MATERIALES INDUSTRIALES II ( ) EJERCICIOS APLICACION CES EDUPACK SEGUNDA PARTE-2DO CUAT. 2012

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MATERIALES INDUSTRIALES II ( 72.13 ) EJERCICIOS APLICACION CES EDUPACK SEGUNDA PARTE-2DO CUAT. 2012 GRUPO N 1 Case Study on a Light, Stiff, Strong Tie (Multiple constraints) 1.

1 MATERIALES INDUSTRIALES II ( ) EJERCICIOS APLICACION CES EDUPACK SEGUNDA PARTE-2DO CUAT GRUPO N 1 Case Study on a Light, Stiff, Strong Tie (Multiple constraints) 1. A tie, of length L loaded in tension, is to support a load F, at minimum weight without failing (implying a constraint on strength) or extending elastically by more than δ (implying a constraint on stiffness, F / δ ). The table summarizes the requirements. Tie rod Must not fail by yielding under force F Must have specified stiffness, F/δ Length L and axial load F specified Minimize mass m Section area A Establish two performance equations for the mass, one for each constraint, from which two material indices and one coupling equation linking them are derived. Show that the two indices are M 1 = and E M 2 = σy and that a minimum is sought for both. Use CES EduPack to produce a graph, which has the indices as axes, to identify candidate materials for the tie when (i) δ /L = 10-3 and (ii) δ /L = Remember y=ax Log(y)=Log(a)+Log(x). 1

2 GRUPO N 2 Y GRUPO N 3 Material index for a light, strong beam. In stiffness-limited applications, it is elastic deflection that is the active constraint: it limits performance. In strength-limited applications, deflection is acceptable provided the component does not fail; strength is the active constraint. Derive the material index for selecting materials for a beam of length L, specified strength and minimum weight, where the beam is secured at one end as in a cantilever. For simplicity, assume the beam to have a solid square cross-section t x t. You will need the equation for the failure load of a beam. It is Ff = Iσ f yml where y m is the distance between the neutral axis of the beam and its outer most surface and I = t 4 / 12 = A 2 / 12 is the second moment of the cross-section. y m = t/2 The table itemizes the design requirements Neutral Axis Beam Length L is specified Beam must support a bending load F without yield or fracture Minimize the mass of the beam Cross-section area, A 2

Example light stiff beam an oar. GRUPO N 4 Y GRUPO N 5 Case Study on a Light, Safe Pressure Vessel When a pressure vessel has to be mobile; its weight becomes important.

$fast fracture. What are the best materials for their construction? The table summarizes the requirements. Pressure vessel Must not fail by yielding Must not fail by fast fracture.$

3 Example light stiff beam an oar. GRUPO N 4 Y GRUPO N 5 Case Study on a Light, Safe Pressure Vessel When a pressure vessel has to be mobile; its weight becomes important. Aircraft bodies, rocket casings and liquid-natural gas containers are examples; they must be light, and at the same time they must be safe, and that means that they must not fail by yielding or by fast fracture. What are the best materials for their construction? The table summarizes the requirements. Pressure vessel Must not fail by yielding Must not fail by fast fracture. Diameter 2R and pressure difference p specified Minimize mass m Wall thickness, t (a) Write, first, a performance equation for the mass m of the pressure vessel. Assume, for simplicity, that it is spherical, of specified radius R, and that the wall thickness, t (the free variable) is small compared with R. Then the tensile stress in the wall is 3

4 σ = pr 2t where p, the pressure difference across this wall, is fixed by the design. The first constraint is that the vessel should not yield that is, that the tensile stress in the wall should not exceed σ y. The second is that it should not fail by fast fracture; this requires that the wall-stress be less than K 1c / π c, where K1c is the fracture toughness of the material of which the pressure vessel is made and c is the length of the longest crack that the wall might contain. (b) Use each of these in turn to eliminate t in the equation for m; use the results to identify two material indices, M 1 = and σy M2 = K1c and a coupling relation between them. It contains the crack length, c. The figure shows the chart you will need with the two material indices as axes. (c) Plot the coupling equation onto this figure for two values of c: one of 5 mm, the other 5 µm. Identify the lightest candidate materials for the vessel for each case. You can add a line and specify its intercept with the y-axis and gradient using the section tab in the dialogue box that opens on clicking the properties icon. However you cannot add more than one line at a time in the software. Remember y=ax+c Log(ax) = Log(a) + Log(x) 4

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Mechanical Engineering Ph.D. Preliminary Qualifying Examination Solid Mechanics February 25, 2002

Mechanical Engineering Ph.D. Preliminary Qualifying Examination Solid Mechanics February 25, 2002 student personal identification (ID) number on each sheet. Do not write your name on any sheet. #1. A homogeneous, isotropic, linear elastic bar has rectangular cross sectional area A, modulus of elasticity