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 2, Problem 2.3.7P

-7 Repeat Problem 2.3-5, but n include the weight of the bar. See Table I-I in Appendix I for the weight density of steel.

Expert Solution & Answer
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To determine

The displacements at point B , C and D .

Answer to Problem 2.3.7P

The displacements at point B 1.135×106in .

The displacements at point C 2.54×106in .

The displacements at point D 23.1×106in .

Explanation of Solution

Given information:

The length from point A to B is 20in , the length from point B to C is 20in , the length from point C to D is 40in , modulus of elasticity is 200GPa , area of bar from point A to B is 6000mm2 , area of bar from point B to C is 5000mm2 , area of bar from point C to D is 4000mm2 .

Write the expression for the elongation at the point B .

  δB=PBL1AE+PCL1AEPDL1AE+γAL122AE+( γA L 2 )L1AE+( γA L 3 )L1AE=L1AE(PB+PCPD)+γL1E( L 1 2+L2+L3)  .........(I)

Here, the elongation at the point B is δB , the load acting at B is PB . The load acting at C is PC , the load acting at D is PD and the modulus of elasticity is E , the length of segment AB is L1 , the length of segment BC is L2, and the length of segment CD is L3 .

Write the expression for the elongation at the point C.

  δC=δB+PCL2AEPDL2AE+γAL222AE+( γA L 3 )L2AE=δB+L2AE(PCPD)+γL2E( L 2 2+L3)   .........(II)

Here, the elongation at the point C is δC .

Write the expression for the elongation at the point D .

  δD=δCPDL3AE+γAL322AE  .........(III)

Here, the elongation at the point D is δD .

Calculation:

Refer to appendix I table 1.1 “Weight and mass densities” to obtain weight density of steel.

  γ=490lb/ft3=490lb/ft3×( 1ft 12in)3=0.2836lb/in3

Substitute 20in for L1 , 20in for L2 , 40in for L3 , 50lb for PB , 100lb for PC and 200lb for PD , 8.24in2 for A , 29000ksi for E and 0.2836lb/in3 for γ in Equation (I).

  δB=[ 20in ( 8.24 in 2 )( 29000ksi )( 50lb+100lb200lb)+ ( 0.2836 lb/in 3 )20in ( 29000ksi )( 20 2 in+20in+40in)]=[ 20in ( 8.24 in 2 )( 29000ksi )( 1000 lb/in 2 1ksi )( 50lb+100lb200lb)+ ( 0.2836 lb/in 3 )20in ( 29000ksi )( 1000 lb/in 2 1ksi )( 20 2 in+20in+40in)]=[ 3000inlb ( 8.24 in 2 )( 29000× 10 3 lb )+ 397.04lb/in ( 29000× 10 3 lb )]=1.135×106in

Substitute 20in for L2 , 40in for L3 , 100lb for PC and 200lb for PD , 8.24in2 for A, 29000ksi for E and 0.2836lb/in3 for γ and 1.135×106in for δB in Equation (II).

  δC=[1.135× 10 6in+{ 20in ( 8.24 in 2 )( 29000ksi ) ×( 100lb200lb ) }+{ ( 0.2836 lb/ in 2 )20in ( 29000ksi ) ( 20 2 in+40in )}]=[1.135× 10 6in+{ 20in ( 8.24 in 2 )( 29000ksi× 1 lb/ in 2 1ksi ) ×( 100lb200lb ) }+{ ( 0.2836 lb/ in 2 )20in ( 29000ksi× 1 lb/ in 2 1ksi ) ( 20 2 in+40in )}]=[1.135× 10 6in+{ 20in 238960lb ×( 100lb )}+{ ( 283.6 lb/ in 2 ) ( 29000 lb/ in 2 ) }]=2.54×106in

Substitute 40in for L3 , 200lb for PD , 8.24in2 for A, 29000ksi for E and 0.2836lb/in3 for γ and and 2.54×106in for δC in Equation (III).

  δD=[2.54× 10 6in 200lb×40in ( 8.24 in 2 )( 29000ksi )+ ( 0.2836 lb/ in 2 ) ( 40in ) 2 2 in 2 ( 29000ksi )]=[2.54× 10 6in 200lb×40in ( 8.24 in 2 )( 29000ksi× 1 lb/ in 2 1ksi )+ ( 0.2836 lb/ in 3 ) ( 40in ) 2 2 in 2 ( 29000ksi× 1 lb/ in 2 1ksi )]=[2.54× 10 6in 8000lbin 238960lb+ 11.344lbin 58000lb]=23.1×106in

Conclusion:

The displacements at point B is 1.135×106in .

The displacements at point C is 2.54×106in .

The displacements at point D is 23.1×106in .

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Chapter 2 Solutions

Mechanics of Materials (MindTap Course List)

Ch. 2 - A small lab scale has a rigid L-shaped frame ABC...Ch. 2 - A small lab scale has a rigid L-shaped frame ABC...Ch. 2 - Two rigid bars are connected to each other by two...Ch. 2 - The three-bar truss ABC shown in the figure part a...Ch. 2 - An aluminum wire having a diameter d = 1/10 in....Ch. 2 - A uniform bar AB of weight W = 25 N is supported...Ch. 2 - A hollow, circular, cast-iron pipe (Ec =12,000...Ch. 2 - The horizon Lai rigid beam A BCD is supported by...Ch. 2 - Two pipe columns (AB, FC) are pin-connected to a...Ch. 2 - A framework ABC consists of two rigid bars AB and...Ch. 2 - Solve the preceding problem for the following...Ch. 2 - The length of the end segments of the bar (see...Ch. 2 - A long, rectangular copper bar under a tensile...Ch. 2 - An aluminum bar AD (see figure) has a...Ch. 2 - A vertical bar consists of three prismatic...Ch. 2 - A vertical bar is loaded with axial loads at...Ch. 2 - Repeat Problem 2.3-4, but now include the weight...Ch. 2 - -7 Repeat Problem 2.3-5, but n include the weight...Ch. 2 - A rectangular bar of length L has a slot in the...Ch. 2 - Solve the preceding problem if the axial stress in...Ch. 2 - A two-story building has steel columns AB in the...Ch. 2 - A steel bar is 8.0 Ft long and has a circular...Ch. 2 - A bar ABC of length L consists of two parts of...Ch. 2 - A woodpile, driven into the earth, supports a load...Ch. 2 - Consider the copper lubes joined in the strength...Ch. 2 - The nonprismalic cantilever circular bar shown has...Ch. 2 - *16 A prismatic bar AB of length L,...Ch. 2 - A flat bar of rectangular cross section, length L,...Ch. 2 - A flat brass bar has length L, constant thickness...Ch. 2 - Repeat Problem 2.3-18, but assume that the bar is...Ch. 2 - Repeat Problem 2.3-18, but assume that the bar is...Ch. 2 - A slightly tapered bar AB of solid circular crass...Ch. 2 - A circular aluminum alloy bar of length L = 1.8 m...Ch. 2 - A long, slender bar in the shape of a right...Ch. 2 - A post AB supporting equipment in a laboratory is...Ch. 2 - The main cables of a suspension bridge (see figure...Ch. 2 - A uniformly tapered lube AB of circular cross...Ch. 2 - A vertical steel bar ABC is pin-supported at its...Ch. 2 - A T-frame structure is torn posed of a prismatic...Ch. 2 - A T-frame structure is composed of prismatic beam...Ch. 2 - Repeat Problem 2.3-29 if vertical load P at D is...Ch. 2 - A bar ABC revolves in a horizontal plane about a...Ch. 2 - The assembly shown in the figure consists of a...Ch. 2 - A cylindrical assembly consisting of a brass core...Ch. 2 - A steel bar with a uniform cross section, is fixed...Ch. 2 - A horizontal rigid bar ABC is pinned at end A and...Ch. 2 - A solid circular steel cylinder S is encased in a...Ch. 2 - Three prismatic bars, two of material A and one of...Ch. 2 - A circular bar ACB of a diameter d having a...Ch. 2 - Bar ABC is fixed at both ends (see figure) and has...Ch. 2 - Repeat Problem 2.4-8, but assume that the bar is...Ch. 2 - A plastic rod AB of length L = 0.5 m has a...Ch. 2 - 2.4-11 Three steel cables jointly support a load...Ch. 2 - The fixed-end bar ABCD consists of three prismatic...Ch. 2 - A lube structure is acted on by loads at B and D,...Ch. 2 - A hollow circular pipe (see figure} support s a...Ch. 2 - The aluminum and steel pipes shown in the figure...Ch. 2 - A rigid bar of weight W = SOO N hangs from three...Ch. 2 - A bimetallic bar (or composite bar) of square...Ch. 2 - S Three-bar truss ABC (see figure) is constructed...Ch. 2 - A horizontal rigid bar of weight If' = 72001b is...Ch. 2 - A rigid bar ABCD is pinned at point B and...Ch. 2 - A rigid bar AB if of a length B = 66 in. is....Ch. 2 - Find expressions For all support reaction forces...Ch. 2 - A trimetallic bar is uniformly compressed by an...Ch. 2 - Find expressions for all support reaction Forces...Ch. 2 - The rails of a railroad track are welded together...Ch. 2 - A circular steel rod of diameter d is subjected to...Ch. 2 - A rigid bar of weight W = 750 lb hangs from three...Ch. 2 - A steel rod. of 15-mm diameter is held snugly (but...Ch. 2 - A bar AB of length L is held between rigid...Ch. 2 - A beam is constructed using two angle sections (L...Ch. 2 - A W 8 × 28 beam of a length 10 ft is held between...Ch. 2 - A plastic bar ACB having two different solid...Ch. 2 - ,5-9 A flat aluminum alloy bar is fixed at both...Ch. 2 - Repeat Problem 2.5-9 for the flat bar shown in the...Ch. 2 - A circular steel rod AB? 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