Mechanics of Materials (10th Edition)
Mechanics of Materials (10th Edition)
10th Edition
ISBN: 9780134319650
Author: Russell C. Hibbeler
Publisher: PEARSON
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Textbook Question
Chapter 14, Problem 14.1RP

A = 2300 mm2, I = 9.5(106) mm4.

Chapter 14, Problem 14.1RP, A = 2300 mm2, I = 9.5(106) mm4. R141

R14–1

Expert Solution & Answer
Check Mark
To determine
The total axial and bending strain energy in the A992 steel beam.

Answer to Problem 14.1RP

The total axial and bending strain energy in the A992 steel beam is U=496J_.

Explanation of Solution

Given information:

The cross-sectional area of the beam is A=2,300mm2.

Moment of inertia of the beam is I=9.5×106mm4.

Assumption:

The modulus of elasticity or Young’s modulus of theA992 steelis E=200GPa=200×109N/m2.

Explanation:

Determine the reactions:

Entire beam:

Show the free body diagram of the entire beam as in Figure 1.

Mechanics of Materials (10th Edition), Chapter 14, Problem 14.1RP , additional homework tip  1

Moment about the point A:

Determine the vertical reaction at point B by taking moment about point A.

MA=0By(10)1.5×10×102=0 (1)

Along the vertical direction:

Determine the vertical reaction at point B by resolving the vertical component of forces.

Fy=0Ay+By1.5×10=0 (2)

Along the horizontal direction:

Determine the horizontal reaction at point A by resolving the horizontal component of force.

Fx=015Ax=0 (3)

Show the calculation of reaction as follows:

Solve Equation (1).

10By75=0By=7.5kN

Substitute 7.5kN for By in Equation (2).

Ay+7.515=0Ay=7.5kN

Solve Equation (3).

Ax=15kN

Region 0x10m:

Show the free-body diagram of the section as in Figure 2.

Mechanics of Materials (10th Edition), Chapter 14, Problem 14.1RP , additional homework tip  2

Moment about the section:

Determine the moment at section by taking moment about the section.

Mx=0M+1.5×x×x2Ay(x)=0 (4)

Along the horizontal direction:

Determine the normal axial force at the section by resolving the horizontal component of forces.

Fx=0NAx=0 (5)

Show the calculation of values as follows:

Substitute 7.5kN for Ay in Equation (4).

M+0.75x27.5(x)=0M=7.5x0.75x2

Substitute 15 kN for Ax Equation (5).

N15=0N=15kN

Strain energy due to axial load:

Determine the strain energy of a bar of constant cross-sectional area A and constant internal axial load N using the equation.

(Ui)a=N2L2AE (6)

Here, N is the axial load, L is the length of beam, E is Young’s modulus or modulus of elasticity, and A is cross-sectional area of the beam.

Substitute 15 kN for N, 10 m for L, 2,350mm2 for A, and 200×109N/m2 for E in Equation (6).

(Ui)a=(15kN×1,000N1kN)2×10m2×2,300mm2×(1m1,000mm)2×200×109=2.4456J

Strain energy due to Bending:

Determine the strain energy in the beam due to bending using the equation.

(Ui)b=0LM2dx2EI (7)

Here, M is the moment in the beam and I is the moment of inertia of the beam.

Substitute 10 m for L, (7.5x0.75x2) for M, 200×109N/m2 for E, and 9.5×106mm4 for I in Equation (7), and integrate.

(Ui)b=010(7.5x0.75x2)2dx2×200×109×9.5×106mm4×(1m1,000mm)4=12×200×109×9.5×106010(56.25x211.25x3+0.5625x4)dx=12×200×109×9.5×106[56.25x3311.25x44+0.5625x55]010=12×200×109×9.5×106[56.25(10)3311.25(10)44+0.5625(10)550+00]

=1,875kN2×1,000N1kN2×200×109×9.5×106=493.421J

Total strain energy:

Determine the total strain energy by adding the strain energy due to axial load and the strain energy due to bending.

U=(Ui)a+(Ui)b (8)

Substitute 2.4456 J for (Ui)a and 493.421 J for (Ui)b in Equation (8).

U=2.4456+493.421=495.87J496J

Thus, the total axial and bending strain energy in the A992 steel beam is U=496J_.

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Chapter 14 Solutions

Mechanics of Materials (10th Edition)

Ch. 14.2 - Determine the total axial and bending strain...Ch. 14.2 - If P = 10 kip, determine the total strain energy...Ch. 14.2 - Determine the maximum force P and the...Ch. 14.2 - Consider the thin-walled tube of Fig.5-26 . Use...Ch. 14.2 - Determine the bending strain energy in the A992...Ch. 14.2 - Determine the bending strain energy in the beam....Ch. 14.2 - Prob. 14.17PCh. 14.2 - Prob. 14.18PCh. 14.2 - Determine the bending strain energy in the 2-in...Ch. 14.2 - Determine the total strain energy in the steel...Ch. 14.2 - Determine the bending strain energy in the beam....Ch. 14.2 - The bolt has a diameter of 10 mm, and the arm AB...Ch. 14.2 - Determine the bending strain energy in the...Ch. 14.2 - Determine the bending strain energy in the simply...Ch. 14.3 - Determine the vertical displacement of joint D. AE...Ch. 14.3 - Determine the horizontal displacement of joint C....Ch. 14.3 - Determine the horizontal displacement of joint A....Ch. 14.3 - AE is constant. Prob. 1428Ch. 14.3 - Determine the vertical displacement of point C of...Ch. 14.3 - Determine the vertical displacement of end B of...Ch. 14.3 - Determine the vertical displacement of point S on...Ch. 14.3 - EI is constant. Prob. 1432Ch. 14.3 - The A992 steel bars are pin connected at C and D....Ch. 14.3 - The A992 steel bars are pin connected at C. If...Ch. 14.3 - Determine the slope of the beam at the pin support...Ch. 14.3 - The cantilevered beam has a rectangular...Ch. 14.3 - The rod has a circular cross section with a moment...Ch. 14.3 - The rod has a circular cross section with a moment...Ch. 14.3 - Determine the vertical displacement of point B on...Ch. 14.3 - Prob. 14.40PCh. 14.3 - Determine the vertical displacement of end B of...Ch. 14.4 - A bar is 4 m long and has a diameter of 30 mm....Ch. 14.4 - Determine the diameter of a red brass C83400 bar...Ch. 14.4 - Prob. 14.44PCh. 14.4 - The collar has a weight of 50 lb and falls down...Ch. 14.4 - The collar has a weight of 50 lb and falls down...Ch. 14.4 - Prob. 14.47PCh. 14.4 - Prob. 14.48PCh. 14.4 - Prob. 14.49PCh. 14.4 - Prob. 14.50PCh. 14.4 - The A-36 steel bolt is required to absorb the...Ch. 14.4 - Prob. 14.52PCh. 14.4 - The composite aluminum 2014T6 bar is made from two...Ch. 14.4 - The composite aluminum 2014-T6 bar is made from...Ch. 14.4 - When the 100-lb block is at h = 3 ft above the...Ch. 14.4 - If the bar has a diameter of 20 mm, determine the...Ch. 14.4 - The collar has a mass of 5 kg and falls dawn the...Ch. 14.4 - The tugboat has a weight of 120 000 lb and is...Ch. 14.4 - The W10 12 beam is made from A-36 steel and is...Ch. 14.4 - The weight of 175 lb is dropped from a height of 4...Ch. 14.4 - The weight of 175 lb, is dropped from a height of...Ch. 14.4 - Determine the maximum height h from which an 80-lb...Ch. 14.4 - The 80-lb weight is dropped from rest at a height...Ch. 14.4 - The 75-lb block has a downward velocity of 2 ft/s...Ch. 14.4 - The 75-lb block has a downward velocity of 2 ft/s...Ch. 14.4 - Prob. 14.66PCh. 14.4 - The overhang beam is made of 2014T6 aluminum....Ch. 14.4 - If the beam is a W1015, determine the maximum...Ch. 14.4 - If the maximum allowable bending stress for the...Ch. 14.4 - A 40-lb weight is dropped from a height of h = 2...Ch. 14.4 - The car bumper is made of...Ch. 14.6 - Determine the vertical displacement of joint A....Ch. 14.6 - Determine the horizontal displacement of joint B....Ch. 14.6 - Determine the vertical displacement of joint B....Ch. 14.6 - Determine the vertical displacement of joint B....Ch. 14.6 - Determine the vertical displacement of joint E....Ch. 14.6 - Determine the horizontal displacement of joint B....Ch. 14.6 - Determine the vertical displacement of joint B....Ch. 14.6 - Determine the horizontal displacement of joint B...Ch. 14.6 - Determine the vertical displacement of joint C of...Ch. 14.6 - Determine the horizontal displacement of joint C....Ch. 14.6 - Determine the vertical displacement of joint D....Ch. 14.6 - Determine the vertical displacement of joint A....Ch. 14.6 - The truss is made from A992 steel rods having a...Ch. 14.6 - Determine the horizontal displacement of joint D....Ch. 14.6 - Determine the horizontal displacement of joint E....Ch. 14.7 - Determine the displacement at point C. El is...Ch. 14.7 - The beam is made of southern pine for which Ep =...Ch. 14.7 - Determine the displacement at point C. El is...Ch. 14.7 - Determine the slope at point C. El is constant....Ch. 14.7 - Determine the slope at point A. El is constant....Ch. 14.7 - Determine the displacement of point C of the beam...Ch. 14.7 - Determine the slope at B of the beam made from...Ch. 14.7 - The beam is made of Douglas fir. Determine the...Ch. 14.7 - Determine the displacement at pulley B. The A992...Ch. 14.7 - The A992 steel beam has a moment of inertia of I =...Ch. 14.7 - The A992 steel beam has a moment of inertia of I =...Ch. 14.7 - The A992 structural steel beam has a moment of...Ch. 14.7 - Determine the displacement at point C of the...Ch. 14.7 - Determine the slope at A of the shaft. El is...Ch. 14.7 - Determine the slope of end C of the overhang beam....Ch. 14.7 - Determine the displacement of point D of the...Ch. 14.7 - Determine the slope at A of the 2014T6 aluminum...Ch. 14.7 - Prob. 14.104PCh. 14.7 - Prob. 14.105PCh. 14.7 - Determine the displacement at point C of the W14 ...Ch. 14.7 - Determine the slope at A of the W14 26 beam made...Ch. 14.7 - Determine the slope at A. El is constant. Prob....Ch. 14.7 - Determine the slope at C of the overhang white...Ch. 14.7 - Determine the displacement at point D of the...Ch. 14.7 - Determine the maximum deflection of the beam...Ch. 14.7 - The beam is made of oak, for which Eo = 11 GPa....Ch. 14.7 - Determine the slope of the shaft at the bearing...Ch. 14.7 - Determine the horizontal and vertical...Ch. 14.7 - Beam AB has a square cross section of 100 mm by...Ch. 14.7 - Beam AB has a square cross section of 100 mm by...Ch. 14.7 - Bar ABC has a rectangular cross section of 300 mm...Ch. 14.7 - Bar ABC has a rectangular cross section of 300 mm...Ch. 14.7 - The L-shaped frame is made from two segments, each...Ch. 14.7 - The L-shaped frame is made from two segments, each...Ch. 14.7 - Determine the vertical displacement of the ring at...Ch. 14.7 - Determine the horizontal displacement at the...Ch. 14.9 - Solve Prob. 1473 using Castiglianos theorem. 1473....Ch. 14.9 - Solve Prob. 1474 using Castiglianos theorem. 1474....Ch. 14.9 - Prob. 14.125PCh. 14.9 - Prob. 14.126PCh. 14.9 - Prob. 14.127PCh. 14.9 - Solve Prob. 1478 using Castiglianos theorem. 1478....Ch. 14.9 - Solve Prob. 1481 using Castiglianos theorem. 1481....Ch. 14.9 - Solve Prob. 1482 using Castiglianos theorem. 1482....Ch. 14.9 - Solve Prob. 1485 using Castiglianos theorem. 1485....Ch. 14.9 - Solve Prob. 1486 using Castiglianos theorem. 1486....Ch. 14.10 - Solve Prob. 1490 using Castiglianos theorem. 1490....Ch. 14.10 - Solve Prob. 1491 using Castiglianos theorem. 1491....Ch. 14.10 - Prob. 14.135PCh. 14.10 - Solve Prob. 1493 using Castiglianos theorem. 1493....Ch. 14.10 - Solve Prob. 1495 using Castiglianos theorem. 1495....Ch. 14.10 - Solve Prob. 1496 using Castiglianos theorem. 1496....Ch. 14.10 - Prob. 14.139PCh. 14.10 - Prob. 14.140PCh. 14.10 - Prob. 14.141PCh. 14.10 - Solve Prob. 14119 using Castiglianos theorem....Ch. 14.10 - Prob. 14.143PCh. 14.10 - Solve Prob. 14105 using Castiglianos theorem....Ch. 14 - A = 2300 mm2, I = 9.5(106) mm4. R141Ch. 14 - If the spring at B has a stiffness k = 200 kN/m....Ch. 14 - The spring at B has a stiffness k = 200 kN/m....Ch. 14 - If they each have a diameter of 30 mm, determine...Ch. 14 - and a length of 10 in. It is struck by a hammer...Ch. 14 - Determine the total axial and bending strain...Ch. 14 - The truss is made from A992 steel rods each having...Ch. 14 - The truss is made from A992 steel rods each having...Ch. 14 - El is constant. Use the method of virtual work....Ch. 14 - using Castiglianos theorem. R149. The cantilevered...

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