## Wednesday, July 16, 2014

### SHEAR STRESS IN STEEL BEAM

This beam, supporting a column point load of 96 k over a door, is a composite beam consisting of a wide-flange base beam with 8x½ in plates welded to top and bottom flanges. The beam is analyzed with and without  plates.  As shown before, for steel beams shear stress is assumed to be resisted by the web only, computed as fv  = V/Av.  The base beam is a W10x49 [10 in (254 mm) nominal depth, 49 lbs/ft (6.77 kg/m) DL] with a moment of inertia Ixx= 272 in^4 (11322 cm^4).  Shear in the welds connecting the plates to the beam is found using the shear flow formula q = VQ/(I).

1  Beam of L= 6 ft (1.83 m) span with P = 96 k point load
2  Composite wide-flange beam W10x49 with 8x½ inch stiffener plates

Shear force V = P/2 = 96/2          V = 48 k
Bending moment M = 48(3)         M = 144 k’

Wide-flange beam

Since the beam would fail in bending, a composite beam is used.

Composite beam

Since the shear force remains unchanged, the web shear stress is still ok.

Shear flow q in welded plate connection

Since there are two welds, each resists half the total shear flow

Note: in this steel beam, bending is stress is more critical than shear stress; this is typical for steel beams, except very short ones.

## Monday, July 7, 2014

### SHEAR STRESS IN WOOD I-BEAM

Since this is not a rectangular beam, shear stress must be computed by the general shear formula.  The maximum shear stress at the neutral axis as well as shear stress at the intersection between flange and web (shear plane As) will be computed.  The latter gives the shear stress in the glued connection.  To compare shear- and bending stress the latter is also computed.

Beam of L= 10 ft length, with uniform load w= 280plf (W = 2800 lbs)
Cross-section of wood I-beam

For the formula v= VQ/(Ib) we must find the moment of inertia of the entire cross-section. We could use the parallel axis theorem of Appendix A.  However, due to symmetry, a simplified formula is possible, finding the moment of inertia for the overall dimensions as rectangular beam minus that for two rectangles on both sides of the web.

Note c= 10/2 = 5 (half the beam depth due to symmetry)

Static moment Q of flange about the neutral axis:

Shear stress at flange/web intersection:

Static moment Q of flange plus upper half of web about the neutral axis

Maximum shear stress at neutral axis:

Note: Maximum shear stress reaches almost the allowable stress limit, but bending stress is well below allowable bending stress because the beam is very short.  We can try at what span the beam approaches allowable stress, assuming L= 30 ft, using the same total load W = 2800 lbs to keep shear stress constant:

At 30 ft span bending stress is just over the allowable stress of 1450 psi.  This shows that in short beams shear governs, but in long beams bending or deflection governs.

## Wednesday, July 2, 2014

### SHEAR STRESS IN WOOD AND STEEL BEAMS

Based on the forgoing general derivation of shear stress, the formulas for shear stress in rectangular wood beams and flanged steel beams is derived here.  The maximum stress in those beams is customarily defined as fv instead of v in the general shear formula.

Shear at neutral axis of rectangular beam (maximum stress),
Note: this is the same formula derived for maximum shear stress before

Shear stress at the bottom of rectangular beam.  Note that y= 0 since the centroid of the area above the shear plane (bottom) coincides with the neutral axis of the entire section. Thus Q= Ay = (bd/2) 0 = 0, hence

v = V 0/(I b) = 0 = fv, thus
fv = 0

Note: this confirms an intuitive interpretation that suggests zero stress since no fibers below the beam could resist shear

3  Shear stress at top of rectangular beam.  Note A = 0b = 0 since the depth of the shear area above the top of the beam is zero.  Thus

Q = Ay = 0 d/2 = 0, hence v = V 0/(I b) = 0 = fv, thus
fv = 0

Note: this, too, confirms an intuitive interpretation that suggests zero stress since no fibers above the beam top could resist shear.

4
Shear stress distribution over a rectangular section is parabolic as implied by the formula Q=b(d^2)/8 derived above.

5  Shear stress in a steel beam is minimal in the flanges and parabolic over the web.

The formula v = VQ/(I b) results in a small stress in the flanges since the width b of flanges is much greater than the web thickness.  However, for convenience, shear  stress in steel beams is computed as “average” by the simplified formula: