28.1. Introduction

Traditionally, evaluations of structural adequacy have been expressed by safety factors SF = C/D, where C is the capacity (i.e. strength) and D is the demand (i.e. load).

In structural engineering, the safety factor (SF) is a method used to ensure that structures can support the loads they are expected to encounter. The formula SF = C/D means that the safety factor can be calculated by dividing the structure's capacity (C) by the load demand (D). Essentially, this tells us how much stronger the structure is compared to the loads it will face. A safety factor greater than 1 means the structure is generally considered safe under expected conditions.

Think of a safety net installed under a high wire act. If the wire can hold 1000 pounds (capacity) but the performer only weighs 200 pounds (demand), the safety factor would be 5 (1000/200). This indicates a strong backup system that would allow for unexpected weights without compromising safety.

This evaluation is quite simple to understand, it suffers from many limitations: it treats all loads equally; does not differentiate between capacity and demands respective uncertainties; is restricted to service loads; and does not allow comparison of relative reliabilities among different structures for different performance modes.

While the traditional evaluation method is straightforward, it has significant limitations. First, it treats all loads the same, ignoring that some loads might be more variable than others. Second, it fails to account for uncertainties in both capacity and demand; for instance, a building might have a load capacity based on ideal conditions that are not realistic. Third, it only considers average service loads and doesn't factor in extreme conditions. Lastly, it doesn’t allow engineers to compare the reliability of different structures under varying service conditions, which can be critical in risk-sensitive applications.

Imagine two bridges: one is newly constructed with modern materials, and the other is an old bridge that has been patched up multiple times over the years. If we use the same safety factor for both, we might overlook that the old bridge cannot handle surprise loads (like heavy trucks) as well as the new one. Thus, while both may appear to meet safety criteria, their actual reliability could be very different.

Another approach, a probabilistic one, extends the factor of safety concept to explicitly incorporate uncertainties in the parameters. The uncertainties are quantified through statistical analysis of existing data or judgmentally assigned.

In contrast to the traditional approach, the probabilistic method acknowledges that there are uncertainties in both the loads acting on a structure and its capacity to bear them. By considering these uncertainties, engineers can use statistical data to better assess the reliability of a structure. This method involves collecting existing data related to loads and material strengths, analyzing them statistically, and incorporating that data into safety assessments.

Consider a weather forecast predicting rainfall. Instead of saying it will definitely rain or it won’t, the forecast might say there's a 70% chance of rain. Similarly, by using probabilistic methods, an engineer might say there’s an 80% chance that a bridge can handle a specific heavy truck load, capturing the uncertainty inherent in the situation.

This chapter will thus develop a procedure which will enable the Engineer to perform a reliability based analysis of a structure, which will ultimately yield a reliability index.

The goal of this chapter is to equip engineers with the tools and procedures necessary to analyze the reliability of structures through a systematic probabilistic approach. The chapter will guide readers through the steps needed to calculate a reliability index, a valuable metric that reflects a structure's ability to perform safely under expected conditions. This will include developing methodologies to assess and quantify reliability based on uncertainty rather than deterministic values.

Using a car as an analogy, consider how modern vehicles include reliability ratings based on extensive testing and data analysis. Engineers might conduct simulations to determine how likely a car is to perform safely in various conditions, similar to how we'll learn to evaluate the performance of structures based on probabilistic analysis.

--