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Unformatted text preview: UNIVERSITY OF CALIFORNIA, BERKELEY Coiiege of Engineering Mechanics of Materials (CElBD—I) Fall 2006 The First Mid—term Examination Problem 1. {25 points} Derive the equilibrium equation for a twodimensional inﬁnitesimal element; in the vertical (Y)
direction. Note iliat the thickness of the element (z—direction) is micen as i {unit length), and X,Y
are the body forces with the unit; of force per unit volume. Figure 1: A 2D inﬁnitesimal element Problem 2 (25 points} Consider the following twonbar system (Figure 2}. Tile fiexibilities of we eiastic bars are given
as f1 and f2, the lengths of the two bars are L1 and L2, and the thermal expansion coefﬁcients of
the two bars are given as a; and cm. The right end of the second bar has a distance A from the well. There is a temperature increase of AT. Find the reaction forces as the point A, i.e. RA, and
at the point C, i.e. R0, for two different cases: (1) A S {CriLi + GeLalAT;
(2)13 > ((11141 + Q2L3)AT . Hints: AP = fl” and AT a: GLAT. Figure 2: A possible statically Indeterminate System Problem 3 (25 points)
Two cylindrical shafts are made of different materials, i.e. shear modulus Ga 75 02, but have the
same length L. One solid cyiinder of diameter 2:: = d, and the other is a holiow cylinder with outer radius co = R and inner radius ci m 0.51%. They are subjected to the same external torque, T9, as
indicated in Figure 3. (I) If the maximum shear stress in botii shafts is the same, ﬁnd the reiationsiiip between c w R ? (2) If the angieoftwist at the free end (where the external torque is applied) (15 is the some for both
shafts, ﬁnd the relationship of d ~ R ? Hints:
. . mi“
For a solid cylinder : J = 73—2— (1)
For a hollow cyiinder : J = gum: — {1?}, d9 = 2cm :1, = 2c1 (2)
Tc TL
7mm: = “'3': = a}? {3} B
/\ / To
L
A
B
/\ In Figure 3: Torsion of two shafts Problem 4 {25 points)
Consider a very long ( 1000 meters in zdirection) concrete biock with its both ends ﬁxed. The cross section of the concrete block (section in xy plane) is a 5 meter square. Suppose that in x—y
plane, the block is subjected biaxial tensile stress load, namely, ox: = 5MP“ and orW = 10M Pa.
This is a typical plane strain state (Cu 2 0}. Let E = 10011an and Poisson’s ratio V = 0.3. Find
on, em, and amp Hints: The equations of the generaiized Hooke’s law are (73:1: 0511; 47:: = “E”— 73””?
.. ‘7“ 52m ‘72:
M “ ""5 E”?
o o (7.
(,z = 1,__ JEL+_::_
E E E ...
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 Spring '07
 Hutchings

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