Steel Grating Concentrated Load Example
***reference document “MBG534-12” METAL BAR GRATING ENGINEERING DESIGN MANUAL”
a = length of partially distributed uniform load or vehicular load, parallel with bearing bars, in.
b = thickness of rectangular bearing bar, in.
c = width of partially distributed uniform load or vehicular load, perpendicular to bearing bars, in.
d = depth of rectangular bearing bar, in.
Ac = distance center to center of main bars, riveted grating, in.
Ar = face to face distance between bearing bars in riveted grating, in.
Aw = center to center distance between bearing bars in welded and pressure locked gratings, in.
C = concentrated load at midspan, pfw
Dc = deflection under concentrated load, in.
Du = deflection under uniform load, in.
E = modulus of elasticity, psi
F = allowable stress, psi
I = moment of inertia, in4
IH20 = moment of inertia of grating under H20 loading, in4
Ib = I of bearing bar, in4
Ig = I of grating per foot of width, in4
In = moment of inertia of nosing, in4
K = number of bars per foot of grating width, 12"/Aw
L = clear span of grating, in. (simply supported)
M = bending moment, Ib-in
Mb = maximum M of bearing bar, Ib-in
Mg = maximum M of grating per foot of width, Ib-in
N = number of bearing bars in grating assumed to carry load
NbH20 = number of main bearing bars under load H20
NcH20 = number of connecting bearing bars under load H20
Pb = load per bar, Ib
Pu = total partially distributed uniform load, Ib
PuH20 = wheel load, H20, Ib
Pw = wheel load, lb
S = section modulus, in3
Sb = S of bearing bar, in3
Sg = S of grating per foot of width, in3
SH20b = section modulus at bottom of grating under H20 loading, in3
Sn = section modulus of nosing, in3
U = uniform load, psf
in. = inch
ft = foot
Ib = pounds
Ib-in = pound-inches
pfw = pounds per foot of grating width
psf = pounds per square foot
psi = pounds per square inch
1. Number of bearing bars per foot of width for welded grating
K = 12/AW
2. Section modulus of rectangular bearing bar
Sb = bd2/6 in3
3. Section modulus of grating per foot of width
Sg = Kbd2/6 in3 = KSb in3
4. Section modulus required for given moment and allowable stress
S = M/F in3
5. Moment of inertia of rectangular bearing bar
Ib = bd3/12 in4 = Sb d/2 in4
6. Moment of inertia of grating per foot of width
Ig = Kbd3/12 in4 = Klb in4
7. Bending moment for given allowable stress and section modulus
M = SF Ib-in
The following formulas are for simply supported beams with maximum moments and deflections occurring at midspan.
8. Maximum bending moment under concentrated load
M = CL/4 Ib-in per foot of grating width
9. Concentrated load to produce maximum bending moment
C = 4M/L Ib per foot of grating width
10. Maximum bending moment under uniform load
M = UL2/(8 x 12) = UL2/96 Ib-in per foot of grating width
11. Uniform load to produce maximum bending moment
U = 96M/L2 psf
12. Maximum bending moment due to partially distributed uniform load
M = Pu (2L - a)/8 Ib-in
13. Maximum deflection under concentrated load
Dc = CL3/48EIg in4.
14. Moment of inertia for given deflection under concentrated load
Ig = CL3/48EDc in4
15. Maximum deflection under uniform load
Du = 5UL4/(384 x 12Elg) = 5UL4/4608EIg in.
16. Moment of inertia for given deflection under uniform load
Ig = 5UL4/4608EDu in4
17. Maximum deflection under partially distributed uniform load
Du = Pu((a/2)3 + L3 - a2 L/2)/48ElbN in.
Required: A welded ASTM A36 steel grating Type W-22-4 to support a concentrated load, C, of 4,000 pounds per foot of width at midspan on a clear span of 8'-0". Deflection, D, is not to exceed the 0.25" recommended for pedestrian comfort.
Allowable stress, F = 20,000 psi
Modulus of elasticity, E = 29,000,000 psi
Span, L = 96in.
Bearing bar spacing, Aw = 1.375 in. K = 12/Aw = 12 / 1.375 = 8.727
For a span of 8'-0", the minimum size bearing bar to sustain a 4,000 pfw load is:
3 x 3/8
Ig = Klb = 8.727 x 0.8438 = 7.364 in4 Sg = KSb = 4.909 in3
C = 4Mg/L = 4 x F x Sg/96 = 4 x 20,000 x 4.909/96 = 4,091 pfw
Dc = CL3 /48Elg = 4,000 x (96)3 /(48 x 29,000,000 x 7.364) = 0.345 in.
Since this exceeds the recommended limitation, a grating with a larger moment of inertia is needed to keep the deflection less than 0.25 in.
Ig = CL3 /48EDc = 4,000 x (96)3 /(48 x 29,000,000 x 0.25) = 10.17 in4
Using the next larger size:
3-1/2 x 3/8
Ig = 8.727 x 1.3398 = 11.693 in4
Sg = 8.727 x 0.7656 = 6.682 in3
C = 4 x 20,000 x 6.682/96 = 5,568 pfw
D = 5,568 x (96)3 /(48 x 29,000,000 x 11.693) = 0.303 in.
Dc = 0.303 x 4,000/5,568 = 0.217 in. ≤ 0.25 in. OK