Influence of Model Accuracy on Load and Resistance Factor Calibration of Steel Strip Walls

Influence of Model Accuracy on Load and

Resistance Factor Calibration of Steel Strip Walls

Yoshihisa MIYATA a , Richard J. BATHURST b and Yoshinori OTANI c a

Department of Civil and Environmental Engineering, National Defense Academy, Japan b

GeoEngineering Centre at Queen’s-RMC, Canada c

Hirose Co., Ltd. Japan

Abstract. This study examines the load and resistance factor design calibration for the ultimate pullout limit state for steel strip reinforced soil walls (SSWs). The tensile loads and pullout capacities of reinforcement strips that were measured during full-scale tests are compared with predicted (nominal) values using analytical models recommended by the Public Works Research Center (PWRC) in Japan. Modified load and resistance models are also proposed in this study. The new load model preserves the general form of the current PWRC load equation. A new pullout capacity model is proposed that has the same number of coefficient terms as the current PWRC equation. The coefficients in both new equations are empirical and are selected by back-fitting to measured load and pullout capacity values to achieve a bias mean equal to one and a low coefficient of variation of bias values. Here, bias is the ratio of measured to predicted value. This study demonstrates the influence of model accuracy on the computed resistance factor in LRFD calibration of SSWs for the pullout limit state.

Keywords. reinforced soil walls, steel strip walls, load and resistance factor design, limit states design, load and resistance factors, reliability analysis

1. Introduction

Steel strip reinforced soil walls (SSWs) were first introduced in Japan in the early 1970s. Currently, there are more than 34,000 of these structures. The current design approach for external and internal stability of reinforced soil wall systems in Japan is based on the classical factor of safety approach (PWRC 2003). Recently, the Public Works Research Center (PWRC) has expressed an interest to move towards a more rigorous reliability-based design approach. This study examines the load and resistance factor design (LRFD) calibration of SSWs for the pullout limit state.

2. Load and Resistance Factor Calibration Model error and variability in model input parameters should be considered in LRFD calibration as demonstrated by Allen et al. (2005) and Bathurst (2014). Load and resistance factors are calculated based on statistical analysis of the bias values that are computed as the ratio of measured value to predicted (nominal) value. A

useful closed-form solution to calculate the reliability index (E) is as follows:

2 2 Q R Q Q R 2 2 Q R ln[ /  / (1+COV ) / (1+COV ) ]  = ln[(1+COV )(1+COV )] M˜ ₍₁₎

The above equation holds for the case where the load (Q) and resistance (R) bias values are uncorrelated and log-normally distributed, and the corresponding ultimate limit state function is linear and expressed as g = R – Q. Here, JQ and M are the load and resistance factors, respectively; PR and COVR are the mean and coefficient of variation (COV) of resistance bias values, respectively; and PQ and COVQ are the mean and COV of the load bias values, respectively. For a given load factor and set of bias mean and COV values, a resistance factor value can be found to satisfy a target reliability index value using Eq. (1). As noted above, load and resistance bias values are assumed to be uncorrelated in this formulation which was shown to be the case for the data in the current study.

© 2015 The authors and IOS Press.

This article is published online with Open Access by IOS Press and distributed under the terms of the Creative Commons Attribution Non-Commercial License.

3. Database of Physical Test Results

The authors created a database of measured reinforcement loads and pullout capacity values from full-scale wall tests and in-situ pullout tests. A summary of the geometry and material properties of the wall tests is provided in Table 1.

All the walls performed well. A total of 159 load measurements were used for LRFD calibration. Details of these walls can be found in

the papers by Bathurst et al. (2009), Allen et al. (2001) and Miyata and Bathurst (2012a). A summary of the in-situ pullout tests is provided in Table 2. Miyata and Bathurst (2012b) reported pullout capacity data for both ribbed and smooth strips. In this study, only data for ribbed strips were considered since smooth strips are no longer used.

Table 1. Summary of full-scale steel strip reinforced soil walls and number of load measurements (data from Bathurst et al., 2009; Allen et al. 2001; Miyata and Bathurst 2012a).

Case study

Wall height, H (m)

Soil unit weight, J (kN/m3₎ Friction angle, I(deg.) Cohesion, c (kPa) Secant friction angle, Isec #2 (deg.)

Number of load measurements Project year, country SS1 6.0 18.1 44 - #1 44 6 1972, USA SS2 6.1 19.8 38 - 38 5 1974, USA SS3 3.7 18.5 36 - 36 12 1976, USA SS4 7.3 19.6 37 - 37 6 1980, USA SS5 8.2 22.6 56 - 56 4 1981, USA SS6 6.0 22.3 48 - 48 9 1981, USA SS7 12.0 17.7 36 19 38 7 1981, Japan SS10 12.6 21.5 50 - 50 5 1985, USA SS11 6.1 20.4 40 2 41 4 1988, USA SS12 12.0 19.0 38 - 38 6 1990, Norway SS13 10.5 16.8 37 - 37 4 1993, USA SS14 10.5 16.8 37 - 37 4 1993, USA SS15 16.5 21.8 38 0 38 5 2001, USA SS17 6.3 16.7 46 41 55 3 1973, USA SS18 10.0 19.6 35 0 35 5 1977, UK SSJ-1 12.0 17.7 36 19 38 10 1975, Japan SSJ-2 7.5 18.0 16 35 28 6 1987, Japan SSJ-3 12.75 19.0 35 0 35 6 1996, Japan SSJ-4 6.0 18.4 40 23 46 9 1994, Japan SSJ-5 6.0 15.2 29 33 42 9 1994, Japan SSJ-6 6.0 13.5 21 3 23 9 1994, Japan SSJ-7 7.5 17.7 40 5 41 1 1994, Japan SSJ-9 4.0 15.7 38 2 39 9 1993, Japan SSJ-11 4.5 16.0 42 0 42 15 1980, Japan Total = 159

#1: –not reported and taken as zero in calculations. #2: Isec=tan-1(c/JH + tanI)

Table 2. Summary of in-situ steel strip pullout tests (data from Miyata and Bathurst 2012b). Case

study

Wall height, H (m)

Number of tests with each soil type (Symbol is based on JGS (2010) soil classification system)

GS G GF SG S SF Other 1 3.0–4.5 2 4 2 1.8–6.3 6 3 2.5–6.5 10 4 2.5–9.2 1 11 8 4 5 0.8–10.2 5 8 21 24 6 0.8–10.2 11 8 13 9 7 7 0.7–8.1 13 24 8 2 Total = 199

Table 3. Coefficient values for current load model and load bias statistics.

Friction angle (deg.) _a Model coefficients _{b No. of data points} Bias statistics _P

Q COVQ

I  35 1.00 1.00 45 0.53 0.48

35  I  45 1.00 1.00 93 1.12 0.33

I! 1.00 1.00 21 2.57 0.44

Table 4. Coefficient values for modified load model and load bias statistics.

Friction angle (deg.) _a Model coefficients _{b No. of data points} Bias statistics _P

Q COVQ I  35 0.62 0.62 45 1.00 0.50 35 < I  45 1.13 1.10 93 1.00 0.33 I! 2.62 2.52 21 1.00 0.42 n 1 B H H4 tan-1 _0.3 H2 Ha H1 0.3 Ha (H+H4)/2 H3 bK0 aKa z zo = 6.0 m

Figure 1. Design chart for earth pressure coefficient K and wall geometry for load model – Coherent Gravity Method (PWRC 2003).

4. Load Models

4.1. Current Load Model

In the current Japanese method for design of steel strip walls (PWRC 2003), the maximum nominal tensile load (Tmax) in a reinforcement layer is calculated as follows:

max v v v

T = S KV S K( (2) where Sv is the reinforcement spacing, K is the coefficient of earth pressure and Vv = Jz + q is the vertical pressure at the elevation of the reinforcement strip. Here, J is the soil unit weight, z is the depth of the layer below the crest of the wall of height H, and q is the equivalent uniform surcharge pressure. The coefficient K is obtained from the bilinear distribution shown in Figure 1. The bilinear distribution for K is expressed as follows: i) z  zo = 6 m o o a o K = bK (1 z z ) + aK ( z z ) (3) ii) z > zo = 6 m a K = bK (4)

where Ka is the classical Rankine active earth pressure coefficient and Ko is the coefficient of earth pressure at rest using the Jaky equation. In the current model for K, the constant coefficients are a = b = 1.

The model coefficients and v Q and COVQ are listed in Table 3. Here, the model accuracy was evaluated for three soil groups based on the friction angle, I. The reason for dividing the soil groups into three is as follows: a) the soils used to originally calibrate the Coherent Gravity Method were reported to be purely frictional soils with 35° < I °; b) soils with I > 45° are believed to generate compaction-induced stresses and thus warrant special interpretation; and c) many of the Japanese case studies used cohesive-frictional soils with a friction angle less than 35°. These soils may be expected to be non-dilatant, resulting in different steel strip tensile load responses.

The computed bias statistics for the load model depend on soil group and value of I. The results show that for cases with 35° < I  45°, the prediction of the maximum tensile load is reasonably accurate on average. However, for the other two soil groups, the bias statistics are much poorer. For I < 35°, the measured loads are about 50% of the predicted values.

Table 5. Model coefficients for current pullout capacity model (PWRC 2003)

Soil type Conditions f*0 \1

A1 soil Dmax 75Pm  _{1.5 36°}

A2 soil 75 mm < Dmax < 300 mm and fines content D75Pm < 25%

B rock Dmax 75Pm < 35% 1.0 25°

On the other hand, for I!45°the measured loads are approximately 2.5 times the predicted values. If these values are used in load and resistance factor calibration, they result in unrealistic load and resistance factors. A more accurate load model is required to improve calibration outcomes.

4.2. Modified Load Model

Miyata and Bathurst (2012a) proposed a modified load model to predict the maximum reinforcement load. In this study, the value of zo was kept constant at 6 m. Coefficients a and b in Eqs. (3) and (4) were determined using the optimization (SOLVER) utility bundled with MS Excel with the objective function PQ = 1 and constrained by  ! "$ A summary of the solved model coefficients and bias statistics using the modified model is provided in Table 4. A model that gives a bias average close to one and a small COV of bias values is desirable. Based on these criteria the modified load model is better than the current model (PWRC 2003), especially for cases with I < 35° and I!45°.

5. Pullout Capacity Models 5.1. Current Pullout Model

In current Japanese design practice, pullout of the reinforcement is considered an ultimate limit state. The design pullout capacity must be larger than the predicted reinforcement load in each reinforcement layer by an acceptable margin. According to PWRC (2003), the unfactored ultimate pullout capacity (Pmax) is calculated using the following equation:

*

max i v e

P = 2 f˜ ˜& ˜ ˜" ' (5)

where fi* is a dimensionless empirical interface shear coefficient, Vv is the vertical pressure at the elevation of the reinforcement strip, b is the strip

width, and Le is the anchorage (pullout) length (Figure 2). The value of fi* is computed using default models that are a function of depth z. The distribution of fi* is bi-linear with depth, as shown in Figure 2, and is expressed as:

i) z  zo = 6 m * * i o o 1 o f = f (1 z z ) + tan ? ( z z )  (6) ii) z > zo = 6 m * i 1 f = tan ? (7)

where \1 is the residual soil friction angle. Coefficient f*i is selected on the basis of soil and steel strip type. Coefficients for three soil types and ribbed strips are shown in Table 5. The model coefficients and corresponding values of R and COVR are listed in Table 6. The model accuracy was evaluated for the two soil types listed in Table 6. The bias statistics for the current design model are very poor. In particular, for the type B soil, the measured pullout capacities are more than three times the predicted values on average. If these values are used in LRFD calibration, they will result in unrealistic resistance factors. Therefore, a more accurate pullout capacity model is required to improve calibration outcomes.

5.2. Modified Pullout Model

Miyata and Bathurst (2012) proposed the following equation as the default pullout capacity model: H2 Ha fi* f0* tan-1 0.3 H Le Htotal 0.3 Ha zo = 6.0 m

Figure 2. Design chart for pullout capacity coefficients and anchorage length (PWRC 2003).

Table 6. Coefficients for current pullout design model and statistics for ratio of measured to predicted pullout capacity (bias).

Soil type _f Model coefficients Bias statistics

0* \1 z0 No. of data points PR COVR

A1 and A2 1.5 36 6 156 1.42 0.50

B 1.0 25 6 43 3.27 0.40

Table 7. Coefficients for modified pullout design model and statistics for ratio of measured to predicted pullout capacity (bias).

Soil type _f Model coefficients Bias statistics

0* \1 c No. of data points  PR COVR

A1 and A2 5.0 36 0.55 156 1.00 0.42

B 4.0 25 0.22 43 1.00 0.43

(a) Soil types A1 and A2

(b) Soil type B

Figure 3. Results of calibration for pullout capacity of ribbed steel strip and soil types A1, A2 and B.

* * o 1 i 1

f

tan ?

f =

+ tan ?

exp(cz)



This equation has the following advantages: a) there are three coefficients, which is the same number as in the current design model (PWRC 2003); b) the minimum value of interface shear coefficient remains fi* = tan \1; and c) the

function is smoothly continuous with depth z. The model coefficients were determined using the same method as the load model.

A summary of the solved model coefficients and bias statistics using the modified model is provided in Table 7. The calibration results for soil types A1, A2 and B are shown in Figure 3(a) and (b). In each figure, corresponding results for the current design model are also shown. The plots show that for the computation of pullout capacity, the modified model is better than the current model (PWRC 2003), especially for the type B soil.

6. Influence of Model Accuracy on Load and Resistance Factors

For a target reliability index E and prescribed load factor JQ, the @   X X Z can be computed using the closed-form solution given by Eq. (1). The results of this calculation are listed in Table 8 using the bias statistics for the current load and pullout capacity models (Tables 3 and 6) and the corresponding revised models (Tables 4 and 7). A target reliability of 2.3 was selected.

Using the current models, the computed resistance factor is more than one in several of the analysis cases. Furthermore these factors depend on friction angle and soil type. Therefore, the current design models give undesirable calibration outcomes. On the other hand, using the modified models, the computed resistance factors depend only on the assumed load factor and target reliability index. The influence of friction angle on computed resistance factors is relatively small. Hence the modified models give better results for LRFD calibration.

Depth, z (m) 0 2 4 6 8 10 12 14 16 18 20 fi * (measured) 0 1 2 3 4 5 6 1.5

Current design model Modified model Depth, z(m) 0 2 4 6 8 10 12 14 16 18 20 f * (measured)i 0 1 2 3 4 5 6 Modified model

Table 8. Computed resistance factors using current and modified models (target reliability index = 2.3). Friction angle

(deg.) Soil type

Current model Modified model

JQ = 1.3 JQ = 2.0 JQ = 3.0 JQ = 1.3 JQ = 2.0 JQ = 3.0 I  35 A1, A2 0.76 1.18 1.76 0.32 0.49 0.74 B 2.10 3.22 4.84 0.32 0.49 0.73 35 < I  45 A1, A2 0.42 0.64 0.96 0.39 0.59 0.89 B 1.17 1.80 2.70 0.38 0.58 0.87 I! A1, A2 0.16 0.25 0.38 0.35 0.54 0.81 B 0.45 0.70 1.04 0.34 0.53 0.79 Range 0.16–2.10 0.25–3.22 0.38–4.84 0.32–0.39 0.49–0.59 0.73–0.89 7. Conclusions

This study examined the effect of the accuracy of current and new load and resistance design models on LRFD calibration of the ultimate pullout limit state for design of SSW structures. The following conclusions can be made:

1. Current design models in Japan (PWRC 2003) to calculate reinforcement loads and pullout capacity were shown to be poor based on bias statistics and model accuracy that varied with soil type.

2. The proposed modified models for prediction of reinforcement load and pullout capacity were demonstrated to be quantitatively more accurate.

3. Reasonable and consistent resistance factor values were computed using the new load and resistance models proposed by the authors.

Acknowledgments

The study described here was carried out with help from funding awarded to the first author by the Japan Ministry of Education, Culture, Sports, Science and Technology (B: 24360195). Additional funding was provided by the collaborative project “Sustainability of Reinforced Soil Walls” supported by the Public Works Research Institute, Nippon Expressway Research Institute Company Limited, National Defense Academy of Japan (NDA), and 11

private companies in Japan. The second author is grateful for their financial support during a sabbatical visit to the NDA in Japan.

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