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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Chapter 2, Problem 2.4.8P

Bar ABC is fixed at both ends (see figure) and has load P applied at B. Find reactions at A and C and displacement SBif P = 200 kN. L = 2 m, t = 20 mm, b, = 100 mm, b2= 115 mm, and E = 96 GPa.

  Chapter 2, Problem 2.4.8P, Bar ABC is fixed at both ends (see figure) and has load P applied at B. Find reactions at A and C

Expert Solution & Answer
Check Mark
To determine

The reaction force at point A and C and displacement at point B.

Answer to Problem 2.4.8P

The reaction force at point A is 80kN .

The reaction force at point C is 120kN .

The displacement at point B is 4.666×104m .

Explanation of Solution

Given:

Load at point B is 200kN , length of the bar is 2m , thickness is 20mm , width at point C and A is 115mm,and the width at point B is 100mm,and modulus of elasticity is 96GPa .

The below figure shows the free body diagram of the bar.

  Mechanics of Materials (MindTap Course List), Chapter 2, Problem 2.4.8P

Figure-(1)

Here, the reaction force at point A is RA , load at point B is Pand reaction force at point C is RA .

Write the expression for the force balance in horizontal direction.

  RA+RC=200kN…… (I)

Write the expression for the total deflection.

  δAB+δBC=0 …… (II)

Here, the deflection of the section AB is δABand deflection of the section BC is δBC .

Write the expression for the deflection for section AB.

  δ=PLAE…… (III)

Here, the cross-section area is A , length is L , load is Pand modulus of elasticity is E .

Write the expression for the cross-section area of the bar AB.

  AAB=t(b2b1)ln( b 2 b 1) …… (IV)

Here, the width of the bar at point A is b2 , the width of the bar at point B is b1and the thickness is t .

Write the expression for the deflection for section AB.

  δAB=RALABAABE…… (V)

Substitute t(b2b1)/ln(b2/b1)for AABin Equation (V).

  δAB=RALAB(t( b2 b1 )/ln( b2/b1 ))E

  δAB=RALABt(b2b1)Eln(b2/b1) …… (VI)

Write the expression for the cross-section area of the bar BC.

  ABC=t(b1b2)ln( b 1 b 2) …… (VII)

Here, the width of the bar at point C is b2 , the width of the bar at point B is b1and the thickness is t .

Write the expression for the deflection for section BC.

  δBC=RCLBCABCE…… (VIII)

Substitute t(b1b2)/ln(b12/b2)for ABCin Equation (VIII).

  δBC=RCLBC(t( b1 b2 )/ln( b1/b2 ))E

  δBC=RCLBCt(b1b2)Eln(b1/b2) …… (IX)

Substitute RCLBCt(b1b2)Eln(b1/b2)for δBCand RALABt(b2b1)Eln(b2/b1)for δABin Equation (II).

  RALABt( b 2 b 1)Eln(b2/b1)RCLBCt( b 1 b 2)Eln(b1/b2)=0RALABt( b 2 b 1)Eln(b2/b1)=RCLBCt( b 2 b 1)Eln(b2/b1)

  RALAB=RCLBC…… (X)

Substitute 3L/5for LABand 2L/5for LBCin Equation (X).

  RA(3L/5)=RC(2L/5)RA=23RC

Calculation:

Substitute 23RCfor RAin Equation (I).

  23RC+RC=200kN53RC=200kNRC=120kN

Substitute 120kNfor RCin Equation (I).

  RA+120kN=200kNRA=200kN120kNRA=80kN

Substitute 3L/5for LABin Equation (VI).

  δAB=RA(3L/5)t(b2b1)Eln(b2/b1) …… (XI).

Substitute 80kNfor RA , 2mfor L , 20mmfor t , 96GPafor E , 100mmfor b1and 115mmfor b2in Equation (XI).

  δAB=(80kN)( (3×2m )/5)(20mm)(115mm100mm)(96GPa)ln(115mm/100mm)=(80kN)( (3×2m )/5)(20mm)(115mm100mm)(96GPa)(0.13976)=(80kN)( (3×2m )/5)(20mm)(15mm)(96GPa)(0.13976)=(96kNm)(300 mm 2( 1 m 2 10 6 mm 2))(96GPa( 10 9Pa1GPa))(0.13976)

  δAB=(96kNm)0.13976(300 mm 2( 1 m 2 10 6 mm 2))(96GPa( 10 9Pa1GPa))=4.666×104m

Conclusion:

The reaction force at point A is 80kN .

The reaction force at point C is 120kN .

The displacement at point B is 4.666×104m .

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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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