7, you learned how the components of stress are transformed by a rotation of the coordinate axes and how to determine the principal planes, principal stresses, and maximum shearing str[r]
(1)MECHANICS OF MATERIALS
Ferdinand P Beer
E Russell Johnston, Jr. John T DeWolf
Lecture Notes: J Walt Oler
Texas Tech University
CHAPTER
Principle Stresses Under a Given
(2)Principle Stresses Under a Given Loading
Introduction
Principle Stresses in a Beam Sample Problem 8.1
Sample Problem 8.2
Design of a Transmission Shaft Sample Problem 8.3
(3)Introduction
• In Chaps and 2, you learned how to determine the normal stress due to centric loads
In Chap 3, you analyzed the distribution of shearing stresses in a circular member due to a twisting couple
In Chap 4, you determined the normal stresses caused by bending couples
In Chaps and 6, you evaluated the shearing stresses due to transverse loads
(4)Principle Stresses in a Beam
• Prismatic beam subjected to transverse loading
It VQ It
VQ I
Mc I
My
m xy
m x
= −
=
= −
=
τ τ
σ σ
• Principal stresses determined from methods of Chapter
• Can the maximum normal stress within the cross-section be larger than
I Mc m =
(5)(6)Principle Stresses in a Beam
• Cross-section shape results in large values of τxy near the surface where σx is also large
(7)Sample Problem 8.1
SOLUTION:
• Determine shear and bending moment in Section A-A’
• Calculate the normal stress at top surface and at flange-web junction A 160-kN force is applied at the end
of a W200x52 rolled-steel beam Neglecting the effects of fillets and of stress concentrations, determine
• Evaluate the shear stress at flange-web junction
(8)Sample Problem 8.1
SOLUTION:
• Determine shear and bending moment in Section A-A’
( )( ) kN 160 m -kN 60 m 375 kN 160 = = = A A V M
• Calculate the normal stress at top surface and at flange-web junction
(9)Sample Problem 8.1
• Evaluate shear stress at flange-web junction
( ) ( )( ) ( )( ) MPa 95 m 0079 m 10 52 m 10 248 kN 160 m 10 248 mm 10 248 96 12 204 6 3 = × × = = × = × = × = − − − It Q V Q A b τ
• Calculate the principal stress at flange-web junction ( ) ( ) 102 102 2 2 max ⎞ ⎛ + +
(10)Sample Problem 8.2
The overhanging beam supports a uniformly distributed load and a
concentrated load Knowing that for the grade of steel to used σall = 24 ksi and τall = 14.5 ksi, select the wide-flange beam which should be used
SOLUTION:
• Determine reactions at A and D.
• Find maximum shearing stress • Find maximum normal stress
• Calculate required section modulus and select appropriate beam section • Determine maximum shear and