Most structural loads vary with time. If a structural element is subjected to only one time-varying load in addition to its dead load, the reliability may be determined simply by considering the combination of the dead load with the maximum time-varying load during some appropriate reference period. It is frequently the case, however, that more than one time-varying load will be acting on a structure at any given time. Conceptually, these load combinations should be dealt with by applying the theory of stochastic processes, which account for the stochastic nature and correlation of the loads in space and time. Loads (or load effects) acting on structural elements typically are represented by various combinations of load process models such as those in Fig. 2.5. Permanent loads (Fig. 2.5a) such as dead loads change very slowly and maintain a relatively constant (albeit random) magnitude.

periods.

Sustained loads (Fig. 2.5b) may change at discrete times but

in between changes remain relatively constant. They may be absent entirely for certain
Occupancy live loads fall in this category. Finally, transient loads of short
Since their durations are so small

duration (Fig. 2.5c) occur relatively infrequently.
relative to permanent and sustained loads, they are modeled as impulses.
earthquake loads are examples.

Extreme wind and

The terminology "arbitrary-point-in-time" load is used frequently in later sections. It is simply the load that would be measured if the load process were to be sampled at some time instant, e.g., in a load survey. The probability densities of the arbitrarypoint-in-time loads are shown in Figs. 2.5a

2.5c. The impulse at zero represents the

probability that the load magnitude is zero at the time samples are taken.

In this report, the analysis of reliability associated with ultimate limit states

requires that the maximum total load during a reference period taken as 50 years be characterized.

When more than one time-varying load acts, it is extremely unlikely that each load will

reach its peak lifetime value at the same moment. This is illustrated in Fig. 2.5d.

Consequently, a structural component could be designed for a total load which is less than

the sum of the peak loads, and in fact this is recognized in Section 4.2 of ANSI Standard A58.1-1972 [2]. The probability factors in that section have evolved on the basis of experience rather than a thorough consideration of the underlying nature of the loads,

however.

For practical reliability analyses, it is necessary to work with random variable representations of the load rather than random process representations.

One such procedure

It is first

is a generalization [14] of a model first proposed by Ferry Borges [6].
assumed that for each time-varying load X ̧ the life T may be divided into a number of
i'

elementary time intervals, t such that the value of load X, is constant within T and
i'
i
i

i

i

values of X, within successive time intervals are statistically independent. The probability of a nonzero value of X within each time interval is P1' The load histories are then arranged in order of decreasing basic time interval (increasing number of load changes) as shown in Figure 2.6. Given that r Fi nonzero values of X occur within interval T distribution of the maximum of X, within interval t is given by

i

1-1

i

the

i-1'

max

F (x) = F1(x)*1

i

(2.25)

i

= distribution of Xi within

in which r1 = T1-1/1 (termed the repetition number) and F1/τ

i

the elementary interval t Using the theorem of total probability and the binomial i'

theorem, the distribution of maximum load within T is given by

Using the procedure for normalizing non-normal random variables explained earlier, this calculation can be handled quite easily. Working down through the set of load histories,

Although this represents a sophisticated approach to load combinations, there are a number of difficulties with its use. The assumption that each peak value of load remains constant within its basic time interval is a conservative one but probably is not unduly so if the basic time intervals are chosen to be reasonably short. A more serious shortcoming is the necessity of making assumptions regarding the number of basic intervals and the probability of a nonzero load value within each one. Information regarding T or r and Pi generally is not available or is not easily recoverable from available load data and as

must be determined artificially. The safety criteria are quite
In short, the quality of data and our

sensitive to the selection of these parameters.

knowledge of the various load processes may not be sufficient to warrant the use of this model in practical reliability analysis and design work.

An alternate way to handle load combinations is through the use of "Turkstra's rule" [17]. This says, in effect, that the maximum of a combination of load effects will occur

when one of the loads is at its lifetime maximum value while others assume their instantaneous

If there are n time-varying loads in the limit state equation, in general it is necessary to consider n distinct load combinations in computing the associated reliability. This tends to be unconservative in certain instances where the probability of a joint occurrence of more than one maximum value is not negligible or in the situation where the maximum combined effect occurs when two variables simultaneously attain "near maximum" values.

Nevertheless,

recent research on load combinations based on the concept of up-crossing rates of random processes show that Turkstra's rule is a good approximation in many practical cases [10,18]. This model will be used for the load combination work in this study because of its simplicity and because it is consistent with the observation that failures frequently occur as a consequence of one load attaining an extreme value.

The following is an example of load combination analysis according to Eq. 2.28. Assume that the loads of interest are dead, live and wind load. As discussed in the following section, the load effects are,

D = permanent or dead load (duration = lifetime T)

According to Eq. 2.28, the calculation of B for reference period T would require the

following load combinations to be considered: