Chapter 6: Sensitivity Analysis of EN 1992 Crack Model: Methodology, Results and Discussion
6.1 Methodology of the Reverse FORM Analysis
6.1.1 Reliability Analysis Formation
The reverse FORM analysis follows much of the same methodology as for the conventional FORM analysis of chapter 5. The major difference in this instance is that the reliability index is selected from the onset and the steel reinforcement (the unknown parameter) required to meet this target reliability index is then calculated. The statistical parameters of the basic variables are as for the conventional FORM calculation presented in chapter 5. The limit state function remains as for the FORM analysis in the previous section and the reverse FORM algorithm used is as follows:
1) The limit state function is first defined (as defined in equation 5.6)
2) Convert the mean and standard deviation of non-normal variates to their normal equivalent.
3) Assume initial failure points- normally taken as the mean values of the random variables in question.
4) Determine the partial derivatives of the performance function with respect to each random variable using MATLAB. Evaluate these derivatives at the failure points. Then substitute the previously determined value for the area of steel required to satisfy the performance function (g(x) = wlim - θw) when evaluating the derivatives.
5) Determine the direction cosines (sensitivity factors) by dividing each partial derivative by the root of the sum of the squared partial derivatives (in other words, normalise the partial derivatives).
6) The failure points may then be determined by substituting the target reliability index in the failure point equation (the normal equivalent of the mean and the standard deviation are used in this equation).
7) The new found failure points are then substituted into the performance function and the amount of area required for the performance function to equal zero is calculated using excel solver. Here the area of steel required is set as the variable cell, whilst the performance function is the objective cell set to meet an objective value of zero in excel solver. The area of steel determined is then substituted into the partial derivatives of the next iteration where the partial derivatives are also evaluated at the failure points determined after the current iteration.
8) Steps 2 through to 7 are repeated until convergence of the required area of steel is reached.
Figure 6.1 illustrates the use of the reverse FORM algorithm for the EN 1992 restrained shrinkage crack model,
Figure 6.1: First and Second Iteration of the Reverse FORM Analysis of EN 1992 Crack Model
1st Iteration
convert non-normals to normal
lognormal: pdf ξ = c.o.v
f ξ λ μN σN λ= lnμ-0.5*(c.o.v)^2
Concrete cover c 0.15 -3.230125825 0.03955 0.006 μN = y*(1-lny*+λ)
Model Uncertainty θ 0.3 -0.045 0.955 0.3 σN = y*ξ
1st iteration logrithm *σ x*=μN-α*σN*β
partial direction failure failure
uncorrelated variables assumed derivatives cosine point equation point
h,c, fctm, θ Xi xi μNxi σNxi (δg/δXi) αxi σ*α
Cross section thickness h 0.25 0.25 0.0025 0 0 0.25+0*β 0.25 0
Concrete cover c 0.04 0.03955 0.006 -2.4942E-05 -0.38385468 0.03955+0.002303128*β 0.043004692 -0.002303128
Model Uncertainty θ 1 0.955 0.3 -6E-05 -0.923393516 0.955+0.277018055*β 1.370527083 -0.277018055
Ʃα^2 1
β 1.5
failure function: g(x)= wlim - θwk = 0.0002-th*wk 0.0002-(0.955+0.277018055*β)*(2*(0.03955+0.002303128*β)+0.25*0.8*1*0.02/(As/(2.5*(0.03955+0.002303128*β+0.01)*b)))*(R*(alpha*(T1+T2)+ eca +ecd))
after 1st iteration A g(x)
0.001854706 0.0002-(0.955+0.277018055*1.5)*(2*(0.03955+0.002303128*1.5)+0.25*0.8*1*0.02/(As/(2.5*(0.03955+0.002303128*1.5+0.01)*1)))*(0.5*(0.000014*(15+23)+ 0.000033 +0.000220))
2nd Iteration
convert non-normals to normal
lognormal: pdf ξ = c.o.v
f ξ λ μN σN λ= lnμ-0.5*(c.o.v)^2
Concrete cover c 0.15 -3.230125825 0.039406069 0.006450704 μN = y*(1-lny*+λ)
Model Uncertainty θ 0.3 -0.045 0.876869535 0.411158125 σN = y*ξ
2nd iteration logrithm *σ x*=μN-α*σN*β
direction failure failure
uncorrelated variables assumed cosine point equation point
h,c, fctm, θ Xi xi μNxi σNxi (δg/δXi) αxi σ*α
Cross section thickness h 0.25 0.25 0.0025 0 0 0.25+0*β 0.25 0
Concrete cover c 0.043004692 0.039406069 0.006450704 -2.56495E-05 -0.393079788 0.039406069+0.002535641*β 0.043209531 -0.002535641 Model Uncertainty θ 1.370527083 0.876869535 0.411158125 -6E-05 -0.919504366 0.876869535+0.378061691*β 1.443962072 -0.378061691
Ʃα^2 1
β 1.5
failure function: g(x)= wlim - θwk = 0.0002-th*wk 0.0002-(0.876869535+0.378061691*β)*(2*(0.039406069+0.002535641*β)+0.25*0.8*1*0.02/(As/(2.5*(0.039406069+0.002535641*β+0.01)*b)))*(R*(alpha*(T1+T2)+ eca +ecd))
A g(x)
after 2nd iteration 0.001996852 0.0002-(0.876869535+0.378061691*1.5)*(2*(0.039406069+0.002535641*1.5)+0.25*0.8*1*0.02/(As/(2.5*(0.039406069+0.002535641*1.5+0.01)*1)))*(0.5*(0.000014*(15+23)+ 0.000033 +0.00022)) Substitute into next
iteration (becomes new 'A' value in derivatives)
These new failure points go into failure equation to find 'new' value for 'A'
Target reliability index β
Solve using excel solver 2
3 4 5 6 7
A total of four iterations were required for the convergence of β (as shown in Figure 6.2). Again, a separate spreadsheet is generated for select data obtained after each analysis. Data of particular interest include the direction cosines (sensitivity factors) of each random variable achieved at the end of each calculation. The sensitivity factors are indicative of the influence that each random variable has on the crack model relative to the other random variables. The closer the sensitivity factors are to the number one (+1 or -1) the more influential the variable. Being a normalised factor, the sum of the square of the sensitivity factor of each random variable should add up to one (Σ (αi*) 2 = 1). The sensitivity factors of each random variable were plotted against the coefficient of variance (CoV) of the model uncertainty.
Figure 6.2: Example of Convergence Achieved After Four Iterations (Edge Restraint, hc,eff
= 2.5(c + ϕ/2))
The theoretical partial safety factors (psf’s) for each random variable were calculated with the eventual failure point and mean of the random variable.
γ𝑖= 𝑋𝑖∗
μX = 1– αi*βwXi (6.1)
The theoretical partial safety factors are indicative of the amounts of adjustment that are required to be made to the input random variables in order for the limit state function to be satisfied and for the given target reliability index to be met. Both the sensitivity factor and the theoretical partial factors were assessed for the crack width limits corresponding to the tightness classes and functions to which liquid retaining structures are designed. The intent of this being that the partial factors and sensitivity factors obtained are meant to represent and work across all expected performance applications.
The restraint crack models assessed were as for the standard FORM analysis, consisting of:
A (mm2) g(x)
from failure function 0.001164 -6.9133E-11
1st iteration 0.001855 2.0535E-12
2nd iteration 0.001997 1.2922E-10
3rd iteration 0.002001 3.1129E-11
4th iteration 0.002001 3.1208E-11
Variable Objective
iterations stopped after convergence is reached
a) Edge restraint with depth of effective tension zone taken to be 2.5(c +φ/2) b) Edge restraint with depth of effective tension zone taken to be h/2
c) End restraint with depth of effective tension zone taken to be 2.5(c +φ/2) d) End restraint with depth of effective tension zone taken to be h/2