Manufacturing Engineering & Technology
7th Edition
ISBN: 9780133128741
Author: Serope Kalpakjian, Steven Schmid
Publisher: Prentice Hall
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Textbook Question
Chapter 2, Problem 65QTP
Using only Fig. 2.5, calculate the maximum load in tension testing of an annealed copper specimen with an original diameter of 5 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.
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.
Chapter 2 Solutions
Manufacturing Engineering & Technology
Ch. 2 - Distinguish between engineering stress and true...Ch. 2 - In a stress-strain curve, what is the proportional...Ch. 2 - Describe the events that take place when a...Ch. 2 - What is ductility, and how is it measured?Ch. 2 - In the equation =Kn, which represents the true...Ch. 2 - What is strain-rate sensitivity, and how is it...Ch. 2 - What test can measure the properties of a material...Ch. 2 - What testing procedures can be used to measure the...Ch. 2 - Describe the differences between brittle and...Ch. 2 - What is hardness? Explain.
Ch. 2 - Describe the features of a Rockwell hardness test.Ch. 2 - What is a Leeb test? How is it different from a...Ch. 2 - Differentiate between stress relaxation and creep.Ch. 2 - Describe the difference between elastic and...Ch. 2 - Explain what uniform elongation means in tension...Ch. 2 - Describe the difference between deformation rate...Ch. 2 - Describe the difficulties involved in conducting a...Ch. 2 - What is Hookes law? Youngs modulus? Poissons...Ch. 2 - Describe the difference between transgranular and...Ch. 2 - What is the reason that yield strength is...Ch. 2 - Why does the fatigue strength of a specimen or...Ch. 2 - If striations are observed under microscopic...Ch. 2 - What is an Izod test? Why are Izod tests useful?Ch. 2 - Why does temperature increase during plastic...Ch. 2 - What is residual stress? How can residual stresses...Ch. 2 - On the same scale for stress, the tensile true...Ch. 2 - What are the similarities and differences between...Ch. 2 - Can a material have a negative Poissons ratio?...Ch. 2 - It has been stated that the higher the value of m,...Ch. 2 - Explain why materials with high m values, such as...Ch. 2 - With a simple sketch, explain whether it is...Ch. 2 - Explain why the difference between engineering...Ch. 2 - Consider an elastomer, such as a rubber band. This...Ch. 2 - If a material (such as aluminum) does not have an...Ch. 2 - What role, if any, does friction play in a...Ch. 2 - Which hardness tests and scales would you use for...Ch. 2 - Consider the circumstance where a Vickers hardness...Ch. 2 - Which of the two tests, tension or compression,...Ch. 2 - List and explain briefly the conditions that...Ch. 2 - List the factors that you would consider in...Ch. 2 - On the basis of Fig. 2.5, can you calculate the...Ch. 2 - If a metal tension-test specimen is rapidly pulled...Ch. 2 - Comment on your observations regarding the...Ch. 2 - Will the disk test be applicable to a ductile...Ch. 2 - What hardness test is suitable for determining the...Ch. 2 - Wire rope consists of many wires that bend and...Ch. 2 - A statistical sampling of Rockwell C hardness...Ch. 2 - In a Brinell hardness test, the resulting...Ch. 2 - Some coatings are extremely thinsome as thin as a...Ch. 2 - Select an appropriate hardness test for each of...Ch. 2 - A paper clip is made of wire 0.5 mm in diameter....Ch. 2 - A 250-mm-long strip of metal is stretched in two...Ch. 2 - Identify the two materials in Fig. 2.5 that have...Ch. 2 - Plot the ultimate strength vs. stiffness for the...Ch. 2 - If you remove the layer of material ad from the...Ch. 2 - Prove that the true strain at necking equals the...Ch. 2 - Percent elongation is always defined in terms of...Ch. 2 - You are given the K and n values of two different...Ch. 2 - A cable is made of two strands of different...Ch. 2 - On the basis of the information given in Fig. 2.5,...Ch. 2 - In a disk test performed on a specimen 1.00 in. in...Ch. 2 - A piece of steel has a hardness of 300 HB....Ch. 2 - A metal has the following properties: UTS = 70,000...Ch. 2 - Using only Fig. 2.5, calculate the maximum load in...Ch. 2 - Estimate the modulus of resilience for a highly...Ch. 2 - A metal has a strength coefficient K = 100,000 psi...Ch. 2 - Plot the true stresstrue strain curves for the...Ch. 2 - The design specification for a metal requires a...Ch. 2 - Calculate the major and minor pyramid angles for a...Ch. 2 - If a material has a target hardness of 300 HB,...Ch. 2 - A Rockwell A test was conducted on a material and...Ch. 2 - For a cold-drawn 0.5% carbon steel, will a...Ch. 2 - A material is tested in tension. Over a 1-in. gage...Ch. 2 - A horizontal rigid bar cc is subjecting specimen a...Ch. 2 - List and explain the desirable mechanical...Ch. 2 - When making a hamburger, you may have observed the...Ch. 2 - An inexpensive claylike material called Silly...Ch. 2 - In tension testing of specimens, mechanical and...Ch. 2 - Demonstrate the impact toughness of a piece of...Ch. 2 - Using a large rubber band and a set of weights,...Ch. 2 - Find or prepare some solid circular pieces of...Ch. 2 - Take several rubber bands and pull them at...Ch. 2 - Devise a simple fixture for conducting the bend...Ch. 2 - By pressing a small ball bearing against the top...Ch. 2 - Describe your observations regarding Fig. 2.14c.Ch. 2 - Embed a small steel ball in a soft block of...Ch. 2 - Devise a simple experiment, and perform tests on...Ch. 2 - Obtain some solid and some tubular metal pieces,...Ch. 2 - Explain how you would obtain an estimate of the...Ch. 2 - Without using the words stress or strain, define...Ch. 2 - We know that it is relatively easy to subject a...
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- Problem 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_forwardProblem 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_forward
- I 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_forwardanswer the questions and explain all of it in words. Ignore where it says screencast and in class explanationarrow_forward
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