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

Chapter 7: Transverse Shear

Shear in Straight MembersShear 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

The Shear FormulaThe 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 memberV = internal resultant shear forceI = moment of inertia of the entire cross-sectional areat = width of the member’s cross-sectional area

Chapter 7: Transverse Shear

Shear Stresses in BeamsShear 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

Example 7.1Example 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

Solution: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

Shear Flow in Built-Up MembersShear 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 flowV = internal resultant shearI = moment of inertia of the entire cross-sectional area

Chapter 7: Transverse Shear

Example 7.4Example 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

Solution: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

Example 7.7Example 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