Steel Grating Partially Distributed Uniform Load Calculation

***reference document “MBG534-12” METAL BAR GRATING ENGINEERING DESIGN MANUAL”

NOMENCLATURE

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.

A_c = distance center to center of main bars, riveted grating, in.

A_r = face to face distance between bearing bars in riveted grating, in.

A_w = center to center distance between bearing bars in welded and pressure locked gratings, in.

C = concentrated load at midspan, pfw

D_c = deflection under concentrated load, in.

D_u = deflection under uniform load, in.

E = modulus of elasticity, psi

F = allowable stress, psi

I = moment of inertia, in⁴

I_H20 = moment of inertia of grating under H20 loading, in⁴

I_b = I of bearing bar, in⁴

I_g = I of grating per foot of width, in⁴

I_n = moment of inertia of nosing, in⁴

K = number of bars per foot of grating width, 12"/A_w

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

ABBREVIATIONS

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

FORMULAS

1. Number of bearing bars per foot of width for welded grating

K = 12/AW

2. Section modulus of rectangular bearing bar

S_b = bd²/6 in³

3. Section modulus of grating per foot of width

S_g = Kbd²/6 in³ = KS_b in³

4. Section modulus required for given moment and allowable stress

S = M/F in³

5. Moment of inertia of rectangular bearing bar

I_b = bd³/12 in⁴ = S_b d/2 in⁴

6. Moment of inertia of grating per foot of width

I_g = Kbd³/12 in⁴ = Kl_b 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 = P_u (2L - a)/8 Ib-in

13. Maximum deflection under concentrated load

D_c = CL³/48EI_g in⁴.

14. Moment of inertia for given deflection under concentrated load

Ig = CL3/48EDc in4

15. Maximum deflection under uniform load

D_u = 5UL4/(384 x 12El_g) = 5UL4/4608EI_g in.

16. Moment of inertia for given deflection under uniform load

Ig = 5UL⁴/4608EDu in⁴

17. Maximum deflection under partially distributed uniform load

D_u = P_u((a/2)³ + L³ - a² L/2)/48El_bN in.

GRATING SELECTION

Example -Partially Distributed Uniform Load

Required: A welded ASTM A1011 CS Type B steel grating Type W-19-4 to support a partially distributed uniform load, Pu, of 1,500 pounds over an area of 6" x 9" centered at midspan on a clear span of 3'-6".

Deflection, D, is not to exceed the 0.25" recommended for pedestrian comfort.

Allowable stress, F = 18,000 psi

Modulus of elasticity, E = 29,000,000 psi

Span, L = 42 in.

Bearing bar spacing, Aw = 1.1875 in.

Since the 6" x 9" load is rectangular, two conditions must be investigated to determine which

condition places the greater stress on the grating:

Condition ‘A’ - 6" dimension parallel to bearing bars

Condition ‘B’ - 9" dimension parallel to bearing bars

Condition ‘A’        a = 6"        c = 9"

Find maximum bending moment with load centered at midspan

M = Pu(2L-a)/8 = 1,500 x (2 x 42 - 6)/8 = 14,625 Ib-in

Find number of bars supporting load

N = c/Aw = 9/1.1875 = 7.58

Maximum bending moment per bearing bar

Mb = M/N =14,625/7.58 = 1,929 Ib-in

Condition ‘B’        a = 9"        c = 6"

Find maximum bending moment with load centered at midspan

M = Pu (2L - a)/8 = 1,500 x (2 x 42 - 9)/8 = 14,063 Ib-in

Find number of bars supporting load

N = c/Aw = 6/1.1875 = 5.05

Mb = M/N = 14,063/5.05 = 2,785 Ib-in

Condition ‘B’ produces greater bending moment and will be used to establish grating size.

Required section modulus, Sb = Mb/F = 2,785/18,000 = 0.155 in³

Select:

2-1/4 x 3/16 bar  Sb = 0.1582 in³  Ib = 0.1780 in⁴

Check deflection:

Du = Pu((a/2)3 + L3 - a2 L/2)/48Elb N

= 1,500 x ((9/2)³ + 42³ - 9² x 42/2)/(48 x 29 x 10⁶ x 0.1780 x 5.05)

= 0.087 in. < 0.25 in.