Practical Load Calculations

Simple Bridge Example: Calculating Dead and Live Loads

Let's work through a practical example to demonstrate how engineers calculate loads for a real bridge. Consider a simple three-span continuous steel girder bridge carrying a four-lane highway. Each span is 120 feet long, the total width is 48 feet (four 12-foot lanes), and the structure uses six parallel steel girders spaced 8 feet apart.

First, we calculate the dead loads. The concrete deck is 8.5 inches thick and weighs 150 pcf (pounds per cubic foot): - Deck weight = 8.5/12 × 48 × 360 × 150 = 765,000 pounds - Asphalt wearing surface (2 inches) = 2/12 × 48 × 360 × 140 = 80,640 pounds - Steel girders (estimated) = 6 × 360 × 150 = 324,000 pounds - Barriers and miscellaneous = 48 × 360 × 50 = 864,000 pounds - Total dead load ≈ 2,034,000 pounds (about 1,017 tons)

For live loads, we use the AASHTO HL-93 loading. The design truck weighs 72,000 pounds with axles of 8,000, 32,000, and 32,000 pounds spaced 14 feet and 14-30 feet apart. The lane load is 640 pounds per linear foot. With four lanes, we must consider multiple presence factors: 1.0 for one lane, 1.0 for two lanes, 0.85 for three lanes, and 0.65 for four lanes.

For maximum moment at mid-span of the center span, critical loading might involve design trucks on multiple spans positioned to create maximum positive moment. Using influence lines and distribution factors, an interior girder might experience a maximum factored moment from live load plus impact of approximately 2,500 kip-feet.

The dead load moment for an interior girder (carrying 1/6 of total dead load) in the center span would be approximately: - Dead load per girder = 2,034,000/6 = 339,000 pounds - Maximum positive moment ≈ 0.08 × 339 × 120² = 3,910 kip-feet

Wind Load Calculations

Wind loads on this same bridge would be calculated using AASHTO specifications. Assuming a basic wind speed of 100 mph (3-second gust, 100-year return period), the design wind pressure would be approximately 25 psf (pounds per square foot) on vertical surfaces.

The bridge presents different areas to wind depending on direction: - Longitudinal wind (along the bridge): acts on barriers and vehicle area - Transverse wind (across the bridge): acts on girders, deck edge, and vehicles

For transverse wind, the total area exposed might be: - Girder area: 6 girders × 4 feet deep × 360 feet = 8,640 sq ft - Deck edge: 1.5 feet × 360 feet × 2 = 1,080 sq ft - Vehicle area: 6 feet × 360 feet = 2,160 sq ft (when loaded) - Total area ≈ 11,880 sq ft

Total transverse wind force = 11,880 × 25 = 297,000 pounds

This force creates overturning moments that must be resisted by the bridge's dead load and foundation systems. Wind loads can control the design of bearings, foundations, and lateral bracing systems.

Seismic Load Example

For seismic design, forces depend on the bridge's location, mass, and dynamic properties. Using simplified procedures for our example bridge in a moderate seismic zone (Site Class D, Ss = 0.5g):

The equivalent static force method gives horizontal forces proportional to the bridge's mass and acceleration coefficients. For our bridge with total dead load of about 1,017 tons:

- Response modification factor (R) for typical girder bridges ≈ 3.5 - Importance factor (I) = 1.0 for typical bridges - Site-modified acceleration coefficient ≈ 0.5g

Seismic force ≈ (0.5 × 1.0 / 3.5) × 1,017 × 2,000 = 290,600 pounds

This force is distributed among the piers based on their relative stiffness. Tall, slender piers attract less force than short, stiff piers. The resulting moments and shears must be combined with other loads using appropriate load combinations.

Load Combinations

Bridge elements must be designed for various combinations of loads that could occur simultaneously. AASHTO LRFD specifies several load combinations with different factors:

Strength I (basic load combination): 1.25 DC + 1.50 DW + 1.75 LL + 1.75 IM

Where: - DC = dead load of structural components - DW = dead load of wearing surface and utilities - LL = vehicular live load - IM = vehicular dynamic load allowance

Strength III (wind loading): 1.25 DC + 1.50 DW + 1.40 WS

Strength IV (high dead load to live load ratio): 1.50 DC + 1.50 DW

For our interior girder example: - Strength I: 1.25(3,910) + 1.75(2,500) = 9,263 kip-feet - Strength III would include wind effects - Strength IV: 1.50(3,910) = 5,865 kip-feet

The girder must be designed for the most critical combination, which in this case would be Strength I at 9,263 kip-feet.