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**Q1:**Q1 Section A Complete both questions in this section in the BLUE answer booklet You are using an AISI 4340 (Fe-0.4wt% C+ alloying additions) steel to produce forged/machined ground anchors for a large mobile phone mast. (a) The steel is supplied in the normalised condition. What fraction of the steel do you expect to be austenite, martensite and pearlite? Justify each answer. (5 marks) (b) Upon examination you find the steel is mostly bainite and martensite with a small amount of proeutectoid ferrite. Determine the range of possible cooling rates this material might have experienced? Would the presence of these microconstituents cause you any concern considering the steel will be hot-forged? (5 marks) (c) The steel needs to be processed in the following manner: (i) Hot forge to the basic shape. (ii) Substantial machining to create threads. (iii) Heat treatment and cooling to create a 100% martensite microstructure. (iv) Tempering to modify the toughness. Sketch a time-temperature history that you would use for this process. Focus on specify- ing the temperatures and the required cooling rates for each stage. Clearly indicate where you have had to use your judgement to estimate a value. (5 marks) (d) You decide that the forging should be a dual-phase steel consisting of 50% ferrite and 50% martensite in order to improve the damage resistance of the anchor. What single change could be made to the above process to produce this desired microstructure? Fully explain your reasoning. (5 marks)/n8-iron (BCC) Temperature, T (°C) Ferrite a (BCC) 1600 1400 1200 1000 800 600 400 200 0 Melting point /of pure Fe 1534°C L+8 Peritectic point 0 Fe Austenite Y (FCC) 910°C 0.8 α+Y 0.035 1 L+Y Liquid, L 2.1 Eutectoid point 2 Eutectic point 3 Ferrite, & + Fe,C Austenite. Y + Fe₂C 723°C 4.3 wt% C 4 1147°C Compound, Cementite Fe₂C 5 6 7/nTemperature (C) 900 800 700 600 500 400 300 200 100 0 10⁰ M₂ M₁ B. Rate (C/s) 20 8 10¹ 10² 10³ time (s) Figure Q1 4 AISI 4340 0.33 0.08 0.023 0.006 104 105/nFORMULAS Emax = VjEj + (1 - V₁) Em 1 [r+rs(f − 2)]¹/2 - agelSee Answer**Q2:**FORMULAS Emax = VjEj + (1 - V₁) Em 1 [r+rs(f − 2)]¹/2 Olgel/nQ2 You are a composites engineer in a company that is currently producing carbon fibre reinforced composites using a blend of a tetrafunctional epoxy and an aromatic diamine with the following characteristics: functionality of the crosslinking groups, f = 4, molar ratio of the epoxy groups to amine hydrogen atoms, r= 1, and the fraction of the amine hydrogen atoms in the reactants, s = 1. The matrix is predicted to undergo gelation at agel of 0.577 (around 58%). (a) Unfortunately, there has been an error in procurement and your materials supplier has provided a trifunctional epoxy for use with the same diamine. You can assume that the fraction of amine hydrogen atoms in the reactant formulation remains the same. Show, using Flory-Stockmayer theory, the polymer conversion at which your new matrix for- mulation reaches gelation. The polymer undergoes a change from a rubbery material to a cross-linked polymer, what do we call this? If you keep the other processing conditions the same, what effect will this have on the manufacturing time for the composite? (5 marks) (b) Fortunately, you realise the mistake before processing the composite, but you have to proceed using the same trifunctional epoxy and diamine. What two steps could you take to shorten the time to reach gelation? Justify your answer by showing the effect on Orgel- (5 marks) (c) A Time-Temperature-Transformation (TTT) diagram for your trifunctional epoxy and diamine blend is shown in Figure Q2. Sketch the TTT diagram and annotate on the dia- gram the isothermal processing temperature that you would propose to get the best ma- terial properties from your resin. Justify your reasoning by describing how your choice influences the manufacturing process. (5 marks) (d) Your cured epoxy resin develops a Young's modulus of 3.4 GPa and is combined with intermediate modulus carbon fibres (250 GPa). If the target fibre volume fraction for your composite is 55%, what is the maximum stiffness (Emaz) that you would expect along the fibre direction in each ply? Sketch a diagram to represent relative stiffness of the laminate against ply angle to show what Emax becomes in: (i) a unidirectional laminate, (ii) a cross-ply laminate, (iii) a quasi-isotropic laminate./nCure temperature (°C) Tg gel To Tso Gelled rubber gelation Liquid Oxidation Log time (min) Figure Q2 Tg Gelled glass Ungelled glassSee Answer**Q3:**FORMULAS Emax = VƒEƒ + (1 - V₁) Em 1 = CX gel [r+rs(f-2)]¹/2/nQ3 A mass of 10 kg is suspended from the ceiling on a steel wire of 1 mm diameter and 2 m length. The mass is now rotated through an angle causing the wire to twist. (a) Determine the shear stress Try in the wire as a function of the angle of twist 0. You may assume that the steel has a shear modulus G = 80 GPa. (6 marks) (b) Estimate the maximum value of 0 if there is to be no plastic deformation. The steel has a yield stress of 300 MPa and you may assume von Mises theory of yielding. (14 marks)See Answer**Q4:**FORMULAS Emax = VƒEƒ + (1 - V₁) Em gel 1 [r+rs(f-2)]¹/2/nQ4 The L-shaped beam of Figure Q5 is subjected to an out-of-plane load F. (a) Show the free-body diagrams for the two segments AB and BC. Comment on the mo- ments and forces transmitted in each segment. (5 marks) (b) Provide expressions for the moments and torques transmitted in segments AB and BC. (3 marks) (c) Determine the strain energy U stored in the L-shaped beam. (3 marks) (d) Use Castigliano's second theorem to find the displacement uc of the end in the direction of the force. (9 marks) Y Fixed end Ac FL C Figure Q5 L BSee Answer**Q5:**FORMULAS Emax = V₁E+ (1-V₁) Em Ogel 1 [r+rs(f-2)]¹/2/nQ5 The fuel rods of a nuclear reactor consist of solid uranium cylinders of diameter 70 mm. During operation, a typical rod experiences a temperature distribution approximated by the equation T(r) = 600 -0.1² °C, where r is the radius in mm. The properties of uranium are E = 172 GPa, v = 0.28, and a = 11 x 10-6 per °C. (a) Find the maximum tensile, compressive and shear stresses in the fuel rod if the outer surface is traction-free and plane strain conditions can be assumed. (14 marks) (b) If the fuel rod is now permitted to expand axially, determine the maximum tensile, com- pressive and shear stresses. (6 marks) [You may assume that the radial and hoop stresses in an axi-symmetric disk in a state of plane strain are Orr 000 = (3-2v)p²r² 8(1-v) (1+2v)p²,² 8(1-v) with the corresponding radial displacement + Ea (1-0)² /rTdr +/ Ea (1-v)r² afr rTdr _ ( 1-20)/(1+1) ²²³³ + 0(1+1) [T rTdr + 8E(1-v) (1-v)r where the symbols have their usual meanings] A+ EaT (1-v) B + A A(12v)(1+ v)r E B (1 + v) B ErSee Answer**Q6:**FORMULAS Emax = VƒEƒ + (1 - Vj) Em 1 = gel [r+rs(f-2)]¹/2/nQ6 A cylindrical pressure vessel with closed ends has a radius R = 1 m and thickness t = 40 mm and is subjected to internal pressure p. The vessel must be designed safely against failure by yielding (according to the von Mises yield criterion) and fracture. Three steels with the following values of yield stress oy and fracture toughness Kic are available for constructing the vessel. Steel Kic(MPa √/m A: 4340 100 B: 4335 70 C: 350 Maraging 55 Fracture of the vessel is caused by a long axial surface crack of depth a. The vessel should be designed with a factor of safety S = 2 against yielding and fracture. (a) By considering equilibrium along the longitudinal (axial) and circumferential (hoop) di- rections determine expressions for the hoop stress and axial stress in terms of the internal pressure, p, the radius, R and the thickness, t. dy (MPa) 860 1300 1550 (4 marks) (b) For the three steels, find the maximum pressure the vessel can withstand without failure by yielding. Note, your calculation should include the factor of safety, S. (4 marks) (c) The fracture toughness for a long axial surface crack of depth a is given by Kic 1.12000 √na. Hence determine an expression for the maximum pressure as a function of crack length a and fracture toughness. Note, your calculation should again include the factor of safety, S. (3 marks) (d) Plot the maximum permissable pressure pe versus crack depth a, for the three steels. (3 marks) (e) Calculate the maximum permissable crack depth a for an operating pressure p = 12 MPa. (3 marks) (f) Calculate the failure pressure p, for a maximum detectable crack depth a = 1 mm. (3 marks)See Answer**Q7:**A [0/+60/-60]s laminate with the ply properties listed in the table below is to be subjected to a temperature change from its initial temperature of 75°F. This temperature change can be expressed as a linear temperature change through the thickness of the laminate, with the temperature at the top of the six-ply laminate set at 225°F and the temperature at the bottom of the six-ply laminate set to -75°F. Therefore, for the temperature distribution defined by the equation AT(2) = AT+T'z, with ATh2=225°F - 75°F = 150°F and AT-2=-75°F-75°F=-150°F, AT. =(AT1/2+AT-1/2)/2= [150+(-150)]/2=0°F and T'=(AT12-AT-1/2)/h=(150-(-150))/6(0.0052) = 9,615.4°F/inch we obtain the distribution expression a) Determine the stresses in the lamina coordinate system at both the top and bottom in each of the 0°, +60° and -60° plies. b) Given the lamina strengths in the table below, determine if the laminate subjected to this temperature change distribution could be expected to survive with no excessive lamina stresses and therefore with no damage to the laminate. c) Assuming the same initial stress-free temperature of 75°F and by subjecting this same [0/+60/-60]s laminate separately to (i) a uniform temperature of 225°F and (ii) a uniform temperature of -75°F, answer the question "Is the through thickness temperature gradient more stressing on the laminate than either the uniform through thickness temperature of 225°F or the uniform through thickness temperature of -75°F?" Property E₁ E₂ G12 V12 α₁ (-200°F to 200°F) α₂ (-200°F to 200°F) 01 0 AT(2) AT+T'z = 9,615.4°F/inch*z TL cu OL σχετι Ply thickness Lamina Value 25 x 10º psi 1.7 x 106 psi 1.3 x 10º psi 0.3 -0.3 x 10 in/in/°F 19.5 x 10 in/in/°F 110 x 10³ psi 4.0 x 10³ psi 9.0 x 10³ psi 110 x 10³ psi 20 x 10³ psi 0.0052 inchSee Answer**Q8:**Problem 1: Stress analysis of a lamina using experimental data from a strain rosette The strain rosette used to measure failure strains at a point yielded the following ratio between the strains in the lamina coordinate axes: &1= n₂/₁2=0 E2 The lamina was loaded by tensile stresses and ₂. Compare the results obtained by the maximum stress, Tsai-Hill and Tsai-Wu criteria to determine the failure stress. Which of these criteria prescribed the safest stress combination? &c Eb 60° Y 60° 30° QU X 45⁰ Figure 1. Typical strain gauge rosettes (eFunda.com). (a)/nNotes: The ratio of strains: See Table 1 for your variant. Material: See Tables 2 and 3 for your variant. Table 1. The strain ratio measured in the lamina at failure. Variant 1 n Variant 10 1.3 n 1.5 Variant 19 1.4 n 2 1.1 11 1.6 20 0.75 3 1.2 12 1.15 4 0.8 13 0.7 5 2.0 14 0.5 6 1.7 15 1.5 7 0.6 16 1.2 8 0.9 17 0.4 9 0.75 18 1.6 1/nTable 2: Material in Problem 1 (see Table 3 for properties) Var. 1 Material 1 Var. 10 Material 1 Var. Material 19 1 2 1 11 1 20 1 3 1 12 1 4 2 13 2 5 2 14 2 6 2 15 2 7 3 16 3 8 3 17 3 9 3 18 3/nTable 3. Composite materials for problems 1 and 2 (material number in your variant refers to the material in Table 1.1 below) From Table 1.1 Typical properties of unidirectional composites E.J. Barbero, "Introduction to Composite Materials" Density ( Longitudinal Model E Tven Mode GP Inplane Shear Modulus G₁ GP P's Radio Longitudinal Tessile Strength F₁, [MPa] Transverse Tensile Strength PMP Inplane Shear Strength (MP Longitudinal Compressive Strength F₁, [MPa] Transverse Compressive Strength F (MP) elaminar Shear Strength (For Fy) [MPa] Longitudinal Tessile Strain 3, 1963 Longitudinal CTE o [10/C) Transverse CTE [10/ Longitudinal moisture Transserse mistare expansice P Fiber Volume Fract V Void Content V. [] Fiber Misalignment Material number E-Glass S-Class Epoxy Epoxy 2.076 12 55 30 0 55 16 0.2 60 82-85SBASERE 0.19 0.28 10:20 1630 40 60 630 140 60 23 37 Jalalalalalalal 16 60 6.90 140 80 29 32 02 60 2 *** 982-30*****338-3822M E-Glas Kevlar Epery ophtal Polyester 1.85 113 40 40 357 24 6.5 3.53 1.380 75.8 5.5 934 13800 34.5 441 586.0 138.0 48.69 1.8 -1.0 60 0.01 02 60 4 Carbo Carbon/ Carbon Carbon Epoxy Epoxy Epey ASA/35-6 T80/1960-2 1551 ASAPCE PEEK 1.58 142 10.3 7.2 0.27 1830 57 71 1096 228 1.29 -49 27 0 02 60 n 1558 19 5.14 0.3 2608 168 0.0095 0.321 151 9.0 5.6 0.3 L64 57.3 DI 1.6 138 102 5.7 0.3 2000 85 186 1360 400 150 1.45 0.5 30 61 6 2 Carbo Polid ASU Avi Note: Carbon/epoxy, Kevlar/epoxy and boron/epoxy (to a smaller extent) are used in aerospace applications, while less expensive glass/epoxy and glass/polyester are preferred by shipbuilders and in civil engineering. 110 83 37 63 1000 0.5See Answer**Q9:**Problem 1 (20 pts): If you are given the fiber and matrix properties below and are asked to estimate the composite elastic constants (Moduli E₁, E2, G₁2 and Poisson's ratio V12) and directional strength (X₁, Xe, Yı, Ye and S). Continuous long fiber reinforcement: Er = 400 GPa; v=0.4; or=3 GPa; Epoxy matrix: Em=3 GPa; vm -0.3; Gm= 80 MPa ; Tm = 60 MPa; Fiber volume fraction: V₁ = 60% Please give your estimate and specify the micromechanics equations you use for the estimate (if you use computer programs your solution must include code and outputs).See Answer**Q10:**Problem 2 (20 pts): A thin-walled tube (closed-cap) is made of a unidirectional composite with fiber oriented at 45⁰ to its axis as shown below. The tube has a radius of r= 50.0 mm and a thickness t = 1.0 mm. The tube is to be loaded under combined internal pressure P = 1.0 MPa and a torque of T = 780,000 N-mm. The strength data for the lamina are: X=2800 MPa; X = 1600 MPa; Y₁ = 80 MPa; Ye280 MPa; S = 90 MPa. Please evaluate whether failure would occur using 1) maximum stress criterion; 2) Tsai-Hill criterion; 3) Tsai-Wu criterion; and 4) Hashin's criterion. (if you use computer programs your solution must include code and outputs) WIESee Answer**Q11:**Problem 3 (20 pts): A unidirectional lamina is subject to a bi-axial stress state as shown in the following figure. The fiber directional is oriented 0-30° to the loading direction x. The stress ay is proportional to ox by a factor of B (B is a positive number). Assuming ox is always positive (meaning tensile stress). Using maximum stress criterion, please discuss the possible change of failure mode by varying the proportional factor B. Determine the range of ß for each possible failure modes. The strength data are X₁ Xc, Y₁ = Y, and S (all are in absolute values). V !!!!! a, Ba,See Answer**Q12:**Problem 4 (20 pts): The two plies as shown in the left figure has the same lamina elastic constants of E₁ = 160 Gpa; E2 10 Gpa; G12= 6 Gpa; V12 = 0.3. The fibers in the top ply (ply 1) is oriented at 600 and the bottom ply (ply 2) is oriented at 120° with respect to the global coordinate x. Both plies are subjected to an in-plane tensile stress ox = 50 MPa. (1) Please calculate the strains (¹), Ey(¹), Yxy (1); Ex (2), Ey (2), Yxy (2)) (2) Please illustrate (hand draw ok) the deformation of the two plies (3) Imagine now the two plies are bonded by an interface and subjected to the same stress ox as shown in the right figure, please comments what the interface needs to provide (i.e., what stress will develop at the interface) to ensure an iso-strain condition at cross-sections. 2 60⁰ 120° Ply 1 Ply 2 ab in -50 MPa - 50 MPa -50 MPa y iso-strain Cross-section deformation -50 MPaSee Answer**Q13:**Problem 5 (20 pts): Using the theories we learned from micromechanics of lamina to estimate the following mechanical properties of a glass-fiber reinforced polymer (GFRP) lamina with fiber volume fraction of Vr= 50%. The elastic and failure properties of glass fiber and epoxy matrix are given below: Glass fiber (isotropic elastic): Ef=70 GPa; vf=0.3; of 1.4 GPa; Epoxy (isotropic elastic): Em = 2 GPa; Vm = 0.3; om = 60 MPa (tensile strength); tm = 35 MPa (shear strength) 1) The elastic/stiffness properties need to be determined are: E1; E2; V12; G12 2) The strength properties need to be estimated are: Xt; Xc; Y₁; Ye; S 3) estimate the critical fiber volume fraction that has fiber strengthening effect. 4) Estimate the minimum fiber volume fraction that matrix can bridge a localized broken fiberSee Answer**Q14:***16-68. Knowing that angular velocity of link AB is WAB = 4 rad/s, determine the velocity of the collar at C and the angular velocity of link CB at the instant shown. Link CB is horizontal at this instant. See Answer**Q15:**Suppose you want to connect a remote speaker to your stereo. It needs to be 100 inches away.You are using copper wire. What should be the diameter of the wire if the resistance of each wire must be less than 0.10 Q? (3 marks)See Answer**Q16:**16-109. Member AB has the angular motions shown.Determine the angular velocity and angular acceleration of members CB and DC. See Answer**Q17:**16-57. At the instant shown the boomerang has an angular velocity o = 4 rad/s, and its mass center G has a velocity vG = 6 in./s. Determine the velocity of point B at this instant. See Answer**Q18:**16-107. At a given instant the roller A on the bar has the velocity and acceleration shown. Determine the velocity and acceleration of the roller B, and the bar's angular velocity and angular acceleration at this instant. See Answer**Q19:**16-67. Determine the velocity of point A on the rim of the gear at the instant shown. See Answer**Q20:***16-88. If bar AB has an angular velocity w AB = 6rad/s determine the velocity of the slider block C at the instant shown.See Answer

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