Steel Girder Bridge Load Combinations

1. The LRFD Philosophy and Limit States

At the heart of the AASHTO LRFD specifications is a single, powerful equation that governs all design checks. It ensures that the structural capacity (resistance) is always greater than the forces acting upon it (loads).

Let's quickly break this down, as it forms the blueprint for everything we will do from this point forward.

• $Q_i$: These are the force effects (moment, shear, axial force) caused by the loads we've studied, like DC, DW, and LL+IM.
• ${\gamma }_i$: These are the load factors. They increase the nominal loads to account for uncertainties, variability, and the statistical probability of their occurrence. A higher factor implies more uncertainty (e.g., live load) than a lower one (e.g., dead load).
• ${\eta }_i$: This is the load modifier, which accounts for ductility, redundancy, and operational importance. For most conventional designs, it is taken as 1.0, and we will proceed with that assumption.
• $\varphi R_n$: This is the factored resistance. It represents the reliable strength of the component, where $R_n$ is the calculated nominal resistance (e.g., the moment capacity of a girder) and $\varphi$ is the resistance factor. The resistance factor is always less than 1.0 and accounts for uncertainties in material properties, dimensions, and analysis models.

This entire framework is applied across different limit states, which are conditions beyond which the bridge no longer meets its design requirements. AASHTO groups these into four main categories:

1. Strength: Relates to safety and preventing collapse under extreme loads.
2. Service: Relates to durability, comfort, and operational performance under normal traffic (e.g., controlling deflections and permanent deformations).
3. Fatigue and Fracture: Relates to damage from repeated loading cycles over the bridge's life.
4. Extreme Event: Relates to catastrophic events like earthquakes, vessel collisions, or major floods.

For an excellent and concise overview of these foundational concepts, please review the introduction to the following document from the National Steel Bridge Alliance (NSBA).

2. The Strength I Load Combination

The Strength I limit state is the primary combination for ensuring a bridge does not fail under the maximum load expected in its 75-year design life. It represents a rare, statistically significant loading event.

The AASHTO LRFD equation for Strength I is:

Where:

• $Q_u$ is the total factored force effect (the demand).
• $\eta$ is the load modifier (we'll use 1.0).
• ${\gamma }_p$ are the load factors for permanent loads.
• 1.75 is the load factor for vehicular live load.

Maximum and Minimum Load Factors for Permanent Loads

A key feature of AASHTO LRFD, which will be very familiar from your experience with continuous structures, is the use of both maximum and minimum load factors for permanent loads (${\gamma }_p$). This is crucial for checking scenarios like uplift at a support or stress reversals.

• To maximize a load effect (e.g., positive moment in a span), you use the maximum factors:
  • ${\gamma }_{p,DC}=1.25$
  • ${\gamma }_{p,DW}=1.50$
• To check for load effects that are counteracted by permanent loads (e.g., uplift), you use the minimum factors where they produce a worse outcome:
  • ${\gamma }_{p,DC}=0.90$
  • ${\gamma }_{p,DW}=0.65$

You, as the engineer, must apply these factors strategically to find the most critical force effects. For most superstructure flexural design, the maximum factors will govern.

3. The Service II Load Combination

The Service limit states are not about preventing collapse, but about ensuring the bridge remains functional and durable under everyday use. The Service II combination is specifically for steel structures. Its primary purpose is to control permanent deformations by limiting stresses in compact steel sections and to prevent slip in slip-critical bolted connections under frequent, heavy traffic.

The AASHTO LRFD equation for Service II is:

Notice the load factors are much lower than for Strength I. The factor of 1.0 on permanent loads represents their nominal value. The live load factor of 1.30 is intended to represent an overloaded vehicle or the simultaneous presence of two design trucks, a more probable event than the extreme load of the Strength I combination.

The reason for these different load factors across combinations is rooted in probability. It's statistically unlikely for the absolute maximum live load (Strength I) to occur at the same time as, for example, the maximum design wind load. Load combinations are structured so that one load is considered "dominant" (at its lifetime maximum value) while others are at their "arbitrary-point-in-time" values.

For a great explanation of this concept, a video by Dr. Mohamed Noureldin on building design codes (which share the same philosophy) is very insightful.

4. The Fatigue I & II Load Combinations

Fatigue is a critical design consideration for steel bridges, as repeated stress cycles from traffic can cause cracks to initiate and grow at structural details. The AASHTO Fatigue limit state is designed to prevent this.

Unlike Strength and Service, fatigue is concerned with the stress range—the difference between maximum and minimum stress at a point—caused by the passage of a truck. AASHTO uses a specific fatigue truck for this check. It's an HL-93 truck but with a fixed rear axle spacing of 30 feet to simplify the analysis. Furthermore, the Dynamic Load Allowance (IM) is reduced to 15% for fatigue, as high-frequency dynamic impacts tend to average out and do not contribute as significantly to fatigue damage.

AASHTO provides two fatigue limit states:

• Fatigue I (Infinite Life): This is the preferred check. It ensures that the stress range from a factored fatigue truck is below the constant-amplitude fatigue threshold (CAFT) for a given detail. If this is met, the detail is considered to have an infinite fatigue life.

  • Load Factor: $\gamma =1.75$
  • Equation: $1.75(\text{Fatigue Truck}+15\%IM)\le \text{Resistance}$

• Fatigue II (Finite Life): If a detail fails the Fatigue I check, it can be checked for a finite design life. This requires an estimate of the total number of truck passages over 75 years.

  • Load Factor: $\gamma =0.80$
  • Equation: $0.80(\text{Fatigue Truck}+15\%IM)\le \text{Resistance}$

The following video provides an outstanding step-by-step guide to applying the fatigue limit states.

5. Summary Example

Let's apply these combinations to a hypothetical girder. Suppose at midspan, we have calculated the following unfactored moments:

• Component Dead Load: $M_{DC}=600$ k-ft
• Wearing Surface Dead Load: $M_{DW}=150$ k-ft
• Live Load (HL-93 + 33% IM): $M_{LL+IM}=1200$ k-ft
• Live Load (Fatigue Truck + 15% IM): $M_{Fatigue}=500$ k-ft

The total factored moments for our key limit states would be:

Strength I (Maximum Effect):
$M_u=1.25(M_{DC})+1.50(M_{DW})+1.75(M_{LL+IM})$
$M_u=1.25(600)+1.50(150)+1.75(1200)=750+225+2100=\text{3075 k-ft}$

Service II:
$M_{ser}=1.0(M_{DC})+1.0(M_{DW})+1.30(M_{LL+IM})$
$M_{ser}=1.0(600)+1.0(150)+1.30(1200)=600+150+1560=\text{2310 k-ft}$

Fatigue I (for stress range check):
The moment range is what's important here. The factored moment range to be used in the stress calculation is:
$(\Delta M{)}_{fat}=1.75(M_{Fatigue})$
$(\Delta M{)}_{fat}=1.75(500)=\text{875 k-ft}$
This factored moment would then be used to calculate a stress range, which is compared to the allowable fatigue resistance of the structural details.

Conclusion

In this lesson, you have learned how to transition from individual loads to the factored load combinations that form the basis of AASHTO LRFD. This is the crucial step that defines the "demand" on your structure.

Key Takeaways:

• LRFD Philosophy: The core principle is that factored resistance must exceed factored loads ($\varphi R_n\ge \sum \eta \gamma Q_i$).
• Limit States: Design checks are organized into limit states (Strength, Service, Fatigue, Extreme Event), each with a specific purpose.
• Strength I: Governs the ultimate safety of the bridge, using the highest load factors (${\gamma }_{LL}=1.75$) to represent a rare, maximum loading event.
• Service II: Governs the operational performance of steel bridges, using moderate load factors (${\gamma }_{LL}=1.30$) to control permanent deformation and connection slip under frequent heavy loads.
• Fatigue I & II: Govern long-term durability by limiting stress ranges from a specific fatigue truck and a lower dynamic allowance (15%). The Fatigue I combination (${\gamma }_{fat}=1.75$) is used to check for infinite life.

We have now firmly established the left-hand side of the LRFD equation. In our next lesson, we will begin exploring the right-hand side—the resistance. We will start by identifying the common structural steel grades used in bridges and the corresponding resistance factors ($\varphi$) that AASHTO specifies for flexure, shear, and connections.