7.1.1 General Form

Generally, the crossing rate of random processes can be calculated by Equation (7.4).

(7.4)

The joint probability density function (PDF) must be obtained before calculation. This can be made by the convolution of and .

(7.5)

(7.6)

(7.7)

where v_i(u) represents the crossing rate of random process X_i(t) at the transcending rate of horizontal u. is the PDF of X_i(t) at any time point t, also known as the PDF at any time point. Equation (7.7) is sometimes called the point crossing formula. The lower limit can also be calculated. The results of calculation can be extended to a nonlinear combination of multiple non-stationary load random processes ^[7-1].

Equation (7.7) offers a solution to the combination of random processes, especially when the random processes X₁(t) and X₂(t) obey a discrete distribution. Equation (7.7) also may apply to the combinations of random processes that satisfy:

(7.8)

Figure 7.1 shows typical random processes that are consistent with this situation.

If both X₁(t) and X₂(t) are stationary normal random processes, with a mean of and , and a standard deviation of and , respectively, then:

The crossing rate of each independent random process can be obtained from:

After the above results are substituted into Equation (7.7), the upper limit of crossing rate of X = X₁ + X₂ is turned into:

where , , . The error is . When , the maximum value is . For most load combinations, the error does not reach ^[7-2].

7.1.2 Discrete Random Process

Let’s consider a combination of two non-negative rectangular update processes. Figure 7.2 shows a typical trajectory (or implementation) chart. The crossing rate of each random process can be calculated:

(7.9)

(7.10)

where the mean arrival rate of the random pulses is v_iq_i = v_mi, the distribution at any time point is , and δ is the Dirac function (see Figure 7.2), q_i = 1 − p_i. Let G_i( ) = 1 − F_i( ); substitute this into Equation (7.7), and let i = 1,2, so

(7.11)

Fortunately, this result is very simple to explain, and can be solved using the most basic theory.

For a random process, if the X(t) value is non-negative whenever it is updated, then the probability value is q. Similarly, when the X(t) value is 0, the probability value is p, regardless of the X(t) value within the current time increment. Therefore, the random process has no aftereffect. Table 7.1 shows various possible states of transcending level a after two random processes are combined.

The effect of each state change (e.g., from X₁ + X₂ < a to X₁ + X₂ > a) on the crossing rate is as follows:

(7.12)

Let Ω = •State Change Δt•. So, the term (a) in Table 7.1 becomes:

or

(7.13)

The above formula is consistent with the second term in Equation (7.11) because v₂q₂ = v_m2.

The term (b) in Table 7.1 becomes:

or

(7.14)

The above formula is consistent with the fourth term in Equation (7.11) (v₂q₂ = v_m2). The term (c) in Table 7.1 is turned into:

or

(7.15)

The above formula is equal to the sixth term in Equation (7.11). Other terms in Equation (7.11) change with the change of states (d)∼(f) in Table 7.1. If each pulse returns to zero before the start of the next pulse, then the term p₂ + q₂F₂(a) in Equation (7.13) is equal to 1 by definition, and the same is true for the term p₂ in Equation (7.14).

If the crossing rate is obtained by calculation, then the cumulative distribution function (CDF) F_X( ) of load X = X₁ + X₂ can be obtained by calculation using Equation (7.1). Some academic researchers have studied the calculation error between Equation (7.1) and Equation (7.2). By comparing with certain known exact solutions ^{[7-3][7-4][7-5][7-6]}, we find that the calculation error for a high-level obstacle a is 20%, while the calculation error for a low-level obstacle is 60%.

7.1.3 Simplified Method

(1) Directional simulation in load effect space

The load in pulse mode can be simplified by Equation (7.11). For example, let

where G₁₂( ) = 1 − F₁₂( ), F₁₂( ) is the CDF between two pulses. So, Equation (7.11) can be simplified to:

(7.16)

(7.17)

This equation was first proposed by Wen ^[7-7]. The first and second terms in Equation (7.17) represent the crossing rate when each random process acts alone, while the third term represents the crossing rate when two random processes act simultaneously (i.e., pulse overlapping). The third term can be ignored if the pulse duration is very short.

The existing research shows that, compared with the simulation results, Equation (7.17) can be used to accurately estimate the crossing rate , even if the height of each random process is as high as 0.2, or in the case of a very high obstacle level a ^[7-6]. The formula also considers other pulse shapes than rectangular ones, as well as the correlation between pulses ^{[7-8][7-9][7-10]}.

(2) Borges processes

The combination of Borges processes is of particular significance for code validation, because it helps to accurately estimate the maximum combination effect of load probability distribution ^[7-11].

When every random process X_i(t) in the random process combination X = X₁ + X₂ + X₃ +⋯ is a Borges process, then the pulse duration is τ₁ < τ₂ < τ₃ < …, and τ_i / τ_i−1 is an integer. See Figure 7.3. Therefore, the occurrence rate is equal to the number n_i of pulses in the random process X_i during the time interval [0, t_L] divided by the duration, i.e., v_i = n_i / t_L. According to the above crossing rate formula, the CDF of the maximum value of X can be obtained by Equation (7.1). However, another method will also be briefly introduced below.

The CDF F_{max X}( ) of maximum X can be calculated by convolution computation in the equation below ^[7-12]:

(7.18)

The above formula is a bivariate random process, τ₂ = τ₁ / m, where m is an integer, and there are n pulses about τ₁ within [0, t_L]. For a combination of two or more loads, this integral term is more complex, but a calculation can be performed using the first-order second-moment method.

For a linear combination of three loads, the maximum value within [0, t_L] can be written as:

(7.19)

(7.20)

where represents the maximum of X₃ in time interval τ₂.

The CDF of Z₂ can be obtained by the convolution integral of Equation (7.18). The maximum value can be expressed as:

(7.21)

(7.22)

where represents the maximum value of Z₂ in time interval τ₁. Similarly, can be obtained from Equation (7.18).

If the supplementary terms to all the random processes in and can be obtained, then the above combination method can be extended to a combination of any number of loads. The calculation steps are summarized below. First, the CDF of each process can be approximated by a normal distribution based on a series of checking points. The transformation starts with X₂, with the same checking point x^*, ; m = τ₂/τ₃ represents the pulse count in pulse X₂ with respect to X₃.

The mean and variance of the equivalent normal distribution X₂ after transformation can be obtained. Similarly, the mean and variance of can be obtained. Therefore, the mean and variance of Z₂(t) can be calculated by means of Equation (7.22), where , . Thus, we can calculate , obtain the normal distribution equivalent to X₁(t) and , and the normal distribution equivalent to Z₁(t), as shown in Equation (7.20). This is the structure that we are seeking, but it will depend on the selection of the initial value of the checking point x^*. A complete distribution function F_{max X} ( ) can be obtained by taking different checking points x = x^* and then repeating the above steps. According to what is presented in Chapter 4, i.e., the joint PDF can reach its maximum if the independent checking point x^* is selected for a random variable X_i(t), it can then use the equivalent normal PDF .

(3) Deterministic load combination—Turkstra’s combination rule

Many simplified methods are used for load combination calculations, but they are still very complicated for conventional structural design and code formulation. The most basic load combination method is to add up the loads directly without considering the uncertainty of these loads. Then, a series of coefficients are proposed to amplify or reduce the loads to an appropriate extent. The key problem is how to ensure the rationality of these coefficients.

Load combination rules can be extracted from a combination of Borges processes ^[7-13] or from Equation (7.7). If the approximation in Equation (7.1) is substituted into Equation (7.7), it can be obtained:

(7.23)

where the maximum value is taken over the entire time interval concerned, such as the service life [0, t_L] of the structure. For combination Z = W + V, where W and V are independent of each other, the CDF G_Z( ) can be obtained by means of the convolution integral:

(7.24)

Therefore, the two integrals in Equation (7.23) represent max and , where represents the X_i value at any time point. Similarly, if Z = max(W, V), then:

(7.25)

(7.26)

(7.27)

This combination is analogous to the load combination rules specified in the current codes, and obviously, is also related to the probability theory requirements in load combination. However, the load level is not specified in Turkstra’s combination rule, but needs to be selected one by one. Chapter 8 will describe that if the random process X_i is stationary, then the value of [max X_i] is usually equal to 95% of the load, and is the mean value at any time point.

Although Turkstra’s combination rule is very simple to apply to standardized engineering applications, it is not very suitable for probability calculations that have high requirements for accuracy.

7.2.1 Peak Superposition Method

This method assumes that all load processes reach a maximum simultaneously, with these maximum values added up to obtain the designed value. This is the most conservative combination model. Actually, it should be considered as a deterministic analysis method. Currently, this method is often used for offshore structure design.

7.2.2 Crossing Analysis Method

The up-crossing theory has been used by many academic researchers to study nonlinear or dynamic load combination problems. Let X(t) be a stationary random process, implemented as x(t) in t₁,t₂. Given a constant threshold value X(t) = r, as shown in the figure, the up-crossing level X(t) is called positive crossing, which is marked in the figure.

The derivative process of the stationary random process X(t) can be defined. If X(t) meets the following condition: the autocorrelation function R_XX(t) has a continuous second derivative , and the above-defined derivative process is also a stationary random process. Because X(t) is a stationary random process, for τ = t₁ – t₂, there is

The step function H(x) needs to be used to derive the up-crossing rate theory, and it is defined by Heaviside, i.e.,

(7.30)

(7.31)

(7.32)

(7.33)

(7.34)

(7.35)

(7.36)

(7.37)

(7.38)

(7.39)

(7.40)

(7.41)

This method is based on a combined model of Poisson processes, first proposed by Hasofer [7-15], who assumed that there should be a linear relationship between load and load effect, as follows:

(7.42)

(2) Wen combination method [7-16][7-17] [7-18] [7-19].

It is basically the same idea as the Hasofer combination method. There is a great difference between Class-a and –b effects in terms of frequency and duration. If the average occurrence rate of the i^th effect is υ_i, and the average duration is , then the size of the product can actually represent the probability of its occurrence within a certain time interval. If the product , this means that the load will not appear for most of the time, making it unnecessary to consider the possibility that this particular load will occur at the same time as other loads. When is large, it becomes necessary to consider the possibility that they will meet. By taking the composite load formed between them as a new process independent of the original loads, and if it is still a Poisson process, we can derive the maximum value distribution after load combination, as follows:

(7.43)

(1) Turkstra’s rule for random load combination [7-20] [7-21]

This is the combination model recommended in American National Standard A58. At present, this method has been adopted in the Unified Standard for Reliability Design of Hydraulic Engineering Structures ^[7-22] (GB50199-94) and the Unified Standard of Reliability of Structure Design for Harbor Engineering ^[7-23] (GB50158-92). According to the first load combination rule to be proposed by Turkstra, the extreme value of a load effect in [0, T] can be combined with the instantaneous value of the remaining load effects in turn, after which the combination with the largest load effect is selected as a control form. The combined maximum is then as follows:

(7.45)

where t₀ represents the moment at which S_i(t) reaches its maximum, as shown in Figure 7.4. Figure 7.4 shows three different loading processes. According to Turkstra’s principle, the combined maximum to be evaluated is determined by the most unfavorable of the three combinations. The distribution function of the combined maximum is as follows:

(7.46)

(2) Ferry Borges-Castanheta rule for random load combinations

The combination rule is an iso-temporal load combination model, proposed in 1972 by Ferry Borges and Castanheta^[7-24], with reference to Turkstra’s rule. According to this combination rule, for every variable load x_i, it can be divided into r_i equal sections τ_i, which are then used as basis intervals throughout the service life of the structure, T. In each time interval, x_i(t) does not change with time, x_i(t) is statistically independent in sequential time intervals, and the probability of occurrence of x_i(t) in each time interval is P_i. r_i represents the number of repetitions of loads during the service life. For selection of τ_i, the maximum reached in sequential time intervals should be used to build an independent hypothesis.

As can be seen from the above, the distribution of the maximum can be described using a stationary binomial process. So

(7.47)

When considering a combination of several variable loads (random processes), we can assume that the random processes are independent of one another, and that the number of time intervals, r_i, divided by time T, is an integer. Also, let be a positive whole number, and arrange it in increasing order of r_i. So

When n=3, the Figure 7.5 shows three rectangular oscillo-grams simplified as equal time intervals.

Starting from the load x_n that is repeated for the largest number of times, we work out the maximum x_n(t, τ_n−1) of x_n(t) in time interval τ_n−1, and obtain

(7.48)

According to Equation (7.48), max(x₁ + x₂ + ⋯ + x_n) can be approximated as max(x₁ + x₂ + ⋯ + x_n−2 + Z_n−1) so as to reduce the problem from a combination of n loads to a combination of (n-1) loads, where the distribution of Z_n−1 is the convolution of the distribution function of x_n(t, τ_n−1) and the density function f_n−1(x) of x_n−1(t). Therefore, the maximum value distribution can be determined by n-1 sequential convolution operations, as follows:

(7.49)

Equation (7.49) is the superposition process of this rule. Because Z_i = x_i+1(τ_i) + x_i obtained by superposition each time is a constant maximum in this time interval, the results obtained are obviously more conservative, especially when n is larger. Besides, the sequential convolution operations in the previous formula are also very tedious. Figure 7.5 shows the superposition process of the F-B rule.

There are three load effects, x₁(t),x₂(t),x₃(t), where r₁ < r₂ < r₃, and the combined maximum is shown in Equation (7.50):

(7.50)

(3) JCSS combination rule for random load combination (pyramid method)

This load effect combination model is recommended in Volume I ^[7-25][7-27] of the International System of Unified Standard for Structures issued by the International Joint Committee on Structural Safety (JCSS), composed of six international organizations. It is also a combined probability model of load effects adopted in China’s Unified Standard for Building Structures ^{[7-26] [7-28]}. This model can be used to combine load effects in accordance with engineering experience. This is compatible with the use of first-order second-moment method considering the probability distribution types of basic variables to analyze structural reliability. The main contents are:

1. Load Q(t) is an isochronous stationary random process.
   
2. There is a linear relationship between load Q(t) and load effect S(t), as follows:
   
   
   
   

   Where C_Q is a coefficient of Q(t).

   
   
3. Mutually exclusive random loads cannot be combined. Only variable loads that may meet in [0,T] are combined. Also, the types of loads concerned must be determined based on experience.
   
4. When a load reaches a maximum in the design reference period or when a certain time interval is applied, other loads involved in the combination only reach a maximum during the duration of this maximum load; otherwise the load is taken at any time point. That is, let n variable loads be involved in the combination, and let the total number of time intervals with respect to various modeled loads Q_i(t) in the design reference period T be r_i to arrange the loads from small to large, i.e., r₁ ≤ r₂ ≤ r₃ ≤ ⋯. Any kind of load effect can be combined to obtain the maximum load effect (combined load effect) of n combinations, , i.e.,

(7.51a)

(7.51b)

Three examples of live load combinations: persistent live load L_i, temporary live load L_r and wind load W, are taken to compare the above random load combination rules. The specific data is shown in Table 7.2.

The load meeting method, F-B combination, TR combination and JCSS combination of these three random loads at different load-effect ratios are calculated according to the above-mentioned random load combination theory. For the convenience of comparison, the combination results are achieved by the dimensionless ratio represents the mean of the maximum distribution of the combined effect; μ_si represents the mean of the truncated distribution of the effect of a single load. μ_sM varies with combination rules, while is only related to the inherent law of the loads involved in the combination, and is not affected by the combination method. The specific calculation results are as follows:

As can be seen from Figure 7.6, the calculation results of various combination rules vary greatly with the load-effect ratio S_L/S_W; when the load-effect ratio is 1, the minimum value is reached. The F-B combination rule boasts the largest calculation results, indicating that this combination method is more conservative. As can also be seen, the JCSS combination rule and TR combination rule are less conservative when the load-effect ratio is low, but become increasingly conservative as the load-effect ratio rises. Therefore, various combination rules differ from one another in terms of risk at different load-effect ratios.

7.3.1 Expression of Design Partial Coefficient

When determining the partial coefficients in the limit state design expression, we must consider the type of load combination, the type of components, extreme working conditions, marine environmental loads, and the relevant statistical characteristics of structural resistance.

Now, let’s show how partial coefficients are determined for the common load combination form, i.e., (dead load + live load). According to the limit state expression:

At checking point X^⋆, the limit state equation is written as:

(7.52)

(7.53)

(7.54)

(7.55)

(7.56)

(7.57)

The size of the I value reflects the difference between the target reliability index and the implicit reliability index of the structural component designed according to the design expression containing this set of γ_G and γ_Q, and various optimized values. Dimensionless relative error is used to achieve a balance in the ratio of the error of each component to the overall error.

This method can be used to determine partial coefficients according to the principle of least squares. Essentially, this method is to fit the standard value of structural component resistance, which changes with the load-effect ratio ρ, using the standard value of the structural component resistance determined by the design expression, given the target reliability index and dead load effect or live load effect. A linear function is used to fit it. If the error between the mean reliability index of components obtained by reverse calculation and the target reliability index is adopted for measurement, the effect is not so good when there are only a few optimization variables, but calculation is simple.

(2) Least squares method for reliability index

By this method, partial coefficients are determined according to the principle that the error between the structural reliability index in the limit state design expression and the target reliability index is minimized.

(7.58)

where is the target reliability index of the i^th type of component; β_ij is the reliability index of the i^th type of component in the limit state design expression at the j^th ρ load effect ratio.

(3) Specification method 1

All the load partial coefficients obtained by Equations (7.57) and (7.58) are results of optimization within a certain range. The load partial coefficients obtained by calculation may be different from the partial coefficients commonly used by designers. Thus, inconvenience is caused for both engineering design and engineering practice. To ensure the continuity between new and old codes, a new method called “Specification Method 1” has been proposed in Figure 7.7: Under extreme conditions, the dead load coefficient is kept unchanged when the dead load and live load are combined, i.e., γ_G = 1.3. The resistance partial coefficients of all structural components are calculated by the principle of least squares. Under the condition that load coefficient is known, the live load coefficient can be changed to calculate the partial coefficients of resistance. This is consistent with both engineering practice and designers’ design habits.

(4) Specification method 2

Specification Method 2 is basically the same as Specification Method 1. To ensure the continuity between the new and old codes, four common resistance partial coefficients are known, i.e., axial compression is 0.85, axial tension is 0.95, bending is 0.95, and shearing is 0.95, to calculate load partial coefficients. Then, other resistance partial coefficients are calculated under the condition that load partial coefficients are known in Figure 7.8. Four common resistance partial coefficients are known in this method to calculate the partial coefficients of load and resistance. This is consistent with accepted engineering practice and design norms.

(5) Method for determining resistance partial coefficients

As mentioned earlier, during the process of selecting an optimal partial coefficient of load, given each set of γ_G and γ_Q, for every component i, under the condition that H_i is minimized, the least squares method can be used to determine an optimal partial coefficient resistance, , under the combination of dead load and live load. The calculation is as follows:

(7.59)

(7.60)

7.4.1 Design Expression Using Combined Value Coefficients

Given multiple variable loads, the standard values and partial coefficients of each variable load in the design expression are equal to what they would be when there is only one variable load. A coefficient less than 1 can be adopted and multiplied by the load effect term. This is the combined value coefficient. According to the multiplication, the combined value coefficient can be divided into the following three types.

(1) All load terms (including dead load) are multiplied by the combination coefficient.

For example, in the U.S. Building Code Requirements for Structural Concrete and Commentary (ACI318-83)^[7-28][7-30], while considering the concurrent occurrence of multiple variable loads, the load effect terms with load coefficients, including dead load, are all reduced by 0.75. They are then compared with dead load plus a variable load term, and the most unfavorable load condition and load effect term selected:

where D, L, W and E are the effects caused by dead load, live load, wind load and earthquake, respectively. T is the effect caused by differential settlement, creep, shrinkage or temperature variation.

(2) Reduction of all variable load terms

For example, the National Building Code of Canada ^[7-29] (2010) adopts the following formula for basic load effect term selection:

(7.61)

where υ represents the effect of the various loads in the brackets; D, L, W and T represent the standard value of dead load, live load, wind and temperature, respectively; γ_D, γ_L, γ_W and γ_T are the corresponding partial coefficients determined under the action of dead load and variable load alone; ψ is the load combination value coefficient, which is set to 1.0, 0.7 and 0.6 when there are one, two or three loads in the internal brackets. In China’s load code (TJ9-74) [7-30][7-32], the following formula is taken as the design expression for load combinations. When wind load is combined with other live loads, the standard value of wind load (W) and all other live loads (Q_i) apart from dead load (G) is multiplied by the combination coefficient ψ. ψ = 0.9 for common buildings.

(7.62)

According to the Unified Rules for Various Structures and Materials[7-31], Volume I of the International System of Unified Standard for Structures formulated by the International Joint Committee on Structural Safety (JCSS), and Code for Design of Concrete Structures^[7-32], Volume II of the International System of Unified Standard for Structures formulated by the Fédération Internationale de la Précontrainte (CEB-FIP), the major variable loads involved in combination are not multiplied by the combination coefficient, while only subordinate variable loads are reduced. For example, in Code for Design of Concrete Structures, the following formula is adopted for basic load combination.

(7.63)

(7.64)

In this kind of design expression, different load partial coefficients or load eigenvalues are specified for variable loads according to their degree of participation in the combination. For example, U.S. national standard Minimum Design Load for Buildings and Other Structures (ANSI A58.1-1982)[7-33] stipulates that, when strength design is adopted for structure and component foundations, the load effect term is the most unfavorable effect to be selected in Equation (7.65).

(7.65)

where D, L, L_r, S, R, W and E represent the standard value of dead load, live load, roof live load, snow load, rain load, wind load and earthquake, respectively. The basic concept of load combination is as follows: except for the dead load that is in permanent action, one of the variable loads is “maximized in the service life”, while the other variable loads are assumed to be at a “value at any time point”. Because the standard load value exceeds the value at any time point, some load coefficients in the above equations are less than 1.

American experts Galambos and Ravindra ^[7-34] suggested taking different standard values of load partial coefficients for variable loads according to different combinations, as follows:

(7.66)

where represents the effect caused by the mean value of dead load; L_T,, represent the effects caused by the mean value of the maximum distribution of live load, wind load and snow load throughout the service life; represents the effect caused by the mean value of the distribution of live load at any time point.

As can be seen from the above, there is a difference from country to country in considering the load combination design expression. The first kind of method, based on the combination coefficient, is simplest to use, with all loads (including dead load) reduced by 0.75. This amounts to increasing the bearing capacity by 1/3 after considering load combination. However, because the dead load is theoretically a random variable that does not change with time, combinational reduction of the dead load in this formula is inconsistent with the statistical characteristics of load combination, making it unsafe to make calculations in some cases. In Equation (7.66, 7.69), all variable loads involved in the combination are multiplied by the combination coefficient in no order, and the method is easy to operate. However, when the effect of one variable load is more significant than that of the others, this load is still reduced, and obviously the value will be on the low side. In Equation (7.67), it is more rational to take all the values of the main loads and reduce subordinate loads only. Besides, there is no combination coefficient in Equation (7.66), and Equation (7.67) intuitively reflects the load combination mode. The mean value and corresponding partial coefficients of variable loads that may meet one another are introduced into this equation, with everything easy to understand, but this method requires that the codes provide several design standard values for each variable load. This is inconsistent with China’s design conventions.

According to the above analysis, as well as China’s design practice and the principle that the standard value and partial coefficients of combined loads must be equal to that of a simple combination, the first type of design expression multiplied by the combination coefficient is adopted in China’s Unified Standard ^[7-25]. In the process of compilation, regarding the design expression for the two types of combination coefficients in the formula below, the combination value coefficients ψ and ψ_c were calculated by probability analysis. After analysis and comparison, the two formulas below were finally adopted for the unified standard, thanks to their advantages of providing a reasonable value and a simple design.

(7.67)

where γ is the load partial coefficient, which has the same value as that in a simple combination; C is the effect coefficient of load conversion into effect; G_K, Q_i, Q_ik represent the standard values of dead load, main variable loads and subordinate variable loads, respectively; ψ and ψ_c represent the combined value coefficients multiplied by all loads and by the subordinate variable loads, respectively.

7.4.3 Method for Determining Load Combination Coefficient in Ocean Engineering

At present, the probability limit state of offshore structures cannot yet be designed directly based on the target reliability index. Instead, an internationally accepted universal practical design expression, represented by the standard value and partial coefficients of basic variables, is adopted for probability limit state design. The standard value of various basic variables is determined according to the high fractional value corresponding to its maximum value distribution, while partial coefficients are, as shown in the previous analysis, determined according to the target reliability index and an optimized simple combination of random loads in the marine environment. A change will take place in the probability distribution of load combination when more environmental loads are involved in the combination, and the probability is extremely low that various load effects involved in the combination meet one another through their standard value. The standard value of each load involved in the combination must be adequately reduced in order that the structural reliability index will not change after combination. This reduction factor is also the load combination coefficient.

In the practical design expression for offshore engineering structures ^[7-35][7-36], the standard value of variable loads is reduced under the premise that the partial coefficients are kept unchanged. The value of the combination coefficient is determined after optimization according to the principle of β equivalence, i.e., load partial coefficients are determined according to the simple combination of environmental marine loads. When two or more variable loads are combined, the standard value of these variable loads can be reduced on the basis of the combination coefficient, so that the reliability index of various components designed in accordance with the limit state design expression remains consistent with that of the simple combination as much as possible.

According to the above principle, when there are several variable loads present, the ultimate state expression for structural bearing capacity can take the following form:

(7.68)

(7.69)

where γ_G, γ_Q, γ_R are all determined according to the simple combination of environmental loads; R_K^* is the value of structural resistance at the checking point. It can be determined by the aforesaid probability limit state design method, in accordance with the reliability index of the structure designed according to the limit state design expression under simple combination and the load combination form that functions as a controller during the design reference period.

Owing to the value ψ of the combination coefficient obtained by the above formula, the reliability index β of the structure designed according to the limit state design expression is the same as that in a simple combination. However, the ψ value varies with the type of components, the type of loads involved in the combination, the effect ratio ρ between variable loads and the dead load, and the ratio ξ between variable loads. Although a reliability index consistent with that in the simple combination can be obtained if the ψ value is obtained under different conditions, it is extremely inconvenient to use. For this reason, the ψ value of an optimal combination coefficient applicable to various components and ρ values can be determined according to the effect ratio ξ between different variable loads under a certain load combination, so that the I value below is minimized:

(7.70)

(7.71)

(7.72)

(7.73)

(7.74)

Reinforcement corrosion of concrete structures in a corrosive environment has long been an issue of great concern for academics throughout the world [7-37]. Reinforcement corrosion is the main cause of deterioration in the resistance and reliability of concrete structures. Throughout their service life, concrete structures are affected by a variety of uncertain factors. According to the damage mechanism of steel bar corrosion, the lifespan of concrete structures can be divided into four stages: destruction of the steel bar passivation film, corrosion and cracking of the protective concrete layer, the width of concrete cracks reaching their upper limit, and the bearing capacity of concrete components dropping below their lower limit.

According to existing research, the corrosion process of steel bars can be divided into two stages ^[7-38][7-39], the initial rust stage and the corrosion propagation stage, both of which are affected by many uncertainties. In the initial corrosion stage, the diffusion coefficient, concentration on the surface and critical concentration of chloride ion are all highly random, which in turn leads to the high randomness of the initial corrosion time. During the extension stage, the corrosion depth is also somewhat random due to the random nature of the corrosion current density.

According to the basic concept of the Path Probability Model ^[7-38], the corrosion process of steel bars can be divided into a series of paths, based on the initial corrosion stage and the corrosion expansion stage, while taking into consideration the influence of uncertain factors in both stages. The unconditional probability distribution for steel bar corrosion under all paths is obtained using Bayes’ probability summation formula, as shown in Figure 7.9.

In the Figure 7.9, T_E is the given lifespan of the structure, t_c is the time when the steel bar initially corrodes, and the corrosion propagation time is T_E − t_c, which defines a possible corrosion path in . When T_E has been determined, the corrosion ratio of any corrosion path is a function of time t_c, and is recorded as ; η is the corrosion ratio of the steel bar.

If is further recorded as the probability density of the initial rusting of the steel bar at time t_c, then the probability of initial rusting of the steel bar at time t_c is . According to Bayes’ probability formula:

(7.75)

(7.76)

Where, is the probability density function of the steel bar corroding under the condition that initial rusting occurs at time t_i; is the probability of initial rusting of the steel at time t_i,

(7.77)

Based on the path probability model mentioned above, the corrosion process for a steel bar is divided into three stages in the literature [7-39], ^[7-40]: Initial stage, rust expansion stage and late stage concrete cracking (from cracking of the protective layer due to corrosion expansion, to the bearing capacity of the structure or component dropping below its limit). The Multipath Probability Model is thus established to predict corrosion ratio, crack width and bearing capacity degradation of steel bars.

In the Figure 7.10, t_c and t_cr are initial rusting time and protective layer cracking time, respectively. There are a total of three conditions for concrete structures in service:

1. t₀ < T_E < t_c, steel bar is not corroded
   
2. t_c < T_E < t_cr, steel bar is corroded, but concrete protective layer has not cracked due to corrosion expansion
   
3. t_c < t_cr < T_E, cracking of concrete protective layer due to corrosion expansion

In this paper, the model is established by the third case, with a corrosion path . It can be seen from the previous section that can be defined as many paths, and that any n time points in can be considered. For any initial rusting time t_i(i = 1,2 … n) and protective layer cracking time , there are corrosion paths in total.

It can be seen from Equation (7.76) that, as long as enough corrosion paths are defined during the lifespan, and as long as t_c and t_cr of each path are uniquely determined, these paths will constitute a group of mutually exclusive events. According to Bayes’ probability summation formula, unconditional probability density functions , f(W) and f (L) for the corrosion ratio of steel bars, crack width due to corrosion expansion, and bearing capacity reduction coefficient in service time can all be calculated.

(7.78)

(7.79)

(7.80)

(7.81)

(7.82)

(7.83)

When the concentration of chloride ion on the surface of a steel bar gradually increases and exceeds critical concentration after a reinforced concrete beam has been used in a chloride-containing environment for a certain period, the passivation film on the surface of the steel bar will be destroyed, and the steel bar will corrode. The transportation modes of chloride ions in concrete include convection, electromigration, and diffusion [7-41]. In general, unsteady diffusion is considered the main transportation mode, which also conforms to Fick’s second law. Then, the chloride ion concentration at a depth of x from the concrete surface at time t can be expressed as ^[7-42]:

(7.84)

Where, is the chloride ion concentration (percentage of concrete mass) at a distance x from the concrete surface kg / m³; C₀ is the concentration of initial chloride ion inside the concrete (usually 0); C_s is the concentration of chloride ion on the concrete’s surface, Val [7-43] considers that its distribution parameters are related to the environment, and it is recommended to take values based on Table 7.3; D_cl is the chloride ion diffusion coefficient mm / s. Stewart believes this is related to time, temperature, humidity and stress level, and assuming a condition of no measured data, it can approximately obey a normal logarithmic distribution with a Mean Value of 2×10^-6 and a coefficient of variation of 0.450 ^[7-44];

When the concentration of chloride ions on the steel bar surface reaches a critical value, the steel bar will corrode. Then the cumulative probability of initial rusting of the steel bar occurring at any time t_i can be expressed as:

(7.85)

Where, C_cr is the threshold value of concentration of surface chloride ion that leads to the initial rusting of the steel bar, which is highly random. As suggested by Stewart, normal distribution with mean 3.35 kg /m³ and coefficient of variation of 0.375[7-45][7-46] should be obeyed.

7.5.4 Probability Prediction Model for Concrete Carbonation

Concrete carbonation generates CaCO₃, reduces porosity, improves concrete compactness and hinders CO₂ diffusion. However, it also reduces Ca(OH₂) concentration and pH value, leading to destruction of the passivation film on the steel surface and thus accelerating corrosion of the steel bar. The variation of concrete carbonation depth is mainly determined by its own variability and the environment^[7-39]. The practical empirical carbonation model considers various influencing factors, and is given in the literature ^{[7-45] [7-47][7-48]}:

(7.86)

Where, k is the carbonation coefficient; K_mc is a calculation model uncertainty variable; k_j is the corner correction coefficient; k_CO2 is the coefficient influence CO₂ concentration; k_p is the correction coefficient of the casting surface; k_b is the coefficient influencing working stress; T is ambient temperature in °C; RH is the relative ambient humidity %; f_c is the standard value MPa of concrete cube strength, which has a certain randomness and obeys a normal distribution. See Table 7.4^[7-47] for the specific distribution.

1% phenolphthalein solution is generally used to measure the pH value of concrete in engineering fields. The pH value from concrete surface to steel bar tends to gradually increase. Based on the change in pH value, concrete carbonation can be divided into three parts: completely carbonized part, partially carbonized part, and uncarbonized part. The partial carbonation area is supposed to change in a linear fashion ^[7-45], as shown in Figure 7.11:

Where, The value of pH varies from 8.5 to 12.5. When the value reaches 11.5, the passivation film on the steel surface will be destroyed and the initial rusting of the steel begins ^[7-39]. The influence of environmental humidity, water to cement ratio, carbonation time, cement dosage and CO₂ concentration on the length of the partial carbonation zone is described in the literature ^[7-49], which also provides a calculation model for partial carbonation zone length.

(7.87)

Where, R_WC is the water-cement ratio, C is the cement dosage (kg / m³), RH is the relative ambient humidity. When RH > 75%, the length of the partial carbonation zone can be ignored.

According to the carbonation curve of the pH value in the partial carbonation zone, the length from pH=11.5 to the front end of the complete carbonation zone can be expressed as:

(7.88)

Therefore, from Equations (7.87) and (7.88), it can be seen that, under a carbonizing environment, the cumulative probability of initial rusting occurring in a concrete steel bar at any time t_i can be expressed as:

(7.89)

According to existing research on the interaction of chloride ions and carbonation [7-45][7-49][7-50], it has been found that carbonation influences chloride ions in two ways. To be specific, carbonation products reduce the porosity of concrete, improve its compactness, and hinder the transmission of chloride ions; the bound chloride ions released by carbonation lead to an increase in free chloride ions at the front end of the carbonation, which in turn accelerates the diffusion of the chloride ions within the concrete.

After studying free chloride ions in concrete, Glass ^[7-50] found that the passivation film on the surface of steel bars grows weaker when the alkalinity and pH value of the concrete are low, or when the concentration of chloride ions is low. Jin Weiliang et al. ^{[7-39][7-40][7-45][7-50]} found the same phenomenon after analyzing concrete engineering works in coastal area of Ningbo City. The concentration of chloride ions on the surface of steel bars in some concrete structures is relatively low, and the concrete carbonation depth is very shallow, but the surface of the steel bar can be significantly corroded. Therefore, it can be seen that the concentration ratio of chloride ion to hydroxide ion serves as an important parameter affecting steel bar corrosion. According to the pH value of the steel bar surface, the functional relationship between pH value and critical chloride ion concentration can be expressed as:

(7.90)

The change curve is shown in Figure 7.12.

It can be seen from the figure that a positive correlation does exist between pH and critical chloride ion concentration; critical chloride ion concentration increases with the increase in pH value. It is assumed in the literature ^[7-39][7-40] that the diffusion behavior of chloride ion and carbon dioxide in concrete do not affect each other. Therefore, in a concrete carbonation environment featuring chloride ion corrosion, a steel bar is considered corroded as long as there is a condition that meets the requirements. That is, when the concentration of the chloride ion on the surface of the steel bar reaches critical chloride ion concentration, or the pH value of the steel surface reaches 11.5. Under the combined action of carbonation and chloride ion, the cumulative probability of the initial rusting of the concrete steel bar at any time can be expressed as:

(7.91)

After the initial corrosion of the steel bar occurs, the corrosion propagation time is T_E − t_c. Song Zhigang [7-38] et al. believe that the probability density of the corrosion ratio in the corrosion propagation stage of a steel bar is . The specific calculation process can be expressed as:

Assuming the steel bar is corroded uniformly, the calculation process for the corrosion ratio can be expressed as:

(7.92)

Where, ΔA_s is the area of the corroded steel bar (mm²), A_s is the initial area of the steel bar (mm²), d is the diameter of the steel bar (mm), and Δd is the corrosion depth of the steel bar (mm).

According to Equation (7.92), the relationship between corrosion ratio and corrosion depth can be expressed as:

(7.93)

ΔD can be determined by the empirical formula (7.94).

(7.94)

Where, λ(t) is the corrosion ratio of the steel bar before cracking of the protective layer (mm/a). According to the literature ^[7-44][7-51], the equation for λ(t) can be expressed as:

(7.95)

Where, i_corr is the corrosion current density of the steel bar μA/cm², t_c is the corrosion time. After discovering the law that the corrosion current density decreases with time, as demontrated by Liu et al., Stewart [7-51] et al. established an empirical formula for corrosion current density:

(7.96)

Where, i_corr(1) is the corrosion current density at the time of initial rusting of the steel bar. In a typical environment, with a temperature of 20°C and a relative ambient humidity of 75%, this is related to the water-cement ratio of concrete and the thickness of the protective layer. The empirical formula can be expressed as [7-51]:

(7.97)

The empirical formula for the water-cement ratio R_WC is: , X_c is the thickness of the concrete protective layer (mm), is the standard value of the compressive strength of the concrete cylinder MPa [7-51], [7-52], and f_c is the standard value of the compressive strength of the concrete cube (MPa).

Thus, the empirical expression for the corrosion depth of the steel bar can be expressed as:

(7.98)

The relationship between corrosion ratio and crack width is expressed by a new model proposed by Vidal [7-53] et al., and used for predicting the loss of steel sections. In this way, the crack width w after cracking due to corrosion expansion can be expressed as:

(7.99)

(7.100)

(7.101)

(7.102)

Based on the existing design calculation theory, the following calculation models [7-54][7-55][7-56][7-57][7-58][7-59] can be used to analyze the bending and shear capacity of corroded rectangular concrete section beams:

(7.103)

(7.104)

(7.105)

(7.106)

(7.107)

(7.108)

(7.109)

(7.110)

(7.111)

(7.112)

The Monte-Carlo numerical simulation flow chart based on the path probability model is as follows in Figure 7.13:

7.5.9 Engineering Example

7.5.9.1 Corrosion of Steel Bars in a Chloride Environment

A certain bridge consists of a simply supported reinforced concrete prestressed structure ^[7-56], as shown in Figure 7.14. The bridge has been used for 16 years. Most piers are located in the water. The main steel bar used in the pier sections is Φ22 while the stirrups are Φ10. The main steel bar of the cap beam is Φ25 while the stirrups are Φ12. See Table 7.5 for related values, such as the probability distribution form and distribution parameters of the influencing parameters. The width of the rust expansion crack of each concrete pier column in wave splash and tide range area is within 0.2∼0.4mm. See Figure 7.15 for the statistical quantities.

Parameter type	Unit	Mean	Coefficient of variation	Distribution type	Source
Cs	kg/m3	7.35	0.7	Logarithmic normal	Stewart[7-8]
Dcl	mm2·s-1	2×10-6	0.45	Logarithmic normal	Stewart[7-8]
Xc	mm	35.2/29.8	0.060/0.066	Logarithmic normal	Measured value
Ccr	kg/m3	3.35	0.375	Logarithmic normal	Stewart[7-8]
fc	MPa	29.9	0.064	Logarithmic normal	Measured value

Predict the probability distribution of corrosion degree and corrosion expansion crack width of the pier and cap beam steel bar in different time periods based on the data given in Table 7.5; estimate the percentage of corrosion samples and cracked samples of steel bar components, as shown in Figure 7.16–Figure 7.19.

It can be seen from Figure 7.16 that about 28% of the steel bar samples have been corroded. According to the test data given in Figure 7.15, if the number of rust expansion cracks on Pier 30 with large-area network cracks (the number of cracks along the longitudinal steel bar is close to that of the longitudinal steel bar), 35, is taken as the sampling number for each pier, then there are 35×64=2,240 pier samples in total, with 367 cracked samples, meaning about 16.4% (<28%) of the samples are cracked. After considering the fact that the corrosion probability of steel bar is greater than the probability of cracking due to corrosion expansion, the prediction result is reasonable. It can also be seen from Figure 7.16 that the corrosion ratio of steel bars in cap beams (5%<25%) is obviously lower than that of piers in the wave splash area, which is also reasonable. The chloride ion content of the bridge pier and superstructure samples shows that the chloride ion content in the pier concrete is high, which will in turn induce steel bar corrosion. In contrast, the chloride ion content in the superstructure is low, which has an uncertain influence on steel bar corrosion. According to the probability distribution Figure 7.17 of the corrosion ratio of the corroded steel bar, the average corrosion ratio can be up to about 5%. According to Figure 7.18, cracking samples account for 25% (<28%, reasonable), but the number of corrosion samples are similar to the number of cracked samples, which also indicate that steel bar corrosion has been taking place. The reasons why the crack rate of 16.4% is lower than 25% can probably be ascribed to: 1) omissions in the test sample data; 2) deviations of model parameters and error of the model itself; and 3) other uncertainties.

The sample distribution of the detected crack width should be classed as a conditional probability problem. Statistical analysis of pier crack width shows that the Mean Value is 0.34mm and the standard deviation is 0.16. From the conditional probability density function for rust expansion crack width shown in Figure 7.19, the mean and variance are 0.3, 642 and 0.1, 879 respectively, which are basically consistent with the measured values.

The probability distribution of the steel bar corrosion ratio and crack width in wave splash zones of piers for every second year within a 10∼24 year period were further analyzed. See Figure 7.20 and Figure 7.21 for the results. It can be clearly seen that with an increase in service time, the corrosion ratio, crack width and variability all increase too. If 20% of crack samples reach 0.5mm, the durability limit state of a structure requiring maintenance, then maintenance measures should be taken after 14 years of service (i.e., 2 years before detection). Therefore, repair measures must be taken immediately for all piers showing cracks, especially for any concrete with seriously corroded steel bars, where crack width exceeds the limit, or where the protective layer has been peeled off. High-performance reinforced concrete should be used for the engineering repair design. For concrete piers with no cracks, or featuring tiny cracks only, durability protection, such as anti-corrosion coating, is recommended, if maintenance funds can be ensured.

7.5.9.2 Corrosion of Steel Bar Under the Combined Action of Carbonation and Chloride Corrosion

A certain bridge is a T-type simply supported structure, as shown in Figure 7.22. The bridge has been used for 36 years since its completion in 1968. However, based on examination, the bridge suffers from serious durability problems, including: 1) a large amount of spalling and exposed steel bars at the bottom of the T-beam and the side concrete, with cracks found everywhere; 2) some concrete sections were damaged at the T-beam resting point at the top of the pier cap beam; and 3) the ends of adjacent beams and slabs were squeezed and damaged due to no expansion joint being included in the design. More than 70 cracks wider than 0.2mm and 28 locations with spalling in the protective layer were found by means of site hole inspection. T-beam cracks mainly consist of vertical cracks, most of which are long cracks running through the whole web with widths up to 2mm. All the observed beam bottoms featured problems such as horizontal cracks, significant concrete spalling areas, exposed stirrups and main steel bars, and serious corrosion.

Parameter type	Unit	Mean	Coefficient of variation	Distribution type	Sources
Cs	kg/m3	2.95	0.7	Logarithmic normal	Stewart[7-8]
Dcl	mm2·s-1	0.167×10-6	0.45	Logarithmic normal	Measured value
Xc	mm	15	–	Constant	Measured value
Ccr	kg/m3	3.35	0.375	Logarithmic normal	Figure 7.15
ds	mm	2Φ20	–	Constant	Design value
dv	mm	Φ6@ 200	–	Constant	Design value
fy	MPa	365	0.05	Logarithmic normal	Design value
fc	MPa	25	0.15	Logarithmic normal	Measured value
T	°C	20	–	Constant	Meteorological survey
RH	%	75	–	Constant	Meteorological survey
	%	0.03	–	Constant	Meteorological survey

Test results showed that the chloride ion content of the bridge is not high, basically within 0.2%. The bridge is in a critical stage of induced steel bar corrosion. The chloride ion diffusion coefficient of the concrete is less than 1.67×10^-9 cm²/s, and the compactness of the concrete is good. According to the carbonation depth test for the concrete core samples, some components are already seriously carbonized. Therefore, the bridge is being attacked by both carbonation and chloride ions. Further analysis showed that the joint action of the concrete carbonation and chloride ion corrosion is the cause of early depassivation and accelerated corrosion of the steel bars. See Table 7.6 for the probability distribution form and the distribution of influencing parameters.

For the convenience of comparison, probability density functions of critical chloride ion concentration, corrosion time of the steel bar, corrosion expansion cracking time of the concrete, corrosion ratio of the steel bar, and crack width at detection time are estimated under the conditions of single chloride ion corrosion and chloride ion corrosion combined with concrete carbonation.

According to Figure 7.23, under the condition of chloride ion attack alone, the critical chloride ion concentration is a time-invariant random variable with a mean of 3.5%. In contrast, under the combined effect of chloride ion corrosion and concrete carbonation, the change of critical chloride ion concentration is a random process as the carbonation progresses, while the probability density function gradually approaches the vertical axis with time. This shows that the critical chloride ion concentration decreases along with the decrease in concrete pH value on the surface of the steel bar, which conforms to the law of Figure 7.12. Because of the synchronous effect of carbonation, the corrosion sample of the steel bar has reached more than 95% after the bridge has been used for 15 years (Figure 7.24). In addition, the concrete protective layer is too thin, and the concrete cracks due to rust expansion (Figure 7.25). However, for chloride ion alone, the steel bar corrosion progresses more slowly, with extensive corrosion of the steel bar specimens and the extensive cracking of the concrete only occurring after the bridge has been used for 40 or 50 years. The corrosion ratio of the main steel bar also shows the same characteristics, that is, the corrosion of the steel bar is fully developed under the combined action, while the corrosion ratio is generally very small under the action of chloride ion corrosion alone (Figure 7.26). Even for corroded specimens, the corrosion of the steel bar under the combined action is more complete, with 95% of corroded specimens showing a corrosion ratio above 20% (Figure 7.27). The same characteristic also applies to crack width (Figure 7.28). 95% of concrete cracks are wider than 2mm (Figure 7.29), and the protective layer has been peeled off. In fact, the actual project verified the investigation results that a considerable area of peeled-off and exposed steel bars and cracks commonly exist at the bottom of the T-beam and at the side protective layer of the concrete.