1.3 Shear Stress
Forces parallel to the area resisting the force cause shearing
stress. It differs to tensile and compressive stresses, which are
caused by forces perpendicular to the area on which they act.
Shearing stress is also known as tangential stress.
𝑽
𝜏=
𝑨𝒗
where V is the resultant shearing force which passes through the
centroid of the area Av being sheared.
ME33 : Fluid Flow
1
Fundamentals
of Deformable Bodies
Chapter 2: Properties of Fluids
Stresses
1.3 Shear Stress
Illustrative Example 4:
What force is required to punch a 20-mm-diameter hole in a plate that is 25
mm thick? The shear strength is 350 MN/m2
Solution:
The resisting area is the shaded area along the
perimeter and the shear force V is equal to the
punching force P.
𝑽
𝑨
𝑁
𝜏 = ; 𝑉 = 𝜏𝐴 = 350 𝑚𝑚2 (𝜋)(25 𝑚𝑚)(20𝑚𝑚)
𝜏 = 𝟓𝟒𝟗 𝟕𝟕𝟖. 𝟕𝟏 𝑵 = 𝟓𝟒𝟗. 𝟖 𝒌𝑵
ME33 : Fluid Flow
2
Fundamentals
of Deformable Bodies
Chapter 2: Properties of Fluids
Stresses
1.3 Shear Stress
Illustrative Example 5:
As shown in the figure, a hole is to be punched out of a plate having a
shearing strength of 40 ksi. The compressive stress in the punch is limited to 50
ksi. (a) Compute the maximum thickness of plate in which a hole 2.5 inches in
diameter can be punched. (b) If the plate is 0.25 inch thick, determine the
diameter of the smallest hole that can be punched.
ME33 : Fluid Flow
3
Fundamentals
of Deformable Bodies
Chapter 2: Properties of Fluids
Stresses
1.3 Shear Stress
Solution:
a.) Maximum thickness of plate:
Based on puncher strength:
𝑙𝑏 𝜋
2.5 𝑖𝑛 2
2
𝑖𝑛 4
= 245 436.93 𝑙𝑏 (equivalent shear force of the plate)
𝑃 = 𝜎𝐴 = 50 000
Based on shear strength of plate:
𝜏=
𝑉
𝑙𝑏
245 436.93 𝑙𝑏
; 40 000 2 =
; 𝒕 = 𝟎. 𝟕𝟖𝟏𝟐𝟓 𝒊𝒏
𝐴
𝑖𝑛
𝜋(2.5 𝑖𝑛)(𝑡)
b.) Diameter of smallest hole:
Based on compression of puncher:
𝑃 = 𝜎𝐴 = 50 000
𝑙𝑏 𝜋
𝑖𝑛2 4
𝑑 2 = 12 500𝜋𝑑2 (equivalent shear force for plate)
Based on shearing of plate:
𝑉
𝑙𝑏
12 500𝜋𝑑2
𝜏 = ; 40 000 2 =
; 𝒅 = 𝟎. 𝟖𝒊𝒏
𝐴
𝑖𝑛
𝜋(𝑑)(0.25 𝑖𝑛)
ME33 : Fluid Flow
4
Fundamentals
of Deformable Bodies
Chapter 2: Properties of Fluids
Stresses
1.3 Shear Stress
Illustrative Example 6:
Find the smallest diameter bolt that can be used in the clevis shown in the
figure if P = 400 kN. The shearing strength of the bolt is 300 MPa.
Solution:
The bolt is subject to double shear.
𝑉
𝜏 = ; 𝑉 = 𝜏𝐴
𝐴
400 𝑘𝑁
1000 𝑁
𝑁
𝜋 2
= 300
(2)
𝑑
1𝑘𝑁
𝑚𝑚2
4
𝒅 = 𝟐𝟗. 𝟏𝟑 𝒎𝒎
ME33 : Fluid Flow
5
Fundamentals
of Deformable Bodies
Chapter 2: Properties of Fluids
Stresses