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Unformatted text preview: UNIVERSITY OF CALIFORNIA, BERKELEY
Mechanics of Materials (CE130) Section I College of Engineering The Second Mid—term Examination Problem 1.
Draw shear & moment diagrams for the following beams (see: Fig. 1 (a) and label the peak
values for the corresponding maximum shear and maximum moment. (30 points (15 each)) ' (I) (b) Figure 1‘. Beams with external loads Problem 2. An Ibeam shown in Fig. 2 is made of three planks, which are connected by nails. Suppose that
each nail can sustain a shear force 1000N. Let t 2 50mm and b = 500mm. Suppose that the beam
cross section is subjected to a shear force V = 5kN. Find the maximum nail spacing. _ VQ _ S _ VQ
T ‘ Izt’ q” A ' Iz
Q = / mm = Ag
A
I z = L,c + dgA parallel axis theorem
3
I u = % for rectangular cross section. (1) (20 points) Figure 2: The Ibeam. Problem 3. (A) (I) Figure 3: A T beam: (a) the geometry of the crosssection; (b) the stressstrain relation. Y,v free end Figure 4: A beam with concentrated moment. A Tbeam shown in Fig. 4 (a) with b = 20mm and h = 200mm, which is made of linear
elasticperfectly plastic material (shown in Fig 4 Find: 1. The position of the elastic nuetral axis ?
Find 11 ?;
Find the yield moment, My ? Find the neutral axis position for plastic bending (no elastic core) ? 9"???” Find the ultimate bending moment, Mun ? (30 points) Problem 4. Consider the cantilever beam with span L = a + b. The beam is subjected with a concentrated
moment at the position a: = a in downward direction. The ﬂexural rigidity of the beam is E1 =
const.. (Recommend using singularity function method). d4v(x) E1 (1554 =q(x) (2) (20 points)
(1) What is the q(:t) ? (a)
(b)
(C)
(d)
(e)
(2)
1. State the four boundary conditions; 2. Find the beam deﬂection v(:1:); 3. Find the beam deﬂection at a: = 0. UNIVERSITY OF CALIFORNIA, BERKELEY
Mechanics of Materials (CE130) Section 11 College of Engineering The Second Midterm Examination Problem 1.
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)) qo U2 LIZ
@) Figure 1: Beams with external loads Problem 2. An T—beam shown in Fig. 2 is made of two planks, which are connected by nails. Suppose that
each nail can sustain a shear force 1000N. Let t = 50mm and b = 500mm. Suppose that the beam
cross section is subjected to a shear force V = 5kN. Find the maximum nail spacing. T _ VQ _ S
‘ Lﬁ q—A
Q = / ydA = A2:
A
I, = I u + dZA parallel axis theorem
3
I zc = {ah—2 for rectangular cross section. (1)
(20 points)
I:
t
z t Figure 2: The Tbeam. Stress (MP8)
100 Strain —50 k
W Figure 3: A I beam: (a) the geometry of the crosssection; (b) the stressstrain relation. Y Figure 4: A beam with concentrated force. Problem 3.
A Ibeam shown in Fig. 4 (a) with t = 20mm and b 2 200mm, which is made of linear elastic— perfectly plastic material (shown in Fig 4 Find: 1. The position of the elastic nuetral axis ? 2. Find I z ?; 3. Find the yield moment, My ? 4. Find the neutral axis position for plastic bending (no elastic core) ? 5. Find the ultimate bending moment, Mun ? (30 points) Problem 4.
Consider the cantilever beam with span L = a + b. The beam is subjected with a concentrated load at position :1: = a in downward direction. The ﬂexural rigidity of the beam is E1 = const..
(Recommend using singularity function method).
d4v(:c) _ EI (13:4 —q(a:) (2) 2 (20 points)
(1) What is the q(a:) ? (a) (b) (C) (d) (e)
(2)
1. State the four boundary conditions;
2. Find the beam deﬂection v(a:); 3. Find the beam deﬂection at a: = 0. q(x)=P<za>o?
q(a:)=P<:z:—a>_1?
q(z)==—P<:1:—a>_1?
q(a:) —P<a:—a>_2?
q(:z:) —P<:c—a >1? ...
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 Spring '09
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