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Unformatted text preview: McGili University
Faculty of Engineering Department of Civil Engineering and Applied Mechanics
CIVE 207 Solid Mechanics FINAL EXAMINATION
Tuesday, December 21, 2004 — 2:00pm ~ 5:00pm Examiner: Dr. C.K. Manatakos Associate Examiner: Professor G. McClure STUDENT NAME: MCGILL ID. NUMBER: Instructions and Notes: 1. This is a CLOSED BOOK examination.
Notes, tests, papers, reports, assignments, etc., are NOT permitted. 2. ONLY a FACULTY STANDARD CALCULATOR is pennitted. 3. This examination consists of FIVE (5) questions.
Attempt ALL questions, which are of equal value for a total of 100%. 4. Write each answer in the space provided. If additional space is needed, use the reverse side
of the page, clearly indicating the number of the question being answered. 5. Show all of the calculations steps in arriving at the ﬁnal answers. 6. Illustrate the answers using. NEAT sketches, as needed, preferably using pencil.
7. State all assumptions clearly. 8. The following APPENDICES are included with this examination: (A) Fundamental Equations of Mechanics of Materials
(B) Geometric Properties of Plane Areas (C) Deﬂections and Slopes of Cantilever and SimplySupported Beams 


 Dr. CK. Manatakos FINAL EXAMINATION Page 1 of 32 General Theory Problem 11 (a) Flexure and Shear in Beams: Deﬁne the ﬂexure formula, brieﬂy explain every term using; sketches and list the assumptions made for its derivation with appropriate Sketches. Sketch the normal strain and stress distributions for [leicure (positive bending) in a. beam of rectangular crosssection. Explain using sketches and equations, how the normal strain and stresses are related? Label all terms and Show proper units. Indicate locations of maximum and minimum values. Dr. C. K. Manatakos FINAL EXAMINATION General Theory Problem 1—1(b) Flexure and Shear in Beams: (cont‘d Deﬁne the shear stress formula, brieﬂy explain every term usingr sketches and list the assumptions made for its derivation with appropriate sketches. Sketch and label the shear stress distribution and the shear ﬂow for: [i] a beam of rectangular crossesection; [ii] an lbearn crosssection. Explain using sketches and equations, how the shear strain and shear stress are related? Label all terms and show proper units. Indicate locations of maximum and minimum values. Dr. C. K. Manatak03 FINAL EXAMINATION "'“""""’““1“”W2‘13ﬁ w: xawﬁmviy = .. Fig. P 1(b) Stress Points A to E on Beam Crosssection General Theory Problem 1—2 Combined Loadings — Stress Distribution in Beams: For the thin wall wideﬂange lsection beam subjected to a concentrated load at midspan as shown
in Fig. 1 (a), ﬁve critical points are identiﬁed and labelled as A, B, C, D and E from the top to the bottom of the crosssection as shown in Fig. 1 Sketch the plane stress elements at these points, showingr the normal and the shear stresses for the following conditions: [i] the x — 3; stress components in the longitudinal and vertical axes of the beam;
[ii] the principal stresses, showing the appropriate element orientations; [iii] the maximum shear stresses and accompanying average normal stresses, showing the appropriate element orientations. Figure P1 (3.) Figure P1 (b) Dr. C. K. Manatakos FINAL EXAMINATION Page 6 of 32 General Theory Problem 13 (a) Column Buckling:
Deﬁne Euler’s equation for the critical buckling load and stress, brieﬂy explain every term using; sketches and list the assumptions made for its derivation with appropriate sketches. How does this formula consider the different column end support conditions? Sketch columns with the different types of end conditions, illustrating the relationship to Euler’s equation. Dr. C. K. Manatakos FINAL EXAMLNATION ————————— General Theory Problem 13(b) Column Buckling: (contid Sketch on ONE diagram, the axial compressive load versus the maximum transverse deflection curves for the buckling response of a, slender column and label the critical load, for the following cases :
[i] the column is perfectly straight with no imperfections
and there is no eccentricity of the load (e : 0); [ii] the column is imperfect with an initial deﬂection (£10) at mid—height and there is no eccentricity of the load (:3 = 0); [iii] the eccentricity of the load increases from c z 0 to e 2 Dr. C. K. Manatakos FINAL EXAMINA'I‘ION Laboratory Theory Problem 2—1(a) Torsion Laboratory:
Sketch on ONE diagram1 the torque versus twist response to failure for specimens of a: [1} mild steel cylindrical coupon; [ii] cast iron cylindrical coupon. Label all regions and basic characteristics of each curve, and identify the ductility characteristics for each specimen. Sketch a typical shear stress versus shear strain diagram
for a perfectly elasticplastic bilinear material response in torsion, explaining brieﬂy the mathematical relationship between the shear stress and shear strain. Dr. C. K. Manatakos FINAL EXAMINATION Page 11 of 32 Laboratory Theory Problem 2~1(b) Torsion Laboratory: (cont’d Sketch the failure mode and crosssectional fracture surfaces, and indicate the torsional failure modes for the Specimens tested: as listed below : Show the shear ﬂow distribution on a typical crosssection, indicating the location of the maximum values. [i] solid cylindrical coupon of mild steel; [ii] solid cylindrical coupon of cast. iron; [iii] solid cylindrical coupon of wood; [iv] a solid rectangular bar of mild steel. Dr. C. K. Manatakos FINAL EXAMINATION Page 1: Laboratory Theory Problem 22 (a) Beam Bending Laboratory:
Sketch 011 ONE diagram, the midispan concentrated load versus the vertical deﬂection response to failure for: [i] the mild steel I—beam tested; [ii] the timber beam tested. Label all regions and basic characteristics of each curve, and identify the ductility characteristics for each beam. Dr. C. K. Manatakos FINAL EXAMINATION Structural Steel IBeam Aluminum IBeam Steel Rectangular Bar Timber Square Beam Fig. P 2 Failure Mode of Beams Tested Laboratory T heory Problem 22 (h) Beam Bending Laboratory: (cont’d Identify the failure mode for each of the four beams tested as shown in each photograph, respectively,
on Fig. P2, for: [i] mild steel Ibeam: [ii] aluminium I—bearm : [iii] mild steel rectangular bar lJCELIIL: [iv] timber square beam: Figure P2 Dr. C. K. Manatakos FINAL EXAMINATION Page 14 of 32 Laboratory Theory Problem 23 (a) Column Buckling Laboratory: Sketch the Southwell plot for a concentrically loaded slender column with an initial curvature Label all the. characteristics of this plot, and identify the critical buckling load. Dr. C. K. Manatakos FINAL EXAMINATION Page 15 Laboratory Theory Problem 23 (1)) Column Buckling Laboratory: (cont’d Sketch the critical stress versus the slenderness ratio curve for
compression or “squashing” failure and buckling failure
of a. concentrically loaded “ideal” column with no imperfections, for all possible column lengths. Identify and mark the regions of the different failure modes, sketch an example for each type, and list the normal stress equation and Euler’s equation in the proper regions Dr. C. K. Manatakos FINAL EXAMINATION Fig. P 3(a) Fixed Ended Beam Oak Board
w = 12 in
d = 4m
I—Beam
df = 8.4 in
tw = 0.4 in Fig. P 3(b) Composite Beam Crosssection Problem 3 For the ﬁxedended beam with dimensions and loading as shown in Fig. P3 (a): a) Using superposition and the ﬁxed end B for the redundant forces, determine the reactions at the ﬁxed ends A and H in Imperial units (kip  ft), assuming that the ﬂexural rigidity BI is constant. Neglect the effects of axial load. Sketch the elastic curve of the deﬂected shape, and the shear force and. the bending moment diagrams, indicating the maxima and minima values and their locations. b) If the beam crosssection is composed of a steel llieam section. alongwith an oak board
attached on top of the I—section between the flanges, as shown in Fig. P3 (b):
Est : 29,000 ksi, GS; : 11.000 ksi, Iyy : 98.2 £72.41 [3; : 20.:3i71.4, A : 8.79 i712
Em = 1,600 ksi, Gw : 660 last Determine the maximum normal stresses in Imperial units (16.327)
that occur in the Ibeam and in the oak board. Identify their locations. Note: ilkip = 1,000le
1ft 2 12m test = hips per square inch Figure P3 (a) Figure P3 (b) Point A /V,
on Surface Eiilii} ﬂ} Fig. P 4 Solid Circular Rod Problem 4 A cantilever solid Circular shaft of dimensions as shown in Fig. P4 and crosssectional radius of
r = 0.75172, is subjected to a vertical force of 800th and a horizontal force of 500er at the free
end. Point A is located at the outer surface of the crosssection, on the inner side of the rod in the direction of the positive :EElXlS from the ﬁxed end support. a) Determine the state of stress at point A. Sketch a freebody diagram showing the calculated resultant internal loadings for the
axial and shear forces, and the bending and twisting moments acting at the cross~section at point A. Calculate the normal and the shear stresses for each resultant forces and moments. and
sketch the distribution of these stresses acting separately. indicating the maximum and minimum values. on appropriate cross—sections and material elements at point A. Sketch 011 ONE element1 the resultant normal and shearing stresses at point A. to) Determine the principal stresses, the maximum inplane shear stress and the accompanying average normal stresses. Show the stress values and the orientation of these planes on appropriate element sketches. Show all details using Mohr’s circle. Give all answers in Imperial units (lb, lb  in and test). Note: k‘S’t 2 tips per square inch
Figure P4
Dr. C. K. Manatakos FINAL EXAMINA’l‘lON Page 22 of 32 Fig. P 5 Cylindrical Pressure Vessel Problem 5 A cylindrical pressure vessel shown in Fig. P5, is constructed from a long, narrow steel plate by
wrapping the plate around a mandrel and buttwelding along the edges of the plate to form a helical
joint. The helical weld makes an angle of a = 55° with the longitudinal axis. The pressure vessel has an inner radius of r = 1.80 m and a wall thickness of t = 20 mm. For steel : E = 200 GPa, G = 80 em,
V = 0.30, 7 = 77 kN/m3, p = 785019g/m3.
“u” 2 850 MM 0y = O20 MP6s as : 260 MPa, and a : 17 x inﬁ/OC. For a maximum internal pressure of pi : 800 131%,. calculate the following quantities for the cylindrical part of the pressure vessel : a) the principal inplane normal stresses, and the maximum iiiplane shear stress and the accompanying average normal stresses; h) the absolute maximum outofplane shear stress and the accompanying average normal stresses;
c) the circumferential hoop and the longitudinal axial. strains (1 and rig. respectively: (1) the normal stress1 the tangential stress and the shear stresses, acting perpendicular and parallel, respectively, to the welded seam line; Identify all the calculated stresses values and angles on Molan circle. Present properly orientated sketches of the different elements showing the stresses. Mandrel = a cylindrical rod, round which metal is forged or shaped. Buttweld : welding together of two ﬂat ends of metal plates. Figure P5 Dr. C. K. Manatakos FINAL EXAMINATION Page 27 of 32 ...
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