Statics and Mechanics of Materials (5th Edition)
Statics and Mechanics of Materials (5th Edition)
5th Edition
ISBN: 9780134382593
Author: Russell C. Hibbeler
Publisher: PEARSON
bartleby

Videos

Textbook Question
Chapter 15.2, Problem 1FP

Determine the minimum dimension a to the nearest mm of the beam’s cross section to safely support the load. The wood has an allowable normal stress of σallow = 10 MPa and an allowable shear stress of τallow = 1 MPa.

Chapter 15.2, Problem 1FP, Determine the minimum dimension a to the nearest mm of the beams cross section to safely support the

Expert Solution & Answer
Check Mark
To determine

Find the minimum dimension a to the nearest beam cross section.

Answer to Problem 1FP

The minimum dimension a to the nearest beam cross section is 139.2mm_.

Explanation of Solution

Given information:

The allowable bending stress is 10 MPa.

The allowable shear stress is 1 MPa.

The width of the cross section is a.

The depth of the cross section is 2a.

Calculation:

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

Statics and Mechanics of Materials (5th Edition), Chapter 15.2, Problem 1FP

Consider a section, VA is the shear force and load acting downward direction.

Apply the equilibrium condition along y –direction.

Fy=0

VA=6+6=12kN

Determine the bending moment at point A.

MA=(6×2)+(6×1)=12+6=18kNm

Calculate the moment of inertia (I) using the relation.

I=bd312

Here, b is the width of the section and d is the depth of the section.

Substitute a for b and 2a for d.

I=a×2a312=23a4

Calculate the dimension (a) value using the relation:

σallow=MmaxcI

Here, Mmax is the maximum moment, I is the moment of inertia, and c is the centroid from the neutral axis.

Substitute 23a4 for I, 18kNm for Mmax, 10 MPa for σallow, and a for c.

10×106=18×103(a)23a4a=0.1392m×1,000mm1m=139.2mm

Use, a=140mm

Hence, the minimum dimension a to the nearest beam cross section is 139.2mm_.

Find the value of I using the relation:

I=23a4

Substitute 140 mm for a.

I=23(140mm×1m1,000mm)4=23(0.14)4=0.256×103m4

Find the value of Qmax using the relation:

Qmax=Ay

Here, A is the area of the cross section and y is the centroid of area.

Substitute (0.14×0.14)m2 for A and 0.07m for y.

Qmax=0.07×0.14×0.14=1.372×103m3

Check:

Calculate the (τmax) using the relation:

(τmax)=VmaxQmaxIt

Here Vmax is the maximum shear force, I is the moment of inertia, and t is the thickness, and (τmax) is the shear stress.

Substitute 12kN for Vmax, 1.372×103m3 for Qmax, 0.256×103m4 for I, and 0.14 m. for a.

(τmax)=12×103(1.372(103))[0.2561(103(0.14))]=16.463.58×105=0.459MPa<τallow=1MPa

Hence it is OK.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Students have asked these similar questions
Determine the minimum dimension b to the nearest mm of the beam’s cross section to safely support the load. The wood has an allowable normal stress of sallow = 12 MPa and an allowable shear stress of tallow = 1.5 MPa.
The beam is constructed from three boards. Determine the maximum loads P that it can support if the allowable shear stress for the wood is tallow = 400 psi. What is the maximum allowable spacing s of the nails used to holdthe top and bottom flanges to the web if each nail can resist a shear force of 400 lb?
Determine the shear stress (in Mpa) in the 23.5-mm-diameter pin at A that support the beam if P = 28.36 kN, a = 2.25 m, and b = 6.56 m.

Chapter 15 Solutions

Statics and Mechanics of Materials (5th Edition)

Knowledge Booster
Mechanical Engineering
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.
Similar questions
    • SEE MORE QUESTIONS
    Recommended textbooks for you
  • Elements Of Electromagnetics
    Mechanical Engineering
    ISBN:9780190698614
    Author:Sadiku, Matthew N. O.
    Publisher:Oxford University Press
    Mechanics of Materials (10th Edition)
    Mechanical Engineering
    ISBN:9780134319650
    Author:Russell C. Hibbeler
    Publisher:PEARSON
    Thermodynamics: An Engineering Approach
    Mechanical Engineering
    ISBN:9781259822674
    Author:Yunus A. Cengel Dr., Michael A. Boles
    Publisher:McGraw-Hill Education
  • Control Systems Engineering
    Mechanical Engineering
    ISBN:9781118170519
    Author:Norman S. Nise
    Publisher:WILEY
    Mechanics of Materials (MindTap Course List)
    Mechanical Engineering
    ISBN:9781337093347
    Author:Barry J. Goodno, James M. Gere
    Publisher:Cengage Learning
    Engineering Mechanics: Statics
    Mechanical Engineering
    ISBN:9781118807330
    Author:James L. Meriam, L. G. Kraige, J. N. Bolton
    Publisher:WILEY
  • Elements Of Electromagnetics
    Mechanical Engineering
    ISBN:9780190698614
    Author:Sadiku, Matthew N. O.
    Publisher:Oxford University Press
    Mechanics of Materials (10th Edition)
    Mechanical Engineering
    ISBN:9780134319650
    Author:Russell C. Hibbeler
    Publisher:PEARSON
    Thermodynamics: An Engineering Approach
    Mechanical Engineering
    ISBN:9781259822674
    Author:Yunus A. Cengel Dr., Michael A. Boles
    Publisher:McGraw-Hill Education
    Control Systems Engineering
    Mechanical Engineering
    ISBN:9781118170519
    Author:Norman S. Nise
    Publisher:WILEY
    Mechanics of Materials (MindTap Course List)
    Mechanical Engineering
    ISBN:9781337093347
    Author:Barry J. Goodno, James M. Gere
    Publisher:Cengage Learning
    Engineering Mechanics: Statics
    Mechanical Engineering
    ISBN:9781118807330
    Author:James L. Meriam, L. G. Kraige, J. N. Bolton
    Publisher:WILEY
    Mechanics of Materials Lecture: Beam Design; Author: UWMC Engineering;https://www.youtube.com/watch?v=-wVs5pvQPm4;License: Standard Youtube License