over sheet and Problem nstructions M 33 FNL M FLL SMSTR 0 Time allowed: hours. There are 4 problems, each problem is of equal value. The first problem consists of three smaller sub-problems. egin each problem in the space provided on the eamination sheets. f additional space is required, use the ellow paper provided. Work on one side of each sheet onl, with onl one problem on a sheet. 3. To obtain maimum credit for a problem, ou must present our solution clearl. ccordingl: a. dentif coordinate sstems b. Sketch free bod diagrams c. State units eplicitl d. larif our approach to the problem including assumptions 4. f our solution cannot be followed, it will be assumed that it is in error.. When handing in the test, make sure to place it under the name of our instructor and separate them b problem number. Prob. Prob. Prob. 3 Prob. 4 Total
quation Sheet Hooke s law: ε (Lf - L i )/L ε ε ε γ G i [ ν ( + )] [ ν ( + )] [ ν ( + )] γ G + α T + α T + α T γ G ( νε ) + νε ( + ε) ( + ν)( ν) ( νε ) + νε ( + ε) ( + ν)( ν) ( νε ) + νε ( + ε) ( + ν)( ν) Stress transformation and Mohr s circle: + ' cos θ ' + + + ' cos θ ' ' ' sin θ ' + cos θ ' p avg R p avg R ma R s s ial deformation, thermal epansion: FL e + Lα T F ( e Lα T ) L F K( e Lα T ) e ucos( θ) + vsin( θ) Torsion: φ Gr L Tr p TL φ G p G p K L T Kφ p _ ircular _ ross _ Section p _ Hollow_ ircular _ ross _ Section 4 π d 3 π d d 3 ( ) M( ) (, ) ρ() 4 4 ( o i ) Stress due to bending moment: 3 bh (rectangle) 4 d π (circle) 64 avg R + + Stress due to shear force: V ( ) Q( ) (, ) b Q( ) η d ' ' '
quation Sheet Failure criteria, factor of safet: von Mises quivalent Stress: + + ( ) ( ) ( ) M p p p p p p 3 3 von Mises Stress (Plane Stress): M p p p p + Failure Stress Yield Strength FS, llowable Stress State of Stress uckling: ritical buckling load for a pinnedpinned beam π L Pcr ritical buckling for fied-fied beam 3V ( ) (rectangle) ( ) ma N.. Stress in pressure vessels: pr spherical t ; pr h t ; pr a t 4V ma (circle) N.. 3 ntegration rules for discontinuit and Macaula functions n 0 for < a a n 0,,,3 n ( a) for a n a d n + v'' M ( v'')' V ( v'')'' p n+ a for n n+ 0 a for n > 0 nerg methods: Strain energ: M F T fv s Utot d + d + d + d Gp G where f s is the shape factor astigliano s theorem for deflection at a point in the direction of the force U where P is the point force astigliano s theorem for slope θ or angle of twist φ in the direction of an applied U moment M or applied torque T is given b tot U θ tot or φ M T Work-energ theorem states that the deflection at the point of application of a force P in the direction of applied force can be calculated b equating P U tot Work-energ theorem states that the slope θ or angle of twist φ in the direction of an applied moment M or applied torque T or can be calculated b equating Mθ Utot or Tφ U tot tot P
over sheet and Problem Problem. (8 Points, short problem) For the beam loaded as shown below, choose the correct shear force and bending moment diagrams from below (ou need to correctl select both the correct shear and the correct moment diagram). The diagrams on the left do not correspond to the diagrams on the right! R 0N w 0 40N/m M 0 N.m 0.m 0.m R 0N () () () (D) () 0 0-0 -0 0 0-0 -0 0 0-0 -0 0 0-0 -0 0 0-0 -0 (F) V() V() V() V() V() Shear Force None of the above ending moment () M() 0. (m). -. 0. (m) - () M() 0. (m). -. 0. (m) - () M() 0.. (m) -. 0. (m) - (D) M() 0.. (m) -. 0. (m) - () M(). 0. (m) (m) 0. -. - (F) None of the above
over sheet and Problem Problem. (8 Points, short problem) Match the stress element in the left column with the correct Mohr s circle on the right T-section beam with internal moment at a section Y Z Thin spherical vessel filled with fluid at gauge pressure p s M (0,0) (0,0) s s s Rod subject to end torque and aial force θ (0,0) T F Thin clindrical vessel filled with fluid at gauge pressure p a (aial) D h (hoop) D (0,0) Mohr s circle is a point! 4
over sheet and Problem Problem.3 (9 Points, short problem) solid clindrical rod of length L, shear modulus G, and polar area moment p is clamped between two walls as shown. known torque T 0 is applied as shown. State whether the following statements are true or false. L/4 3L/4 () () T 0 ircle either True or False for each of the statements,,, D below.. Reaction torques at and must be equal in magnitude True False. ( ) 3 ( ) (magnitude of ma. shear stress in () is three times larger than ma ma ()) True False. ( ) 3 ( ) (magnitude of ma. shear stress in () is three times larger than ma ma ()) True False D. φ φ (magnitudes of twist angles in () and () are the same) True False. T T (internal torques in () and () are identical) True False 6
Problem ( Points) The internal resultants of a particular cross section of a structure are: - ial force in + direction of magnitude 3000 lb - Shear force in direction of magnitude 900 lb - ending moment in + direction of magnitude 00 lb-in - ending moment in + direction of magnitude 700 lb-in 700 lb.in 00 lb.in 900 lb 3000 lb a. learl sketch the stress distribution on the cross section that results from each internal resultant on the diagram below. f applicable, indicate the neutral ais and maimum and minimum stress location. Shear Force in - ial Force in + ending Moment in + ending Moment in + ross-sectional rea: in in b. Determine the state of stress at points,, and and sketch it on the stress elements below. ndicate stress magnitudes on the stress element. State of State of State of Stress at : Stress at : Stress at :
Problem 3 ( Points) The slender L-shaped beam consisting of two members () and () is welded to the wall (cantilevered) and is subject to a vertical force P as shown. Neglecting gravit and the contribution of shear strain to the total strain energ, derive an epression for the downward deflection at the point the force is applied at. The final answer for needs to have all the integrals worked out and the answer should be given in terms of the applied load P as well as,, L, and,, L, which are respectivel the Young s modulus, rea moment, length, and cross sectional areas of the two members. You can either use the work-energ theorem or astigliano s theorem.,, L, (),, L, () P
Problem 4 ( Points) The figure shows a double-walled carbon nanotube (DWNT) that consists of two concentric tubes with eternal diameters d and d with the same wall thickness, t. The dimensions, Young modulus and thermal epansion coefficient are L00 0-9 m, t0.0 0-9 m, d.0 0-9 m, d. 0-9 m, 0 Pa and α.6 0-6 /. The double-walled carbon nanotube has rigid stoppers at the ends (fied-fied boundar conditions). ach tube can be regarded as a thin beam with hollow cross section with the following areas and area moments of inertia (subscript is for inner tube, subscript is for outer tube): 4 (d (d t) ) 0.30 0 8 m 4 (d (d t) ) 0.8 0 8 m 64 (d 4 (d t) 4 ) 0. 0 36 m 4 64 (d 4 (d t) 4 ) 0.030 0 36 m 4 (a) alculate the force P applied to these stoppers that prevents elongation of the tubes when the temperature is increased b ΔT00. (b) Under the conditions of (a), calculate the (normal) aial stress in each tube. (c) Under the conditions of (a), check if the sstem would fail b buckling. Which tube would buckle first?