Stability safety assessment of long-span continuous girder bridges in cantilever construction

Abstract

The overturning stability issue of continuous girder bridges is critical so that it is necessary to obtain the true overturning stability performance. At present, the parameters uncertainties in the structure were neglected in the stability evaluation method of the long-span continuous girder bridges, which leads to the unknown safety level of the continuous girder bridges during the cantilever construction. Therefore, a calculating method for overturning stability safety factors of long-span continuous girder bridges in cantilever construction based on inverse reliability theory is presented in this paper. The proposed method is extended from the traditional deterministic form of safety factor, which considered influence of uncertainty factors among structure parameters was used to obtain safety factors through target reliability index based on inverse reliability theory. Overturning stability safety factor of long-span continuous girder bridges in cantilever construction and parameter sensitivity were assessed using the proposed method, as well as the reasonableness of longitudinal overturning stability safety factors was discussed. The results show that parameter uncertainties have a major effect on overturning stability safety factors of long-span continuous girder bridges in cantilever construction, ignoring parameter uncertainties will result in overestimation of overturning stability safety factors of long-span continuous girder bridges in cantilever construction, reasonable safety factor should be obtained based on target performance. The sum of the self-weight of the travelling form and the pouring segment has the most significant effect on the safety factor. It’s critical to ensure a reasonable situation of the travelling form during the construction stage in case of falling. The resistant moment of the temporary support and the eccentric distance of the support also need to be handled carefully because of the remarkable effect. The proposed method is stable and reliable, which will be convergent to the same result from different initial value in spite of different iteration progress.

Keywords

Bridge engineering overturning stability safety factor inverse reliability theory cantilever construction long-span continuous girder bridges uncertainty target reliability index

1 Introduction

The overturning risk of bridge structure caused by the unbalanced load increases along with increasing of the cantilever length during the cantilever construction stage of long-span pre-stressed concrete continuous girder bridge. To ensure the structural safety, a temporary reinforcement measure is usually located at the pier, which is implemented by arranging several rows rebar symmetrically on both sides of the support to anchor the 0# block on the pier in order to resist the overturning moment from unbalanced load. Although the cantilever construction method of continuous girder bridge is widely used, the longitudinal overturning stability issue during construction can also be considered in case of the overturning risk of the girder. For example, at about 19:10 on March 1st 2017, at the Longyan Stage in Xiarong Freeway, the right line of Longtan Bridge at A7 section of the expansion project overturned along the axis of the cantilever pouring girder (seen in Fig. 1). Additionally, a continuous girder bridge in specific Shenyang Line overturned longitudinally during cantilever construction stage.

Fig.1

Bridge overturning of Longyan Stage in Xiarong Freeway.

Research on overturning safety issue of continuous girder bridge during the cantilever construction is mainly divided into the deterministic method and the reliability method. At present, the parameter uncertainties in the structure are neglected in the stability evaluation method using safety factor based on deterministic theory, as indicated in Equation (1), which leads to the unknown level of overturning stability safety of the continuous girder bridges during the cantilever construction from the risk point of view. $K = \frac{M_{W}}{M_{Q}}$ (1) where K = safety factor, M_W= stabilizing moment, M_Q= overturning moment.

In home and abroad bridge codes, the overturning stability factor of the bridge structure was introduced in both the specifications of the China’s highway steel bridges and the railway bridges. At the section 1.2.12 of 〈〈Code for Design of Steel Structure and Wood Structure of Highway Bridges and Culverts (JTJ 025-86)〉〉 [1], the bridge structure should ensure the lateral and longitudinal overturning stability during construction stage, and the stability factor should not be less than 1.3. However, the resource of the equation definition was not clear enough and the factor was determined mainly according to the experience of humans. At the section 5.1.1 of 〈〈Basic Specifications of Railway Bridges and Culverts Design (TB 10002.1-2005)〉〉 [2], the overturning stability factor of the reinforced concrete girder bridge structure should not be less than 1.3 when the corresponding stress exceeds 30% of the allowable value under vertical live load. The expression of safety factor comes from the Equation (1) based on the deterministic model neglecting the parameter uncertainties. In addition, a clear overturning stability factor during the cantilever construction stage is not provided in Chinese design codes of long-span pre-stressed concrete continuous girder bridge.

The reliability theory provides an effective method for safety analysis of cantilever construction considering the parameter uncertainties. Zhang [3] and Casas [4, 5] respectively used a pre-stressed concrete continuous girder bridge as an example to analyze the stability of long-span continuous girder bridge during the cantilever construction based on the reliability method where the structure overturning stability safety is evaluated through the reliability index. The design has to be modified, as well as the reliability index recalculated until the specification was satisfied if the structural overturning stability reliability index did not meet the requirements of the specification.

The current bridge design standards have been developed to the performance-based design concept. In other words, the structure was designed according to the prescribed target reliability index of the structure in order to ensure the structural safety. In this way, the overturning stability safety factor was calibrated to guarantee the prescribed reliability of overturning stability in the cantilever construction. However, the present methods of calculating the overturning stability safety factor based on the deterministic model are unable to deal with the calibration problem. The inverse reliability theory provides an effective way to calculate the safety factor under the premise of specifying the target reliability index. Jiang and Lv [6] studied the stability safety factor of concrete filled steel tube (CFST) arch using the inverse reliability theory. Cheng et al. [7 –9] studied the safety factor of main cable of suspension bridge using the inverse reliability theory. In order to evaluate the overturning stability of continuous girder bridge during cantilever construction, several researches have been carried out. Zhang and Huang [10], Lou and Dong [11], Wu [12] respectively studied the overturning stability safety factor of continuous girder bridge at the cantilever construction stage using the inverse reliability theory. However, the common drawback of above-mentioned studies are as follows: (1) the limit state function was not definitely established; (2) the influence factors of random variables were not comprehensive; (3) the overturning stability analysis model needs to be further improved; (4) parameter sensitivity analysis was not enough; (5) the engineering meaning is not clear.

Therefore, aiming at the limitations of previous studies, a calculation method of overturning stability safety factor using inverse reliability theory is proposed based on the existing research. Firstly, mechanical analysis of the cantilever girder bridge during the construction stage was carried out. After that, the overturning stability analysis model and the limit state function were established based on the mechanical model of continuous girder bridge. Then, the overturning stability safety factor was calculated and the parameter sensitivity was analyzed using the proposed method. Finally, the proposed method in this paper is applied to calculate the overturning stability safety factor and to discuss the reasonable value for an example continuous girder bridge during cantilever construction. For the construction organization, the safety factor studied in this paper can be determined according to the construction quality to achieve the prescribed target. For the supervision organization, the safety factor during construction stage can be verified according to the target reliability index.

2 Proposed method

2.1 Inverse reliability theory

Der Kiureghian, Zhang and Li [13] defined the inverse reliability problem by the following set of equations: $∥ u ∥ - β_{T} = 0$ (2) $u + \frac{∥ u ∥}{∥ \nabla_{u} G (u, θ) ∥} \nabla_{u} G (u, θ) = 0$ (3) $G (u, θ) = 0$ (4) where u= vector of standard normal variables; β_T= target reliability index of structures; G (u, θ)= limit state function; ∇_u= gradient operator with respect to u; θ= the design variable.

The basic idea of calculating algorithm for solving parameter is as follows: given β_T, find $\bar{θ}$ (or mean value of θ), subject to $min (u^{T} u) = β_{T}^{2}$ and G = G (u, θ) =0.

As shown in the development of the forward reliability procedure FORM, the vector u at the design point must satisfy $u = [\frac{(\nabla_{u} G)^{T} u}{(\nabla_{u} G)^{T} \nabla_{u} G}] \nabla_{u} G$ (5)

The reliability index β_T is then given by $β_{T} = \frac{- (\nabla_{u} G)^{T} u}{[(\nabla_{u} G)^{T} \nabla_{u} G]^{1 / 2}}$ (6)

Combining Equations (5 and 6), one obtains: $u = \frac{- β_{T} (\nabla_{u} G)^{T}}{[(\nabla_{u} G)^{T} \nabla_{u} G]^{1 / 2}}$ (7)

By using a truncated Taylor expansion of G on $\bar{θ}$ , at ${\bar{θ}}_{0}$ and conditional on u = u₀,

$\begin{matrix} G (u_{0}, \bar{θ}) & = & G (u_{0}, {\bar{θ}}_{0}) \\ + {\frac{\partial G (u_{0}, \bar{θ})}{\partial \bar{θ}} |}_{{\bar{θ}}_{0}} (\bar{θ} - {\bar{θ}}_{0}) = 0 \end{matrix}$ (8)

From which, $\bar{θ} = {\bar{θ}}_{0} - \frac{G (u_{0}, {\bar{θ}}_{0})}{{\frac{\partial G (u_{0}, \bar{θ})}{\partial \bar{θ}} |}_{{\bar{θ}}_{0}}}$ (9)

The convergence criterion used is, $\frac{({∥ u^{i + 1} - u^{i} ∥}^{2} + {∥ θ^{i + 1} - θ^{i} ∥}^{2})^{1 / 2}}{({∥ u^{i + 1} ∥}^{2} + {∥ θ^{i + 1} ∥}^{2})^{1 / 2}} ⩽ ɛ$ (10) where, ɛ is a small control parameter assigned by the user. From the writers’ previous experience, a value of 0.0001 usually provides a satisfactory θ estimation.

2.2 Analysis model of the safety factor

The continuous girder bridge in the state of the longest cantilever construction is shown in Fig. 2, where G₁= weight of travelling form and pouring segment at the left side; G₂= weight of travelling form and pouring segment at the right side; Q= concentrated load of live load during construction; q₁= uniformly distributed load per unit length of live-load during construction; q₂= wind load per unit length; e= bearing eccentricity at the limit state of instability; l= span length; l_g= length of the linking segment between cantilevers; l_p= length of the segment being poured.

Fig.2

Cantilever construction state.

The continuous girder bridge in the cantilever construction stage, due to the asymmetric load arrangement, there is the danger of the overall overturning instability. The overturning stability safety factor of pre-stressed concrete continuous girder bridge was calculated in the situation of the longest cantilever construction (as shown in Fig. 2). In the reliability analysis of structures, the limit state of structures was expressed as performance function. The performance function of overturning stability is as follows, $Z = g (X_{1}, X_{2}, \dots, X_{n})$ (11) where, X_i (i = 1, 2, …, n)= basic random variables.

The performance function of overturning stability can be expressed as $Z = M_{W} - {KM}_{Q}$ (12)

From Fig. 1, the stabilizing moment M_W and overturning moment M_Q can be respectively expressed as follows: $M_{W} = G_{2} (\frac{l - l_{p} - l_{g}}{2} + e) + \int_{0}^{(l - l_{g} - 2 l_{p}) / 2} q (x) (x + e) dx + \int_{0}^{e} q (x) (e - x) dx + M$ (13) $\begin{matrix} M_{Q} & = & (Q + G_{1}) (\frac{l - l_{g} - l_{p}}{2} - e) + \frac{q_{1} b}{2} \times {(\frac{l - l_{g}}{2} - e)}^{2} + \frac{q_{2} b}{2} \times {(\frac{l - l_{g}}{2} + e)}^{2} \\ + \int_{e}^{(l - l_{g} - 2 l_{p}) / 2} (1 + v) q (x) (x - e) dx \end{matrix}$ (14)

2.3 Procedure

The procedures of solving overturning stability safety factor using the above-mentioned inverse reliability method can be summarized as follows:

Step 1. Assume the initial values of the random variables and unknown deterministic design parameter. The initial values of the random variables can be their mean values.

Step 2. Initialize the iterative counter i = 1.

Step 3. Calculate u from Equation (7).

Step 4. Substitute u in to Equation (9); the values of $\bar{θ}$ (safety factor K) are determined.

Step 5. Check the convergence criterion using Equation (10); if unsatisfied, set i = i + 1 and go to Step 3; otherwise, STOP (end of calculation).

3 Application

The example bridge studied here is a long-span pre-stressed concrete continuous girder bridge with the span arrangement (72 + 120 + 72) m. The corresponding values related to the example bridge at longest cantilever stage are as follows: G₁ = G₂ = 3298 kN, Q = 80 kN, M = 193770 (kN • m), q₁ = 0.3 kPa, q₂ = 0.2 kPa, v = 0.025, e = 1.8 m, l = 120 m, l_g = 3 m, l_p = 4.5 m, b = 15 m (half bridge).

The statistical values of random variable G₁, G₂, Q, M, q₁, q₂, v, e in the example bridge were presented in Table 1.

Table 1
Statistics of basic random parameters

Variables Probability distribution type Mean value Coefficient of variation References

G₁/kN Normal distribution 3298 0.10 Assumed

G₂/kN Normal distribution 3298 0.10 Assumed

Q/kN Normal distribution 80 0.10 [4]

M/kN •m Log-Normal distribution 193770 0.15 [3]

q₁/kPa Normal distribution 0.30 0.15 [3]

q₂/kPa Extreme type-I distribution 0.20 0.15 [4]

v/% Normal distribution 2.5 0.15 [3]

e/m Normal distribution 1.8 0.15 [3]

Variables	Probability distribution type	Mean value	Coefficient of variation	References
G₁/kN	Normal distribution	3298	0.10	Assumed
G₂/kN	Normal distribution	3298	0.10	Assumed
Q/kN	Normal distribution	80	0.10	[4]
M/kN •m	Log-Normal distribution	193770	0.15	[3]
q₁/kPa	Normal distribution	0.30	0.15	[3]
q₂/kPa	Extreme type-I distribution	0.20	0.15	[4]
v/%	Normal distribution	2.5	0.15	[3]
e/m	Normal distribution	1.8	0.15	[3]

Substitute the corresponding value of parameters into Equations (13 and 14), the performance function of structural overturning stability can be obtained from Equation (12).

$\begin{matrix} Z & = & M + G_{2} (56.25 + e) + 27508 e + 536105 \\ - K [(G_{1} + Q) (56.25 - e) + 7.5 q_{1} (58.5 - e)^{2} \\ + 7.5 q_{2} (58.5 + e)^{2} + (1 + v) \\ (536105 - 22448 e)] \end{matrix}$ (15)

3.1 Result of safety factor

To estimate the overturning stability safety factor using the proposed method, the target reliability level needs to be specified for the limit state considered in this study. A target reliability index of 3.5 has been used in the calibration of the OHBD code and the AASHTO code for bridges. The most European codes suggested that the target reliability index for bridge structures should lie in an approximate range of 3.2–5.2. Based on these data, a target reliability index of 3.5 and initial value of safety factor K of 1.5 are used in this study unless otherwise stated.

A calculated overturning stability safety factor is 1.1697, which is less than the allowable value of 1.3, with the target reliability index of 3.5. According to the OHBD code and the AASHTO code, the overturning stability safety factor of 1.1697 was suggested according to the reliability index of 3.5.

3.2 Influence of target reliability index

A parameter study was conducted to investigate how the target reliability index affect the overturning stability safety factor. Six different values of the target reliability index are used: 2.5, 3.0, 3.5, 4.0, 4.5, 5.0. The results are given in Table 2. In addition, the results are confirmed by forward reliability analysis, which was given in Table 3.

Table 2
Influence of reliability index on overturning stability safety factor

Target reliability index 2.5 3.0 3.5 4.0 4.5 5.0

Failure probability 0.62×10^–2 1.3×10^–3 2.33×10^–4 3.17×10^–5 3.40×10^–6 2.87×10^–7

Safety factor 1.2239 1.1964 1.1697 1.1437 1.1184 1.0938

Table 3

Verification of overturning stability safety factors

Target reliability index	2.5	3.0	3.5	4.0	4.5	5.0
Safety factor	1.2239	1.1964	1.1697	1.1437	1.1184	1.0938
Reliability index using JC method	2.4999	2.9999	3.5001	4.0000	4.5001	4.9999

It can be seen from Table 2 that the overturning stability safety factor of continuous girder bridge decreases obviously when the target reliability index increases. The estimated safety factor has no difference according to the adjacent target reliability index while the corresponding failure probabilities differ greatly.

The estimated safety factor obtained from the Equation (1) is 1.3376 which is larger than the results in Table 2. Thus, neglecting the parameter uncertainties results in a significant overestimation of the overturning stability safety factor. In order to obtain an accurate overturning stability safety factor, it is necessary to consider the parameter uncertainties in estimating the safety factor.

The current bridge design code has been developed to the performance-based design concept [14 –19], whose purpose is to design bridge structures at different levels of importance under various risk levels, so that it is able to meet corresponding performance requirements [20 –22]. The risk level can be determined according to the probability of failure, as well as the reliability index. Therefore, target reliability index can be utilized to characterize performance requirements, and then a reasonable safety factor can be selected according to the performance requirements [23 –27]. For example, for the organization with good construction quality, a smaller safety factor should be selected to achieve the prescribed reliability, while for the organization with lower construction quality, a larger safety factor should be selected.

3.3 Influence of parameter uncertainties

According to the estimated safety factor based on the deterministic model, a larger overturning stability safety factor can be obtained due to neglecting the parameter uncertainties. In order to analyze the influence of parameter uncertainties on overturning stability safety factor, the estimated overturning safety factor was shown in Fig. 3 from the range 0.05–0.3 of variation coefficients of random variables and Fig. 4 from 10% – +10% of mean value of random variables.

Fig.3

Influence of variation coefficient on overturning stability safety factor.

Fig.4

Influence of mean value on overturning stability safety factor.

It can be seen from Figs. 3 and 4 that the safety factor of overturning stability is most sensitive to the sum of the dead weight of the travelling form and the pouring segment, the overturning moment and bearing eccentricity of the temporary support have a great influence on the safety factor, and the deviation of the weight of cantilever segment, the live load during construction, the concentrated load at cantilever end and the wind load during construction stage have little effect on the safety factor.

The sum of the weight of travelling form and the pouring segment was considered as a random parameter in this paper, which has a great influence on the overturning stability safety factor. In actual construction of long-span continuous girder bridge, it is considered as an accidental condition to describe the situation that the travelling form and pouring segment fall together. As for the overturning stability model of continuous girder bridge presented in this paper, at the worst condition G₂ = 0 when travelling form and pouring segment fall together, the estimated overturning stability safety factor K is 0.9408 under the prescribed target reliability index of 3.5. On the contrary, the overturning stability safety factor K is 1.1697. Thus, it can be seen that the falling of the travelling form has a great influence on the overturning stability of the continuous girder bridge. It is extremely important to ensure the normal working condition of the travelling form during the construction stage.

Due to the limited overturning resistant moment provided by the temporary reinforcement measures in cantilever construction, the reasonableness of temporary reinforcement measures should be considered specially (including the overturning resistant moment of the temporary support and bearing eccentricity) in cantilever construction stage, so as to ensure structural overturning stability.

As shown in Figs. 3 and 4, the uncertainties of self-weight deviation at both cantilever ends have little effect on the overturning stability safety factor. However, special attention should be considered to ensure the quality of pouring concrete during cantilever construction stage so as to ensure the symmetry.

The stacking cases were not same exactly under different situations. In this paper, the stacking load during construction is used as a concentrated force Q. Considering the worst condition of overturning stability, the concentrated load Q is arranged at the end of the cantilever structure. Figures 2 and 3 show that the uncertainties of stacking load have little effect on the overturning stability safety factor. If the construction organization arrange the stacking load in a different position from that presented in this paper during actual construction, the concentrated force Q can be applied according to the actual construction situation, and the proposed method in this paper can also be used to calculate the safety factor.

Although the wind load and live load of construction have almost no effect on the overturning stability, the travelling form should be fixed well, and the live load and the stack load of construction should be well managed to ensure that the entire bridge structure stable and safe when there comes a pause due to the strong wind during the construction stage.

3.4 Effect of distribution type of random variables

In order to study the effect of distribution type of random variables on the safety factor, the distribution type of random variables are divided into three types: eight normal distributions, eight log-normal distributions, and eight different type distributions (the distribution of M is log-normal, the distribution of q₂ is extreme type-I, and the distribution of G₁, G₂, Q, q₁, v, and e are normal.). The corresponding results related to three cases of distribution type of random variables are given in Table 4.

Table 4
Influence of distribution type on overturning stability safety factor

Case Safety factor K

Eight normal distributions Eight log-normal distributions Eight different distributions

β = 2.5 1.2194 1.2377 1.2239

β = 3.5 1.1595 1.1802 1.1697

β = 4.5 1.1013 1.1135 1.0938

Case	Safety factor K
β = 2.5	1.2194	1.2377	1.2239
β = 3.5	1.1595	1.1802	1.1697
β = 4.5	1.1013	1.1135	1.0938

It can be seen form Table 4 that the distribution type of random variables has a major effect on the result of overturning stability safety factor. Therefore, the safety factor related to the practical engineering application can be obtained only the distribution type of random variables was determined reasonably.

3.5 Effect of initial value of K

Since the initial value of K⁰ used in the proposed method is chosen arbitrarily, it is necessary to investigate the effect of initial value of K on the estimated safety factor. For this purpose, five different initial value of K are used: 1.5, 2.5, 3.5, 4.5, 5.5. The variations of the overturning stability safety factor with the iteration number are shown in Fig. 5 where horizontal ordinate is iteration number.

Fig.5

Influence of different initial values of K on calculation results.

It can be seen from Fig. 5 that the initial value of K only affect the convergence of overturning stability safety factor while have no influence on the estimated result. Therefore, the proposed method has a high quality of stability and can be used to estimate the overturning stability safety factor of long-span concrete continuous girder bridge when the target reliability level is specified for the limit state considered in the design.

4 Conclusions

In this paper, an estimation method for overturning stability safety factor of long-span pre-stressed concrete continuous girder bridge based on inverse reliability theory is proposed. Through the previous analysis, the following conclusions can be drawn:

The parameter uncertainties have a great influence on the overturning stability safety factor. And the random load will reduce the safety of overturning stability when the constructed structure was designed according to the deterministic safety factor.

The overturning stability safety factor of continuous girder bridge decreases obviously when the target reliability index increases. For the organization with good construction quality, a smaller safety factor should be selected to achieve the prescribed reliability, while for the organization with lower construction quality, a larger safety factor should be selected, so as to meet the performance requirements.

During cantilever construction stage, setting of temporary reinforcement measures and the weight of travelling form and the pouring segment should be considered specially. When considering the travelling form and pouring segment falling, live load of construction arranging, position of stack load changing in practical engineering, the proposed in this paper can also be used to estimate the overturning stability safety factor to meet the prescribed performance requirements.

The distribution type of random variables has a major effect on the analysis result of overturning stability safety factor. Therefore, the safety factor related to the practical engineering application can be obtained only the distribution type of random variables was determined reasonably through sampling statistics of actual variables so as to meet the engineering requirement well.

The initial value of K only affect the convergence of overturning stability safety factor while have no influence on the estimated result. Therefore, the proposed method has good applicability to determining the overturning stability safety factor.

Footnotes

Acknowledgments

This work presented herein has been supported by the National Natural Science Foundation of China under grant number 51278064 and the Science and Technology Plan Project of Henan Provincial Department of Transportation (2013K30). These supports are gratefully acknowledged.

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Target reliability index	2.5	3.0	3.5	4.0	4.5	5.0
Failure probability	0.62×10^–2	1.3×10^–3	2.33×10^–4	3.17×10^–5	3.40×10^–6	2.87×10^–7
Safety factor	1.2239	1.1964	1.1697	1.1437	1.1184	1.0938

Case	Safety factor K
	Eight normal distributions	Eight log-normal distributions	Eight different distributions
β = 2.5	1.2194	1.2377	1.2239
β = 3.5	1.1595	1.1802	1.1697
β = 4.5	1.1013	1.1135	1.0938

Stability safety assessment of long-span continuous girder bridges in cantilever construction

Abstract

Keywords

1 Introduction

2.1 Inverse reliability theory

3 Application

3.2 Influence of target reliability index

Table 2 Influence of reliability index on overturning stability safety factor Target reliability index 2.5 3.0 3.5 4.0 4.5 5.0 Failure probability 0.62×10–2 1.3×10–3 2.33×10–4 3.17×10–5 3.40×10–6 2.87×10–7 Safety factor 1.2239 1.1964 1.1697 1.1437 1.1184 1.0938

Table 4 Influence of distribution type on overturning stability safety factor Case Safety factor K Eight normal distributions Eight log-normal distributions Eight different distributions β = 2.5 1.2194 1.2377 1.2239 β = 3.5 1.1595 1.1802 1.1697 β = 4.5 1.1013 1.1135 1.0938

Footnotes

Acknowledgments

References

Table 2
Influence of reliability index on overturning stability safety factor

Target reliability index 2.5 3.0 3.5 4.0 4.5 5.0

Failure probability 0.62×10^–2 1.3×10^–3 2.33×10^–4 3.17×10^–5 3.40×10^–6 2.87×10^–7

Safety factor 1.2239 1.1964 1.1697 1.1437 1.1184 1.0938

Table 4
Influence of distribution type on overturning stability safety factor

Case Safety factor K

Eight normal distributions Eight log-normal distributions Eight different distributions

β = 2.5 1.2194 1.2377 1.2239

β = 3.5 1.1595 1.1802 1.1697

β = 4.5 1.1013 1.1135 1.0938