Mechanics of Materials, 7th Edition
Mechanics of Materials, 7th Edition
7th Edition
ISBN: 9780073398235
Author: Ferdinand P. Beer, E. Russell Johnston Jr., John T. DeWolf, David F. Mazurek
Publisher: McGraw-Hill Education
bartleby

Videos

Question
Book Icon
Chapter 4.6, Problem 92P

(a)

To determine

Find the residual stress at y=2in.

(a)

Expert Solution
Check Mark

Answer to Problem 92P

The residual stress is σres=13.36ksi_.

Explanation of Solution

Given information:

The yield stress for the beam is σY=42ksi.

The Young’s modulus of steel is E=29×106psi.

Calculation:

Show the cross-section of the beam as shown in Figure 1.

Mechanics of Materials, 7th Edition, Chapter 4.6, Problem 92P , additional homework tip  1

Refer to Figure 1.

Calculate the area of the cross section (A) as shown below.

A=bd (1)

Here, b is the width of the cross section and d is the depth of the cross section.

Calculate the area of the portion (1) (A1) as shown below.

Substitute 3in. for b and 1in. for d in Equation (1).

A1=3×1=3in.2

Calculate the area of the portion (2) (A2) as shown below.

Substitute 1in. for b and 1in. for d in Equation (1).

A2=1×1=1in.2

Calculate the moment of inertia (I) as shown below.

I=bd312+Ah2 (2)

Calculate the moment of inertia of portion (1) (I1) as shown below.

Substitute 3in. for b, 1in. for d, 3in.2 for A and 1.5in. for h in Equation (2).

I1=3×1312+3×1.52=7in.4

Hence, I3=7in.4

Calculate the moment of inertia of portion (2) (I2) as shown below.

Substitute 1in. for b, 2in. for d, 2in.2 for A and 0 for h in Equation (2).

I2=1×2312+0=0.667in.4

Calculate the total moment of inertia (I) as shown below.

I=I1+I2+I3

Substitute 7in.4 for I1, 0.667in.4 for I2, and 7in.4 for I3.

I=7+0.667+7=14.667in.4

Calculate the centroid (c) as shown below.

c=h2

Substitute 4in. for h.

c=42=2in.

Sketch the stress acting on the cross-section of the beam as shown in Figure 2.

Mechanics of Materials, 7th Edition, Chapter 4.6, Problem 92P , additional homework tip  2

Refer Figure 2.

Calculate the area of the portion (2) (A2) as shown below.

A2=12bh

Substitute 1in. for b and 1in. for d.

A2=12×1×1=0.5in.2

Calculate the reaction applied to portion (1) (R1) as shown below.

R1=σYA1

Substitute 42ksi for σY and 3in.2 for A1.

R1=42×3=126kips

Calculate the reaction applied to portion (2) (R2) as shown below.

R2=σYA2

Substitute 42ksi for σY and 0.5in.2 for A2.

R2=42×0.5=21kips

Calculate the moment (M) as shown below.

M=2(R1y1+R2y2)

Substitute 126kips for R1, (1+12)in. for y1, 21kips for R2, and (23×1)in. for y2.

M=2(126×(1+12)+21×(23×1))=406kipin.

Calculate the stress (σ) as shown below.

σ=McI

Substitute 406kipin. for M, 2in. for c, and 14.667in.4 for I.

σ=406×214.667=55.362ksi

Calculate the stress (σ) as shown below.

σ=MyI

Substitute 406kipin. for M, 1in. for y, and 14.667in.4 for I.

σ=406×114.667=27.681ksi

Calculate the residual stress at y=c as shown below.

σres=σσY

Substitute 55.362ksi for σ and 42ksi for σY.

σres=55.36242=13.362ksi

Calculate the residual stress at y=yY as shown below.

σres=σσY

Substitute 27.681ksi for σ and 42ksi for σY.

σres=27.68142=14.319ksi

Sketch the stress distribution as shown in Figure 3.

Mechanics of Materials, 7th Edition, Chapter 4.6, Problem 92P , additional homework tip  3

Hence, the residual stress is σres=13.36ksi_.

(b)

To determine

Find the point where the residual stress is zero.

(b)

Expert Solution
Check Mark

Answer to Problem 92P

The point where the residual stress is zero is y0=1.517in., 0, 1.517in_.

Explanation of Solution

Given information:

The yield stress for the beam is σY=42ksi.

The Young’s modulus for steel is E=29×106psi.

Calculation:

Consider that the residual stress is σres=0.

Calculate the yield stress (σ) as shown below.

σ=My0I

Calculate the point where the residual stress is zero as shown below.

σres=σσY

Substitute My0I for σ and 0 for σres.

0=My0IσYσY=My0Iy0=σYIM

Substitute 42ksi for σY, 406kipin. for M, and 14.667in.4 for I.

y0=42×14.667406=1.517in.

Therefore, the point where the residual stress is zero is y0=1.517in., 0, 1.517in_.

(c)

To determine

Find the radius of curvature corresponding to the permanent deformation of the bar.

(c)

Expert Solution
Check Mark

Answer to Problem 92P

The radius of curvature is ρ=168.8ft_.

Explanation of Solution

Given information:

The yield stress for the beam is σY=42ksi.

The Young’s modulus of steel is E=29×106psi.

Calculation:

Refer to part (a).

The residual stress σres=14.32ksi.

Calculate the radius of curvature (ρ) as shown below.

σ=My0I

Calculate the point where the residual stress is zero as shown below.

ρ=Eyσ

Substitute 29×106psi for E, 14.32ksi for σ and 1in. for y.

ρ=29×106psi×1ksi1,000psi×1in.14.32ksi=2,025.14in.×1ft12in.=168.8ft

Therefore, the radius of curvature is ρ=168.8ft_.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Students have asked these similar questions
In a laboratory test of a beam loaded by end couples, the fiber at layer AB as shown are found to increase 50 x 10 3 mm while those at CD decrease 70 x 103 mm in the 230- mm-gage length. Using E = 170 GPa, determine the flexural stress at the bottom fiber. Answer must be in MPa. Given: a = 50 mm, b = 140 mm, and C = 80 mm. %3D gage length a mm b mm c mm B.
Please answer part a, b,c
Question 1: A member having the dimensions shown is used to resist an internal bending moment of M kNm. Determine the maximum stress in the member if the moment is applied (a) about the z axis (as shown) (b) about the y axis. Sketch the stress distribution for each case. Take: M= 98 kNm mm A= 208 mm B= 158 mm B mm Solution: The moment of inertia of the cross-section about z and y axes are 1 AB³ 12 |(10-) m* 1 ВАЗ — 12 I, |(10) m* = For the bending about z axis, c = m Mc O pax MPа Iz For the bending about y axis, c = m Mc MPа Iy max z MPa KN=M Omax Y MPa. M KN-M MPa O max Z Omax Y MPa

Chapter 4 Solutions

Mechanics of Materials, 7th Edition

Ch. 4.3 - 4.9 through 4.11 Two vertical forces are applied...Ch. 4.3 - Knowing that a beam of the cross section shown is...Ch. 4.3 - Knowing that a beam of the cross section shown is...Ch. 4.3 - Solve Prob. 4.13, assuming that the beam is bent...Ch. 4.3 - Knowing that for the extruded beam shown the...Ch. 4.3 - The beam shown is made of a nylon for which the...Ch. 4.3 - Solve Prob. 4.16, assuming that d = 40 mm.Ch. 4.3 - Knowing that for the beam shown the allowable...Ch. 4.3 - 4.19 and 4.20 Knowing that for the extruded beam...Ch. 4.3 - 4.19 and 4.20 Knowing that for the extruded beam...Ch. 4.3 - Straight rods of 6-mm diameter and 30-m length are...Ch. 4.3 - A 900-mm strip of steel is bent into a full circle...Ch. 4.3 - Straight rods of 0.30-in. diameter and 200-ft...Ch. 4.3 - A 60-Nm couple is applied to the steel bar shown,...Ch. 4.3 - (a) Using an allowable stress of 120 MPa,...Ch. 4.3 - A thick-walled pipe is bent about a horizontal...Ch. 4.3 - A couple M will be applied to a beam of...Ch. 4.3 - A portion of a square bar is removed by milling,...Ch. 4.3 - In Prob. 4.28, determine (a) the value of h for...Ch. 4.3 - For the bar and loading of Concept Application...Ch. 4.3 - Prob. 31PCh. 4.3 - It was assumed in Sec. 4.1B that the normal...Ch. 4.5 - 4.33 and 4.34 A bar having the cross section shown...Ch. 4.5 - 4.33 and 4.34 A bar having the cross section shown...Ch. 4.5 - 4.35 and 4.36 For the composite bar indicated,...Ch. 4.5 - Prob. 36PCh. 4.5 - 4.37 and 4.38 Wooden beams and steel plates are...Ch. 4.5 - 4.37 and 4.38 Wooden beams and steel plates are...Ch. 4.5 - 4.39 and 4.40 A copper strip (Ec = 105 GPa) and an...Ch. 4.5 - 4.39 and 4.40 A copper strip (Ec = 105 GPa) and an...Ch. 4.5 - 4.41 and 4.42 The 6 12-in. timber beam has been...Ch. 4.5 - 4.41 and 4.42 The 6 12-in. timber beam has been...Ch. 4.5 - 4.43 and 4.44 For the composite beam indicated,...Ch. 4.5 - Prob. 44PCh. 4.5 - Prob. 45PCh. 4.5 - Prob. 46PCh. 4.5 - A concrete slab is reinforced by 58-in.-diameter...Ch. 4.5 - Solve Prob. 4.47, assuming that the spacing of the...Ch. 4.5 - The reinforced concrete beam shown is subjected to...Ch. 4.5 - Prob. 50PCh. 4.5 - Knowing that the bending moment in the reinforced...Ch. 4.5 - A concrete beam is reinforced by three steel rods...Ch. 4.5 - The design of a reinforced concrete beam is said...Ch. 4.5 - For the concrete beam shown, the modulus of...Ch. 4.5 - 4.55 and 4.56 Five metal strips, each 0.5 1.5-in....Ch. 4.5 - 4.55 and 4.56 Five metal strips, each 0.5 1.5-in....Ch. 4.5 - The composite beam shown is formed by bonding...Ch. 4.5 - A steel pipe and an aluminum pipe are securely...Ch. 4.5 - The rectangular beam shown is made of a plastic...Ch. 4.5 - Prob. 60PCh. 4.5 - Knowing that M = 250 Nm, determine the maximum...Ch. 4.5 - Knowing that the allowable stress for the beam...Ch. 4.5 - Semicircular grooves of radius r must be milled as...Ch. 4.5 - Prob. 64PCh. 4.5 - A couple of moment M = 2 kNm is to be applied to...Ch. 4.5 - The allowable stress used in the design of a steel...Ch. 4.6 - The prismatic bar shown is made of a steel that is...Ch. 4.6 - Prob. 68PCh. 4.6 - Prob. 69PCh. 4.6 - Prob. 70PCh. 4.6 - The prismatic rod shown is made of a steel that is...Ch. 4.6 - Solve Prob. 4.71, assuming that the couples M and...Ch. 4.6 - 4.73 and 4.74 A beam of the cross section shown is...Ch. 4.6 - 4.73 and 4.74 A beam of the cross section shown is...Ch. 4.6 - 4.75 and 4.76 A beam of the cross section shown is...Ch. 4.6 - Prob. 76PCh. 4.6 - 4.77 through 4.80 For the beam indicated,...Ch. 4.6 - Prob. 78PCh. 4.6 - Prob. 79PCh. 4.6 - 4.77 through 4.80 For the beam indicated,...Ch. 4.6 - 4.81 through 4.83 Determine the plastic moment Mp...Ch. 4.6 - Prob. 82PCh. 4.6 - Prob. 83PCh. 4.6 - Determine the plastic moment Mp of a steel beam of...Ch. 4.6 - Determine the plastic moment Mp of the cross...Ch. 4.6 - Determine the plastic moment Mp of a steel beam of...Ch. 4.6 - Prob. 87PCh. 4.6 - Prob. 88PCh. 4.6 - Prob. 89PCh. 4.6 - Prob. 90PCh. 4.6 - Prob. 91PCh. 4.6 - Prob. 92PCh. 4.6 - Prob. 93PCh. 4.6 - Prob. 94PCh. 4.6 - Prob. 95PCh. 4.6 - Prob. 96PCh. 4.6 - Prob. 97PCh. 4.6 - Prob. 98PCh. 4.7 - Knowing that the magnitude of the horizontal force...Ch. 4.7 - A short wooden post supports a 6-kip axial load as...Ch. 4.7 - Two forces P can be applied separately or at the...Ch. 4.7 - A short 120 180-mm column supports the three...Ch. 4.7 - As many as three axial loads, each of magnitude P...Ch. 4.7 - Two 10-kN forces are applied to a 20 60-mm...Ch. 4.7 - Portions of a 1212-in. square bar have been bent...Ch. 4.7 - Knowing that the allowable stress in section ABD...Ch. 4.7 - A milling operation was used to remove a portion...Ch. 4.7 - A milling operation was used to remove a portion...Ch. 4.7 - The two forces shown are applied to a rigid plate...Ch. 4.7 - Prob. 110PCh. 4.7 - Prob. 111PCh. 4.7 - A short column is made by nailing four 1 4-in....Ch. 4.7 - A vertical rod is attached at point A to the cast...Ch. 4.7 - A vertical rod is attached at point A to the cast...Ch. 4.7 - Knowing that the clamp shown has been tightened...Ch. 4.7 - Prob. 116PCh. 4.7 - Three steel plates, each of 25 150-mm cross...Ch. 4.7 - A vertical force P of magnitude 20 kips is applied...Ch. 4.7 - The four bars shown have the same cross-sectional...Ch. 4.7 - Prob. 120PCh. 4.7 - An eccentric force P is applied as shown to a...Ch. 4.7 - Prob. 122PCh. 4.7 - Prob. 123PCh. 4.7 - Prob. 124PCh. 4.7 - A single vertical force P is applied to a short...Ch. 4.7 - The eccentric axial force P acts at point D, which...Ch. 4.9 - 4.127 through 4.134 The couple M is applied to a...Ch. 4.9 - 4.127 through 4.134 The couple M is applied to a...Ch. 4.9 - 4.127 through 4.134 The couple M is applied to a...Ch. 4.9 - 4.127 through 4.134 The couple M is applied to a...Ch. 4.9 - 4.127 through 4.134 The couple M is applied to a...Ch. 4.9 - 4.127 through 4.134 The couple M is applied to a...Ch. 4.9 - Prob. 133PCh. 4.9 - Prob. 134PCh. 4.9 - 4.135 through 4.140 The couple M acts in a...Ch. 4.9 - 4.135 through 4.140 The couple M acts in a...Ch. 4.9 - Prob. 137PCh. 4.9 - 4.135 through 4.140 The couple M acts in a...Ch. 4.9 - 4.135 through 44.140 The couple M acts in a...Ch. 4.9 - 4.135 through 4.140 The couple M acts in a...Ch. 4.9 - Prob. 141PCh. 4.9 - 4.141 through 4.143 The couple M acts in a...Ch. 4.9 - 4.141 through 4.143 The couple M acts in a...Ch. 4.9 - The tube shown has a uniform wall thickness of 12...Ch. 4.9 - Prob. 145PCh. 4.9 - Knowing that P = 90 kips, determine the largest...Ch. 4.9 - Knowing that a = 1.25 in., determine the largest...Ch. 4.9 - A rigid circular plate of 125-mm radius is...Ch. 4.9 - Prob. 149PCh. 4.9 - A beam having the cross section shown is subjected...Ch. 4.9 - Prob. 151PCh. 4.9 - Prob. 152PCh. 4.9 - Prob. 153PCh. 4.9 - Prob. 154PCh. 4.9 - Prob. 155PCh. 4.9 - Prob. 156PCh. 4.9 - Prob. 157PCh. 4.9 - Prob. 158PCh. 4.9 - A beam of unsymmetric cross section is subjected...Ch. 4.9 - Prob. 160PCh. 4.10 - For the curved bar shown, determine the stress at...Ch. 4.10 - For the curved bar shown, determine the stress at...Ch. 4.10 - Prob. 163PCh. 4.10 - Prob. 164PCh. 4.10 - The curved bar shown has a cross section of 40 60...Ch. 4.10 - Prob. 166PCh. 4.10 - Prob. 167PCh. 4.10 - Prob. 168PCh. 4.10 - The curved bar shown has a cross section of 30 30...Ch. 4.10 - Prob. 170PCh. 4.10 - Prob. 171PCh. 4.10 - Three plates are welded together to form the...Ch. 4.10 - 4.173 and 4.174 Knowing that the maximum allowable...Ch. 4.10 - Prob. 174PCh. 4.10 - Prob. 175PCh. 4.10 - Prob. 176PCh. 4.10 - Prob. 177PCh. 4.10 - Prob. 178PCh. 4.10 - Prob. 179PCh. 4.10 - Knowing that P = 10 kN, determine the stress at...Ch. 4.10 - Prob. 181PCh. 4.10 - Prob. 182PCh. 4.10 - Prob. 183PCh. 4.10 - Prob. 184PCh. 4.10 - Prob. 185PCh. 4.10 - Prob. 186PCh. 4.10 - Prob. 187PCh. 4.10 - Prob. 188PCh. 4.10 - Prob. 189PCh. 4.10 - Prob. 190PCh. 4.10 - For a curved bar of rectagular cross section...Ch. 4 - Two vertical forces are applied to a beam of the...Ch. 4 - Prob. 193RPCh. 4 - Prob. 194RPCh. 4 - Determine the plastic moment Mp of a steel beam of...Ch. 4 - In order to increase corrosion resistance, a...Ch. 4 - The vertical portion of the press shown consists...Ch. 4 - The four forces shown are applied to a rigid plate...Ch. 4 - Prob. 199RPCh. 4 - Prob. 200RPCh. 4 - Three 120 10-mm steel plates have been welded...Ch. 4 - A short length of a W8 31 rolled-steel shape...Ch. 4 - Two thin strips of the same material and same...
Knowledge Booster
Background pattern image
Mechanical Engineering
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.
Similar questions
SEE MORE QUESTIONS
Recommended textbooks for you
Text book image
Elements Of Electromagnetics
Mechanical Engineering
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Oxford University Press
Text book image
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:9780134319650
Author:Russell C. Hibbeler
Publisher:PEARSON
Text book image
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:9781259822674
Author:Yunus A. Cengel Dr., Michael A. Boles
Publisher:McGraw-Hill Education
Text book image
Control Systems Engineering
Mechanical Engineering
ISBN:9781118170519
Author:Norman S. Nise
Publisher:WILEY
Text book image
Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:9781337093347
Author:Barry J. Goodno, James M. Gere
Publisher:Cengage Learning
Text book image
Engineering Mechanics: Statics
Mechanical Engineering
ISBN:9781118807330
Author:James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:WILEY
Mechanics of Materials Lecture: Beam Design; Author: UWMC Engineering;https://www.youtube.com/watch?v=-wVs5pvQPm4;License: Standard Youtube License