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Unformatted text preview: UNIVERSITY OF CALIFORNIA, BERKELEY College of Engineering Statics (E36) Final Examination Problem 1. (20 points) Draw the shear and moment diagrams for the beams shown in Figure l ; (A) we 2 ION/m, o, = 1.0m: (a1) Find the reactions: {a2} Find the expressions for 17(3) and Draw the shear diagram, and (a3)
Find the expressions for ﬁ/I(;r) and Draw the moment diagram ; (B) M =10 N—m and L := 5m.:
(131) Find the reactions, (b2) Find the expressions for V[z) and Draw the sheer diagram, and (b3)
Find the expressions for M and Draw the moment diagram . Hints: dV MM
2— r d _=V ‘
me), an dx (x) E A Milt. B O
7 (B) (b) Figure l: A simply supported beam with diﬁerent load conditions Problem 2. (15 points}
{1] F ind the centroid of the cross—section shown in Figure 2 and set up the centroids] axes. __ z giAi
y 2‘41 :2_. Find IE with respect to the global centroids} axes. W 'rlrlltl‘o
IAWIIIII’? \ \ ,._.
§ Y ?
e
41
t
Figure 2: The cross section of e. T—beein
Hints:
IS = / yZdA, Iy = / ﬁrm (1)
A A
Is 2 [me + (FA, 4: Parallel Axis Theorem (2)
M3
Lectmgugm = E <2 (The genetic formula for local centroidal axis) {3) Problem 3 (15 points)
A 2.4—mlong boom is held by a ball—end—socket joint at C and by two cables AD and 3E. An external load W = —880j is acting at point A. Determine the tension in each cable and the reaction
at the point C. Fig. 91.116 Figure 3: A three—dimensional structure (a) Draw free—body diegram for her AC;
(b) Write down the vector expressions for rA, r3, rep = 1—D — rm and ['35 = r3 — r3; (c) Find the forces in the vector form, W, TAD, and T55; ((1) Write the vector form moment equilibrium equation, 2M0 = Eric x Fl :0 and ﬁnd TAD and T33; (f) Write the vector form force equilibrium equation Zia=0 and ﬁnd 01, Cy, and Cz Hint: ,
[1 j k
er=[rx ry rz
in F3, Fz Problem 4. (20 points) Denoting by its the coefﬁcient of static fric— tion between the block attached to rod ACE and the horizontal surface, derive expressions in terms of P, as, and 6‘ for the largest and smallest magnitudes of the force Q for which equilibrium is maintained. (3.) Draw the free—body diagram for the whole system; {b} Find the ground support force As, and the ._ . friction force acting on block A; \ / {c} Find IA: 351;: and the virtual displacements
it dig and 6gp;
{d} VVIite down EU and let 5U = O to ﬁnd max mini
Fig. moss and 1910.51 Q an Q Figure 4: Friction and Virtual Work Method Problem 5. (15 points) A ﬁoor truss is loaded as shown. Determine the forces in members Fl, Hi3 and H3. (1] Find the reactions at the point A and the point K; ('2) Use method of section making a cut, draw the free—body diagram of the remaining sub—
SIructura and then solve for internal forces FF}, Fm? and PH; . 2501b 5709. lb s00 lb 3751!; 253:5 2501b r2511:
4f; mire ' ' r—i Figure 5; A Truss System with External Loads. Problem 6. (15 points) Bar AC is attached to a hinge at A and to
a spring at the point B. The spring constant
is k, and it is undeformed when the bar is
vertical. Find the range of values of P for
which the equilibrium of the system is stable
at shown the position 9 = 0. {a) Find me in terms of 6 and (I) and find the
relationship between 8 and qb when 9, a} << 1;
(b) Find 3:3 and yg and write down the po—
tential function for the system in terms of
8, P, k and a; (2) Show that 6 = O is an equilibrium position
by using the equilibrium condition dV _
a9 _ U ; (3] Find the range of values of P such that
dQV
(£92 the equilibrium at 6' = 0 is stable > 0. Figure 6: Equilibrium of a two—bar system. ...
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 Spring '07
 Lee
 Force, Friction, vector form, equilibrium equation, form moment equilibrium

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