26/04/99 Reliability Applications Department of Civil Engineering The University of British Columbia, Vancouver, Canada Ricardo O. Foschi

Outline Introduction Performance Function Definition Methods to Estimate Reliability Software Example: A Beam Design Problem Example: A Fire Application Example: Calibrating a Code

The response of a structure, as that of any other engineering system, is influenced by a set of intervening variables. These enter into a model for the system, and some will control the system capacity while others will be associated with the demands. Example: In beam bending, the bending strength is a variable associated with the capacity, while the bending stress produced by the applied loads is the demand.

All variables will have a degree of uncertainty. Therefore, both the capacity and the demands will be uncertain or random. There is then a probability that the variables will combine, randomly, to produce a demand greater than the capacity. In such a situation the system will “fail” or will not perform as intended. We can then calculate the “probability of failure” of the system, and then adjust its design to bring this probability to an acceptable level.

Definition of Performance or Limit State Software: RELAN Performance function: X C, X D : Random Variable Vectors d C, d D : Design Parameter Vectors Probability of failure = P f = Prob(G<0) Reliability = P f

Why bother? Reliability-based design does not mean that older, traditional approaches were “unsafe”. However, they were not based on a specific quantification of safety, and also did not lead to nearly uniform safety levels across materials or structural applications. Using reliability-based procedures, wood structures can be designed to be “as safe” as those made with steel or concrete, since we can quantify safety and make them all equal in this regard. Reliability in performance forms the basis of new “Performance-Based Design Code Guidelines”.

Reliability m ethods allow a clear introduction of quality control procedures into the design process. These methods also allow a better use of materials, particularly of mixes, and thus promote innovation and a better resource utilization.

What is it required for each of the random variables? Basic statistics for each one (mean, standard deviation, cumulative probability distributions) Should they be modified for extremes? Should they be modified for lower bounds? Should they be modified for upper bounds? Are they correlated? If so, what is the correlation structure? Usual cumulative probability distributions are: Normal, Lognormal, Weibull, Gumbel, Uniform, Gamma, Beta.

Fit of data with cumulative distribution function F(x)

Reliability Estimation Methods Simulation (Standard MonteCarlo): P f = N f / N where N f is the number of times that G<0 out of a sample of N trials. Disadvantage: Too large an N required for small probabilities, time consuming. Advantage: Approaches correct probability as N   (can be used as a benchmark)

Approximate Methods FORM: First Order Reliability Method This is a very efficient method based on the calculation of a “reliability index  ”. It is only approximate. Three conditions must be met for FORM to give exact answers: All variables have to be Normal All variables must be un-correlated The G function must be linear

Basic Problem: FORM Consider the following performance function: G = X 1 – X 2 This is a linear function. Let X 1 and X 2 be Normals, un-correlated variables. Let x 1 and x 2 be normalized variables using the mean and standard deviations of X 1 and X 2 : x 1 =(X 1 -M 1 )/  1 x 2 =(X 2 -M 2 )/  2 Therefore, G < 0 can be expressed as:

x 2 > (M 1 -M 2 )/  2 + x 1  1 /  2 Thus, the failure zone is that above the straight line G = 0. Rotating the axes, the failure zone is also completely described by y 2 > 

Since it can be shown that y 2 is also a Normal (Standard) variable, the probability of failure P f can be obtained from the Normal distribution function  : P f =  (-  ) Thus, FORM only requires the calculation of the distance . This is the minimum distance from the origin of coordinates to the line G = 0, (failure surface separating failure from survival).

26/04/99 Algorithms have been developed to transform all variables to Normals, and to un-correlate them if they are correlated. However, the linearity of G is a feature of the specific problem and cannot be modified. These transformations and the general nonlinearity of G contribute to the approximate character of the FORM approach. Normally, the FORM results are very good, with a large gain in computational efficiency.

Algorithm to obtain  General algorithms for the calculation of the reliability index  are well developed (e.g., Rackwitz and Fiessler, 1978). These are quasi- Newton, iterative procedures based on the gradient of G.

Once  is found, the point P on the failure surface G = 0 is determined. This point is called the “design point” or “most likely failure combination point ”. The vector joining the origin O with P is called the sensitivity vector n, because its direction cosines give the derivatives of  with respect to the variables x: n i =  /  x i These are very useful to determine the most important variables in a problem.

Multiple failure modes: A system may fail in a variety of ways. For each failure mode, a separate G function must be written. If failure of the system is triggered by the failure of at least one mode, then the system is called series or brittle. Two modes can be correlated if they share some of the variables (e.g., snow loads can produce simultaneous beam failure in bending and in shear).

Example: Beam Design Mode 1: Failure in bending G 1 = X(2)- 6 [X(3)+X(4)]L /(4.0 X(5) X(6) 2 ) Mode 2: Deflection limit G 2 = L/K – 48 [X(3)+X(4)]L 3 /(X(1) X(5) X(6) 3 / 12) P+D

Random variables: X(1) = modulus of elasticity E (Mpa) X(2) = bending strength f b (Mpa) X(3) = concentrated dead load D at midspan (kN) X(4) = concentrated live load P at midspan (kN) X(5) = beam width (m) X(6) = beam depth (m) Deterministic parameters: L = beam span (m) K = deflection criterion limit

Note that the G functions are quite nonlinear and, in addition, some of the random variables may not be Normal, requiring transformation to Normality. One would then expect that FORM would have some error, to be assessed by carrying out a simulation. Let us run the software RELAN for this case.

Another Example: Reliability Considering Fire P+Q L B H e Consider a glulam beam of dimensions B, H and span L, with fire exposure creating a burnt or charred section of thickness e. The burning rate is a constant , so that after time T the amount burnt is e =  T. P and Q are two distributed loads (one permanent, the other live)

Random VariableMeanStd. Deviation Distribution f, bending strength (kN/m 2 ) 35,0005,250Lognormal P (kN/m) Normal Q (kN/m)8.02.0Extreme Type I  (burn rate, mm/minute) Lognormal E, modulus of elasticity (kN/m2) 12,000 x10 3 1,800x10 3 Lognormal

1) Bending failure under normal conditions: [ 2) Deflection under normal conditions: 3) Bending failure under fire conditions:

Corresponding target (desired) reliability levels are:  1 = 4.0 for failure under normal circumstances  2 = 2.0 for deflection under normal circumstances  3 = 2.5 for failure under fire conditions after T= 60 minutes of exposure The problem is to find optimum values H and B for the cross-section that will achieve the specified reliabilities or deviate a minimum from them.

H (m)B (m)  1 Achieved  2 Achieved  3 Achieved Optimization Strategy Unconstrained Forcing  3 = 2.50 Target  Optimization Results (Performance-Based Design)

Another Example: Code Calibration Let’s assume that a Code design equation is of the form: where  d,  L = Load factors  = Resistance factor D N, L N = Nominal, design load effects R N = Nominal resistance S = Geometric design parameter

We use the design equation to calculate the parameter S, e.g., in bending, S = bh 2 /6. The load and resistance factors must be such that the designed beam achieves a target level of reliability under the actual, random loads. In order to calculate the achieved reliability we must use the performance function G:

Introducing the calculated S into the function G, we can write it as where d = D/D N, normalized dead load l = L/L N, normalized live load r = nominal dead/live load ratio, D N /L N

Using the last form of G, we can find the reliability index  as a function of the load and resistance factors, choosing the combination which provides for the desired target. Notice that the selected combination will only provide the desired  for the specific properties used for the loads d and l. If the live load is snow, and the location changes, the statistics of l will change and the reliability will change. Thus, a few factors in the design equation cannot provide for constant reliability across conditions and applications. The factors must be optimized to minimize the variation in reliability.

A customized approach avoids the Code and estimates the reliability achieved in each particular case. A customized approach would permit the use of more sophisticated structural analysis software instead of the simpler methods in the design equations. This would also promote a system analysis rather than only components. A Limit States Design Code is only a short-cut and not a proper reliability analysis.