclic loading conditions. Consider a
material element on the bar surface. The coordinate system of the material element is
shown. The material element is subjected to constant amplitude loading conditions from the minimum normal stress of O to the maximum normal stress 0m in the x1 direction as shown. Consider a damage plane for the material element with the normal
making an angle of g1) with respect to the x1 axis. A damage parameter D for this plane can be deﬁned as a linear combination of the shear stress range Arm to the plane and the maximum normal stress (0' rm )max to the plane in a cycle as
D = ATM + k(0',m)max where k is determined as 0.4 to ﬁt the experimental fatigue life data. The shear stress
range ATM is deﬁned as the difference between the maximum shear stress and the minimum shear stress in a cycle. (a) Determine the value of (I) for the maximum value of D based on the stress
transformation rule (or Mohr’s circle). This value of ¢ should deﬁne the damage
plane orientation under constant amplitude cyclic loading conditions. (b) In experiments, however, small shallow surface cracks are usually observed near
the direction where the maximum shear stress range occurs. Determine the value of ¢ where the shear stress range ATM is maximum using the stress transformation rule (or Mohr’s circle).
X2.
S‘l' Yes 5 2. Consider a fcc single crystal material element with the coordinate axes parallel to the
axes of the unit cell. The material element is subject to combined normal and shear
stresses as shown. The tensile normal stress has a magnitude of 100 MPa and the
shear stress has a magnitude of 100 MPa as shown. We consider the (1 1 0) slip plane and two possible slip directions. The slip direction sA is in the direction of [1 l l]
and the slip direction SE is in the direction of [1 1 l]. (a) Determine the traction on the (l 1 0) plane.
(b) Determine the magnitude of the normal stress on the (1 l 0) plane. (0) Determine the shear stress on the (1 1 0) plane as a vector. (d) Determine the magnitude of the resolved shear stress in the sA or [—1 1 1]
direction on the (l l 0) plane as shown. (e) Determine the magnitude of the resolved shear stress in the 5B or [1 1 1]
direction on the (l 1 0) plane as shown. (f) Which direction of the slip will be ﬁrst activated if the combined normal and shear
stresses increases proportionally? l 3. A thinwalled cylindrical pressure vessel has the mean radius r of 100 mm and the wall thickness t of 10 mm. The vessel is under the internal pressure p and a torque T .1 Two strain gages A and B are located on the outer wall surface in the middle part
of the vessel as shown. For the material element near the strain gages, the local  coordinate parallel to the longitudinal direction is denoted as x , the coordinate in the circumferential direction is denoted as y , and the coordinate in the radial or outof plane direction is denoted as z. The strain gages are used to measure the normal
strains in the x direction and the direction making 45° with respect to the x direction
as shown, respectively. The measured strains are used to infer the pressure p and the torque T . The strain gage A gives a tensile strain of 100 y and the strain gage B
gives a tensile strain of 400 y. Note that Young’s modulus E is 210 GPa and Poisson’s ratio v is 0.3 for the vessel material. Use the formulae based on the thin—
walled pressure vessel assumption. (a) Determine the pressure p inside the vessel, (b) Determine the torque T , (c) Determine the principal stresses. Make a sketch of a material element and indicate
the principal directions, (d) Determine the maximum shear stress. Make a sketch of a material element and
indicate the planes of the maximum shear stress. ‘ “P MP0... /
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lm1+ML 10~ PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN PHILIP PARK CIVIL 8: ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN ﬁ @, Midterm Exam
3:40PM — 5:10 PM, October 23, 2006 ME572 (Prof. Pan) Name , A formula sheet is allowed. Box your answers and include units. 1. We consider a tensile bar under uniaxial cyclic loading conditions. Consider a material element on the bar surface. The coordinate system of the material element is
shown. The material element is subjected to constant amPlitude loading conditions from the minimum normal stress of 0 to the maximum normal stress am in the x1 direction as shown. Consider a damage plane for the material element with the normal
making an angle of ¢ with respect to the x1 axis. A damage parameter D for this plane can be deﬁned as a linear combination of the shear stress range AT,u to the
plane and the maximum normal stress (0,”, )1m to the plane in a cycle as D_ = ATM + k(o‘,m )m _ r where k is determined» to ﬁt the experimental fatigue life data. The shear stress range A1," is deﬁned as the difference between the maximum shear stress and the
minimum shear stress in a cycle. (a) Determine the value of q) for the maximum value of D based on the stress
transformation rule (or Mohr’s circle). This value of (I) should deﬁne the damage '
plane orientation under constant amplitude cyclic loading conditions. ' (b) In experiments, however, small shallow surface cracks are usually observed near
the direction Where the maximum shear stress range occurs. Determine the value of ¢ where the shear stress range A1,“ is maximum using the stress transformation rule (or
Mohr’s circle). ' x2. 2. Consider a fee single crystal material element with the coordinate axes parallel to the
' axes of the unit cell. The material element is subject to combined normal and shear
stresses as shown. The tensile normal stress has a magnitude of 100 MPa and the
shear stress has a magnitude of 100 MPa as shown. We consider the (1 1 O) slip plane and two possible slip directions. The slip direction sA is in the direction of [1 1 1]
and the slip direction sB is in the direction of [1 1 1]. (a) Determine the traction on the (1 1 0) plane.
(b) Determine the magnitude of the normal stress on the (1 1 0) plane.
(c) Determine the shear stress on the (1 1. 0) plane as .a vector. (d) Determine the magnitude of the resolved shear stress in the sA or [1 1 1]
direction on the (l 1 0) plane as shown. (e) Determine the magnitude of the resolved shear stress in the 5B or [1 1 1]
direction on the (1 l 0) plane as shown. (t) Which direction of the slip will be ﬁrst activated if the combined normal and shear
messes increases proportionally? )) 3..A thin—walled cylindrical pressure vessel has the (mean radius r of 100 mm and the wall thickness t of 10 mm. The vessel is under the internal pressure p and a torque T .' Two strain gages A and B are located on the outer wall surface in the middle part
of the vessel as shown. For the material element near the strain gages, the local  coordinate parallel to the longitudinal direction is denoted as x , the coordinate in the circumferential direction is denoted as y ,’ and the coordinate in the radial or outof— plane direction is denoted as 2:. The strain gages are used to measure the normal
strains in the x direction and the direction making 45° with respect to the x direction
as shown, respectively. The measured strains are used to infer the pressure p and the torque T. The strain gage A gives a tensile strain of 100 p and the strain gage B
gives a tensile strain of 400 ,u. Note that Young’s modulus E is 210 GPa and Poisson’s ratio V is 0.3 for the vessel material. Use’the formulae based on the thin
walled pressure vessel assumption. (a) Determine the pressure p inside the vessel, (b) Determine the torque T , .
(c) Determine the principal stresses. Make a sketch of a material element and indicate
the principal directions, ' ' ' ((1) Determine the maximum shear stress. Make a sketch of a material element and
indicate the planes of the maximum shear stress. _ ' at) = 0~ on“
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This note was uploaded on 10/13/2009 for the course ME 572 taught by Professor Pan during the Spring '07 term at University of MichiganDearborn.
 Spring '07
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