Page 1 of 46 Exam 1. Exam 1 Past Exam Problems without Solutions NAME: Given Formulae: Law of Cosines: C. Law of Sines:

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1 NAME: EXAM 1 PAST PROBLEMS WITHOUT SOLUTIONS 100 points Tuesday, September 26, 2017, 7pm to 9:30 You are allowed to use a calculator and drawing equipment, only. Formulae provided 2.5 hour time limit This massive compilation of problems represents about 7 years of past exams Given Formulae: Law of Cosines: C b a Law of Sines: A c B Formulae: Normal stress = F/A, where F is the normal force on the cut and A is the area of the cut. Shear stress avg = V/A, where V is the shear force on the cut and A is the area of the cut. Normal strain = L/L, where L is the change in length and L is the original length (AKA, gauge length) Shear strain xy = the change in angle of x-y, where x-y are initially perpendicular or xy = /2, where is the deformed angle between x-y, measured in radians and is obviously. Hooke s Law = E, where E = modulus of elasticity (AKA Elastic Modulus or Young s Modulus) Elastic Deformation of an Axially Loaded bar: Page 1 of 46 Exam 1

2 1. (20 points). Determine the force in members CD, CH, DH, GH. Clearly specify whether the force is compression (C) or tension (T). 1. (25 points). Determine the horizontal and vertical components of force that the linkage exerts on plate DEIJH at points D and E. Given: The lifting machine hoists a weight of 10 kn that has a center of mass at G. The lifting machine exerts only vertical reactions at K and L. Page 2 of 46 Exam 1

3 1. (12 points). Two pieces of wood are cut at the 30 angle shown, connected by a 3/8 diameter bolt, and subjected to the 1 kip force, shown. The cross-section of the wood is 2 x 4. The bolt is perpendicular to the cut-plane. Determine the shear and normal stress on the bolt cross-section (cut a-a). 1 kip a a kip 4 3/8 Diameter Bolt 1 kip 1 kip Final Answers Stresses on Bolt Cross-Section bolt = ksi bolt = ksi Page 3 of 46 Exam 1

4 1. (12 points). Determine the maximum force P that may be applied to the bolted connection without failure, considering: a. Failure of the bolts, in shear. b. Failure of a plate, in normal stress at holes (on net area) => Given: b. Plate Fracture at holes (6) ½ diameter bolts in 9/16 diameter holes: ult = 36 ksi, ult = 60 ksi (3) Plates. Each plate has a cross-section that is ¼ x 4, as shown and ult = 30 ksi, ult = 50 ksi P Typical Plate P ¼ Final Answer: Maximum P (kips) Pmax = kips Page 4 of 46 Exam 1

5 2. (25 points). Determine the internal normal force N, shear force V, and bending moment M at the midpoint of AB, indicating the signs, for each. Given: BE and CD are pin-ended members. A and F are pinned supports. Final Answers (circle the correct sign) N = lbs V = lbs T or C or Page 5 of 46 Exam 1 M = ft-lbs

6 1. (22 points). Given: allowable normal stress allow = 3ksi, allowable shear stress allow = 4ksi 1. Determine the minimum required pin diameter at pin C. 2. Determine the minimum required cross-section area for bar BC at point F. 3. Determine the internal N, V, M acting on the cross-section at D and indicate signs Detail of Pin C Answers: a. Minimum pin diameter: inches b. Minimum cross-section area: in 2 Signs c. N = V = M = Page 6 of 46 Exam 1

7 2. (10 points). A steel nail with a diameter of 0.15 was driven 1 into wood. Determine the force P that is needed to pull the nail out of the wood, then determine the normal stress that is in the nail when this force is applied Given: the maximum shear stress that the interface between wood and steel can resist is 60psi. the maximum normal stress that the interface between wood and steel can resist in tension is 0psi (this interface resists no tensile normal stress). P 1 Answer P = lbs in the nail = Page 7 of 46 Exam 1

8 3. (17 points) Two pieces of wood with a cross-section of 2 x2 are glued together at the 30º angle shown and subjected to the force P. The glue fails if the normal stress in the glue exceeds 1 ksi. Determine the maximum force P that can be applied without glue-failure o P 30º P Answer P = lbs Page 8 of 46 Exam 1

9 1. (3 points). What is the magnitude of internal shear force V on the cross-section through link BC? Answer: Internal V in Link BC: 2. (3 points). TRUE or FALSE. The external reactions for the beam below can be solved by Statics. Given: Support is fixed at A and has a roller support at B. w=1 kip/ft A B Page 9 of 46 Exam 1

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11 3. (25 points). Determine the internal normal force N, shear force V, and bending moment M on sections a-a and b-b. Indicate the sign of N as either C or T. Indicate the sign of V and M with the icons in the boxes below. Answer - Section a-a: N = lbs ( C or T?) V = lbs or? M = in-lbs or? Answer - Section b-b: N = lbs ( C or T?) V = lbs or? M = in-lbs or? Page 11 of 46 Exam 1

12 4. (12 points). Determine the minimum height h that is required to prevent the small ledge at the bottom of member A from breaking off vertically, in shear. Given: Member A and Member B are both 3 thick (i.e., the dimension, going into the page is 3 ). The contact between Member A and B is smooth. The maximum shear stress that the wood can resist is = 100 psi. Members A and B both have a thickness (dimension into the page) of lbs This small ledge will break off vertically, in shear if h is insufficient Answer: Minimum dimension h: inches Page 12 of 46 Exam 1

13 5. (15 points) Determine the internal normal force N, shear force V, and bending moment M for the cross-section at point E. A and C are pinned supports. B is a pinned connection between members AB and CB. Indicate the signs in the box provided. Answer for the cross-section at E: N = N ( C or T?) V = N or? M = N-m or? Page 13 of 46 Exam 1

14 1. (17 points) Determine the magnitude of internal bending moment that is present at the location of the 60 kip point load and specify whether this moment results in compression on the outside of the arch or on the inside of the arch (see below). Given: A and B are pinned supports. C is a hinge (pinned connection). Indicate Either: M M causes compression on the outside of the structure M M causes compression on the inside of the structure 20 ft A C B Page 14 of 46 Exam 1

15 2. (15 points) A 6 diameter solid wooden pole is embedded 40 inches into the ground and is subjected to a vertical force P that will pull the pole out of the ground. It is known that the average shear stress between the pole and the dirt will be 8 psi (lb/in 2 ) when the pole pulls out of the ground. The overall length of the pole is 180 inches. Determine the normal stress on the cross-section of the pole when it is pulled out and report the answer in psi (lb/in 2 ) Assume that the interface between the bottom end of the bar and the concrete does not support any tensile stress. P = Vertical Force that pulls out the pole P 6 Diameter Pole 40 The Ground 40 Embedment The Ground 3D View Side View Page 15 of 46 Exam 1

16 3. (18 points). Determine the largest distributed load w (units: N/m) that can be supported IF beam AB fails when the magnitude of internal moment at D reaches 800 N-m. Given: Pinned supports at A and C. Pinned connection at B. Page 16 of 46 Exam 1

17 1. (20 points) Determine the magnitude of the internal moment M at point F and specify the sign of the internal moment by circling the appropriate symbol below and writing the magnitude in the space provided. Given: Member ABFC is a continuous member that is connected to member CDE by a pin at C. There are pinned supports at A and E. There is a uniformly-distributed loading of 1 kip/ft from A to B due to horizontal wind pressure, as shown. 1 kip/ ft B F C D 10 ft A E 5 ft 5 ft 10 ft Solution: Circle the correct sign, cross out the incorrect sign, and label the magnitude, as indicated. Magnitude Magnitude Page 17 of 46 Exam 1

18 2. (20 points) Determine the magnitude of force P, then determine the shear stress on plane AB. Given: The rectangular bar has a width w = 3.00 in. and a thickness of t = 2.00 in. The normal stress on plane AB is 6 ksi when the normal force P is applied. Page 18 of 46 Exam 1

19 3. (10 points). A 1 diameter vertical shaft is supported by a 1.5 diameter thrust collar that rests on the bearing plate as shown. The average shear stress between the collar and the shaft cannot exceed 18 ksi. Determine the maximum axial load P that can be applied to the shaft. Thrust Collar Bearing Plate Vertical Shaft Page 19 of 46 Exam 1

20 1. (20 points) Determine the magnitude of the internal moment M at point F and specify the sign of the internal moment by circling the appropriate symbol below and writing the magnitude in the space provided. Given: Member ABFC is a continuous member that is connected to member CDE by a pin at C. There are pinned supports at A and E. There is a uniformly-distributed loading of 1 kip/ft from B to C and a uniformly-distributed loading of 2 kips/ft from C to D. 2 kip/ft 1 kip/ft B F C D 10 ft A E 5 ft 5 ft 10 ft Solution: Circle the correct sign, cross out the incorrect sign, and label the magnitude, as indicated. Magnitude Magnitude Page 20 of 46 Exam 1

21 2. (15 points) Determine the normal stress a-a and shear stress a-a on the vertical cut a-a if the cross-sectional area is 1 in Cross Section of Member AB D 6 in lb. a B 6 in. A a C 8 in. Page 21 of 46 Exam 1

22 3. (15 points) A 3/8 diameter steel rod is embedded 10 inches into concrete and is subjected to a vertical force P that will pull the bolt out of the concrete when the average shear stress between the concrete and the steel rod reaches 400 psi (i.e., the concrete grips the bar with a shear stress of up to 400 psi). Determine the maximum normal stress that can be present in the bar, without pulling the rod out of the concrete. Assume that the interface between the bottom end of the bar and the concrete does not support any tensile stress. p 10 in. CONCRETE Page 22 of 46 Exam 1

23 1. (2 points) TRUE or FALSE. The cross-section at point F may have internal normal force, but cannot have internal moment or internal shear force. 2. (2 points) How many internal unknowns are there at cross-section B? Page 23 of 46 Exam 1

24 1. (20 points) Determine the internal normal force N, the internal shear force V, and the internal moment M at cross-section F, due to the applied loads. Given: Member ABC is connected to member CDE by a pin. Pinned supports at A and E Loadings and dimensions as shown. B 5 kips C 5 kips 2.5 D 2.5 A F E Page 24 of 46 Exam 1

25 4. (25 points). Determine the average normal stress and the average shear stress acting at sections a-a and b-b. Given: Two-member frame, subjected to loading shown. AB and BC are connected with pins. A and C are pinned supports. Member CB has a square cross section of 2 inches on each side. 1. (4 points). Pure Gold is the most ductile metal. If it has an ultimate tensile strength of about 200MPa, estimate its ultimate shear strength: Page 25 of 46 Exam 1

26 1. (15 points) Given: Rigid beam AB supports a uniformly-distributed load w = 2 kips/ft. Rope BC has a diameter of ¼. If the loading causes position B to go downward by 1, determine the Modulus of Elasticity E for the rope. BE CAREFUL WITH UNITS (1 foot = 12 inches). The material is elastic. C 6 ft. w = 2 kips/ft A B 8 ft. Page 26 of 46 Exam 1

27 1. (20 points). The jib crane is pinned at A and supports the 900-lb force shown. Beam AC is considered rigid. Pinended bar BC has a diameter of ½. When the 900-lb load is applied, position C deforms downward by inches. Determine the elastic modulus E of rod BC. BE CAREFUL WITH UNITS (Feet, inches, lbs, kips). 2. (5 points). Consider the previous jib crane. If the diameter of Pin B is ¼, determine the shear stress in the pin, due to the loading. Page 27 of 46 Exam 1

28 1. (2 points) Rank the following materials in order of INCREASING elastic stiffness from 1 (lowest elastic stiffness) to 4 (highest elastic stiffness). Typical Concrete Typical Aluminum Alloy (such as the 6061 alloy) Typical Wood (such as Southern Yellow Pine) Structural Steel 2. (3 points). Circle all that are true a. The elastic modulus of typical concrete is about 3 times larger than the elastic modulus of typical wood. b. The ultimate compressive strength of typical concrete is about 3 times larger than the ultimate compressive strength of typical wood. c. The elastic modulus of aluminum alloy (6061) is about 3 times larger than the elastic modulus of structural steel. d. The ultimate tensile strength of aluminum alloy (6061) is about 3 times larger than the ultimate tensile strength of structural steel. e. Structural steel has lesser elastic strain than aluminum alloy and lesser plastic strain than aluminum alloy. f. Structural steel has lesser elastic strain than aluminum alloy, but greater plastic strain than aluminum 3. (2 points). Pure Gold is the most ductile metal. If it has an ultimate tensile strength of about 200MPa, estimate its ultimate shear strength: Page 28 of 46 Exam 1

29 4. (15 points). The timber pile is made of Southern Yellow Pine (wood) with E = 1500 ksi; assume it remains elastic. It has a diameter of 12-inches and a length of 500-inches. It is subjected to a downward load at point A of 100 kips. The vertical force is resisted by 25 kip end-bearing on rock at point B and by side friction that varies linearly from f = 0 kip/in at point A to its maximum value at point B. Determine the vertical displacement of A, assuming that B does not move. 100 kips A 500 B Friction: Zero at A. Max at B. Linear variation Sand Rock 25 kips Final Answer: B = inches Page 29 of 46 Exam 1

30 5. (18 points). Beam ABC is rigid. Hanger BD (E=1000ksi) is a pin-ended member with a ½ x ½ cross-section and the stress-strain curve, shown. When 10-kips is applied at C, determine the vertical movement of point C. ½ ½ Cross-Section BD D kips 80 Stress, (ksi) A B C Strain, (in/in) Final Answer: C = inches Page 30 of 46 Exam 1

31 6. (10 points). Determine the elongation of the steel bar (E=30000 ksi) due to a 25-kip axial load. The 48-inch long bar has a 2 x ½ cross-section. The middle 24 of the bar has a 1 slot cut in it, as shown. Assume the steel remains elastic and ignore any local (St. Venant s) effects. 25 kips 25 kips k Total Elongation is: inches Page 31 of 46 Exam 1

32 The following questions are based upon your lab experience: 1. (1 point). Which material is tougher (circle one)? a. Structural Steel b. Aluminum 6061 Alloy 2. (2 points) Structural Steel is stiffer than Wood (Southern Yellow Pine, tested with the loading parallel to the grain) by about what factor (circle the closest answer. Assume one digit of precision)? a. About 1 to 2 b. About 2 to 5 c. About 5 to 10 d. About 10 to 50 e. About 50 to (1 points) TRUE or FALSE. Concrete is stiffer than Structural Steel 4. (1 points) TRUE or FALSE. Ordinary Strength Concrete is stronger than Wood (Southern Yellow Pine), in compression. 5. (1 points) TRUE or FALSE. Aluminum is stiffer than Structural Steel. 6. (1 points) TRUE or FALSE. Because natural rubber may be stretched up to 800% of its initial length without ever becoming permanently deformed, it is considered to be an extremely ductile material. 7. (2 points) Referring to the graph below, circle the correct statement. A. Material 1 is less ductile and has a larger yield stress y. B. Material 1 is less ductile and has a smaller yield stress y. C. Material 1 is more ductile and has a smaller yield stress y. D. Material 1 is more ductile and has a larger yield stress y. Stress, (psi) rupture rupture 2 1 Figure 1 Strain,, (in/in) Page 32 of 46 Exam 1

33 8. (15 points). A metal tension specimen with a diameter of in. was tested to failure while measuring its deformation over a 2.00-in. gage length, resulting in the data, below. A two-bar pin-ended truss is made using the same metal from the tension test, but with in diameter bars. Determine the vertical movement of point B due to the 84-kip vertical force at B. Note: graph paper is attached. A 100-in. C B 84 kips 100-in. Test Specimen Load (kips) Deformation (inches) Page 33 of 46 Exam 1

34 1. (10 points) Two 1 x3 x6 rubber blocks are bonded to rigid supports and to rigid steel plate AB. When 1 kip is applied to the plate, it moves by = 0.1. Determine the shear modulus of elasticity, G, for the rubber. The material is elastic. 3 in. 1 in. 1 in. 1 kip 6 in. Page 34 of 46 Exam 1

35 1. (2 points) Rank the following materials in order of INCREASING elastic stiffness from 1 (lowest elastic stiffness) to 4 (highest elastic stiffness). Typical Aluminum Alloy (such as the 6061 alloy tested in lab) Typical Concrete (such as that tested in lab) Structural Steel (such as that tested in lab) Typical Wood (such as the Southern Yellow Pine tested in lab) 2. (1 points) Based on the laboratory testing, circle the material that is more brittle: STRUCTURAL STEEL or ALUMINUM ALLOY (6061) 3. (2 points). Circle the closest answer a. The yield strength of structural steel is about 10 times larger than the yield strength of aluminum alloy (6061). b. The yield strength of structural steel is about the same as the yield strength of aluminum alloy (6061). c. The yield strength of aluminum alloy (6061) is about 10 times larger than the yield strength of structural steel. d. The yield strength of structural steel is about 3 times larger than the yield strength of aluminum alloy (6061). 4. (2 points). Circle the closest answer a. The elastic modulus of typical concrete is about 3 times larger than the elastic modulus of typical wood. b. The elastic modulus of typical concrete is about the same as the elastic modulus of typical wood. c. The elastic modulus of typical wood is about 3 times larger than the elastic modulus of typical concrete. d. The elastic moduli of typical concrete and typical wood were not determined in lab. 5. (2 points). Circle the correct answer, based on the materials tested in lab: a. Structural steel has greater elastic strain than aluminum alloy, but lesser plastic strain than aluminum alloy. b. Structural steel has lesser elastic strain than aluminum alloy and lesser plastic strain than aluminum alloy. c. Structural steel has lesser elastic strain than aluminum alloy, but greater plastic strain than aluminum alloy. d. Structural steel has greater elastic strain than aluminum alloy and greater plastic strain than aluminum alloy. 6. (2 points). Gold is the most ductile metal. It has an ultimate tensile strength of about 200MPa. Estimate its ultimate shear strength: Page 35 of 46 Exam 1

36 1. (10 points). Determine the force P that is required to stretch a structural steel bar to a final length of 530 inches, if its initial length was 500 inches, prior to applying the load. Given: The stress-strain curves for the steel are given below (2 different horizontal scales provided) The steel bar has cross sectional dimensions of: 6 inch width, ½ thickness Prior to Applying the Load Length = 500 inches P =? After Load P is Applied P =? 6 ½ Length = 530 inches, when P is applied Cross Section: 6 x ½ Page 36 of 46 Exam 1

37 2. (2 points). Using your knowledge and experience from lab, you know that concrete has a higher Modulus of Elasticity than wood. Circle the answer that is closest to the truth: a. Concrete E is 1.1 times higher than Wood E b. Concrete E is 3 times higher than Wood E c. Concrete E is 30 times higher than Wood E d. Concrete E is 300 times higher than Wood E 3. (2 points). Given equal-sized bars of 6061 Aluminum alloy and Structural Steel (use your knowledge of these materials from lab), which material will deform more when subjected to the same axial tension, if it is known that the bars are loaded elastically (i.e., to a stress that is less than the yield stress)? (circle the correct answer) a. Aluminum 6061 alloy b. Structural Steel Page 37 of 46 Exam 1

38 4. (20 points) Determine the rotation of Rigid Beam ABC and report your answer in degrees. Given: Beam ABC is considered to be rigid. It is supported by pin-ended wires AE and CD, as shown Wire AE is made of aluminum (E=10000 ksi) and has a diameter of 0.2 inches. Wire CD is made of steel (E=30000 ksi) and has a diameter of 0.5 inches. With the load applied, it is known that the wires remain elastic (stresses are below yield) E D 300 in. 1 kip 200 in. A B 100 in. 100 in. C Page 38 of 46 Exam 1

39 6. (13 points). Wood has radically different properties when loads are parallel to the grain versus perpendicular to the grain, as shown: Wood Grain Wood Grain When shear is parallel to grain, it fails at 90psi When shear is perpendicular to grain, it fails at 900psi Given: A 5/8-in.-diameter Wooden rod AB is fitted into a round hole near end C of the wooden member CD, at a distance of 3 from the center of the hole to the edge C. Determine the maximum force P that may be applied without either: A). Failing the 5/8 -diameter wooden rods in shear, or B). Tearing out the dashed area. P/2 P/2 P 3 Page 39 of 46 Exam 1

40 1. (2 points) TRUE or FALSE. The stress-strain behavior of all engineering materials is consistent with Hooke s Law. 2. (2 points) What is the typical tensile strength of ordinary structural steel, such as that tested in lab? (circle the closest answer): a. 50 psi b. 500 psi c psi d psi 3. (2 points) For the stress-strain curves below, circle the correct statement. a. Material 1 has greater toughness, greater elastic modulus, greater ultimate stress u, and greater yield stress y than Material 2. b. Material 1 has less toughness, lower elastic modulus, lower ultimate stress u, and lower yield stress y than Material 2. c. Material 1 has greater toughness, lower elastic modulus, lower ultimate stress u, and lower yield stress y than Material 2. d. Material 1 has less toughness, lower elastic modulus, higher ultimate stress u, and lower yield stress y than Material 2. e. Material 1 has greater toughness, lower elastic modulus, higher ultimate stress u, and lower yield stress y than Material 2. Stress, (psi) rupture rupture 2 1 Strain,, (in/in) 4. (2 points). Which material has a higher Modulus of Elasticity, Concrete or Steel? (circle the correct answer) a. Concrete b. Steel 5. (2 points). Which material has a higher Modulus of Elasticity, Aluminum (such as 6061 alloy) or Steel? (circle the correct answer) a. Aluminum (such as 6061 alloy) b. Steel Page 40 of 46 Exam 1

41 1. (13 points) Determine the shear modulus G for the material below. Given: A 100mm-thick sample of the material below (hence, the material s dimensions are 100mm x 200mm x 150mm) is subjected to a shearing force of 30 kn, as shown, causing the material to distort in shear, moving from its initial position, indicated by the solid lines, to its final position, indicated by the dotted lines. 30 kn 30 kn Page 41 of 46 Exam 1

42 2. (13 points) Determine the elastic modulus, yield strength, ultimate strength, maximum strain, and toughness for the following engineering material and plot its stress-strain curve on the graph paper below. Given: The material is a 4 x 4 block that was tested in compression, recording load and deformations over its 8- inch length. Plot compressive stress and compressive strain as positive. Load (lbs) Deformation (inches) Evenly distributed compressive load on 4 x4 cross-section Page 42 of 46 Exam 1

43 3. (15 points) Determine the force P. Given: Position B moves 0.5 inches to the right, due to force P. Rigid pole BC is supported by a pin at C and a steel wire (E=29000 ksi) at point B. The diameter of the wire is 0.2 inches. P Page 43 of 46 Exam 1

44 1. (18 points). Determine the vertical displacement of point B. Given: Both members have area A=1in 2 and E=500ksi (a plastic material). Connections are pins. Materials are elastic. A C B 30 inches 40 inches 6 kip 40 inches FINAL ANSWER: Vertical displacement at B, B = inches Page 44 of 46 Exam 1

45 2. (12 points). A ½ diameter steel (E=30000ksi) anchor bolt is embedded in concrete. When the upward load P is applied, the wedge-shaped base of the anchor resists a force of 1 kip, while the constant shear stress between the concrete and the 8-inch-long bolt is ksi. Determine the extension of the bolt length AB. Materials are elastic. P Concrete 8 A B 1 kip Resisted by Wedge Constant Shear Stress Page 45 of 46 Exam 1

46 3. (18 points). Determine the horizontal displacement of point B. Given: Both members have area A=1in 2 and E=500ksi (a plastic material). Connections are pins. Materials are elastic. A 30 inches B 8 kip C 30 inches 40 inches Page 46 of 46 Exam 1