Shigley's Mechanical Engineering Design (McGraw-Hill Series in Mechanical Engineering)
10th Edition
ISBN: 9780073398204
Author: Richard G Budynas, Keith J Nisbett
Publisher: McGraw-Hill Education
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Chapter 3, Problem 35P
3-35 to 3-38 For the beam illustrated in the figure, find the locations and magnitudes of the maximum tensile bending stress due to M and the maximum shear stress due to V.
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Note : Exam is open notes &tables / Answer all questions.
Q.1. The press shown for Figure.1 has a rated load
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threads, a diameter of 50 mm, and a pitch of 6 mm.
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Problem 2 (55 pts). We now consider the FEM solution of Problem 1.(a) [5pts] Briefly describe the 4 steps necessary to obtain the approximate solution of thatBVP using the Galerkin FEM. Use the minimum amount of math necessary to supportyour explanations.(b) [20pts] Derive the weak form of the BVP.(c) [10pts] Assuming a mesh of two equal elements and linear shape functions, sketch byhand how you expect the FEM solution to look like. Also sketch the analytical solutionfor comparison. In your sketch, identify the nodal degrees of freedom that the FEMsolution seeks to find.(d) [10pts] By analogy with the elastic rod problem and heat conduction problem considered in class, write down the stiffness matrix and force vector for each of the twoelements considered in (c).(e) [10pts] Assemble the global system of equations, and verbally explain how to solve it.
Chapter 3 Solutions
Shigley's Mechanical Engineering Design (McGraw-Hill Series in Mechanical Engineering)
Ch. 3 - 31 to 34 Sketch a free-body diagram of each...Ch. 3 - 31 to 34 Sketch a free-body diagram of each...Ch. 3 - Sketch a free-body diagram of each element in the...Ch. 3 - 3-1 to 3-4 Sketch a free-body diagram of each...Ch. 3 - 35 to 38 For the beam shown, find the reactions at...Ch. 3 - 35 to 38 For the beam shown, find the reactions at...Ch. 3 - 35 to 38 For the beam shown, find the reactions at...Ch. 3 - For the beam shown, find the reactions at the...Ch. 3 - For the beam shown, find the reactions at the...Ch. 3 - Repeat Prob. 36 using singularity functions...
Ch. 3 - Repeat Prob. 37 using singularity functions...Ch. 3 - Repeat Prob. 38 using singularity functions...Ch. 3 - For a beam from Table A9, as specified by your...Ch. 3 - A beam carrying a uniform load is simply supported...Ch. 3 - For each of the plane stress states listed below,...Ch. 3 - Repeat Prob. 315 for: (a)x = 28 MPa, y = 7 MPa, xy...Ch. 3 - Repeat Prob. 315 for: a) x = 12 kpsi, y = 6 kpsi,...Ch. 3 - For each of the stress states listed below, find...Ch. 3 - Repeat Prob. 318 for: (a)x = 10 kpsi, y = 4 kpsi...Ch. 3 - The state of stress at a point is x = 6, y = 18, z...Ch. 3 - The state of stress at a point is x = 6, y = 18, z...Ch. 3 - Repeat Prob. 320 with x = 10, y = 40, z = 40, xy =...Ch. 3 - A 34-in-diameter steel tension rod is 5 ft long...Ch. 3 - Repeat Prob. 323 except change the rod to aluminum...Ch. 3 - A 30-mm-diameter copper rod is 1 m long with a...Ch. 3 - A diagonal aluminum alloy tension rod of diameter...Ch. 3 - Repeat Prob. 326 with d = 16 mm, l = 3 m, and...Ch. 3 - Repeat Prob. 326 with d = 58 in, l = 10 ft, and...Ch. 3 - Electrical strain gauges were applied to a notched...Ch. 3 - Repeat Prob. 329 for a material of aluminum. 3-29...Ch. 3 - The Roman method for addressing uncertainty in...Ch. 3 - Using our experience with concentrated loading on...Ch. 3 - The Chicago North Shore Milwaukee Railroad was an...Ch. 3 - For each section illustrated, find the second...Ch. 3 - 3-35 to 3-38 For the beam illustrated in the...Ch. 3 - 3-35 to 3-38 For the beam illustrated in the...Ch. 3 - 3-35 to 3-38 For the beam illustrated in the...Ch. 3 - 3-35 to 3-38 For the beam illustrated in the...Ch. 3 - The figure illustrates a number of beam sections....Ch. 3 - A pin in a knuckle joint canning a tensile load F...Ch. 3 - Repeat Prob. 3-40 for a = 6 mm, b = 18 mm. d = 12...Ch. 3 - For the knuckle joint described in Prob. 3-40,...Ch. 3 - The figure illustrates a pin tightly fitted into a...Ch. 3 - For the beam shown, determine (a) the maximum...Ch. 3 - A cantilever beam with a 1-in-diameter round cross...Ch. 3 - Consider a simply supported beam of rectangular...Ch. 3 - In Prob. 346, h 0 as x 0, which cannot occur. If...Ch. 3 - 348 and 349 The beam shown is loaded in the xy and...Ch. 3 - The beam shown is loaded in the xy and xz planes....Ch. 3 - Two steel thin-wall tubes in torsion of equal...Ch. 3 - Consider a 1-in-square steel thin-walled tube...Ch. 3 - The thin-walled open cross-section shown is...Ch. 3 - 3-53 to 3-55 Using the results from Prob. 3-52,...Ch. 3 - 3-53 to 3-55 Using the results from Prob. 3-52,...Ch. 3 - 3-53 to 3-55 Using the results from Prob. 3-52,...Ch. 3 - Two 300-mm-long rectangular steel strips are...Ch. 3 - Using a maximum allowable shear stress of 70 Mpa,...Ch. 3 - Repeat Prob. 357 with an allowable shear stress of...Ch. 3 - Using an allowable shear stress of 50 MPa,...Ch. 3 - A 20-mm-diameter steel bar is to be used as a...Ch. 3 - A 2-ft-long steel bar with a 34-in diameter is to...Ch. 3 - A 40-mm-diameter solid steel shaft, used as a...Ch. 3 - Generalize Prob. 3-62 for a solid shaft of...Ch. 3 - A hollow steel shaft is to transmit 4200 N m of...Ch. 3 - The figure shows an endless-bell conveyor drive...Ch. 3 - The conveyer drive roll in the figure for Prob....Ch. 3 - Consider two shafts in torsion, each of the same...Ch. 3 - 3-68 to 3-71 A countershaft two V-belt pulleys is...Ch. 3 - 3-68 to 3-71 A countershaft two V-belt pulleys is...Ch. 3 - 3-68 to 3-71 A countershaft two V-belt pulleys is...Ch. 3 - A countershaft carrying two V-belt pulleys is...Ch. 3 - A gear reduction unit uses the countershaft shown...Ch. 3 - Prob. 73PCh. 3 - Prob. 74PCh. 3 - Prob. 75PCh. 3 - Prob. 76PCh. 3 - Prob. 77PCh. 3 - Prob. 78PCh. 3 - Prob. 79PCh. 3 - The cantilevered bar in the figure is made from a...Ch. 3 - Repeat Prob. 3-80 with Fx = 0, Fy = 175 lbf, and...Ch. 3 - Repeat Prob. 3-80 with Fx = 75 lbf, Fy= 200 lbf,...Ch. 3 - For the handle in Prob. 3-80, one potential...Ch. 3 - The cantilevered bar in the figure is made from a...Ch. 3 - Repeat Prob. 3-84 with Fx = 300 lbf, Fy = 250 lbf,...Ch. 3 - Repeat Prob. 3-84 with Fx = 300 lbf, Fy = 250 lbf,...Ch. 3 - Repeat Prob. 3-84 for a brittle material,...Ch. 3 - Repeat Prob. 3-84 with Fx = 300 lbf, Fy = 250 lbf,...Ch. 3 - Repeat Prob. 3-84 with Fx = 300 lbf, Fy = 250 lbf,...Ch. 3 - The figure shows a simple model of the loading of...Ch. 3 - Develop the formulas for the maximum radial and...Ch. 3 - Repeat Prob. 391 where the cylinder is subject to...Ch. 3 - Develop the equations for the principal stresses...Ch. 3 - 3-94 to 3-96 A pressure cylinder has an outer...Ch. 3 - 3-94 to 3-96 A pressure cylinder has an outer...Ch. 3 - 3-94 to 3-96A pressure cylinder has an outer...Ch. 3 - 3-97 to 3-99 A pressure cylinder has an outer...Ch. 3 - 3-97 to 3-99 A pressure cylinder has an outer...Ch. 3 - 3-97 to 3-99 A pressure cylinder has an outer...Ch. 3 - An AISI 1040 cold-drawn steel tube has an OD = 50...Ch. 3 - Repeat Prob. 3-100 with an OD of 2 in and wall...Ch. 3 - Prob. 102PCh. 3 - Prob. 103PCh. 3 - A thin-walled cylindrical Steel water storage tank...Ch. 3 - Repeat Prob. 3-104 with the tank being pressurized...Ch. 3 - Find the maximum shear stress in a 512-in-diameter...Ch. 3 - The maximum recommended speed for a...Ch. 3 - An abrasive cutoff wheel has a diameter of 5 in,...Ch. 3 - A rotary lawnmower blade rotates at 3500 rev/min....Ch. 3 - 3110 to 3115 The table lists the maximum and...Ch. 3 - Prob. 111PCh. 3 - Prob. 112PCh. 3 - 3110 to 3115 The table lists the maximum and...Ch. 3 - Prob. 114PCh. 3 - Prob. 115PCh. 3 - 3116 to 3119 The table gives data concerning the...Ch. 3 - Prob. 117PCh. 3 - Prob. 118PCh. 3 - 3116 to 3119 The table gives data concerning the...Ch. 3 - A utility hook was formed from a round rod of...Ch. 3 - A utility hook was formed from a round rod of...Ch. 3 - The steel eyebolt shown in the figure is loaded...Ch. 3 - For Prob. 3122 estimate the stresses at the inner...Ch. 3 - Repeat Prob. 3122 with d = 14 in, Ri = 12 in, and...Ch. 3 - Repeat Prob. 3123 with d = 14 in, Ri = 12 in, and...Ch. 3 - Shown in the figure is a 12-gauge (0.1094-in) by...Ch. 3 - Repeat Prob. 3126 with a 10-gauge (0.1406-in)...Ch. 3 - Prob. 128PCh. 3 - The cast-iron bell-crank lever depicted in the...Ch. 3 - Prob. 130PCh. 3 - Prob. 131PCh. 3 - A cast-steel C frame as shown in the figure has a...Ch. 3 - Two carbon steel balls, each 30 mm in diameter,...Ch. 3 - A carbon steel ball with 25-mm diameter is pressed...Ch. 3 - Repeat Prob. 3134 but determine the maximum shear...Ch. 3 - A carbon steel ball with a 30-mm diameter is...Ch. 3 - An AISI 1018 steel ball with 1-in diameter is used...Ch. 3 - An aluminum alloy cylindrical roller with diameter...Ch. 3 - A pair of mating steel spur gears with a 0.75-in...Ch. 3 - 3140 to 3142 A wheel of diameter d and width w...Ch. 3 - 3140 to 3142 A wheel of diameter d and width w...Ch. 3 - 3140 to 3142 A wheel of diameter d and width w...
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- An aluminum rod of length L = 1m has mass density ρ = 2700 kgm3 andYoung’s modulus E = 70GPa. The rod is fixed at both ends. The exactnatural eigenfrequencies of the rod are ωexactn =πnLqEρfor n=1,2,3,. . . .1. What is the minimum number of linear elements necessary todetermine the fundamental frequency ω1 of the system? Discretizethe rod in that many elements of equal length, assemble the globalsystem of equations KU = ω2MU, and find the fundamentalfrequency ω1. Compute the relative error e1 = (ω1 − ωexact1)/ωexact1.Sketch the fundamental mode of vibration.arrow_forwardProblem 1 (65 pts, suggested time 50 mins). An elastic string of constant line tension1T is pinned at x = 0 and x = L. A constant distributed vertical force per unit length p(with units N/m) is applied to the string. Under this force, the string deflects by an amountv(x) from its undeformed (horizontal) state, as shown in the figure below.The PDE describing mechanical equilibrium for the string isddx Tdvdx− p = 0 . (1)(a) [5pts] Identify the BCs for the string and identify their type (essential/natural). Writedown the strong-form BVP for the string, including PDE and BCs.(b) [10pts] Find the analytical solution of the BVP in (a). Compute the exact deflectionof the midpoint v(L/2).(c) [15pts] Derive the weak-form BVP.(d) [5pts] What is the minimum number of linear elements necessary to compute the deflection of the midpoint?(e) [15pts] Write down the element stiffness matrix and the element force vector for eachelement.arrow_forwardProblem 1 (35 pts). An elastic string of constant line tension1 T is pinned at x = 0 andx = L. A constant distributed vertical force per unit length p (with units N/m) is appliedto the string. Under this force, the string deflects by an amount v(x) from its undeformed(horizontal) state, as shown in the figure below.Force equilibrium in the string requires thatdfdx − p = 0 , (1)where f(x) is the internal vertical force in the string, which is given byf = Tdvdx . (2)(a) [10pts] Write down the BVP (strong form) that the string deflection v(x) must satisfy.(b) [2pts] What order is the governing PDE in the BVP of (a)?(c) [3pts] Identify the type (essential/natural) of each boundary condition in (a).(d) [20pts] Find the analytical solution of the BVP in (a).arrow_forward
- Problem 2 (25 pts, (suggested time 15 mins). An elastic string of line tension T andmass per unit length µ is pinned at x = 0 and x = L. The string is free to vibrate, and itsfirst vibration mode is shown below.In order to find the frequency of the first mode (or fundamental frequency), the string isdiscretized into a certain number of linear elements. The stiffness and mass matrices of thei-th element are, respectivelyESMi =TLi1 −1−1 1 EMMi =Liµ62 11 2 . (2)(a) [5pts] What is the minimum number of linear elements necessary to compute the fundamental frequency of the vibrating string?(b) [20pts] Assemble the global eigenvalue problem and find the fundamental frequency ofvibration of the stringarrow_forwardI need part all parts please in detail (including f)arrow_forwardProblem 3 (10 pts, suggested time 5 mins). In class we considered the mutiphysics problem of thermal stresses in a rod. When using linear shape functions, we found that the stress in the rod is affected by unphysical oscillations like in the following plot E*(ux-a*T) 35000 30000 25000 20000 15000 10000 5000 -5000 -10000 0 Line Graph: E*(ux-a*T) MULT 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Arc length (a) [10pts] What is the origin of this issue and how can we fix it?arrow_forward
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