EBK MECHANICS OF MATERIALS
EBK MECHANICS OF MATERIALS
7th Edition
ISBN: 8220100257063
Author: BEER
Publisher: YUZU
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Question
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Chapter 4.10, Problem 166P

(a)

To determine

Find the error in the computation of maximum stress by assuming the bar as straight.

(a)

Expert Solution
Check Mark

Answer to Problem 166P

The error in the computation of maximum stress by assuming the bar as straight is 34.36%_.

Explanation of Solution

Given information:

The value of h is 40mm.

The inner (r1) and outer radius (r2) of the curved bar is 20mm and 60mm.

The width and depth of the bar are b=60mm and h=40mm.

The moment (M) is 120Nm.

Calculation:

Calculate the cross-section area (A) of the bar as follows:

Substitute 60mm for b and 40mm for h.

A=bh=60×40=2,400mm2

Calculate the moment of inertia (I) of the cross-section of the bar using the relation:

I=112bh3 (1)

Substitute 60mm for b and 40mm for h in Equation (1).

I=112×60×403=320,000mm4

Calculate the distance (c) between the neutral axis and the extreme fiber using the relation:

c=h2 (2)

Substitute 40mm for h in Equation (2).

c=402=20mm

Calculate the stress (σs) using the relation:

σs=McI (3)

Substitute 120Nm for M, 20mm for c, and 320,000mm4 for I in Equation (3).

σs=120Nm×20mm×(1m1000mm)320,000mm4×(1m41012mm4)=2.40.32×106=7.5×106Pa×(1MPa106Pa)=7.5MPa

Calculate the radius (R) of the neutral surface using the relation:

R=hlnr2r1 (4)

Substitute 40mm for h, 20mm for r1, and 60mm for r2 in Equation (4).

R=40ln(6020)=40ln(3)=401.09861=36.4096mm

Calculate the mean radius (r¯) of the curved bar using the relation:

r¯=12(r1+r2) (5)

Substitute 20mm for r1 and 60mm for r2 in Equation (5).

r¯=12(20+60)=40mm

The distance (e) between the neutral axis and the centroid of the cross-section using the relation:

e=r¯R (6)

Substitute 40mm for r¯ and 36.4096mm for R in Equation (6).

e=4036.4096mm=3.5904mm

Calculate the actual stress using the relation:

σa=M(r1R)Aer1 (7)

Substitute 120Nm for M, 36.4096mm for R, 2,400mm2 for A, 3.5904mm for e, 20mm for r1 in Equation (7).

σa=120Nm×(2036.4096)mm×(1m1,000mm)2400mm2×(1m2106mm2)×3.5904mm×(1m1,000mm)×20mm×(1m1,000mm)=120×(16.4096)×1032400×106×3.5904×103×20×103=1.96910.172339×10611.426×106Pa×1MPa106Pa

σa=11.426MPa

Calculate the error in the computation of maximum stress by assuming the bar as straight using the relation:

%error=(σaσsσa)×100 (8)

Substitute 11.426MPa for σa and 7.5MPa for σs in Equation (8).

%error=(11.426(7.5)11.426)×100=3.92611.426×100=34.36%

Thus, the in the computation of maximum stress by assuming the bar as straight is 34.36%_.

(b)

To determine

Find the error in the computation of maximum stress by assuming the bar as straight.

(b)

Expert Solution
Check Mark

Answer to Problem 166P

The error in the computation of maximum stress by assuming the bar as straight is 6%_.

Explanation of Solution

Given information:

The value of h is 40mm.

The inner (r1) and outer radius (r2) of the curved bar is 200mm and 240mm.

The width and depth of the bar are b=60mm and h=40mm.

The moment (M) is 120Nm.

Calculation:

Calculate the radius (R) of the neutral surface using the relation:

Substitute 40mm for h, 200mm for r1, and 240mm for r2 in Equation (4).

R=40ln(240200)=40ln(1.2)=400.182321=219.39326mm

Calculate the mean radius (r¯) of the curved bar using the relation:

Substitute 200mm for r1 and 240mm for r2 in Equation (5).

r¯=12(200+240)=220mm

The distance (e) between the neutral axis and the centroid of the cross-section using the relation:

Substitute 220mm for r¯ and 219.39326mm for R in Equation (6).

e=220219.39326mm=0.60674mm

Calculate the actual stress using the relation:

Substitute 120Nm for M, 219.39326mm for R, 2,400mm2 for A, 0.60674mm for e, 200mm for r1 in Equation (7).

σa=120Nm×(200219.39326)mm×(1m1,000mm)2400mm2×(1m2106mm2)×0.60674mm×(1m1,000mm)×200mm×(1m1,000mm)=120×(19.39326)×1032400×106×0.60674×103×200×103=2.3270.291235×1067.990×106Pa×1MPa106Pa

σa=7.990MPa

Calculate the error in the computation of maximum stress by assuming the bar as straight using the relation:

Substitute 7.99MPa for σa and 7.5MPa for σs in Equation (8).

%error=(7.990(7.5)7.990)×100=0.4907.990×1006%

Thus, the in the computation of maximum stress by assuming the bar as straight is 34.36%_.

(c)

To determine

Find the error in the computation of maximum stress by assuming the bar as straight.

(c)

Expert Solution
Check Mark

Answer to Problem 166P

The error in the computation of maximum stress by assuming the bar as straight is 0.6%_.

Explanation of Solution

Given information:

The value of h is 40mm.

The inner radius (r1) is 2m.

Show the unit conversion of inner radius as follows:

r1=2m×1,000mm1m=2000mm

The inner (r1) and outer radius (r2) of the curved bar is 2,000mm and 2,040mm.

The width and depth of the bar are b=60mm and h=40mm.

The moment (M) is 120Nm.

Calculation:

Calculate the radius (R) of the neutral surface using the relation:

Substitute 40mm for h, 2,000mm for r1, and 2,040mm for r2 in Equation (4).

R=40ln(2,0402,000)=40ln(1.02)=400.01980262=2019.934mm

Calculate the mean radius (r¯) of the curved bar using the relation:

Substitute 2000mm for r1 and 2040mm for r2 in Equation (5).

r¯=12(2000+2040)=2020mm

The distance (e) between the neutral axis and the centroid of the cross-section using the relation:

Substitute 2020mm for r¯ and 2019.934mm for R in Equation (6).

e=20202019.934mm=0.066mm

Calculate the actual stress using the relation:

Substitute 120Nm for M, 2019.9mm for R, 2,400mm2 for A, 0.1mm for e, 200mm for r1 in Equation (7).

σa=120Nm×(2,0002019.934)mm×(1m1,000mm)2400mm2×(1m2106mm2)×0.1mm×(1m1,000mm)×2,000mm×(1m1,000mm)=120×(19.934)×1032400×106×0.066×103×2,000×103=2.392080.3168×1067.550×106Pa×1MPa106Pa

σa=7.550MPa

Calculate the error in the computation of maximum stress by assuming the bar as straight using the relation:

Substitute 7.55MPa for σa and 7.5MPa for σs in Equation (8).

%error=(7.550(7.5)7.550)×100=0.0507.550×1000.6%

Thus, the in the computation of maximum stress by assuming the bar as straight is 0.6%_.

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Chapter 4 Solutions

EBK MECHANICS OF MATERIALS

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