from barber chapter 2; questions: 2.3, 2.7, 2.9, 2.17 from boresi chapter 1; questions: 1.24, 1.28, 1.29, 1.33 The questions are from the pdf's that I have attached and there's one more pdf file attt
22 CHAPTER 1 INTRODUCTION C. material parameter when failure occurs by brittle fracture. Instead, notch toughness, the ability of a material to absorb energy in the presence of a notch (or sharp crack), is the parameter that governs the failure mode. Dynamic loading and low tempera- tures also increase the tendency of a material to fracture in a brittle manner. Failure by brittle fracture is discussed in Chapter 15. Progressive Fracture (Fatigue). If a metal that ordinarily fails by general yield- ing under a static load is subjected to repeated cycles of stress, it may fail by fracture without visual evidence of yielding, provided that the repeated stress is greater than a value called the fatigue strength. Under such conditions, minute cracks start at one or more points in the member, usually at points of high localized stress such as at abrupt changes in section, and gradually spread by fracture of the material at the edges of the cracks where the stress is highly concentrated. The progressive fracture continues until the member finally breaks. This mode of failure is usually called a fatigue failure, but it is better designated as failure by progressive fracture resulting from repeated loads. (See Chapter 16.) 4. Failure by Instability (Buckling) Some members may fail by a sudden, catastrophic, lateral deflection (instability or buck- ling), rather than by yielding or crushing (Chapter 12). Consider an ideal pin-ended slen- der column (or strut) subjected to an axial compressive load P. Elastic buckling of the member occurs when the load P reaches a critical value P,, = z2EIJL2, where E is the modulus of elasticity, I is the moment of inertia of the cross section, and L is the member length. PROBLEMS 1.1. What requirements control the derivation of load-stress relations? 1.2. Describe the method of mechanics of materials. 1.3. How are stress-strain-temperature relations for a material established? 1.4. Explain the differences between elastic response and inelastic response of a solid. 1.5. What is a stress-strain diagram? 1.6. Explain the difference between elastic limit and propor- tional limit. 1.7. Explain the difference between the concepts of yield point and yield stress. 1.8. What is offset strain? 1.9. How does the engineering stress-strain diagram differ from the true stress-strain diagram? 1.10. What are modes of failure? 1.11. What are failure criteria? How are they related to modes of failure?1.12. What is meant by the term factor of safety? How are fac- tors of safety used in design? 1.13. What is a design inequality? 1.14. How is the usual design inequality modified to account for statistical variability? 1.15. What is a load factor? A load effect? A resistance factor? 1.16. What is a limit-states design? 1.17. What is meant by the phrase “failure by excessive deflec- tion”? 1.18. What is meant by the phrase “failure by yielding”? 1.19. What is meant by the phrase “failure by fracture”? 1.20. Discuss the various ways that a structural member may fail. 1.21. Discuss the failure modes, critical parameters, and failure criteria that may apply to the design of a downhill snow ski. 1.22. For the steels whose stress-strain diagrams are repre- sented by Figures 1.8 to 1.10, determine the following proper- ties as appropriate: the yield point, the yield strength, the upper yield point, the lower yield point, the modulus of resilience, the ultimate tensile strength, the strain at fracture, the percent elon- gation. 1.23. Use the mechanics of materials method to derive the load-stress and load-displacement relations for a solid circular rod of constant radius r and length L subjected to a torsional moment T as shown in Figure P1.23. PROBLEMS 23 T FIGURE P1.23 Solid circular rod in torsion. 1.24. Use the mechanics of materials method to derive the load-stress and load-displacement relations for a bar of con- stant width b, linearly varying depth d, and length L subjected to an axial tensile force P as shown in Figure P1.24. P FIGURE P1.24 Tapered bar in tension. A Longitudinal section (rods not shown) End plate Pipe: OD = 100 mm ID = 90 mm 1.25. A pressure vessel consists of two flat plates clamped to the ends of a pipe using four rods, each 15 mm in diameter, to form a cylinder that is to be subjected to internal pressure p (Figure P1.25). The pipe has an outside diameter of 100 mm and an inside diameter of 90 mm. Steel is used throughout (E = 200 GPa). During assembly of the cylinder (before pressuriza- tion), the joints between the plates and ends of the pipe are sealed with a thin mastic and the rods are each pretensioned to 65 IcN. Using the mechanics of materials method, determine the internal pressure that will cause leaking. Leaking is defined as a state of zero bearing pressure between the pipe ends and the plates. Also determine the change in stress in the rods. Ignore bending in the plates and radial deformation of the pipe. 1.26. A steel bar and an aluminum bar are joined end to end and fixed between two rigid walls as shown in Figure P1.26. The cross-sectional area of the steel bar is A, and that of the alumi- num bar is A,. Initially, the two bars are stress free. Derive gen- eral expressions for the deflection of point A, the stress in the steel bar, and the stress in the aluminum bar for the following conditions: a. A load P is applied at point A. b. The left wall is displaced an amount 6 to the right. 1.27. In South African gold mines, cables are used to lower worker cages down mine shafts. Ordinarily, the cables are made of steel. To save weight, an engineer decides to use cables made of aluminum. A design requirement is that the stress in the cable resulting from self-weight must not exceed one-tenth of the ulti- mate strength o, of the cable. A steel cable has a mass den- sity p = 7.92 Mg/m3 and o, = 1030 MPa. For an aluminum cable, p = 2.77 Mg/m3 and o, = 570 MPa. -Steel rod (typical) Diameter = 15 mm Section A-A FIGURE PI .25 Pressurized cylinder. Steel Aluminum FIGURE P1.26 Bi-metallic rod. a. Determine the lengths of two cables, one of steel and the other of aluminum, for which the stress resulting from the self- weight of each cable equals one-tenth of the ultimate strength of the material. Assume that the cross-sectional area A of a cable is constant over the length of the cable. b. Assuming that A is constant, determine the elongation of each cable when the maximum stress in the cable is 0.100,. The steel cable has a modulus of elasticity E = 193 GPa and for the aluminum cable E = 72 GPa. c. The cables are used to lower a cage to a mine depth of 1 km. Each cable has a cross section with diameter D = 75 mm.
Determine the maximum allowable weight of the cage (includ- ing workers and equipment), if the stress in a cable is not to exceed 0.200,. 1.28. A steel shaft of circular cross section is subjected to a twisting moment T. The controlling factor in the design of the shaft is the angle of twist per unit length (y/L; see Eq. 1 S). The maximum allowable twist is 0.005 rad/m, and the maximum shear stress is z,,, = 30 MPa. Determine the diameter at which the maximum allowable twist, and not the maximum shear stress, is the controlling factor. For steel, G = 77 GPa. 1.29. An elastic T-beam is loaded and supported as shown in Figure P1.29~. The cross section of the beam is shown in Fig- ure P1.296. 24 CHAPTER 1 INTRODUCTION 0 0 3.1 0.01 6.2 0.02 9.3 0.03 12.4 0.04 15.5 0.05 18.6 0.06 21.7 0.07 24.7 0.08 25.8 0.09 26.1 0.10 29.2 0.15 31.0 0.20 34.0 0.30 IP 10 kNlm 36.3 0.40 38.8 0.50 41.2 0.60 44.1 1.25 48.1 2.50 50.4 3.75 51.4 5.00 52.0 6.25 52.2 7.50 52.0 8.75 50.6 10.00 45.1 11.25 43.2 11.66 - - t I X &A FIGURE P1.29 T-beam. a. Determine the location j of the neutral axis (the horizontal centroidal axis) of the cross section. b. Draw shear and moment diagrams for the beam.
c. Determine the maximum tensile stress and the maximum compressive stress in the beam and their locations. 1.30. Determine the maximum and minimum shear stresses in the web of the beam of Problem 1.29 and their locations. Assume that the distributions of shear stresses in the web, as in rectangu- lar cross sections, are directed parallel to the shear force V and are uniformly distributed across the thickness (t = 6.5 mm) of the web. Hence, Eq. 1.9 can be used to calculate the shear stresses. 1.31. A steel tensile test specimen has a diameter of 10 mm and a gage length of 50 mm. Test data for axial load and corre- sponding data for the gage-length elongation are listed in Table P1.31. Convert these data to engineering stress-strain data and determine the magnitudes of the toughness U, and the ultimate strength 0,. TABLE P1.31 Elongation Elongation Load (kN) (mm) Load (kN) (mm) AISC (2001). Manual of Steel Construction-Load and Resistance Factor Design, 3rd ed. Chicago, IL: American Institute of Steel Construction.
AISC (1989). Specification for Structural Steel Buildings-Allowable Stress Design and Plastic Design. Chicago, IL: June 1. AMERICAN SOCIETY OF CIVIL ENGINEERS (ASCE) (2000). Minimum Design Loads for Buildings and Other Structures, ASCE Std. 7-98.
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