Mechanics of Solids (VDB1063)

Shear Formula

Lecturer: Dr. Montasir O. Ahmed

Learning Outcomes

• To evaluate the shear stress by applying the shear formula

LECTURE OUTLINES

Shear in Straight Members

The Shear Formula

Copyright © 2011 Pearson Education South Asia Pte Ltd

• Transverse shear stress always has its associated longitudinal shear stress acting

along longitudinal planes of the beam.

Shear in Straight Members

• Effects of Shear Stresses:

Shear in Straight Members

• As a result of shear stress, shear strain will be developed and these

will tend to distort the cross section in a complex manner (warping ).

• When beam is subjected to bending as well as shear, the cross

section will not remain plane as assumed in the application of the

flexure formula. However, for slender beams, this cross sectional

warping is small and can be neglected.

The Shear Formula

Shear Formula for longitudinal and transverse shear stress:

Q =

𝐴′ = Is the area of the top/bottom portion of the member’s cross sectional area, above/below

the section plane where t is measured.

= Is the distance from the NA to the centroid of 𝐴′

𝜏 = 𝑉𝑄

𝐼𝑡

𝑦′

𝑦′ 𝐴′

𝑦′

where

𝜏 = shear stress in a member at point located a distance from

the neutral axis.

V = the internal resultant shear force, determined from the method of

sections and the equations of equilibrium.

I = the moment of inertia of the entire cross sectional area calculated about the NA.

t = width of the member’s cross sectional area, measured at the point where 𝜏 is to be determined.

Flat Sections

7

• Limitation on the use of the shear formula:

The Shear Formula

• Flexure formula was used in the derivation of the shear formula. Therefore, it is

necessary that the material behave in a linear elastic manner.

Flange-web junction Cross section with an

irregular or

nonrectangular boundary

The Shear Formula

Internal Shear• Section the member

perpendicular to its axis andobtain V

Section Properties

• Determine the location of NA

• Determine I for the entire cross section about the NA

• Pass an imaginary horizontal section through the point

where the 𝜏 is to be determined

• Measure t

• Determine A/ , Q = y-/ A/

Shear Stress• Apply the shear

stress formula

Procedure for application the shear formula

EXAMPLE 1

A steel wide-flange beam has the dimensions shown in Fig. 7–11a.

If it is subjected to a shear of V = 80kN, plot the shear-stress

distribution acting over the beam’s cross-sectional area.

Copyright © 2011 Pearson Education South Asia Pte Ltd

EXAMPLE 1 (cont)

• The moment of inertia of the cross-sectional area about the neutral axis is

• For point B, tB’ = 0.3m, and A’ is the dark

shaded area shown in Fig. 7–11c

Copyright © 2011 Pearson Education South Asia Pte Ltd

Solutions

4623

3

m 106.15511.002.03.002.03.012

12

2.0015.012

1

I

MPa 13.1

3.0106.155

1066.01080

m 1066.002.03.011.0''

6

33

'

''

33

'

B

BB

B

It

VQ

AyQ

EXAMPLE 1 (cont)

• For point B, tB = 0.015m, and QB = QB’,

• For point C, tC = 0.015m, and A’ is

the dark shaded area in Fig. 7–11d.

• Considering this area to be composed of two rectangles,

• Thus,

Copyright © 2011 Pearson Education South Asia Pte Ltd

Solutions

MPa 6.22015.0106.155

1066.010806

33

B

BB

It

VQ

33 m 10735.01.0015.005.002.03.011.0'' AyQC

MPa 2.25015.0106.155

10735.010806

33

max

C

cC

It

VQ

Important Points in this Lecture

• There is longitudinal shear stress associated with the transverse shear stresses.

• Shear stresses produce non uniform shear strain.

• The shear Formula for longitudinal and transverse shear stress is:

• The shear formula can’t be applied in predicting shear stress for:

1- Flat Sections

2- Flange-web junction

3- Cross section with an irregular or nonrectangular boundary

𝜏 = 𝑉𝑄

𝐼𝑡

Next Class

Stress Caused by Combined Loadings

Thank You