Mechanics of Materials (MindTap Course List)
Mechanics of Materials (MindTap Course List)
9th Edition
ISBN: 9781337093347
Author: Barry J. Goodno, James M. Gere
Publisher: Cengage Learning
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Textbook Question
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Chapter 6, Problem 6.2.6P

A r o lukI f/frm f «m t ub e of ou t sid e d ia met er ^ and a copper core of diameter dxare bonded to form a composite beam, as shown in the figure,

(a) Derive formulas for the allowable bending moment M that can be carried by the beam based upon an allowable stress <7Ti in the titanium and an allowable stress ( u in the

copper (Assume that the moduli of elasticity for the titanium and copper are Er- and £Cu, respectively.)

(b)

If d1= 40 mm, d{= 36 mm, ETl= 120 GPa, ECu= 110 GPa, o-Ti = 840 MPa, and

ctqj = 700 MPa, what is the maximum bending moment Ml (c)

What new value of copper diameter dtwill result

in a balanced design? (i.e., a balanced design is

that in which titanium and copper reach allow-

able stress values at the same time).

  Chapter 6, Problem 6.2.6P, A r o lukI f/frm f «m t ub e of ou t sid e d ia met er ^ and a copper core of diameter dxare bonded

i.

Expert Solution
Check Mark
To determine

The formula for the allowable bending moment M for titanium tube and copper core of the composite beam

Answer to Problem 6.2.6P

  Theallowablebendingmomentfortitianium,MallowableTi=σTiπ(E Ti*( d 2 4 d 1 4 )+E Cud14)32*ETi*d2Theallowablebendingmomentforcopper,MallowableCu=σCuπ(E Ti*( d 2 4 d 1 4 )+E Cud14)32*ECu*d1

Explanation of Solution

Given:

Allowable stress for titanium is sti

Allowable stress for titanium is scu

Diameter of the copper rod = d1

Outer Diameter of the titanium rod = d2

Concept Used:

  Themomentofinertiaofhollowtube=π(d24d14)64Themomentofinertiaofarod=π(d14)64Allowablebendingmoment=σ1(E1I1+E2I2)E1*( d 2 2)

Calculation:

   Themomentofinertiaoftitaniumtube= π( d 2 4 d 1 4 ) 64

   Themomentofinertiaofthecopperrod= π( d 1 4 ) 64

   Allowablebendingmomentoftitanium= σ Ti ( E Ti I Ti + E Cu I Cu ) E Ti *( d 2 2 )

   M allowableTi = σ Ti ( E Ti *( π( d 2 4 d 1 4 ) 64 )+ E Cu ( π( d 1 4 ) 64 ) ) E Ti *( d 2 2 )   M allowableTi = σ Ti π( E Ti *( d 2 4 d 1 4 )+ E Cu d 1 4 ) 32* E Ti * d 2

   Allowablebendingmomentofcopper= σ Cu ( E Ti I Ti + E Cu I Cu ) E Cu *( d 2 2 )   M allowableCu = σ Cu ( E Ti *( π( d 2 4 d 1 4 ) 64 )+ E Cu ( π( d 1 4 ) 64 ) ) E Cu *( d 2 2 )

   M allowableCu = σ Cu π( E Ti *( d 2 4 d 1 4 )+ E Cu d 1 4 ) 32* E Cu * d 1

Conclusion:

  Theallowablebendingmomentfortitianium,MallowableTi=σTiπ(E Ti*( d 2 4 d 1 4 )+E Cud14)32*ETi*d2Theallowablebendingmomentforcopper,MallowableCu=σCuπ(E Ti*( d 2 4 d 1 4 )+E Cud14)32*ECu*d1

ii.

Expert Solution
Check Mark
To determine

The allowable bending moment for titanium and copper.

Answer to Problem 6.2.6P

Allowable bending moment for titanium, Mallowableti = 4989 N-m

Allowable bending moment for Copper, MallowableCu =5039.6 N-m

Explanation of Solution

Given:

Allowable stress for titanium, sti = 840 MPa

Allowable stress for titanium, scu= 700 MPa

Diameter of the copper rod, d1= 36 mm

Outer Diameter of the titanium rod, d2 = 40 mm

Eti= 110 GPa

Ecu= 120 GPa

Concept Used:

  Theallowablebendingmomentfortitianium,MallowableTi=σTiπ(E Ti*( d 2 4 d 1 4 )+E Cud14)32*ETi*d2Theallowablebendingmomentforcopper,MallowableCu=σCuπ(E Ti*( d 2 4 d 1 4 )+E Cud14)32*ECu*d1

Calculation:

  Theallowablebendingmomentfortitianium,MallowableTi=σTiπ(E Ti*( d 2 4 d 1 4 )+E Cud14)32*ETi*d2MallowableTi=840π(110* 103*( 40 4 36 4 )+120* 103* 364)32*110*103*36MallowableTi=4989NmTheallowablebendingmomentforcopper,MallowableCu=σCuπ(E Ti*( d 2 4 d 1 4 )+E Cud14)32*ETi*d2MallowableTi=700π(110* 103*( 40 4 36 4 )+120* 103* 364)32*120*103*36MallowableTi=5039.6NmConclusion:

Allowable bending moment for titanium, Mallowableti = 4989 N-m

Allowable bending moment for Copper, MallowableCu =5039.6 N-m

iii.

Expert Solution
Check Mark
To determine

The value of the diameter of the copper rod for a balanced design

Answer to Problem 6.2.6P

The diameter of the copper for a balanced design is 36.4 mm

Explanation of Solution

Given:

Allowable stress for titanium, sti = 840 MPa

Allowable stress for titanium, scu= 700 MPa

Outer Diameter of the titanium rod, d2 = 40 mm

Eti= 110 GPa

Ecu= 120 GPa

Concept Used:

  Allowablebendingmomentoftitanium=AllowablebendingmomentofcopperσTiπ(E Ti*( d 2 4 d 1 4 )+E Cud14)32*ETi*d2=σCuπ(E Ti*( d 2 4 d 1 4 )+E Cud14)32*ECu*d1

Calculation:

  Allowablebendingmomentoftitanium=AllowablebendingmomentofcopperσTiπ(E Ti*( d 2 4 d 1 4 )+E Cud14)32*ETi*d2=σCuπ(E Ti*( d 2 4 d 1 4 )+E Cud14)32*ECu*d1σTiETi*d2=σCuECu*d1840120*40=700110*d1d1=36.4mm

Conclusion:

The diameter of the copper for a balanced design is 36.4 mm

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Chapter 6 Solutions

Mechanics of Materials (MindTap Course List)

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