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practice_final - UNIVERSITY OF CALIFORNIA BERKELEY FALL...

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Unformatted text preview: UNIVERSITY OF CALIFORNIA, BERKELEY FALL SEMESTER 2002 FINAL EXAMINATION (CE130—I Mechanics of Materials) Problem 1: (15 points) A pin-jointed 3-bar structure is shown in the Figure 1. There is an external force, F, acting on the point C. All three bars are two—force members. (1) find the vertical displacement as well as horizontal displacement at nodal point C, (2) ﬁnd the critical load PM for this system. (Hint: use Castigliano’s second Figure 1: Schematic illustration of problem 1 P2L 2EA’ is the axial internal force, L is the length of the bar, E is the Young’s modulus, and A is the cross section of the bar. ) where P theorem, the strian energy for an axially deformed bar is, U = Problem 2 (15 points) A L-shaped beam is made of a rectangular section and a solid cylinder section with radius R = 0.1m. The span of the both section is L = 2.0 m. There is a concentrated load, P = 300N, acting on the free-end of the rectangular (as shown in Figure 7.). (1) Draw the moment diagram, shear diagram, and internal torque diagram; (2) Find the normal stress 09,, shear stresses Tmy and 7'“ at point A; (3) Find the normal stress am and shear stress sz and Tm; at point B; Hints: T=—, Iz=§Ip=— (1) Figure 2: Problem 2 For semi—circle Q(y) l = 3R3 (2) y=0 3 Problem 3 (15 points) The Cauchy stress tensor at a point is given in a matrix form (2D problem) as follows —2 —3 a" — < _3 —6 ) (MPa) a. ﬁnd principal stresses 01, 02; b. ﬁnd the maximum shear stress; c. draw Mohr’s circle; d. ﬁnd the angle between the right face of the initial inﬁnitesimal element and the plane on which principal stress 01 acting upon, and Show the results on properly oriented element. e. ﬁnd the angle between the right face of the initial inﬁnitesimal element and the plane on which the positive maximum stress 7' acting upon, and show the results on properly oriented element. (Hint: 01 = Uw+0y+ (022 )2+T\$2y 0'2 2 2+ (Ty: )/2 tan20 = ,tan2082 3 (am ay)/2 Hay H O'Y=— C Figure 3: An inﬁnitesimal triangle element (problem 4) Problem 4 (10 points) An inﬁnitesimal triangular element is shown in the Figure 3. Let 02:0, ay=—or, sz=Tyx=T, (4) and 0 = 60". Find the normal stress 09 and the shear stress 79 on the inclined plane in terms of a and 7'. Figure 4: Simply supported beam with distributed load and concentrated mo- ment (problem 5). Problem 5 (10 points) An over-hanging simply supported beam subjected a concentrated moment, P0 = 500.0 N, and a distributed load qo = —100 g, shown in Figure 4. Draw shear and moment diagrams. Problem 6 (10 points) Determine the critical buckling load PC, for a cantilever elastic column with span L and constant stiffness (rigidity) EI. Figure 5: Problem 6 (Hint: the ﬁrst root of transedental equation, tan()\L) = AL, (5) is AL 8 4.4934.) A 13,1 3 L E,I L Mo C Figure 6: Problem 7 Problem 7 (10 points) A planar frame ABC is subjected a concentrated moment, M0, at point C as shown in Figure 7. (a) draw moment diagram; (b) ﬁnd the horizontal displacement at point C, i.e. 12H '? (Hint: use Castigliano’s second theorem _ aU _ 8P’ P’=0 ”H where P’ is a fictitious force placing at point C. ) Problem 8. A T-beam shown in Fig. 7 (a) is made of linear elastic-perfectly plastic material (shown in Fig 7 (b)). Find: (a) the position of elastic neutral axis ? (b) yield moment Myp ? (c) neutral axis position for plastic bending (no elastic core) 7 (d) ultimate bending moment, Mutt ? (15 points) (A) (3) Figure 7: A T beam: (a) the geometry of the cross-section; (b) the stress—strain relation. ...
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