# ME Final Exam. PROBLEM NO. 4 Part A (2 points max.) M (x) y. z (neutral axis) beam cross-sec+on. 20 kip ft. 0.2 ft. 10 ft. 0.1 ft.

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1 ME Final Exam Name December 15, 2015 Instructor (circle) PROEM NO. 4 Part A (2 points max.) Krousgrill 11:30AM-12:20PM Ghosh 2:30-3:20PM Gonzalez 12:30-1:20PM Zhao 4:30-5:20PM M (x) y 20 kip ft 0.2 ft 10 ft x z (neutral axis) 0.1 ft 30 kip ft 3.5 ft 5 ft beam cross-sec+on The longitudinal axis of a beam is aligned with the x axis. The beam has a triangular cross-section, as shown above. The loading on the beam produces the bending moment diagram for M(x) shown above. a) Provide the x-y coordinates of the point on the beam that experiences the largest magnitude compressive normal stress. b) Provide the x-y coordinates of the point on the beam that experiences the largest magnitude tensile normal stress. Page 1 of 6

2 ME Final Exam Name December 15, 2015 Instructor (circle) PROEM NO. 4 Part D (6 points max.) Krousgrill 11:30AM-12:20PM Ghosh 2:30-3:20PM Gonzalez 12:30-1:20PM Zhao 4:30-5:20PM H y (3) x (1) θ P (2) 2 C D The truss shown above is made up of truss elements (1), (2) and (3). A horizontal force P is applied to joint C. (i) Draw a free body diagram (FD) for joint C. (ii) et σ 3 be the axial stress in element (3). Circle the correct description of σ 3 below (consider your FD from above): a) σ 3 > 0 (tension) b) σ 3 = 0 c) σ 3 < 0 (compression) (iii) et v C be the vertical component of displacement of joint C. Circle the correct description of v C below: a) v C > 0 (UP) b) v C = 0 c) v C < 0 (DOWN ) Page 4 of 6

3 ME Final Exam Name December 15, 2015 PROEM NO. 4 Part E (6 points max.) Instructor (circle) Krousgrill 11:30AM-12:20PM Ghosh 2:30-3:20PM Gonzalez 12:30-1:20PM Zhao 4:30-5:20PM Rod\$I\$ (1) 2A C A (2) D Rod\$II\$ (1) 2A C A (2) D i) Rod I shown above is made up of a material with a Young s modulus of E and thermal expansion coefficient α. The cross-sectional areas of elements (1) and (2) are given by 2A and A, respectively. oth elements are heated in such a way that each has a temperature increase of ΔT. et σ 1 and σ 2 represent the stress in elements (1) and (2), respectively. Circle the correct description below of these two stresses: a. σ 1 > σ 2 b. σ 1 = σ 2 c. σ 1 < σ 2 ii) Rod II is exactly the same as Rod I, except its right end is attached to a rigid wall. Again, both elements are heated to the same temperature increase ΔT. Circle the correct description below of the stresses in the two elements: a. σ 1 > σ 2 b. σ 1 = σ 2 c. σ 1 < σ 2 Page 5 of 6

4 ME Final Exam Name December 15, 2015 PROEM NO. 4 Part F (3 points max.) Instructor (circle) Krousgrill 11:30AM-12:20PM Ghosh 2:30-3:20PM Gonzalez 12:30-1:20PM Zhao 4:30-5:20PM rigid cap of weight W t g r A thin-walled tank (having an inner radius of r and wall thickness t) constructed of a ductile material contains a gas with a pressure of p. A rigid cap of weight W = 3π pr 2 rests on top of the tank. Ignore the weight of the tank. Determine the principal components of stress in the wall of the tank. Page 6 of 6

5 ME 323 Final examination Name (Print) (ast) (First) December 14, 2016 Instructor Problem 1 (28 points): The member shown below is fixed to ground at end C, is supported by a roller at end A, and is subject to a distributed load. It is known that the member has a uniform cross section area!, second area moment!, and a shear shape factor!!. The member is composed of a material whose Young s modulus is! and shear modulus!. Using Castigliano s theorem, determine the reaction on the beam at A. Please include the shear, normal and flexural strain energies in your solution. eave your answer in terms of!,!,!,!,!,!,!, and!!.

6 ME 323 Final examination December 14, 2016 Name (Print) (ast) (First) Instructor Problem 2 (28 points): A composite shaft of length! is composed of an inner shaft (1) and an outer shaft (2). The shaft is fixed to a wall at A, and to a rigid connector at which ensures the same axial and torsional deformation for the two shafts. A torque! is applied to the rigid connector. Simultaneously, the temperature of shaft (1) is increased from the ambient by Δ!, while shaft (2) is maintained at ambient temperature. The load and material properties are the following:! = 1000!",! = 1!,! = 50!"#,! = 100!"!,! = 20!!,! = 5 10!! / C, while Δ! is unknown, and is desired to be determined. a) Using an appropriate FD, write down the equilibrium conditions that relate the reactions of the two shafts. b) Write down the compatibility conditions that relate the elongations and the rotations of shafts (1) and (2). c) Determine the location on the cross section of shafts (1) and (2) where both the axial and shear stresses are maximum. Also, find the magnitude of the stresses. eave your answers in terms of Δ!. d) Calculate the maximum rise in temperature Δ! (Celsius), such that the absolute maximum shear stress!!"#,!"# in either shaft does not exceed 50!"#. cut-away view Cross section! 3" \$,%,& (1) 2" # 2\$,3% (2) (1) (2) A

7 ME 323 Final examination December 14, 2016 Name (Print) (ast) (First) Instructor Problem 3 PART A (10 points): A rod is made up of two rectangular cross-section segments C and CD, with each segment having a length of and a constant depth dimension of the cross section d. Segment CD has a thickness of h, whereas segment C is tapered with its thickness going from 3h at to 2h at C. The segments are joined by a rigid connector C, and the rod is connected to fixed walls at ends and D. A load P acts at connector C. A finite element model of the rod using a single element per segment produces the following equilibrium equations:!! K u = F For this problem, you are asked to: a) determine the stiffness matrix K and the loading vector enforcement of the boundary conditions for the problem.! F after the b) determine the axial displacement of connector C using the finite element equilibrium equations in a) above. C D 3h P h d cross sec'on of rod

8 ME 323 Final examination December 14, 2016 Name (Print) (ast) (First) Instructor Problem 3 PART (10 points): A homogeneous beam (of length, having a square cross section and made up of a material with an elastic modulus of E) is fixed to a rigid wall at end and is supported by a roller at location C. A line load of constant magnitude p 0 acts along the length of the beam. Determine the shear and normal components of stress on the bottom side of the beam at point a. HINT: You may use the superposition method in the deflection analysis for this problem. p 0 a D h h C beam cross sec*on / 3 / 3 / 3

9 ME 323 Final examinationa December 14, 2016 Name (Print) (ast) (First) Instructor Problem 4 PART A (6 points): A steel cylindrical specimen is subjected to tension until ductile failure is observed. Circle the correct answer in the following statements: (a) The state of stress of a point on the top fiber of the specimen is represented by the following Mohr s circle: (i) (ii) (iii) (iv) (b) The state of stress of a point on the bottom fiber of the specimen is represented by the following Mohr s circle: (i) (ii) (iii) (iv) (c) efore / After necking occurs, a crack oriented at 0 / 45 / 90 from the (i) (ii) (iii) (iv) (v) direction of loading will develop.

10 ME 323 Final examinationa December 14, 2016 Name (Print) (ast) (First) Instructor Problem 4 PART (6 points): A concrete slab is subjected to bending until brittle failure is observed. As in most brittle materials, the ultimate compressive strength is larger than the ultimate tensile strength. M M Circle the correct answer in the following statements: (a) The state of stress of a point on the top face of the specimen is represented by the following Mohr s circle: (i) (ii) (iii) (iv) (b) The state of stress of a point on the bottom face of the specimen is represented by the following Mohr s circle: (i) (ii) (iii) (iv) (c) The first cracks develop on the top face / bottom face and are oriented vertically/ (i) (ii) (iii) obliquely / horizontally. (iv) (v)

11 ME 323 Final examinationa Name (Print) (ast) (First) December 14, 2016 Instructor Problem 4 PART C (8 points): M eam (i) Steel M eam (ii) Aluminum eams (i) and (ii) shown above are identical, except that beam (i) is made up of steel and beam (ii) is made up of aluminum. Note that!!"##\$!!"#\$%&#\$. et!!"#,(!) and!!"#,(!!) represent the maximum magnitude of flexural stress in beams (i) and (ii), respectively. Circle the correct relationship between these two stresses: a)!!"#,(!) >!!"#,(!!) b)!!"#,(!) =!!"#,(!!) c)!!"#,(!) <!!"#,(!!) et!!"#,(!) and!!"#,(!!) represent the maximum magnitude of the xy-component of shear stress in beams (i) and (ii), respectively. Circle the correct relationship between these two stresses: a)!!"#,(!) >!!"#,(!!) b)!!"#,(!) =!!"#,(!!) c)!!"#,(!) <!!"!,(!!) et!!"#,(!) and!!"#,(!!) represent the maximum magnitude of deflection in beams (i) and (ii), respectively. Circle the correct relationship between these two stresses: a)!!"#,(!) >!!"#,(!!) b)!!"#,(!) =!!"#,(!!) c)!!"#,(!) <!!"#,(!!) et!! (!) and!! (!!) represent the vertical reaction at in beams (i) and (ii), respectively. Circle the correct relationship between these two stresses: a)!! (!) >!! (!!) b)!! (!) =!! (!!) c)!! (!) <!! (!!)

12 ME 323 Final Examination August 6, 2015 Name PROEM NO points max. y d a P R x z 2P A circular cross-section shaft (of diameter d and length ) is built into a fixed wall at end. A rigid pulley is attached to the other end of the shaft. A cable (not shown) wrapped around the pulley applies a pair of tension forces of P and 2P on the pulley, as shown in the figure (note that these two forces both act in the positive z-direction). et E and G represent the Young s modulus and shear modulus, respectively, for the ductile material of the shaft. Also, σ Y is the yield strength of the material. a) Determine the state of stress at point a on the shaft (point a is at end on the z-axis). b) If the shaft is to fail at point a, determine the value for P at which failure will occur. Use the maximum distortional energy theory for failure prediction. Use / R = 3in your calculations. Page 3 of 16

13 ME 323 Final Examination August 6, 2015 Name PROEM NO points max. P D y C x A thin, -shaped bracket is rigidly attached to a fixed wall at. A concentrated force P is applied to end D of the bracket. The material of the bracket has a Young s modulus of E, a shear modulus of G, and the circular cross-section of the bracket has an area of A and second area moment I everywhere. Ignore the weight of the bracket. Use Castigliano s theorem to: a) Determine the vertical deflection of end D for this loading. b) Determine the horizontal deflection of end D for this loading Page 5 of 16

14 ME 323 Final Examination August 6, 2015 Name PROEM NO. 3 8 points max. y D ( 1) 0.5 C ( 2) x A truss is made up of members (1) and (2) as shown above, with each member being composed of a material with a Young s modulus of E and thermal expansion coefficient of α, and having a crosssectional area of A. With the members being initially unstressed and unstrained, the temperature of (1) is increased by an amount of ΔT and the temperature of (2) is decreased by an amount of 2ΔT. a) Using equations, show that members (1) and (2) remain unstressed after the temperature change described above. b) What is the strain in each member after the temperature change?! Page 7 of 16

15 ME 323 Final Examination August 6, 2015 Name PROEM NO points max. 2 / 3 / 3 d d T (1) C (2) D The shaft shown above is made up of solid components (1) and (2), with each component having the same diameter d. The components are joined by the rigid connector C as shown above, with a torque T being applied to connector C. a) If the two components are made up of the same material (i.e., having the same shear modulus, G 1 = G 2 = G ) what is the value of T 1 / T 2 where T 1 and T 2 are the torques carried by components (1) and (2), respectively? b) If both components are replaced with a material having half of the original shear modulus (i.e., G 1 = G 2 = 0.5G ), what is the new value of T 1 / T 2? c) If, instead, the material for component (1) is kept the same as in (a) (G 1 = G ) and the shear modulus for component (2) is changed to G 2 = 0.5G, what is the new value of T 1 / T 2? Page 8 of 16

16 ME 323 Final Examination July 2, 2015 Name PROEM NO. 7 8 points max. y σ y x axis τ xy 60 σ x x 20 MPa σ τ 30 MPa The Mohr s circle for a stress state is presented above. a) Show the location of the y-axis in the Mohr s circle above. b) Determine the principal stresses and the absolute maximum shear stress for this state. c) Determine the σ x,σ y and τ xy for this stress state. Page 11 of 16

17 ME 323 Final Examination July 2, 2015 Name PROEM NO. 8 8 points max. A cantilevered beam is loaded with concentrated couples and concentrated/distributed transverse forces. Although this loading is not shown below, a plot of the resulting internal bending moment M(x) is provided. Determine the reactions acting on the beam by the wall at its right end (at x = 10 ft ). Express your answers in terms of their xyz components. y x M (x) (kip ft) C x ( feet ) 50 linear quadratic quadratic linear linear Page 12 of 16

18 ME 323 Final Examination July 2, 2015 Name PROEM NO. 9 8 points max. p 0 C d D A structure is made up of a horizontal rigid bar C and a vertical elastic member CD, where CD has a circular cross section of diameter d and is made up of a material having a Young s modulus of E. A distributed loading having a transverse force/length of p 0 acts on bar C. Ignore the weight of the members. Determine the critical load value p 0 that corresponds to the buckling of member CD. Page 13 of 16

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