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CIVE_207_december2006

# CIVE_207_december2006 - McGill University Faculty of...

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Unformatted text preview: McGill University Faculty of Engineering SOLID MECHANICS CIVE-207A01 Final Examination: 2:00 — 5:00 PM, DECEMBER 8, 2006 Examiner: Prof. Shao Co-examiner: Prof. McClure bFEQ :\ [hat/wk STUDENT NAME: MCGILL ID. NUMBER: INSTRUCTIONS: 1) This is a closed book examination. N0 crib sheet is allowed. 2) Only faculty standard calculators are permitted. _ 3) This examination consists of SIX problems of a total of 5 pages, including cover page. 4) The examination will be marked out of 100. CIVE-207A Final Examination December 8, 2006 p.2 Problem 1 110 marks): An inﬂatable structure used by a traveling circus has the shape of a half-circular cylinder with closed ends (see Fig. 1). The fabric and the plastic structure is inﬂated by a small blower and has a radius of 40 ft when fully inﬂated. A longitudinal seam runs the entire length of the “ridge” of the structure. If the seam tears open when it is subjected to a tensile load of 540 pounds per inch of seam, what is the factor of safety (F S.) against tearing when the internal pressure is 0.5 psi and the structure is fully inﬂated? Longitudinal seam Fig. 1: Problem 2 (15 marks): An elastic prismatic beam with an overhang is loaded with a concentrated end moment Mo and supported by a spring at B (see Fig. 2). Determine the deﬂection of the free end A. The spring constant k = 48EI/L3, where E1 is the ﬂexural stiffness of the beam. Express the deﬂection in terms of M0, L, a, and E1. [Hintz P=kA, where P=spring force, A=spring displacement] CIVE-207A Final Examination December 8, 2006 p.3 Problem 3 (15 marks): A thin bar of stainless steel (6-mm wide and 3-mm thick) is axially precompressed 100 N between two plates that are ﬁxed at a constant distance of 150 mm apart (see Fig. 3). This assembly is made at 20 0C. How high can the temperature of the bar rise so as to avoid buckling? Assume factor of safety = 2, E = 200 GPa and coefﬁcient of thermal expansion or = 15 x 10'6 per 0C. Fig. 3a is a front view and Fig. 3b a side view. Dimension is in mm. (Hint: For column with pin-pin ends, L6 = L; for column with ﬁxed-ﬁxed ends, L6 = 0.5L.) Fig. 3: T 3 1 o 6 i (a) (b) Problem 4: (15 marks! A 4-in. by 4-in. by 2-in. (thick) concrete block is subjected to biaxial compression by force PX and Py acting through loading pads, as shown in Fig. 4a. (1) Determine the compressive axial load PX (in kips) if the vertical force is Py = 8 kips and compressive normal stress on the x’ and y’ planes have the value shown in Fig. 4b. (2) Construct a Mohr’s circle of stress at point A. (3) Use this Mohr’s circle to determine the shear stress, Ixeyv, on x’ plane. CIVE-207A Final Examination December 8, 2006 p.4 Problem 5: [20 marks! A cantilever beam has a cross section shown in Fig. 5 and is loaded at the free end. Knowing that the vertical force P = 20 kN acts at the left corner of the section, determine the torque T that would cause the section to twist. All members are to be considered thin walled and calculation should be based on the centerline dimensions. Moment of inertia of the section I = 13.43 x106 mm4. Problem 6: 125 marks! A horizontal L—shaped rod is connected by a taut wire to a cantilever beam, as shown in Fig. 6. If a drop in temperature of 100 0C takes place in wire and a downward force P = 250 N is applied at the end of the cantilever beam, determine the maximum normal stress and maximum shear stress occurred in the structure. Assume the wire carries no stress before the temperature is dropped and the load is applied. All dimensions shown in the ﬁgure are in mm. The diameter of the bent rod, as well as that of the cantilever beam, is 20 mm. The cross sectional area of the wire is 0.4 m2. The assembly is made from steel having E = 200 GPa, G = 80 GPa and coefﬁcient of thermal expansion (1 = 11.7 x 10'6 per 0C. [Hint tan20p=2IXy/(ox-oy)] CIVE-207A Final Examination December 8, 2006 p.5 Useful formulas for section properties: Appendix D. Beam Deflections and Slopes _ Maximum I Beam and Loading Elastic Curve Deflection Slope at End Equation of Elastic Curve _ PP PLZ p 3 3E] _2_EI y = Ea — 3Lx2) WL4 wL3 = — w ( 4 — 4Lx3 2 SE] 6E1 y 24E] x + 6L x2) ...
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