UNIVERSITY OF CLIFORNI, ERKELEY FLL SEMESTER 001 FINL EXMINTION (CE130- Mechanics of Materials) Problem 1: (15 points) pinned -bar structure is shown in Figure 1. There is an external force, W = 5000N, acting on the point C. (1) find internal axial forces for bar C, C; () find the normal stresses in bar C and C ( 1 = = 0.01 m, and E = 00 MPa); (3) find the vertical displacement at nodal point C. 01 θ 1 = o 60 θ 01 = 30 o L = 1 m C W = 5000 N Figure 1: Schematic illustration of problem 1 (Hint: (1) Statics and equilibrium equations; () σ = P ; (3) use Castigliano s second theorem, v = U, or the energy method to find the vertical displacement at point C. The elastic potential energy in a single bar is: U = P L W E, where P is the internal axial force, L is the length of the bar, E is the Young s modulus, and is the cross section of the bar, and W e = 1 W v. ) 1
Problem (15 points) three-bar system as shown in Figure. The external force F is acting at the point C, i.e. the interface between the nd bar and the third bar. The system is statically indeterminate of rank one. (1) Find the reaction force at point and, i.e. R and R ; () Find the displacement at point C. (Hint: use Castigliano s second theorem, or superposition method. The elastic potential energy in a single bar is: U = P L, where P is the internal axial E force, L is the length of the bar, E is the Young s modulus, and is the cross section of the bar. ) R 0011 L,, E C F 0011 R L,, E L,, E Figure : Schematic illustration of problem
Problem 3 (15 points) The Cauchy stress tensor at a point is given in a matrix form (D problem) as follows ( ) 8 3 σ = (MPa) 3 4 a. find principal stresses σ 1, σ ; 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 σ 1 acting upon, and show the results on properly oriented element. (Hint: σ 1 = σ x + σ y (σx σ ) y + + τ xy σ = σ x + σ (σx y σ ) y + τ xy τ xy (σ x σ y )/ tanθ p = ) 3
Problem 4 (10 points) n infinitesimal triangular element is shown in the Figure 3. Let σ x = σ (1) σ y = 3σ () τ xy = τ yx = τ (3) and θ = 30 o. Find the normal stress σ θ and the shear stress τ θ on the inclined plane in terms of σ and τ. t n σ X = σ τ θ τ θ σ θ θ d σ Y = 3 σ τ Figure 3: n infinitesimal triangle element 4
Problem 5 (10 points) n over-hanging simply supported beam subjected a concentrated moment, M 0 = 100.0 N-m, and a distributed load q 0 = 100 N, shown in Figure 4. m Draw shear and moment diagrams. M 0 q 0 0011 0011 0011 0011 1 m m 1 m Figure 4: Simply supported beam with distributed load and concentrated moment. 5
Y T Z L X Y P Z b a Figure 5: Illustration of Problem 6 Problem 6 (15 points) cantilever beam is made of a hollow cylinder with inner radius a = 0.1m and outer radius b = 0.m. The span of the beam L =.0 m. There is a concentrated load, P = 1000N, and a concentrated torque, T = 500 N-m, acting on the freeend (as shown in Figure 5.). (1) Draw the moment diagram, shear diagram, and internal torque diagram; () Find the normal stress σ x, shear stresses τ xy and τ xz at point ; (3) Find the normal stress σ x and shear stress τ xy at point ; Hints: (1) 1. The flexural formula σ x = M zy I z ;. The torsion formula τ = T ρ I p ; 3. The shear formula τ = V (x)q(y) I z t () For circular cross section I p = πr4, where R is the radius of the circle (4) I z = 1 I p (5) (3) For half annular cross section area (see Figure 5.) Q(y) = 3 (b3 a 3 ) = 3 (b a)(b + ba + a ) (6) 6
Y, v P 00000 11111 x v(x) H 000111 P X Figure 6: Problem 6 Problem 7 (10 points) Using the neutral equilibrium condition, δ Π = 0, determine the approximated critical buckling load P cr for a simply supported (pinned) elastic column of constant EI. ssume that the deflected shape for a slightly bent column in a neighboring equilibrium position can be approximated as v(x) = x(x L), where is the amplitude of the buckling mode, or a free variable controlling deflection. Compare your approximated result with the exact solution P cr = π EI L. (Hint: (1) Π = U W e ; () External work W e = P v, and v = 1 L 0 ( dv ) dx dx The moment of the cantilever Euler column is M(x) = P v(x) under the chosen coordinate system, and U = 1 L M (x)dx EI ) 0 7
E, I L E, I L C M 0 Figure 7: Problem 7 Problem 8 (10 points) planar frame C is subjected a concentrated moment, M 0, at point C as shown in Figure 7. (a) Draw moment diagram; (b) Find the cross section rotation at point C, i.e. θ C? (Hint: (1) use virtual force method, 1 θ C = where s is a local coordinate; or () use Castigliano s second theorem ) m c (s) M(s) EI ds θ C = U M 0 8