Learning Objectives:

• Review simple beam theory.

• Consider effects of bending on shear forces in beams.

Shear stresses in beams is given by:

\[\tau = \dfrac{VQ}{Ib}\]

where Q is the first moment of area \(\int ydA\) of the surface section of the cross-section.

For the rectangular beam, one has the maximum shear stress in the cross section arises at the neutral surface: \[\tau_{max} = \dfrac{3V}{2bh} = \dfrac{3V}{2A}\] and the shear stress dies away towards the upper and lower surfaces. Note that the average shear stress over the cross-section is \(\dfrac{V}{A}\) and the maximum shear stress is 150% of this value.

Solved Example: 44-1-01

A symmetrical I-section is subjected to shear force. The shear stress is induced across the section is maximum at which location?

A. Extreme fibers

B. At the bottom of the flanges in flanges

C. At the bottom of the flanges in web portion

D. At the neutral axis.

Shear stress in beams is maximum at the neutral axis and it is zero at the extreme top and bottom layers.

Correct Answer: D

Solved Example: 44-1-02

Shear stress is zero at the:

A. Outermost fiber

B. Central fiber

C. Neither outermost nor central fiber

D. None of these.

Correct Answer: A

Solved Example: 44-1-03

Shear stress in the beam acting on the cross section is:

A. Normal to the cross section

B. Tangential to the cross section

C. Neither normal nor tangential

D. None of these.

Correct Answer: B

Solved Example: 44-1-04

A symmetric I-section (with width of each flange = 50 mm, thickness of each flange = 10 mm, depth of web = 100 mm, and thickness of web = 10 mm) of steel is subjected to a shear force of 100 kN. Find the magnitude of the shear stress in the web at its junction with the top flange.

A. 71.11 MPa

B. 61.11 MPa

C. 32.23 MPa

D. 43.12 MPa

\[I =\dfrac {50\times 120^{3}}{12}-\dfrac {40\times 100^{3}}{12} =3.866\times 10^{6}\ mm^{4}\] \[q =\dfrac {SA\overline {y}}{Ib} =\dfrac {100\times 10^{3}\times 50\times 10\times 55}{3.866\times 10^{6}\times 10} =71.12\ \mathrm{N/mm^{2}}\]

Correct Answer: A

Solved Example: 44-1-05

An inverted T-section is subjected to a shear force F. The maximum shear stress will occur at: (TNTRB ME 2017)

A. Top of the section

B. Junction of web and flange

C. Neutral axis of the section

D. Bottom of the section

Correct Answer: C