Structural Reliability Assessment

Abstract

This chapter discusses the method and application in practical engineering of structural reliability assessment. Reliability theory is the basis for measuring structural safety level under a probability-based framework. This chapter starts from the definition of structural reliability and reliability index, followed by the computation methods of reliability index, including first-order second-moment method, first-order reliability method, simulation-based method and moment-based method. Subsequently, some simple models for system reliability are presented, namely series system, parallel system and k-out-of-n system. The probability-based modelling of structural resistance and external loads are also addressed, followed by the probability-based limit state design approaches. The reliability theory can be further used to optimize structural design and conduct life-cycle cost analysis. Finally, the evaluation of structural reliability in the presence of imprecisely-informed random variables is considered.

Problems

4.1

In the presence of the limit state function of $Z=R-S$, where both R and S are normal variables, let $\theta$ be the central factor of safety, $\nu _R,\nu _S$ the COVs of R and S respectively. Show that conditional on $\nu _R,\nu _S$, the reliability index $\beta$ is an increasing function of $\theta$.

4.2

The annual maximum peak ground acceleration (PGA) is modeled by a Fréchet distribution with scale parameter $\varepsilon$ and shape parameter k. Let $A_{\mathrm {n}}$ be the nominal PGA with an exceeding probability of $\alpha$ in T years. Derive $A_{\mathrm {n}}$.

4.3

If the reliability index is 2.5 for a reference period of 50 years, what is the corresponding annual failure probability?

4.4

The structural resistance R follows a Gamma distribution with shape parameter $a > 0$ and scale parameter $b > 0$; the load effect, S, which is independent of R, is an exponential random variable with mean $\frac{1}{\lambda }$. Let $Y=\frac{R}{S}$ be the overall factor of safety, and $F_Y(y)$ the CDF of Y. Show that

$$\begin{aligned} F_Y(y)=\frac{1}{\left( 1+\frac{b\lambda }{y}\right) ^a},\quad y\ge 0 \end{aligned}$$

(4.314)

4.5

The structural resistance R follows a lognormal distribution with mean 3 and standard deviation 0.75; the load effect S follows a Weibull distribution with mean 1 and standard deviation 0.5.

• (1) What is the failure probability if R and S are statistically independent?

• (2) Modeling the correlation between R and S with Gumbel–Hougaard copula, what is the failure probability if the Kendall’s tau for R and S is 0.5?

4.6

Consider a limit state function of $G=R-S$, with which the structural failure is deemed to occur when $G < 0$. The resistance R has a mean value of 3 and a standard deviation of 0.5, while the load effect S has a mean value of 1 and a standard deviation of 0.4.

• (1) Find the checking point in the normalized space and the corresponding one in the original space.

• (2) If the nominal resistance and load effect are as in Eq. (4.12), compute the resistance factor $\varphi _R$ and the load factor $\gamma _S$.

4.7

Tail sensitivity problem. Consider a limit state function of $G=R-S$. Suppose that the resistance, R, is lognormally distributed with a mean value of 3 and a COV of 0.2. The load effect S has a mean value of 1 and a COV of 0.3. Compute the failure probability for the following two cases: (1) S is a normal variable; (2) S follows an Extreme Type I distribution.

4.8

For a structure subjected to live load L and dead load D, the limit state function is $G=R-L-D$, where R is the structural resistance. If $\mu _R=3,\mu _D=1,\mu _L=0.6,\sigma _R=0.45,\sigma _D=0.1, \sigma _L=0.3$, where $\mu _{\bullet }$ and $\sigma _{\bullet }$ are the mean and standard deviation of random variable $\bullet$ respectively, determine the resistance and load factors (use the nominal resistance and nominal loads as defined in Eq. (4.12)).

4.9

Consider a 2-span beam as shown in Fig. 4.36. The maximum deflection for the left span (with a length of $l_1$) is $u_{\max }=\frac{Fl_1^2l_2}{9\sqrt{3}EI}$, where EI is the bending rigidity. If F has a mean of 10 kN and a COV of 0.4, EI has a mean of 500 kN$\cdot$m² and a COV of 0.2, compute the Hasofer–Lind reliability index. Assume that $l_1=l_2=5$ m. The beam fails if $u_{\max }>l_1/75$.

Fig. 4.36

A 2-span beam subjected to concentrated load F

4.10

Consider a simply supported beam with a length of L. Suppose that the moment resistance at the mid-span, $M_{\text {mid}}$, has a mean of 200 kN$\cdot$m and a COV of 0.15. The beam is subjected to uniform load W along the length, which has a mean of 10 kN/m and a COV of 0.2. The length L has a mean of 8 m and a COV of 0.125. Suppose that each variable ($M_{\mathrm {mid}},L,W$) is independent mutually. Focusing on the moment at the mid-span of the beam, calculate the failure probability using Eq. (4.69).

4.11

Recall Problem 4.10. Recalculate the reliability index $\beta _{\mathrm {HL}}$ using Eq. (4.82) or (4.90). Set $\varepsilon =0.01$. Compare the result with that in Problem 4.10.

4.12

In Problem 4.10, if we additionally know that $M_{\mathrm {mid}}$ is a lognormal variable, L is a Gamma variable, and W follows an Extreme Type I distribution, recalculate the probability of failure using FORM.

4.13

In Problem 4.12, use Monte Carlo simulation method to find the failure probability and the reliability index $\beta$.

4.14

Consider a simply supported beam with a length of $l=1$ and a square cross section (the side length is a). A vertical concentrated load F, which is a normal variable with a mean of 0.3 and a standard deviation of 0.2, is applied at the middle of the beam. The normal yield stress is a normal variable with a mean of 1 and a COV of 0.15, while the shear yield stress is a normal variable with a mean of 1 and a COV of 0.1. If the target reliability index is 2.5, what is the minimum value for a?

4.15

Consider a five-meter beam as in Fig. 4.37, which is subjected to uniformly distributed load W. Suppose that W is an Extreme Type I variable with a mean value of 45 kN/m and a COV of 0.3, the yield stress of the beam material, $F_{\mathrm {y}}$, is a lognormal variable with mean 300 MPa and COV 0.15, the section modulus $Z_x$ is deterministically 800 cm³. What is the probability of failure of the beam by plastic collapse?

Fig. 4.37

A beam subjected to uniform load W

4.16

Consider a limit state function of $G=R-S$. Suppose that the resistance, R, is a lognormal variable with a mean value of 3 and a COV of 0.2, and the load effect S is a Rayleigh variable with a mean value of 1. Calculate the probability of failure using Eq. (4.132).

4.17

Reconsider Problem 4.16. If we use a simulation-based approach to estimate the failure probability, show that the use of importance sampling can increase the computational efficiency.

4.18

Consider a structure subjected to the combined effect of live load L and dead load D. The structure was designed using the allowable stress design according to $\frac{R_{\mathrm {n}}}{\mathrm {FS}}\ge D_{\mathrm {n}}+L_{\mathrm {n}}$, where FS is the factor of safety, $R_{\mathrm {n}},D_{\mathrm {n}},L_{\mathrm {n}}$ are the nominal resistance, nominal dead load and nominal live load respectively. Suppose that R, D, L are normal variables with $\mu _R=1.10R_{\mathrm {n}},\sigma _R=0.15\mu _R$, $\mu _D=1.05D_{\mathrm {n}},\sigma _D=0.05\mu _D$, $\mu _L=0.9L_{\mathrm {n}}, \sigma _L=0.3\mu _L$, where $\mu _{\bullet }$ and $\sigma _{\bullet }$ are the mean and standard deviation of $\bullet =R,D,L$ respectively. Assume that $\mathrm {FS}=\frac{5}{3}$. When $\frac{L_{\mathrm {n}}}{D_{\mathrm {n}}}$ varies within a range of [0.5, 4], determine the reliability index $\beta$ for different values of $\frac{L_{\mathrm {n}}}{D_{\mathrm {n}}}$.

4.19

Consider a structure subjected to wind load, as shown in Fig. 4.38. The height and width of the surface normal to wind load are $H=8$ m and $L=3$ m respectively. The wind speed varies linearly with the height, and equals zero at the ground level. The wind pressure p is calculated by $p=\frac{1}{2}\rho V^2 C_{\mathrm {fig}}$, where $\rho$ is the air density (1.2 kg/m³), V is the wind speed, $C_{\mathrm {fig}}$ is a constant that reflects the structural shape and aerodynamics. Take $C_{\mathrm {fig}}$ as 0.45 in the following.

• (1) When the design wind speed at height H, $V_{\mathrm {m}}$, is 40 m/s, with a resistance factor of 0.85, what is the design resisting moment of the structure at the bottom?

• (2) If the wind speed at height H has a mean of 40 m/s and a COV of 0.3, and the structural resisting moment at the bottom is a lognormal variable with a COV of 0.15, determine the mean of the resisting moment with a target reliability index of 2.5.

• (3) If we additionally know in (2) that the wind speed at height H is a Weibull variable, recalculate the mean of the resisting moment using FORM.

Fig. 4.38

A structure subjected to wind load

4.20

Consider an eight-bar truss as shown in Fig. 4.39, which is subjected to a horizontal load F. Suppose that F follows an Extreme Type I distribution with a mean value of 1 and a standard deviation of 0.3. Each bar has a statistically independent and identically lognormally distributed resistance of axial force (either compression or tension), with a mean value of 3 and a standard deviation of 0.5. Compute the failure probability of the truss.

Fig. 4.39

An eight-bar truss

4.21

In Problem 4.20, if the resistances of any two bars are correlated with a linear correlation coefficient of 0.8, what is the failure probability of the truss?

4.22

In Problem 4.20, if the occurrence of the horizontal load F is a Poisson process with rate $\lambda =0.2$/year, and the magnitude of each load, conditional on occurrence, follows an Extreme Type I distribution with a mean value of 1 and a standard deviation of 0.3, what is the failure probability of the truss for a reference period of 10 years?

4.23

Consider an n-component parallel system as shown in Fig. 4.40. Suppose that each component has an identical failure probability of 0.2. The performance of component i is only dependent on that of component $i-1$ for $i=2,3,\ldots , n$. Provided that component $i-1$ fails, the probability of failure for component i is $1-\frac{1}{n}$. What is the system failure probability as $n\rightarrow \infty$?

Fig. 4.40

An n-component parallel system

4.24

Reconsider Example 4.29. If the capacities of each rope, $R_1$, $R_2$ and $R_3$, are mutually correlated with an identical linear correlation coefficient of 0.6 for any two capacities,

• (1) Compute the system failure probability.

• (2) Compare the result from (1) and the bounds obtained in Example 4.29.

4.25

Consider a 5-out-of-8 system with a failure probability of $p=0.01$ for all components. What is the system failure probability?

4.26

Consider a three-component system as shown in Fig. 4.41. Each box represents a component. The system is deemed as survival if points A and B are connected by any path. We introduce a Bernoulli random variable for each of the three components, denoted by $B_1,B_2,B_3$, which returns 1 if the component fails and 0 otherwise. If $\mathbb {P}(B_i=1)=0.01$ for $i=1,2,3$, and the correlation coefficient of $B_i$ and $B_j$ is 0.5 for $i,j\in \{1,2,3\}$, what is the system failure probability?

Fig. 4.41

A three-component system

4.27

Consider an eight-component system as shown in Fig. 4.42. Each box represents a component. The system is deemed as survival if points A and B are connected by any path. If each component behaves independently and has an identical failure probability of $p=0.01$, what is the system failure probability?

Fig. 4.42

An eight-component system

4.28

In Example 4.28, if V and H are correlated with a linear correlation coefficient of 0.7, recompute the system failure probability.

4.29

Consider a 5-out-of-8 system, where each component has an identically lognormally distributed resistance with mean 2 and COV 0.3. If all the components are subjected to a deterministic load effect of 1, what is the system failure probability if any two of the component resistances are correlated with a linear correlation coefficient of 0.7?

4.30

In Example 4.34, let $W(x)=\overline{W}+\varepsilon (x)$, where $\overline{W}$ has a mean value of 1 kN/m and a COV of 0.5, and $\varepsilon (x)$ has a zero-mean, a standard deviation of 0.3 kN/m and a correlation structure of $\rho (\varepsilon (x_1),\varepsilon (x_2))=\exp (-(x_1-x_2)^2)$. Compute the mean value and standard deviation of EUDL. Assume $l=2$ m. Compare the result with that in Example 4.34 and comment on the difference.

4.31

Reconsider Example 4.41. If the load effect S is associated with a reference period of 40 years, taking into account the impact of the discount rate, $r=5\%$, what is the optimal annual failure probability?

4.32

Reconsider Example 4.42. If the load effect S is associated with a reference period of 40 years, taking into account the impact of the discount rate, $r=5\%$, what is the optimal annual failure probability?

4.33

We consider the reliability of a scaffolding system during concrete replacement [34]. Both the live load L and the dead load D are present. The live load follows an Extreme Type I distribution with a mean-to-nominal ratio of 0.85 and a COV of 0.6, the dead load is normally distributed having a mean-to-nominal value of 1.05 and a COV of 0.3, and the scaffolding resistance is lognormally distributed with a mean-to-nominal value of 1.10 and a COV of 0.15. The target reliability index is set as 3.0.

• (1) Using a LRFD-based safety check as follows, $\varphi R_{\mathrm {n}}=\gamma _D D_{\mathrm {n}}+\gamma _L L_{\mathrm {n}}$, calculate $\varphi ,\gamma _L,\gamma _D$ for different values of $\frac{L_{\mathrm {n}}}{D_{\mathrm {n}}}\in [0.1,0.3]$.

• (2) Determine a LRFD-based design criterion for the scaffolding system with a typical range of $\frac{L_{\mathrm {n}}}{D_{\mathrm {n}}}\in [0.1,0.3]$.

4.34

Consider the serviceability of a lining structure subjected to water seepage, as discussed in Problem 2.32. The lining structure has a surface of 10 m $\times$ 10 m and a thickness of 0.15 m. Assume that the hydraulic conductivity K and the water pressure p are mutually independent. At any single location of the surface, the hydraulic conductivity K is lognormally distributed having a mean value of $5.67\times 10^{-13}$ m/s and a COV of 0.3, and the water pressure is a Gamma variable with a mean value of 0.06 MPa and a COV of 0.2. The water pressure p is constant for the whole surface, while the hydraulic conductivity is a random filed. The K’s at any two locations with a distance of x (in meters) have a correlation coefficient of $\exp (-x)$. At time t, the structural performance is deemed as “satisfactory” if the water seepage depth does not exceed the thickness for at least 99% area of the surface. What are the probabilities of satisfactory structural performance at the end of 20 and 50 years respectively?

4.35

Recall Example 4.43. We additionally consider the following case: H has a mean value within [1.87, 1.93], a standard deviation of 0.45, and is strictly defined within [1.0, 3.0].

• (1) Compute the CDF envelope of H with linear-programming-based approach.

• (2) Find the lower and upper bounds of failure probability.

4.36

In Example 4.44, if we additionally know that $C_1\in [0.5,1.5]$, recompute the interval for system failure probability.