Purpose of this Guide: To thoroughly prepare students for the exact types of problems that will be on Exam 3.

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1 ES230 STRENGTH OF MTERILS Exam 3 Study Guide Exam 3: Wednesday, March 8 th in-class Updated 3/3/17 Purpose of this Guide: To thoroughly prepare students for the exact types of problems that will be on Exam 3. Exam Format: Closed-book, closed-notes. The formulae below will be given. Students should have a calculator, pencils, and a straight-edge (if a graph is required). Given: Required material properties will be given. The following formulae and resources will also be given on exam: Law of Cosines: C b a Law of Sines: c Formulae: Normal stress = P/, where P is the normal force on the cut and is the area of the cut. Shear stress avg = V/, where V is the shear force on the cut and is the area of the cut. Normal strain = L/L, where L is the change in length and L is the original length (K, 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 (K Elastic Modulus or Young s Modulus) Poisson s Ratio = - ( lat / long ), where the specimen is loaded in the long direction, resulting in long strain, as well as lat strain. Hooke s Law for Shear: = G, where G is the shear modulus of elasticity (K, the modulus of rigidity) The elastic properties E, G, and are related by: Lesson Coverage, Objectives, and Example Problems: The exam covers Lessons 12 through 17. The objectives are given below, with example problems/questions for each objective. Compute the axial forces in an indeterminate member for which a compatibility expression that describes the geometry of deformation, must be employed. There are three basic classes of problems: Problems in which a particular deformation is known; e.g., = 0 or C = 0.001m. Problems in which two deformations must be equal or are related to one another; e.g., = or = + C. Problems in which a structure or machine is known to undergo rigid-body movement according to a geometric rule; positions,, and C will move but they will remain along a line. 1. (30 points) 1.2-m section of aluminum pipe of cross-sectional area 1100 mm 2 rests on a fixed support at. The 15-mm-diameter steel rod C hangs from a plate that rests on the top of the pipe at. The plate is considered to be rigid. Given: E=200 GPa for steel and E=70 GPa for aluminum. Determine the vertical movement of point C if P=60 kn. ll materials are known to be elastic.

2 2. (30 points) Determine the force in each rod Given: ar is considered a rigid member EF and DC are made of solid aluminum rounds, (E = ksi) each with a diameter of 1 inch. ll materials are known to be elastic. 3. (30 points) Determine the stress in the concrete and the stress in the steel due to the 80kN loading. Given: the steel pipe (E st =200 GPa) is filled with concrete (E c =24 GPa). The pipe has an outer diameter of 80-mm and an inner diameter of 70-mm. ll materials are known to be elastic. 4. (40 points) Determine the average normal stress in each post due to the applied loading. Given: The horizontal end-beams are rigid. The center post of the assembly has an original length of mm, whereas posts and C have a length of 125-mm (hence, there is initially a 0.3-mm gap between post and the rigid beam). ll of the posts are made of aluminum (E al = 70 GPa) with a cross-sectional area of 400 mm 2. ll materials are known to be elastic. ssume the gap closes, under load.

3 5. (35 points) Determine the maximum force P that can be applied without yielding either the steel bolt or the aluminum sleeve and determine the corresponding change in length at that load. Given: The steel bolt and nut are initially in loose contact with the aluminum sleeve. Steel olt: 3/8 Diameter, = in 2, E=29000 ksi, Yield Stress y = 50 ksi luminum Sleeve: 3/8 Inside Diameter, ½ Outside Diameter, = in 2, E=10000 ksi, Yield Stress y = 42 ksi luminum Sleeve Steel olt P P 5 inches Compute forces and stresses on indeterminate axial problems in which thermal forces are created. 6. (40 points) Compute the average stress in the bolt, after heating. Steel olt luminum Sleeve Given: steel bolt fits inside an aluminum sleeve, as shown. The nut at is adjusted so that it just fits the sleeve at T 1 = 20 C. Final Temperature T 2 = 100 C Steel bolt diameter = 7mm luminum sleeve outer diameter = 10mm luminum sleeve inner diameter = 8mm E steel = 200 GPa E alum = 70 GPa steel = 14(x10-6 )/ C alum = 23(x10-6 ) C 100 mm 7. (40 points) Determine the largest temperature increase that the assembly can sustain without the normal stress in the longitudinal direction of the brass shell exceeding 80 MPa. Given: The brass shell (E=115GPa, =18.6x10-6 / C) has a cross sectional area of 0.001m 2 ; The ceramic core (E=230GPa, =3.1x10-6 / C) has a cross sectional area of 0.001m 2. t the initial temperature of 15 C, the assembly is unstressed. The assembly is fully bonded together. ll materials are known to be elastic. 8. (30 points) Determine the compressive force that the concrete driveway slab exerts on the rigid abutment when it reaches a temperature of 120ºF.

4 Given: The concrete (E c = 3200 ksi, =6.0x10-6 /ºF) slab is 20-ft long, 20-ft wide, and 5 inches thick. It has a gap of between it and its rigid abutment at 20ºF. ll materials are known to be elastic. ssume the gap closes after heating. Rigid butment Rigid butment 20 feet 5 thick concrete slab inch gap at 20F 20 feet 9. (30 points). reinforced concrete slab 24 wide, 6 deep, and it is L inches long. The slab is free to expand or contract due to temperature, but the steel and concrete always remain bonded to one another. It contains (4) ½ diameter steel bars. Determine the stress in the concrete due to a temperature increase of 100 degrees Fahrenheit. Given: Concrete (E = 4000 ksi, = 5.5 x 10-6 / F), Steel (E = ksi, = 6.6 x 10-6 / F) L (40 points). eam C is rigid. Pin-ended rods D and CE are both steel (E = ksi, = 6.6 x 10-6 / F) with cross-sectional areas of 1 in 2. If rod D is heated 100 degrees Fahrenheit, while the other rod remain at the same temperatures, determine the forces in the rods and the reaction at. D E 25 C Compute the moment, shear, and normal force at any point in a beam, frame, or other statically determinate body.

5 11. (20 points) Compute the internal bending moment at Point D in the building structure and indicate whether this internal bending moment causes compression on the inside of the frame at this point or on the outside. Given: Points and C are pinned supports. Point is a hinge. 1 kip/ft 15 ft D C 5ft 20 ft 20 ft Derive M(x) and V(x) expressions for beams. 12. (30 points) Write the shear function V(x) and the moment function M(x) for the beam shown, between and, only using the standard sign conventions. It is subjected to a distributed load that increases linearly from 0 at to 2 kip/ft at. The load is uniform between and C. 2 kip/ft 10 feet 5 feet C x 13. (5 points). beam moment diagram is shown, below. Write the moment function from to if the beam is 10 ft long. M (kip-ft) (5 points). Write the shear function from to for the previous beam. Write and describe the differential and integral relationships between w(x), V(x), and M(x). 20-foot-long, simply-supported beam is shown, but no loads are given. The shear diagram, however, is given. ased on this, answer the following four questions by circling the correct answer. 15. (4 points). The internal moment at is: a. 3 kip-ft

6 b. 15 kip-ft c. -15 kip-ft d. -3 kip-ft e. Cannot be determined 16. (4 points). The maximum moment occurs at: a. Point b. Point c. Point C d. Point D e. Point E 17. (4 points). How is the beam loaded? a. It has a uniformly-distributed downward load between and and a lesser uniformly-distributed downward load between and E. b. It has a downward point load of 3 kips at point. c. It has an upward point load of 3 kips at point. d. It has a downward point load of 4 kips at point. 18. (4 points). TRUE or FLSE. Referring to the previous, the slope of the moment diagram, between and is 3. Memorize and apply the Six Rules for Constructing Shear and Moment Diagrams 19. (5 points) Select the rule that should be used to compute the next key point on the moment diagram. Concentrated force creates a jump in the shear diagram.. Concentrated moment creates a jump in the moment diagram C. Change in shear equals area under the load diagram. D. Change in moment equals area under the shear diagram Construct scaled shear and moment diagrams that give the correct value and shape at all points. 20. (4 points). Consider the beam and its corresponding moment diagram, shown. For position, specify whether the top of the beam contains compressive internal bending stresses or tensile internal bending stresses. nswer (circle one) a). Compressive internal bending stresses b). Tensile internal bending stresses

7 21. (30 points) Draw the shear and moment diagrams for beam CD. For each local maximum moment, give the magnitudes and locations, and indicate whether the top or bottom of the beam is in compression. Given: is a pinned support, in an internal pin, C is a roller support, D is a roller support 22. (30 points) Draw the shear and moment diagrams for beam. For each local maximum moment, give the magnitudes and locations, and indicate whether the top or bottom of the beam is in compression. Given: is a fixed support 23. (30 points) Draw the shear and moment diagrams for beam C. For each local maximum moment, give the magnitudes and locations, and indicate whether the top or bottom of the beam is in compression. Given: is a roller support. is a pinned support. Note that the 20 kip horizontal load creates a moment about the centroid of the beam (see dimension at C). 24. (40 points). Draw the shear and moment diagrams for the beam below, being careful to show the proper shape for each and drawing roughly to scale.. Determine the values of the maximum magnitude of positive and negative moments and clearly specify these locations. C. Indicate whether the top of the beam at is in compression or tension, due to bending. Given: is a pinned support, while C is a roller support. 3 kip/ft 2 kip/ft 2 kip/ft

8 5 ft 10 ft 12 ft Recommended problems from Philpot:P5.19, P5.40, P7.14, P7.49 LESSON 17: FUNDMENTLS OF FLEXURE Derive relationships between curvature and strain, in bending Compute stresses (including extreme-fiber stresses) in a beam, given the curvature (or radius of curvature) and beam thickness. Compute the internal moment in a beam, given the extreme-fiber stresses and the beam dimensions. Compute the curvature (or radius of curvature) of a beam, given the extreme-fiber stresses. 25. (10 points) 5 thick wooden (E=1000 ksi) beam with a rectangular cross-section is bent to a minimum radius of curvature of Determine the maximum normal stress due to bending. ssume that the beam remains elastic (15 points). The cross-section of a beam that is in positive bending about the horizontal axis is shown. It is known to have a maximum compressive strain magnitude of in/in. Determine the maximum tensile strain magnitude. 8 x 1 Flange 8 x 1 Web EM CROSS-SECTION 27. (25 points). Given: Steel (E=30000 ksi) beam with a 6 x6 cross-section is subjected to equal end-moments, M. It is known that the strain at Position (1 inch down from the centroid) is = in/in. ssume the material remains elastic. Determine: i. The strain at Position (3 inches down from the centroid), ii. The stress at Position, iii. The applied end-moments, M iv. The radius of curvature M M 1 Side View of Steel eam Cross Section of Steel eam

9 28. (25 points). Determine the force P at the center of the beam. Given: The beam (E=1000ksi) is 10-ft long and subjected to a centerpoint load P. The extreme-fiber strain has been found to be = in/in. P Strain-gage measures = in/in 12 x 12 Cross-Section eam Cross Section

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