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, Problem 134RP

The aluminum test specimen shown is subjected to two equal and opposite centric axial forces of magnitude P. (a) Knowing that E = 70 GPa and σall = 200 MPa, determine the maximum allowable value of P and the corresponding total elongation of the specimen, (b) Solve part a, assuming that the specimen has been replaced by an aluminum bar of the same length and a uniform 60 × 15-mm rectangular cross section.

Chapter 2, Problem 134RP, The aluminum test specimen shown is subjected to two equal and opposite centric axial forces of

Fig. P2.134

(a)

Expert Solution
Check Mark
To determine

Find the maximum allowable load value P and corresponding total elongation length of the specimen.

Answer to Problem 134RP

The maximum allowable load value P is 92.3kN_.

The total elongation length of the specimen is 0.791mm._

Explanation of Solution

Given information:

The modulus of elasticity (E) is 70GPa.

The allowable stress (σall) is 200MPa.

The width (D) of the specimen is 75mm.

The width (d) of the fillet is 60mm

The radius of the fillet is 6mm

The uniform rectangular cross section is (60×15)mm.

Calculation:

Determine the area (A) of the cross section as follows:

A=bt (1)

Here, b is the width of the specimen and t is thickness of specimen.

Substitute 60mm for b and 15mm for t in Equation (1).

A=60×15=900mm2(1m103mm)2=900×106m2

Test specimen:

Calculate the ratio of (Dd) as follows:

Substitute 75mm for D and 60mm for d.

Dd=7560=1.25

Calculate the ratio of rd as follows:

Substitute 6mm for r and 60mm for d.

rd=660=0.10

Refer to Figure 2.52b (flat bars with fillets), “Stress concentration factors for flat bars under axial loading” in the textbook.

Get the stress concentration factor (K) using the ratio of Dd and rd.

Get the value of K is 1.95 for the corresponding value of Dd=1.25 and rd=0.10

Calculate the maximum allowable load using the expression as follows:

σmax=KPAPall=AσmaxK (2)

Here, K is stress concentration factor and σmax is maximum stress.

Substitute 900×106m2 for A, 200MPa for σmax, and 1.95 for K in Equation (2).

Pall=900×106×200MPa×(106Pa1MPa)1.95=900×106×200×1061.95=92.308×103N(1kN103N)=92.3kN

Thus, the maximum allowable value P is 92.3kN_.

Determine the wide area (Aw) of the cross section as follows:

Aw=bt (3)

Substitute 75mm for b and 15mm for t in Equation (3).

Aw=75×15=1125mm2(1m103mm)2=1.125×103m2

Determine the total elongation of the specimen using the relation:

δ=PiLiAiEi=PELiEi (4)

Substitute 92.308×103N for P, 70GPa for E, [0.1501.125×103+0.300900×106+0.1501.125×103] for LiEi in Equation (4).

δ=92.308×10370GPa(106Pa1GPa)[0.1501.125×103+0.300900×106+0.1501.125×103]=92.308×10370×109×[0.1501.125×103+0.300900×106+0.1501.125×103]=7.91×104m(103mm1m)=0.791mm

Thus, the total elongation of the specimen is 0.791mm._

(b)

Expert Solution
Check Mark
To determine

Find the maximum allowable load value P of an aluminium bar and total elongation length of the aluminum bar.

Answer to Problem 134RP

The maximum allowable load value P is 180.0kN_.

The total elongation length of the aluminum bar is 1.714mm._

Explanation of Solution

Calculation:

Calculate the maximum allowable load of uniform bar as follows:

P=AσAll (5)

Substitute 900×106m2 for A and 200MPa for σmax in Equation (5).

P=900×106×200MPa(106Pa1MPa)=900×106×200×106=180×103N(1kN103N)=180kN

Thus, the maximum allowable value P is 180.0kN_.

Calculate the total elongation of uniform bar as follows:

δ=PLAE (6)

Substitute 180kN for P, 600mm for L, 900×106m2 for A and 70GPa for E in Equation (6).

δ=180kN(103N1kN)×600mm(1m103mm)900×106×70GPa×109Pa1GPa=180×103×0.6900×106×70×109=1.714×106m((103mm1m))=1.714mm

Thus, the total elongation of the uniform bar is 1.714mm._

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Chapter 2 Solutions

Mechanics of Materials, 7th Edition

Ch. 2.1 - A block of 10-in. length and 1.8 1.6-in. cross...Ch. 2.1 - A square yellow-brass bar must not stretch more...Ch. 2.1 - Rod BD is made of steel (E = 29 106 psi) and is...Ch. 2.1 - The 4-mm-diameter cable BC is made of a steel with...Ch. 2.1 - A single axial load of magnitude P = 15 kips is...Ch. 2.1 - A 250-mm-long aluminum tube (E = 70 GPa) of 36-mm...Ch. 2.1 - The specimen shown has been cut from a...Ch. 2.1 - The brass tube AB (E = 105 GPa) has a...Ch. 2.1 - Both portions of the rod ABC are made of an...Ch. 2.1 - The rod ABC is made of an aluminum for which E =...Ch. 2.1 - For the steel truss (E = 200 GPa) and loading...Ch. 2.1 - For the steel truss (E = 29 106 psi) and loading...Ch. 2.1 - Members AB and BC are made of steel (E = 29 106...Ch. 2.1 - The steel frame (E = 200 GPa) shown has a diagonal...Ch. 2.1 - Link BD is made of brass (E = 105 GPa) and has a...Ch. 2.1 - Members ABC and DEF are joined with steel links (E...Ch. 2.1 - Each of the links AB and CD is made of aluminum (E...Ch. 2.1 - The length of the 332-in.-diameter steel wire CD...Ch. 2.1 - A homogenous cable of length L and uniform cross...Ch. 2.1 - The vertical load P is applied at the center A of...Ch. 2.1 - Denoting by the "engineering strain'' in a...Ch. 2.1 - The volume of a tensile specimen is essentially...Ch. 2.3 - An axial centric force of magnitude P = 450 kN is...Ch. 2.3 - An axial centric force of magnitude P = 450 kN is...Ch. 2.3 - The 4.5-ft concrete post is reinforced with six...Ch. 2.3 - The 4.5-ft concrete post is reinforced with six...Ch. 2.3 - An axial force of 200 kW is applied to the...Ch. 2.3 - The length of the assembly shown decreases by 0.40...Ch. 2.3 - A polystyrene rod consisting of two cylindrical...Ch. 2.3 - Three steel rods (E = 29 106 psi) support an...Ch. 2.3 - Fig. P2.41 2.41 Two cylindrical rods, one of steel...Ch. 2.3 - Solve Prob. 2.41, assuming that rod AC is made of...Ch. 2.3 - Each of the rods BD and CE is made of brass (E =...Ch. 2.3 - The rigid bar AD is supported by two steel wires...Ch. 2.3 - The rigid bar ABC is suspended from three wines of...Ch. 2.3 - The rigid bar AD is supported by two steel wires...Ch. 2.3 - The aluminum shell is fully bonded to the brass...Ch. 2.3 - The aluminum shell is fully bonded to the brass...Ch. 2.3 - The brass shell (b = 11.6 10-6/F) is fully bonded...Ch. 2.3 - The concrete post (Ec = 3.6 106) psi and c = 5.5 ...Ch. 2.3 - A rod consisting of two cylindrical portions AB...Ch. 2.3 - A rod consisting of two cylindrical portions AB...Ch. 2.3 - Fig. P2.52 2.52 A rod consisting of two...Ch. 2.3 - The steel rails of a railroad (rack (Es = 200GPa,...Ch. 2.3 - Two steel bars (Es = 200 GPa and s = 11.7 10-6/C)...Ch. 2.3 - Determine the maximum load P that can be applied...Ch. 2.3 - An aluminum rod (Ea = 70 GPa, a = 23.6 10-6/C)...Ch. 2.3 - Knowing that a 0.02-in. gap exists when the...Ch. 2.3 - Determine (a) the compressive force in the bars...Ch. 2.3 - At room temperature (20C) a 0.5-mm gap exists...Ch. 2.9 - A standard tension test is used to determine the...Ch. 2.9 - A 2-m length of an aluminum pipe of 240-nun outer...Ch. 2.9 - A line of slope 4:10 has been scribed on a...Ch. 2.9 - A 2.75-kN tensile load is applied to a test coupon...Ch. 2.9 - Fig. P2.65 2.65 In a standard tensile test a steel...Ch. 2.9 - The change in diameter of a large steel bolt is...Ch. 2.9 - The brass rod AD is fitted with a jacket that is...Ch. 2.9 - A fabric used in air-inflated structures is...Ch. 2.9 - A 1-in. square was scribed on the side of a large...Ch. 2.9 - The block shown is made of a magnesium alloy for...Ch. 2.9 - The homogeneous plate ABCD is subjected to a...Ch. 2.9 - For a member under axial loading, express the...Ch. 2.9 - In many situations it is known that the normal...Ch. 2.9 - In many situations physical constraints prevent...Ch. 2.9 - The plastic block shown is bonded to a rigid...Ch. 2.9 - The plastic block shown is bonded to a rigid...Ch. 2.9 - Two blocks of rubber with a modulus of rigidity G...Ch. 2.9 - Fig. P2.77 and P2.78 2.78 Two blocks of rubber...Ch. 2.9 - An elastomeric bearing (G = 130 psi) is used to...Ch. 2.9 - 2.80 For the elastomeric bearing In Prob. 2.79...Ch. 2.9 - A vibration isolation unit consists of two blocks...Ch. 2.9 - Prob. 82PCh. 2.9 - Prob. 83PCh. 2.9 - Prob. 84PCh. 2.9 - Prob. 85PCh. 2.9 - A 2.75-kN tensile load is applied to a test coupon...Ch. 2.9 - A vibration isolation support consists of a rod A...Ch. 2.9 - Prob. 88PCh. 2.9 - Prob. 89PCh. 2.9 - Show that for any given material, the ratio G/E of...Ch. 2.9 - Prob. 91PCh. 2.9 - Prob. 92PCh. 2.13 - Knowing that, for the plate shown, the allowable...Ch. 2.13 - Knowing that P = 38 kN, determine the maximum...Ch. 2.13 - A hole is to be drilled in the plate at A. 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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