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 1.5, Problem 41P

Members AB and BC of the truss shown are made of the same alloy. It is known that a 20-mm-square bar of the same alloy was tested to failure and that an ultimate load of 120 kN was recorded. If bar AB has a cross-sectional area of 225 mm2, determine (a) the factor of safety for bar AB, (b) the cross-sectional area of bar AC if it is to have the same factor of safety as bar AB.

Fig. P1.40 and P1.41

Chapter 1.5, Problem 41P, Members AB and BC of the truss shown are made of the same alloy. It is known that a 20-mm-square bar

(a)

Expert Solution
Check Mark
To determine

The factor of safety for bar AB.

Answer to Problem 41P

The factor of safety for bar AB is 3.97_.

Explanation of Solution

Given information:

The ultimate load PU is 120kN.

The factor of safety F.S is 3.2.

The area (a) of square cross section is 20mm.

Calculation:

Refer to Figure P1.40 in the text book.

Find the length of member AB using the relation:

lAB=0.752+0.42=0.5625+0.16=0.85m

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

Mechanics of Materials, 7th Edition, Chapter 1.5, Problem 41P , additional homework tip  1

Here, Ax is the horizontal component of reaction at point A and Ay is the vertical component of reaction.

Refer to Figure 1.

Calculate the horizontal reaction A by using equilibrium Equation as follows:

MC=0

1.4Ax(0.75×28)=01.4Ax21=01.4Ax=21Ax=15kN

Calculate the vertical reaction (Ay) using the relation:

Fy=0

Ay28=0Ay=28kN

Sketch the free body diagram of joint A as shown in Figure 2.

Mechanics of Materials, 7th Edition, Chapter 1.5, Problem 41P , additional homework tip  2

Refer to Figure P1.40 in the text book.

Refer to Figure 2.

Fx=0

0.750.85FABAx=00.8823FABAx=0 (1)

Substitute 15kN for Ax in Equation (1).

0.8823FAB(15kN)=00.8823FAB=15FAB=150.8823FAB=17kN

Refer to Figure 2.

Fy=0

AyFAC0.40.85FAB=0 (2)

Substitute 28kN for Ay and 17kN for FAB in Equation (2).

28FAC0.40.85(17)=028FAC8=0FAC+20=0FAC=20kN

Find the area of test bar (A) using the relation:

A=a2 (3)

Substitute 20mm for a in Equation (3).

A=202=400mm2(1m103mm)2=400×106m2

Find the ultimate load for the material using the formula:

σU=PUA (4)

Here, PU is ultimate load.

Substitute 120kN for PU and 400×106m2 for A in Equation (4).

σU=120kN(103N1kN)400×106=120×103400×106=300×106Pa

Determine the factor of safety for bar AB using the relation:

F.S=FUFAB (5)

Here, FAB is force in member AB.

Modify Equation (5).

F.S=σUAABFAB (6)

Substitute 300×106Pa for σU, 17kN for FAB and 225mm2 for AAB in Equation (6).

F.S=300×106×225mm2(1m2106mm2)17kN(103N1kN)=300×106×225×10617×103=67,50017×103=3.97

Thus, the factor of safety for bar AB is 3.97_.

(b)

Expert Solution
Check Mark
To determine

The cross sectional area of bar AC.

Answer to Problem 41P

The cross sectional area of bar AC is 265mm2_.

Explanation of Solution

Calculation:

Show the expression factor of safety for bar AC using the relation:

F.S=FUFAC (7)

Here, FAC is force in member AC.

Modify Equation (7).

F.S=σUAACFACAAC=F.S(FAC)σU (8)

Substitute 300×106Pa for σU, 20kN for FAB and 3.97 for FS in Equation (8).

AAC=20kN(103N1kN)×3.97300×106=79,400300×106=264.67×106m2(106m1mm)2=265mm2

Thus, the cross sectional area of bar AC is 265mm2_.

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Chapter 1 Solutions

Mechanics of Materials, 7th Edition

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