Reliability Analysis for Eccentrically Loaded Columns

ACI Structural Journal, Sep/Oct 2005 by Szerszen, Maria M, Szwed, Aleksander, Nowak, Andrzej S

LOAD MODEL

The new load model incorporated into ACI 318-05 Code,1 being consistent with the load model specified in the ASCE 7-02 Standard,4 was used in the reliability analysis. The following design load (factored load or required strength) U was considered as the basic load combination of the ultimate limit state

... (6)

The statistical parameters for dead load D, considered as a time-invariant component, were selected depending on the type of construction. Statistical parameters (bias and coefficient of variation) of load components were taken as follows: λ^sub D^ = 1.05 and V^sub D^ = 0.10 for cast-in-place; λ^sub D^ = 1.03 and V^sub D^ = 0.08 for plant-cast, according to Nowak and Collins.5 The statistical parameters for live load L were selected for the maximum 50-year live load and for 40 m^sup 2^ influence area: λ^sub L^ = 1.0 and V^sub L^ = 0.18.

Reliability indexes for eccentrically loaded columns

According to the ACI 318-05 Code,1 the combined nominal axial force and bending moment strength (P^sub n^, M^sub n^) of a column, multiplied by the strength reduction factor to obtain the design strength (φP^sub n^, φM^sub n^), must be at least equal to the required strength

(φP^sub n^, φM^sub n^) ≥ (P^sub u^, M^sub n^) (7)

The required strength (P^sub u^, M^sub u^) is specified by Eq. (6) and defines maximum factored load effect.

A general format of a limit state function (performance function) g consistent with the ACI design format, is defined as

g = R - Q (8)

where g can also be called a safety margin; R is the resistance; and Q is the load effect. A positive sign of the safety margin defines desired (safe) performance, g ≥ 0, and undesired (not safe) performance occurs when g

... (9)

where m^sub R^ is the mean value of resistance; m^sub Q^ is the mean value of load effect; σ^sub R^ is the standard deviation of resistance; and σ^sub Q^ is the standard deviation of the load effect.5

In general, the reliability index is defined as a function of the probability of failure P^sub f^ , which is equal to the probability of occurrence of undesired performance. If resistance and load are normal random variables, then the probability of failure is expressed as

P^sup [function of]^ = Φ(-β) (10)

where Φ is the standard normal cumulative distribution function (CDF) defined by the integral

...

For example, if the reliability index is 3.5 to 4.0, then the probability of failure is 2 × 10^sup -4^ to 3 × 10^sup -5^.

The reliability indexes were calculated using Eq. (9) for the considered design cases and three preselected resistance factors (φ = 0.70, 0.75, 0.85). The range for possible values of the φ-factor was selected based on specified values given by the ACI 318-051 shown in Fig. 7. Some of the obtained results are presented in Table 3 through 6 for cast-in-place or plant-cast tied columns with different size and aspect ratios, four concrete strengths, and four reinforcement ratios. From among 30 analyzed eccentricities, four are reported in the tables as representative for compression-controlled failure, start of transition zone (balanced failure), end of transition zone, and tension-controlled failure.