Problem 1: Stress analysis of a lamina using experimental data from a strain rosette

Questión 4:For a bronze alloy, the stress at which plastic deformation begins is 275 MPa and the modulus of elasticity is 115 GPa. (6 marks – 3 marks each part) (a) What is the maximum load that may be applied to a specimen with a cross-sectional area of 325 mm without plastic deformation? (b) If the original specimen length is 115 mm, what is the maximum length to which it may be stretched without causing plastic deformation?

*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.

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.

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)

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.

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 inch

Today most turbo charged car engines have their rotors made of silicon nitride (Si3N4)rather than the traditional nickel alloy (Figure 1). i)Justify this material switch from metals to ceramics by comparing FOUR relevant material properties of silicon nitride with that of the previously used nickel alloy. ii)Draw a flow chart to outline the main process steps of the Si3N4 ceramic turbocharger rotor manufacture and assembly, and discuss two micro structural features that must be controlled to within strict limits during manufacture.

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 Er

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 - agel

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 MPa