1. STRUCTURAL MECHANICS: CE203
Chapter 6
Transverse Shear
Notes are based on Mechanics of Materials: by R. C. Hibbeler, 7th Edition, Pearson
Dr B. Achour & Dr Eng. K. El-kashif
Civil Engineering Department, University of Hail, KSA
(Spring 2011)
Chapter 7: Transverse Shear

2. Shear in Straight Members
When a shear V is applied, non-uniform shear-strain distribution over the cross section will cause the cross section to warp.
The relationship between moment and shear is
Chapter 7: Transverse Shear

3. The Shear Formula
The shear formula is used to find the transverse shear stress on the beam’s cross-sectional area.
τ = the shear stress in the member
V = internal resultant shear force
I = moment of inertia of the entire cross-sectional area
t = width of the member’s cross-sectional area
Chapter 7: Transverse Shear

4. Shear Stresses in Beams
For rectangular cross section, shear stress varies parabolically with depth and maximum shear stress is along the neutral axis.
Chapter 7: Transverse Shear

5. Example 7.1
The beam is made of wood and is subjected to a resultant internal vertical shear force of V = 3 kN. (a) Determine the shear stress in the beam at point P, and (b) compute the maximum shear stress in the beam.
Solution:
(a) The moment of inertia of the cross sectional area computed about the neutral axis is
Applying the shear formula, we have
Chapter 7: Transverse Shear

6. Solution:
(b) Maximum shear stress occurs at the neutral axis, since t is constant throughout the cross section,
Applying the shear formula yields
Chapter 7: Transverse Shear

7. Shear Flow in Built-Up Members
For fasteners it is necessary to know the shear force by the fastener along the member’s length.
This loading is referred as the shear flow q, measured as a force per unit length.
q = shear flow
V = internal resultant shear
I = moment of inertia of the entire cross-sectional area
Chapter 7: Transverse Shear

8. Example 7.4
The beam is constructed from four boards glued together. If it is subjected to a shear of V = 850 kN, determine the shear flow at B and C that must be resisted by the glue.
Solution:
The neutral axis (centroid) will be located from the bottom of the beam,
The moment of inertia computed about the neutral axis is thus
Since the glue at B and holds the top board to the beam
Chapter 7: Transverse Shear

9. Solution:
Likewise, the glue at C and C’ holds the inner board to the beam
Therefore the shear flow for BB’ and CC’,
Since two seams are used to secure each board, the glue per meter length of beam at each seam must be strong enough to resist one-half of each calculated value of q’.
Chapter 7: Transverse Shear

10. Example 7.7
The thin-walled box beam is subjected to a shear of 10 kN. Determine the variation of the shear flow throughout the cross section.
Solution:
The moment of inertia is
For point B, the area thus q’B = 0.
Also,
For point C,
The shear flow at D is
Chapter 7: Transverse Shear