# Hoek Bridge Anchor

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ANALYSIS OF THE STABILITY OF AN ANCHOR BLOCK FOR A SUSPENSION BRIDGE

By Dr Evert Hoek Vancouver February 2003

1. Introduction

The analysis discussed in the following notes deals with one of the anchors for a 500 m long suspension bridge proposed for a major highway project. A detailed analysis was carried out as part of the design of the suspension bridge and my role was to review this analysis on behalf of the owner. In fact, the bridge was not built since an alternative design was chosen for the final construction. However, this does not alter the validity of the design and the valuable lessons that can be learned from this analysis. The layout of the bridge is illustrated in Figure 1 that shows that the suspension cables are anchored directly in the rock. Different options for this anchor, located on a steep rocky outcrop, were considered. Three of these options, one for a gravity anchor block two for tunnelled socket anchors, are discussed here.

Figure 1: Overall view of suspension bridge layout. 2. Geological factors influencing the stability of the anchorage

A plan of the gravity anchor block and of the tunnelled socket anchorage is shown in Figure 2. A section through the proposed gravity anchor block is shown in Figure 3. The anchorages are located on a promontory of thick-bedded sandstone, which is underlain by thinly bedded sandstone/siltstone outcropping to the west of the anchorage. Due to a fold structure that forms the nose of the promontory, the structural geology of the East-facing slope is significantly different from that in the South-facing slope. This difference results in a stable East-facing slope and a less stable South-facing slope. The outline of a landslip on the South-facing slope is shown in yellow in Figure 2 and this landslip has an influence of the layout of the anchorage alternatives. 3. Methods used to assess the stability of anchorage options

The Designer assessed the stability of the gravity anchorage by means of a two-dimensional limit equilibrium analysis (using the program TALREN) and two-dimensional numerical analyses (using the programs SAFE, VISAGE and UDEC). A

Suspension bridge anchor block analysis Page 3

three dimensional study, using the program VISAGE, was also carried out. No analyses were performed for the tunnelled socket options. In reviewing the design report I found that the easiest way for me fully to understand the information presented was to carry out my own analyses, using different software from that used by the Designer. In preparing the models for these analyses I have also reinterpreted the basic data and this has resulted in some slight differences in the parameters included in the analyses. I consider that this is an advantage since it provides an independent view of the stability of the anchor block and an assessment of the behaviour of one of the tunnelled socket options.

N

Figure 2: Layout of alternative anchorages for the bridge cable.

Suspension bridge anchor block analysis Page 4

Backfill

Figure 3: Section through the gravity anchor block. 4. Limit equilibrium analysis

The program SLIDE1 was used to check the limit equilibrium analysis carried out by the Designer. The details of the model are illustrated in Figure 4 and the properties of the various materials used in the model, assumed to be fully drained, are given in Table 1.

Backfill Concrete anchor block front open, back filled with rockfill

Excavation damaged rock

5000 kN cable force on anchor

2588 kN reaction on bearing

Thin bedded sandstone

Thick bedded sandstone

Figure 4: Details of limit equilibrium model in SLIDE. The cable force of 150 MN is applied over a 30 m wide gravity block base, giving a force of 5000 kN/m on the cable anchor and a vertical reaction of 2588 kN/m on the saddle bearing. T able 1: Summary of rock mass properties used in the SLIDE analysis.

1 Details are available from www.rocscience.com

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Material Properties Comments Thick bedded sandstone Unit weight = 26 kN/m3

Anisotropic strength representing the combined properties of the rock mass (based on Hoek Brown criterion) and two sets of joints (B) or one set of joints and one set of bedding planes (C).

Thin bedded sandstone/siltstone

Unit weight = 26 kN/m3

Anisotropic strength representing the combined properties of the rock mass (based on Hoek Brown criterion) and two sets of joints (B) or one set of joints and one set of bedding planes (C).

Excavation damaged rock Unit weight = 26 kN/m3 c = 0, = 36

Backfill Unit weight = 16.9 kN/m3 c = 0, = 36

Concrete anchor block (concrete construction with rock backfill)

Equivalent Unit weight = 19 kN/m3 (back), = 12 kN/m3 (front) Assumed to be elastic (infinite strength)

Equivalent unit weight based on cross-sectional area and total mass of 380 MN given in the Designers report.

A B/C

A

B A

-75 -90

-45

25

45 90

A : c = 352 kPa, = 54 B : c = 0 kPa, = 30.2 C : c = 0 kPa, = 18

A

A

B A

-75 -90

-45 B/C

25

45 90

A : c = 182 kPa, = 41 B : c = 0 kPa, = 29.3 C : c = 0 kPa, = 16

The program SLIDE has a number of options for automatic non-circular failure surface searches and one of these options was used to produce the results shown in Figure 5 and 6. The General Limit Equilibrium method is a very powerful non-circular analysis method that was used in this study. A wide range of material models, including the Hoek-Brown failure criterion and user-defined anisotropic criteria, can be included in the model. Hence, I believe that the failure surfaces shown in Figure 5 and 6 are realistic for this slope. The structural geology of the rock mass forming this slope varies significantly across the width of the slope and, in order to cover this variation, I have carried out two sets of analyses. The first of these assumes that the structure in the plane of the analysis is controlled by two sets of orthogonal joints (B in Table 1, with the results shown in Figure 5) while the second assumes one set of joints and a set of bedding planes dipping approximately parallel to the slope (C in Table 1, with the results shown in Figure 6). The failure surfaces shown in Figure 5 indicate that potential sliding is controlled by a combination of sliding along joints and failure through the rock mass. The global factor of safety of 3.5 involves the entire rock mass from the crest to the toe of the slope. The local factor of safety of 3.9 is typical of the numerous failure surfaces generated in the immediate vicinity of the anchor block. These factors of safety were checked for a range of cable load factors and were found to be almost independent of this load up to a cable load of six times the design load.

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Figure 5: Most critical failure surfaces for the East-facing slope on which the proposed gravity anchor block is located. Factors of safety are shown for three of the surfaces. In the case of the model with the bedding planes dipping approximately parallel to the slope, Figure 6 shows a number of failure surfaces with a minimum factor of safety of 2.81. The bedding planes in the thick bedded sandstone are assumed to have a shear strength defined by c = 0 and = 18 while, for the thin bedded sandstone, I have assumed c = 0 and = 16, based on some laboratory shear test results. Once again, the stability of the slope is independent of the load applied to the cable anchor block. I believe that these two models represent the extremes of the behaviour of the slope and that the actual situation probably lies somewhere between these extremes. These studies confirm the conclusion reached by the Designer that the factors of safety of the slope determined by two-dimensional limit equilibrium analyses, exceed the factor of safety of 1.25 required by Euro Code 7 by a substantial margin.

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Figure 6: Most critical failure surfaces for the model with bedding planes dipping approximately parallel to the slope. Sliding along the base of the block controls stability of the gravity anchor. The factor of safety for this failure mode was not calculated since it depends upon the material properties assumed and the geometry of the shear key on the base of the block. It is a simple matter to change this geometry and I am in complete agreement with the Designer that an adequate factor of safety against sliding on the base can be achieved. I have not carried out any detailed seismic loading studies but a few checks gave results consistent with those reported by the Designer. 5. Numerical analyses of gravity anchorage stability

The limit equilibrium analyses described above suffer from the disadvantages that some form of predefined failure path must be assumed and that displacements in the slope are not taken into account. These disadvantages can be overcome by using a numerical analysis method in which the progressive failure and deformation of the entire system can be simulated. The Desugners used the continuum finite element models SAFE and VISAGE and the discrete element finite difference UDEC to carry out two-dimensional analyses on the gravity anchorage. The program VISAGE was used for three-dimensional studies. In

Suspension bridge anchor block analysis Page 8

order to understand the behaviour of these models and to check the results of the analyses, I have carri

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