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Principles of Limit State Design

1.1 Structural Design Philosophies

While in service, structures are likely to be subjected to various types of loads (or actions) and load effects (or action effects) due to operational and environmental conditions that are usually normal but are sometimes extreme or even accidental. The mission of the structural designer is to design a structure that can withstand the operational and environmental requirements designated throughout its expected lifetime.

The load effects or maximum load-carrying capacities or limit states of a structure are affected by a variety of factors that essentially involve a great deal of uncertainty, which include the following:

• Geometric factors associated with structural characteristics, buckling, large deformation, crushing, or folding
• Material factors associated with chemical composition, mechanical properties, yielding or plasticity, or fracture
• Fabrication related initial imperfections, such as initial distortion, welding induced residual stress, or softening
• Temperature factors, such as low temperatures associated with operation in cold waters or low-temperature cargo and high temperatures due to fire and explosions
• Dynamic or impact factors (e.g., strain rate sensitivity or inertia effect) associated with freak waves and impact pressure actions that arise from sloshing, slamming, or green water; overpressure actions that arise from explosion; and impact from collisions, grounding, or dropped objects
• Age related degradation factors, such as corrosion or fatigue cracking
• Accident induced damage factors, such as local denting, collision damage, grounding damage, fire damage, or explosion damage
• Human factors related to unusual operations (e.g., ship's operational speed compared with maximum permitted speed or acceleration, ship's heading, or loading or unloading conditions)

Uncertainties can comprise two groups: inherent uncertainties and modeling uncertainties. Inherent uncertainties are caused by natural variabilities in environmental actions and material properties, and modeling uncertainties arise from inaccuracy in engineering modeling associated with the evaluation and control of loads, load effects (e.g., stress, deformation), load-carrying capacities, or limit states and from variations in building and operational procedures. In design, a structure is thus required to have an adequate margin of safety against service requirements because of such inherent and modeling uncertainties.

A "demand" is analogous to load, and a "capacity" is analogous to the strength necessary to resist that load, both measured consistently (e.g., as stress, deformation, resistive or applied load or moment, or energy either lost or absorbed). In this regard, a performance function G of a structure can be given as follows:

where Cd represents the "design" capacity and Dd represents the "design" demand. The terminology "design" implies that both demand and capacity are determined by accounting for the inherent and modeling uncertainties.

Because both Cd and Dd in Equation (1.1a) are a function of the basic variables, , the performance function G can be rewritten as follows:

When , the structure is in the desired state. When , the structure is in the undesired state. In industry practice, the performance function of a structure is sometimes defined in an opposite manner to Equation (1.1a) as follows:

where G* is the performance function of a structure. In this case, the structure is in the desired state when , and it is in the undesired state when . Figure 1.1 illustrates the two performance functions associated with the desired and undesired states.

Figure 1.1 The performance functions associated with the desired and undesired states: (a) a performance function G, Equation (1.1a); (b) a performance function G*, Equation (1.2).

1.1.1 Reliability-Based Design Format

The reliability-based design format usually involves the following tasks:

1. Definition of a target reliability
2. Identification of all unfavorable failure modes of the structure
3. Formulation of the limit state (performance) function for each failure mode identified in item (2)
4. Identification of the probabilistic characteristics (mean, variance, probability density distribution) of the random variables in the limit state function
5. Calculation of the reliability against the limit state with respect to each failure mode of the structure
6. Evaluation of the predicted reliability whether or not it is greater than the target reliability
7. Redesign of the structure otherwise
8. Evaluation of the reliability analysis results with respect to a parametric sensitivity consideration

Each of the basic variables in the reliability-based design format is dealt with in a probabilistic manner as a random parameter, where each random variable must be characterized by the corresponding probability density function that has a mean value and standard deviation. If the first-order approximation is adopted, the performance function G(X) can be rewritten by the Taylor series expansion as follows:

where µxi is the mean value of the variable xi, is the mean value of the basic variables = (µx1, µx2, ., µxi, ., µxn), and is the partial differentiation of G(X) with respect to xi at .

The mean value of the performance function G(X) is then given by

where µG represents the mean value of the performance function G(X).

The standard deviation of the performance function G(X) is calculated by

where sG is the standard deviation of G(X), is the standard deviation of the variable xi, is the covariation of xi and xj, and E[] is the mean value of [ ].

When the basic variables are independent of each other, covar(xi, xj) = 0. In this case, Equation (1.5a) is simplified to

If the so-called first-order second-moment method (Benjamin & Cornell 1970) is adopted, the reliability index for this case can be determined as follows:

where ß represents the reliability index.

For a simpler case with a performance function G(X) of two parameters, for example, capacity C and demand D, that are considered to be statistically independent, the reliability index ß can be calculated as follows:

where µC or µD are the mean values of C or D, sC or sD are the standard deviations of C or D, and ?C or ?D are the coefficients of variation (i.e., the standard deviation divided by the mean value) of C or D.

To achieve a successful design, the reliability index should be greater than a target reliability index:

where ßT is the target reliability.

The target reliability or the required level of structural reliability may vary from one industry to another depending on various factors such as the type of failure, the seriousness of its consequence, or public and media sensitivity. Appropriate values of target reliability are not readily available and are usually determined by surveys or by examinations of the statistics on failures although the fundamental difference between a risk assessment and a reliability analysis needs to be acknowledged when interpreting such results. The methods to select the target safeties and reliabilities may be categorized into the following three groups (Paik & Frieze 2001):

• "Guesstimation": A "reasonable" value as recommended by a regulatory body or professionals on the basis of successful prior experience. This method may be employed for the new types of structure for which statistical database on failures does not exist.
• Calibration of design rules: The level of reliability is estimated by calibrating a new design rule to an existing successful one. This method is normally used for the revisions of existing design rules.
• Economic value analysis: The target reliability is selected to minimize total expected costs during the service life of the structure.

For elaborate descriptions in reliability analysis, interested readers may refer to Benjamin and Cornell (1970), Nowak and Collins (2000), Melchers (1999a), and Modarres et al. (2016), among others.

1.1.2 Partial Safety Factor-Based Design Format

In the partial safety factor-based design format, the design capacity or demand is defined by considering the corresponding partial safety factors...