Mechanics of Materials, 7th Edition
Mechanics of Materials, 7th Edition
7th Edition
ISBN: 9780073398235
Author: Ferdinand P. Beer, E. Russell Johnston Jr., John T. DeWolf, David F. Mazurek
Publisher: McGraw-Hill Education
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Chapter 2.3, Problem 41P

Chapter 2.3, Problem 41P, Fig. P2.41 2.41 Two cylindrical rods, one of steel and the other of brass, are joined at C and

Fig. P2.41

2.41 Two cylindrical rods, one of steel and the other of brass, are joined at C and restrained by rigid supports at A and E. For the loading shown and knowing that Es = 200 GPa and Eb = 105 GPa, determine (a) the reactions at A and E, (b) the deflection of point C.

(a)

Expert Solution
Check Mark
To determine

Find the reaction at point A and E.

Answer to Problem 41P

The reaction at point A is 62.8kN_.

The reaction at point E is 37.2kN_.

Explanation of Solution

Given information:

The length (lAB) of rod AB is 180mm.

The length (lBC) of rod BC is 120mm.

The length (lCD) of rod CD is 100mm.

The length (lDE) of rod DE  is 100mm.

The young’s modulus for steel (Es) is 200GPa.

The young’s modulus for steel (Eb) is 105GPa.

Calculation:

Find the area (AAc) of rod AC using the relation:

AAC=π4(dAC) (1)

Here, dAC is diameter of rod AC.

Substitute 40mm dAC in Equation (1).

AAC=π4(40)2=1.25664×103mm2(1m103mm)2=1.25664×103m2

Find the value of EA for AC section as follows:

Substitute 200GPa for E and 706.86×106m2 for ACE.

EA=1.25664×103(200GPa(109Pa1GPa))=1.25664×103×200×109=251.327×106N

Find the area (ACE) of rod CE using the relation:

ACE=π4(dCE) (2)

Here, dCE is diameter of rod CE.

Substitute 30mm dCE in Equation (2).

ACE=π4(30)2=706.85mm2(1m103mm)2=706.86×106m2

Find the value of EA for CE section as follows:

Substitute 105GPa for E and 706.86×106m2 for ACE.

EA=706.86×106((105GPa(109Pa1GPa)))=706.86×106×105×109=74.220×106N

Sketch the free body diagram of cylindrical rod as shown in Figure 1.

Mechanics of Materials, 7th Edition, Chapter 2.3, Problem 41P

Refer to Figure 1.

Take section A to B.

Find the change in length AB using the relation:

δAB=PLEA (3)

Here, P is the load, L is the length of rod, E is the young’s modulus, and A is the area of the section AB.

Substitute RA for P, 180mm for L, and 251.327×106N for EA in Equation (3).

δAB=RA×180mm(1m103mm)251.327×106=RA×0.180251.327×106=716.20×1012RA

Take section B to C.

Find the change in length BC using the relation:

δBC=PLEA (4)

Here, A is the area of the section BC.

Substitute RA(60×103) for P, 120mm for L, and 251.327×106N for EA in Equation (4).

δBC=RA(60×103)×120mm(1m103mm)251.327×106=(RA60×10)3×0.120251.327×106=447.47×1012RA26.848×106

Take section C to D.

Find the change in length CD using the relation:

δCD=PLEA (5)

Here, A is the area of the section CD.

Substitute RA(60×103) for P, 100mm for L, and 74.220×106 for EA in Equation (5).

δCD=RA(60×103)×100mm(1m103mm)74.220×106=(RA60×103)×0.10074.220×106=1.347×109RA80.841×106

Take section D to E.

Find the change in length DE using the relation:

δDE=PLEA (6)

Here, A is the area of the section CD.

Substitute RA(100×103) for P, 100mm for L, and 74.220×106 for EA in Equation (6).

δDE=RA(100×103)×100mm(1m103mm)74.220×106=(RA100×103)×0.10074.220×106=1.347×109RA134.735×106

Take section A to E.

δAE=δAB+δBC+δCD+δDE (7)

Here, δAB is rotation of angle AB, δCB is rotation of angle BC, δCD is rotation of angle CD, δDE is rotation of angle DE.

Substitute (1.347×109RA134.735×106) for δDE, 1.347×109RA80.841×106 for δCD, 447.47×1012RA26.848×106 for δCB, and 716.20×1012RA for δAB in Equation (7).

δAE=[716.20×1012RA+447.47×1012RA26.848×106+1.347×109RA80.841×106+1.347×109RA134.735×106]=3.85837×109RA242.424×106

Since the point E cannot move relative to A.

δAE=0

Find the reaction at point A as follows:

3.85837×109RA242.424×106=03.85837×109RA=242.424×106RA=62.831×103N(1kN103N)RA=62.8kN

Thus, the reaction at point A is 62.8kN_.

Find the reaction at point E as follows:

RE=RA(100×103) (8)

Substitute 62.831×103N for RA in Equation (8).

RE=62.831×103(100×103)=37.2×103N(1kN103N)=37.2kN

Thus, the reaction at point E is 37.2kN_.

(b)

Expert Solution
Check Mark
To determine

Find the deflection of point C.

Answer to Problem 41P

The deflection of point C is 46.3μm_

Explanation of Solution

Calculation:

Determine the deflection of point C using the relation:

δC=δAB+δBC (9)

Substitute 447.47×1012RA26.848×106 for δCB and 716.20×1012RA for δAB in Equation (9).

δC=447.47×1012RA26.848×106+716.20×1012RA

Substitute 62.831×103N for RA in Equation (9).

δC=(447.47×1012(62.831×103)26.848×106+716.20×1012(62.831×103))=(2.8111×100526.848×106+4.499×105)=46.3×106m(1μm106m)=46.3μm

Thus, the deflection of point C is 46.3μm_.

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Chapter 2 Solutions

Mechanics of Materials, 7th Edition

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The...Ch. 2.13 - Fig. P2.95 and P2.96 2.96 (a) For P = 13 kips and...Ch. 2.13 - 2.97 Knowing that the hole has a diameter of 9 mm,...Ch. 2.13 - For P = 100 kN, determine the minimum plate...Ch. 2.13 - Prob. 99PCh. 2.13 - A centric axial force is applied to the steel bar...Ch. 2.13 - The cylindrical rod AB has a length L = 5 ft and a...Ch. 2.13 - Fig. P2.101 and P.102 2.102 The cylindrical rod AB...Ch. 2.13 - Rod AB is made of a mild steel that is assumed to...Ch. 2.13 - Prob. 104PCh. 2.13 - Rod ABC consists of two cylindrical portions and...Ch. 2.13 - Prob. 106PCh. 2.13 - Prob. 107PCh. 2.13 - Prob. 108PCh. 2.13 - Each cable has a cross-sectional area of 100 mm2...Ch. 2.13 - Prob. 110PCh. 2.13 - Two tempered-steel bars, each 316 in. thick, are...Ch. 2.13 - Prob. 112PCh. 2.13 - Prob. 113PCh. 2.13 - Prob. 114PCh. 2.13 - Prob. 115PCh. 2.13 - Prob. 116PCh. 2.13 - Prob. 117PCh. 2.13 - Prob. 118PCh. 2.13 - Prob. 119PCh. 2.13 - For the composite bar in Prob. 2.111, determine...Ch. 2.13 - Prob. 121PCh. 2.13 - Bar AB has a cross-sectional area of 1200 mm2 and...Ch. 2.13 - Bar AB has a cross-sectional area of 1200 mm2 and...Ch. 2 - The uniform wire ABC, of unstretched length 2l, is...Ch. 2 - The aluminum rod ABC (E = 10.1 106 psi), which...Ch. 2 - Two solid cylindrical rods are joined at B and...Ch. 2 - Prob. 127RPCh. 2 - Prob. 128RPCh. 2 - Prob. 129RPCh. 2 - A 4-ft concrete post is reinforced with four steel...Ch. 2 - The steel rods BE and CD each have a 16-mm...Ch. 2 - Prob. 132RPCh. 2 - Prob. 133RPCh. 2 - The aluminum test specimen shown is subjected to...Ch. 2 - Prob. 135RP
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