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Unformatted text preview: UNIVERSITY OF CALIFORNIA, BERKELEY
Mechanics of Materials (CE130) Section I The Second Midterm Examination College of Engineering Problem 1. (a) Find the reaction forces at both ends; (b) Draw shear & moment diagrams for the following beams (see: Fig. 1 (a)(b)) and label the
peak values for the corresponding maximum shear and maximum moment, (30 points (15 each)) (a) (b) Figure l: Beams with external loads Problem 2. An Lbeam shown in Fig. 2 is made of three planks, which are connected by nails. Suppose that
each nail can sustain a shear force lOOON, Let t = 50mm and b : 500mm. Suppose that the beam
cross section is subjected to a shear force V : 101CN, Find the maximum nail spacing. VQ _§_VQ
Izt’ q"A‘ I! Q = fydAzAz;
A T = 12 2 In + dill parallel axis theorem
bha i
In = E for rectangular cross section (1)
(20 points)
Y
l:
l
I
Z an «r b L l l Figure 2: The I—beam, h
(A) (E) Figure 3: A T beam: (a) the geometry of the crossisection; (b) the stress~strain relation. ‘ Y,v /wa free and Figure 4: A beam with a concentrated moment. Problem 3.
A Tbeam shown in Fig. 3 (a) with b = .50an and IL : 200mm, which is made of linear elasticiperfectly plastic material with U’Tﬂlaf] = 100 MP5 (shown in Fig 4 Find: (ﬁf (306 M e) 1. The position of the elastic bending neutral axis, or the centroidal axis '?
2‘ Find Iz '?; 3. Which surface yields ﬁrst 7 and Find the yield moment, M'y 7 4‘ Find the neutral axis position for plastic bending (no elastic core) 7 5‘ Find the ultimate bending moment, Mult 7 (30 points) Problem 4. Consider the cantilever beam with span L = a + b. The beam is subjected with a Concentrated
moment at the position x z a in counterclock direction The ﬂexural rigidity of the beam is
E1 = const. (Recommend using the singularity function method). 4
Elddsz) = —w(a:) (2)
(20 points)
(I) What is the 10(1) '?
(a) 211(37) = M < z W (1 >0?
(1)) w(z):M<I—a>:1?
(c) w(x):1\/I<m—a>:27
((1) 111(1) : ill/I <1—a>:2?
(6) w(z)=71\/I<I—u>1? (2)
1. State the {our boundary conditions; 2, Find the beam deﬂection 11(1); 3‘ Find the beam deﬂection at z : 0, ie y(0) .
(3) Draw moment and shear diagrams i ...
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 Spring '07
 Hutchings

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