Fifteenth Geomechanics Lecture

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Published 01 June 2016
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Fifteenth Geomechanics Lecture

Lecture Summary

There is a growing emphasis on Displacement Based Earthquake Design (DBD) analysis for buildings, retaining walls and bridge structures. DBD is specified as the preferred design method for highway structures in a draft revision of Section 5 of the Bridge Manual (NZ Transport Agency) expected to be adopted in 2016.
For bridges and major retaining wall structures, the damping and deformations within their foundations and backfilling have a major impact on their displacement response. In the past, the geotechnical input for the design of structures has focused on investigating and defining the soil strength parameters. To implement DBD methods there is now a need to investigate and assess soil stiffness as well as strength and to focus more on soil-structure interaction analysis.
The lecture highlighted the influence of soil stiffness and damping on the earthquake response of retaining walls and bridges and discussed the effects of the uncertainty in these parameters. DBD design procedures were illustrated by examples from the presenter’s research background on soil-structure interaction.
The lecture was presented in the following three parts:

  1. Rigid and flexible including outward sliding retaining walls
  2. Stiff retaining walls and bridge abutments
  3. Bridge DBD and pile foundations

This paper presents material discussed in Part 2 of the lecture. Material presented in Part 1 was published in the June 2015 issue of NZ Geomechanics News. Material in Part 3 will be published in the December 2016 issue of NZ Geomechanics News.

1. Stiff Retaining Walls

It is usual to simplify the complex problem of the interaction of earthquake generated elastic waves in the soil with wall structures by assuming that the earthquake ground motions are equivalent to dynamic inertia forces acting in the backfill mass. Dynamic pressures on the wall can then be estimated by analysing the wall and backfill modelled as an elastic continuum or failure wedge subjected to both gravity and horizontal body forces. The pressures that develop are very sensitive to the elastic flexibility of the structural components of the wall and the ability of the wall to move outward (rotation or translation) because of either permanent deformations in the foundation soils or inelastic behaviour of the structure.
The behaviour of wall structures during earthquakes can be broadly classified into three categories related to the maximum strain condition that develops in the soil near the wall. The soil may remain essentially elastic, respond in a significantly nonlinear manner or become fully plastic. The rigidity of the wall and its foundations will have a strong influence on the type of soil condition that develops.
Many low walls are of cantilever type construction. In this type of wall, lateral pressures from vertical gravity and earthquake forces will often produce sufficient displacement within the wall structure to induce a fully plastic stress state in the retained soil. In more rigid free-standing walls, such as gravity (e.g. reinforced earth and crib block walls) counterfort walls and building basement walls, a fully plastic stress state may develop as the result of permanent outward movement from sliding or rotational deformations in the foundation. In cases where significant nonlinear soil behaviour or a fully plastic stress state occurs in the soil during earthquake loading, the well-known Mononobe-Okabe, (M-O) method (Mononobe and Matsuo, 1929) can be used to compute earthquake pressures and forces acting on the wall. Details of the M-O theory and its application were presented in Part 1 (NZ Geomechanics News, June 2015).
Retaining structures that are not free-standing or have rigid foundations (piles or footings on rock or stiff soil) may not displace sufficiently, even under severe earthquake loading, for a fully plastic stress state to develop in the soil backfill. Particular examples of these types of walls include; bridge abutments that may be rigidly attached to the bridge superstructure or founded on piles, basement walls that are an integral part of a building on a firm foundation, and closed culvert or tank structures embedded in the ground. For many of these types of walls, the assumptions of the M-O method are not satisfied, and pressures and forces are likely to be significantly higher than given by application of this method.
In some cases, the wall may be sufficiently rigid for the soil to remain elastic under combined earthquake and gravity loads. More generally, there will be sufficient deformation for nonlinear soil effects to be important or for wall pressures to be significantly lower than for a fully rigid wall. These intermediate cases or stiff walls are more difficult to analyse than the limiting cases of fully plastic or rigid elastic behaviour.
In Part 1, theory of elasticity methods for estimating earthquake induced pressures on rigid walls were presented. This present section addresses the issue of walls that are flexible but not sufficiently flexible for the M-O assumptions to be valid (called stiff walls). The top deflection under gravity and earthquake loads of stiff walls is typically less than 0.3% of the wall height. Elastic theory solutions are presented for walls that deform by flexure in a cantilever stem and for rigid walls that rotate about their base. Theory of elasticity approximations are an informative method of assessing the importance of the wall deformations and whether more refined non-linear finite element analyses are necessary for stiff walls that have intermediate flexibility.

1.1 Flexure in Cantilever Wall Stem

Theory of elasticity solutions for cantilever fixed base walls that deform in flexure and for rigid walls that deform by base rotation were presented by Wood (1973) and Wood (1991) respectively. Veletsos and Younan (1997) presented approximate theory of elasticity solutions for both these types of walls.
The finite element model used by Wood (1973)for the flexural analysis of the fixed base wall is shown in Figure 1. Earthquake loading was represented by a static horizontal body force assumed to be uniform throughout the soil layer and have magnitude Coγ, where Co is an acceleration coefficient and γ the soil unit weight. Gravity loading was represented by a body force of magnitude γ acting in the vertical direction. The wall height was divided into 20 equal elements and the soil mass was divided into 29 elements along the length of the layer with a square mesh used for the six elements closest to the wall. Plane strain rectangular elements with a second order displacement field were used. The model was verified against the analytical theory of elasticity solution for a rigid smooth wall (Wood, 1973). Good agreement between the analytical and numerical results confirmed that the mesh and element stiffness theory were satisfactory.

 

Fig 01
Figure 1. Wood, 1973 FEA model for fixed base cantilever wall deforming in flexure

Minor differences in assumptions were made by Wood, and Veletsos and Younan to evaluate the pressures on the wall, force actions within the wall and the displacements of the wall. These are summarised in Table 1 for both the fixed base walls deforming in flexure and the rotating base rigid stem walls (see Section 1.2).

Table 1. Assumptions for Cantilever Wall Analysis

Tab 1

The results of Wood have been reworked for the present study by adopting the Veletsos and Younan relative flexibility parameters dw and dθ and using a Poison’s Ratio of 0.333. The results for the wall formed with rectangular plane strain elements (see Figure 1) were also checked against a model with beam elements representing the wall.
The normal pressures acting on the wall, σx from the Wood analyses for the smooth and bonded wall assumptions are shown in Figures 2 and 3 respectively. A comparison between the Wood, and Veletsos and Younan wall pressures for the bonded wall case is shown in Figure 4.
The pressure plots are presented in dimensionless parameters to enable them to be conveniently evaluated for any soil stiffness (shear modulus), soil unit weight, horizontal acceleration and wall height. For a given flexibility ratio dw , the normal pressures are dependent on the acceleration coefficient C0, soil unit weight γ, and wall height H but are independent of the soil stiffness directly.

Fig 02

Figure 2. Normal pressures on smooth fixed base cantilever wall

 

Fig 03

Figure 3. Normal pressures on bonded fixed base cantilever wall
Fig 04
Figure 4. Comparison of Wood, and Veletsos and Younan normal pressures.

Pressures calculated by Wood for a smooth wall under gravity loading are shown in Figure 5. In view of the usual layered construction method of placing soil the bonded wall assumption is not relevant for the gravity load case.
In practical applications it is necessary to combine earthquake and gravity load pressures. Gravity pressures for rigid walls can be calculated by the conventional at-rest assumption (pressure coefficient Ko = 1-sin(θ), where θ is the soil friction angle) or by assuming elastic theory which gives Ko = ν/(1-ν) where ν is the soil Poison’s ratio. For flexible walls active Rankine pressure can be assumed but for stiff walls it is helpful to have results based on the elastic soil assumption consistent with the assumptions made for the earthquake pressures.

Fig 05

Figure 5. Normal pressures on smooth fixed base wall from gravity load.

The total force acting on the wall (or stem base shear) and the bending moment at the base of the stem were obtained by integrating the pressure distributions shown in Figures 2, 3 and 4. Plots of these shears and moments for both smooth and fully bonded walls for earthquake load are shown in Figures 6 and 7 respectively and for the smooth wall gravity load case in Figure 8.
The shears and moments are plotted in dimensionless terms so that, as was the case for the normal pressures, they can be used to evaluate solutions for any values of Co,  and H. For comparison, the values of these force actions presented by Veletsos and Younan are also plotted. Superimposed on the plots are M-O forces and moments calculated for a smooth wall assuming a soil friction angle  = 35o. M-O values are plotted for acceleration coefficient values of Co = 0.2 and 0.5. Since the M-O actions do not vary linearly with Co and are independent of dw, they are drawn as separate horizontal lines over a range of typical Co values used in design.
The normal pressure plots show that significant tension stress develops in the soil near the top of the wall for dw values greater than five. Significant tension is unlikely to occur in most soils so for dw > 5 integration of the pressures results in base shear and moment values that are lower than might occur in practice. Corrections can be made by subtracting the shear and moments calculated for the tension areas. (More refined finite element analyses could iteratively modify the element properties to eliminate the tension areas.)
For the bonded wall case Figures 6 and 7 show good agreement between the Wood, and the Veletsos and Younan base shears and moments. Comparison of the elastic theory base moments with the M-O values indicates that for dw greater than approximately 15, walls are likely to be sufficiently flexible for the M-O base moment predictions to be satisfactory for wall design. (The tension normal pressures indicated by the elastic solutions are relatively minor for dw < 15 suggesting that the theory might be satisfactory for the design of walls of intermediate stiffness, that is for 0 < dw < 15.)

Fig 06

Figure 6. Wall base shear from earthquake load pressures.

 

Fig 07

Figure 7. Wall base moment from earthquake load pressures.

 

Fig 08

Figure 8. Wall base shear and moment from gravity load pressures.

Plots of the earthquake load displacement at the top of the wall, ut for both smooth and bonded walls are shown in Figure 9. The Veletsos and Younan curve for wall top displacement is shown for comparison and agrees closely with the bonded wall case. The factor required to express the deflections in dimensionless form includes the soil shear modulus G in addition to Co,  and H.

Fig 09

Figure 9. Wall top displacement from earthquake load case.

1.2 Base Rotation of Rigid Wall

The model used by Wood (1991) for the analysis of the rigid wall rotating about its base is shown in Figure 10. As for the rigid base cantilever wall deforming in flexure, earthquake loading was represented by a horizontal body force of magnitude, Co assumed to act throughout the soil layer. Gravity loading was represented by a vertical body force of magnitude . The finite element mesh was the same as described in Section 1.1 for the rigid base cantilever wall.

Fig 10

Figure 10. Rigid rotating wall model.

The difference in assumptions and relative stiffness parameters used by Wood, and Veletsos and Younan to evaluate the pressures on the wall, force actions within the wall and the displacements of the wall are summarised in Table 1 (Section 1.1). Wood’s analysis method used superposition of the pressures on a rigid wall and a forced rotating wall resulting in the adoption of the more complex stiffness parameter than used by Veletsos and Younan. As shown in Figure 10 the rotational stiffness was represented by a linear spring of stiffness kw located at height h. The Veletsos and Younan rotational stiffness parameter R is equivalent to kwh2.
The results of Wood (1991) have been reworked for the present study by adopting the Veletsos and Younan relative flexibility parameter d (Table 1) and using a Poison’s Ratio of 0.333.
The rotating wall normal pressures, base shear, base moment and top displacement from the Wood analyses for smooth and bonded assumptions are shown in Figures 11 to 18. The same dimensionless parameters used to present the flexible stem rigid base wall results have been used.

Fig 11

Figure 11. Normal pressures on smooth rotating wall.

Fig 12

Figure 12. Normal pressures on bonded rotating wall.

Fig 13

Figure 13. Wood, and Veletsos and Younan normal pressures for bonded rotating wall.

Fig 14
Figure 14. Normal pressures on smooth rotating wall from gravity load.

Figure 12 shows good agreement between the Wood, and Veletsos and Younan normal pressures acting on the bonded rotating wall.

Fig 15

Figure 15. Wall base shear from earthquake load pressures on rotating wall.

Fig 16

Figure 16. Wall base moment from earthquake load pressures on rotating wall.

Fig 17
Figure 17. Wall base shear and moment from gravity load pressures on rotating wall.

Fig 18
Figure 18. Wall top displacement from earthquake load on rotating wall.

1.3 Experimental Verification of Rotating Wall Theory

Wood (1991) carried out shaking table tests on a model rotating wall consisting of a 25 mm thick aluminium alloy plate, 0.6 m high by 2.24 m wide, mounted at one end of a 2.15 m long sand box. The depth of the sand layer was 0.55 m. The wall was supported horizontally by eight load cells arranged in two horizontal rows. Swivel joints were used on the load cells and adjustable length cantilever springs were added to the top row of load cells to allow rotation about the bottom load cells. The load cell units were very rigid resulting in negligible horizontal translation at the base of the wall. The arrangement of the sand box and wall are shown schematically in Figure 19.

Fig 19

Figure 19. Rotating wall shaking table test set-up.

Adjusting the cantilever spring lengths allowed the rotational stiffness of the wall to be varied. Measuring the forces on the wall with load cells at two different heights enabled the centre of pressure to be determined as well as the total force on the wall.
Uniform medium sand was placed into the sand box by showering from a screw conveyor. The density of the dry sand was increased by shaking the table at peak accelerations of up to 0.6 g for periods between 1 and 2 minutes. Measured densities after compaction by this method ranged from 16.0 kN/m3 to 16.5 kN/m3.
Two methods were used to measure Young’s modulus for the sand. The first involved comparing the measured natural frequency of the compacted sand with analytical predictions. This analysis showed that a Young’s modulus of 1.9 MPa was required to match the measured frequency of 12.5 Hz for the sand retained by a rigid wall. The second method involved rotating the wall by static forcing and measuring and comparing the force-displacement relationship with the theory of elasticity solutions for the forced rotating wall. Several load cycles were carried out before and after the dynamic testing. A Young’s modulus of 2.1 MPa was obtained taking an average of the initial parts of the inward and outward loading curves.
The sand box and wall were dynamically shaken with a 4 Hz sinusoidal input at acceleration amplitudes stepping from 0.1 g to 0.6 g in increments of 0.1 g for each cantilever position.
In order to compare the measured wall forces with elastic theory, a finite element model was used to compute force and moment coefficients for wall boundary and soil elastic constant assumptions simulating those used in the testing. The model represented the wall plate as beam elements and therefore included the flexibility of the plate between the load cell locations and the flexure in the cantilever, above the top load cells. A fully bonded contact was assumed using a uniform soil with a Poisson’s Ratio of 0.3.
For comparison with the theory, the experimental responses were based on an average of the peak values recorded for shaking table nominal accelerations of 0.2 g and 0.3 g.

The experimental and theoretical dimensionless dynamic outward force increments, Fd are plotted in Figure 20 against the Veletsos and Younan rotating wall stiffness parameter, d (see Table 1).

Fig 20
Figure 20. Comparison of theoretical and experimental wall forces

There is reasonable agreement between the measured and theoretical forces for d < 4. At higher d values, the experimental forces become significantly greater than the theoretical solution. The M-O dimensionless horizontal force increment for  = 350 and wall friction of 2/3 is also shown in Figure 20 and varies from 0.32 to 0.37 as the peak acceleration increases from 0.2 g to 0.3 g. The dynamic force increment on the wall for d > 4 agrees quite closely with the M-O solution suggesting significant nonlinear behaviour of the soil at this level of flexibility.
Errors in the experimental forces will arise because of side wall friction effects and amplification in the wall/soil model. Theoretical finite element studies indicated that the lowest natural frequency of the most flexible wall/soil model was about 10 Hz. Therefore the amplification of the highly damped soil model at 4 Hz would be expected to be small. Side wall friction effects could not be measured but from the comparisons of the rigid wall results with theoretical predictions, it is thought that side friction might reduce the wall forces by 10 to 15%.

1.4 Combined Flexure of Stem and Base Rotation

Veletsos and Younan (1997) presented base shear, center-of-pressure heights and top of wall displacement for cantilever walls with combined deformation from flexure in the stem and rotation of the base of the wall. These combined results are not available from the Wood analyses since superposition is not strictly valid when the wall shape differs for the two separate deformation cases. Superposition may give approximate results but it is better to use the Veletsos and Younan results when there is significant deformation from both types of wall flexibility.
Veletsos and Younan did not present base moment results but these can be derived from their tabulated values of base shear and center-of-pressure. Plots of base shear, base moment and top of wall deflection are shown in Figures 21, 22 and 23 respectively. The relative flexibility factors and dimensionless parameters are the same as used above for the separate wall types.

Fig 21
Figure 21. Base shear on cantilever wall; from Veletsos and Younan.

Fig 22
Figure 22. Base moment on cantilever wall; from Veletsos and Younan.

Fig 23

Figure 23. Top displacement of cantilever wall; from Veletsos and Younan.

1.5 Flood Control Channel Example

To illustrate the application of the results presented for the fixed base cantilever wall an analysis of a U-section flood control channel is presented in this section. The channel section is based on the Wilson Canyon, San Fernando, California channel analysed by Wood (1973). Sections of this channel were seriously damaged in the 9 February 1991, magnitude 6.6 San Fernando earthquake. A typical section is shown in Figure 24 and the soil cracking that developed in the backfill soil as a consequence of the failure of a length of wall is shown in Figure 25.

Fig 24

Figure 24. Wilson Canyon flood channel

Fig 25

Figure 25. Soil cracking indicating wall stem failure.

In the as-designed walls the main flexural reinforcement in the stem was reduced in area and lapped at approximately 1 m above the base. Failure probably occurred at this lap area which was the most critical section. To simplify the example presented here, it was assumed that the main flexural steel was unlapped and that the base section of the wall was the most critical section.
The assumed input parameters and a summary of the computed results are given in Table 2.

Table 2. Flood Control Channel Example

Tab 2

Moments at the base of the flood channel wall stem and the deflection of the top of the wall are plotted over the design range of the acceleration coefficient (0 to 0.55) in Figure 26. The inertia moment from the wall stem has been added to the combined gravity and earthquake moments (G + E moments). This is a conservative approximation as the “free” inertia load from the wall will be reduced by interaction with the backfill.

Fig 26

Figure 26. Moment and deflection performance curves

 

The performance curves show that the wall becomes cracked at an acceleration coefficient of 0.11 and reaches its flexural capacity at an acceleration coefficient of 0.4. Over this range the G + E base moment is approximately 13% greater than the base moment calculated using M-O. After the flexural capacity is reached the wall will deform plastically with rotation at a plastic hinge at the base of the wall. Outward displacement beyond the critical acceleration coefficient of 0.4 was estimated using the Newmark, 1965 sliding block theory and adopting the Jibson, 2007 correlation equation (see Part 1) for a displacement probability of exceedance of 16 %. A top displacement of approximately 0.65% of the wall height was estimated at the design acceleration coefficient level of 0.55 indicating a displacement ductility demand on the wall of less than 2. This is a relatively small demand compared to the ductility capacity of approximately 5 estimated from consideration of strain limits in the stem base section (explained in Part 3).
In applying elastic theory for this example the effects of the tension stresses at the top of the wall have been ignored. This will lead to the moments in the wall being underestimated by a moderate margin. On the other hand, the assumptions made regarding a rigid base and uniform body forces to represent the earthquake load are expected to be very conservative in many applications. The charts presented above and the procedure used for the flood channel example should provide a convenient preliminary design method for stiff walls.

2. Bridge Abutments

2.1 Reinforced Earth Abutment Walls

Many recently constructed rail and highway overpass bridges in New Zealand use reinforced earth (RE) abutment walls with spread footing type abutments resting on the wall backfill. The State Highway 1, Seddon Railway Overbridge, shown in Figures 27 and 28 is a typical single span railway overbridge with high RE abutment walls.

Fig 27

Figure 27. Seddon Railway Overbridge.

Fig 28

Figure 28. Typical section of overbridge abutment wall.

2.1.1 Earthquake Design

The earthquake design method specified in the NZ Transport Agency’s Bridge Manual (BM) is based on a limiting equilibrium (LE) analytical analysis developed by Bracegirdle, 1980 and verified by model shaking table tests carried out at the University of Canterbury (Fairless, 1989). A bilinear failure surface is assumed to develop at the toe of the wall and to propagate up through the reinforced block (RB) and the retained soil behind the RB. An upper-bound failure criterion is applied to find the critical failure surface inclination angles and the acceleration at which sliding develops.
Forces acting on the failure wedge are shown in Figure 29. The disturbing forces acting on the sliding block are the imposed forces from the bridge, Fi and R, RB wedge weight W, RB wedge inertia force, Pi, and the M-O pressure on the back of the RB, FAE (Wood and Elms, 1980). These are resisted by soil friction and cohesion (usually zero) on the failure plane and the tension forces in the reinforcing strips that cross the failure surface. Horizontal and vertical equilibrium equations are solved for the critical acceleration coefficient , kc to initiate failure of the wedge resulting in the following expression,
k_c=(T-F_i+(W+R)tan⁡(φ-α)+F_AE [sinφ_b tan⁡(φ-α)-cosφ_b ])/W (1)

Where is the failure plane angle to the horizontal and the other symbols are as defined in Figure 29.
The failure plane angle is varied iteratively to give the minimum critical acceleration. The design response acceleration acting on the sliding block is obtained by reducing the peak ground acceleration (PGA) by a performance reduction factor Sp to allow for wave scattering effects and an acceptable limit of outward movement.
An external stability analysis is also undertaken as part of the wall design using horizontal equilibrium equations for sliding on a horizontal plane through the base of the wall. The base vertical pressures are estimated using moment equilibrium equations. Gravity and earthquake forces acting on the RB for the external stability analysis are shown in Figure 30. For bridge abutment walls the friction angle on the back of the block is usually taken as 10o.

Fig 29
Figure 29. Forces acting on RB failure wedge.

Fig 30

Figure 30. External stability analysis.

The earthquake inertia forces from the bridge superstructure and abutment footings are assumed to be distributed equally to the abutment walls at either end of the bridge, and are applied uniformly across the wall with the line of action at the same height as the centre of gravity of the combined superstructure and abutment footing.
Thee passive resistance of the soil behind the abutment structure can often resist the total of the superstructure and abutment inertia loads but relatively large displacements are required to develop the full passive resistance and therefore providing resistance at both abutments is considered necessary to limit displacement damage to the wall facings.
Transfer of the bridge inertia forces to the RB is often critical at the “pull” abutment (outward loading) and is checked by calculating the pressures on the top facing panels by an empirical or finite element analysis. The panels need to be reinforced to resist these pressures and additional strips may need to be provided to anchor the top panels. Sliding of the abutment on top of the block is also checked with additional resistance provided by a shear key on the underside of the footing or anchor strips attached to the footing.
For the earthquake load case, unfactored gravity loads are combined with earthquake loads with no live load on the bridge and approach carriageways.
The LE analyses can be verified using the STARES software program (Balaam, 2006). The analysis method used in STARES is based on the Bishop, 1995 simplified procedure for unreinforced slopes and is modified specifically for investigating the stability of Reinforced Earth structures. A circular rupture surface is assumed and the limiting equilibrium of the sliding mass considered taking into account the stabilising influence of the tensions developed in the reinforcing strips. The analysis is repeated for a large number of trial failure circles to estimate the minimum critical acceleration to initiate sliding of the mass. The method is similar in principle to the LE analysis method but the LE method assumes a bilinear failure surface rather than a circular surface.

2.1.2 Dead and Live Load Design

For the static load case of dead plus live load (G + Q) design analyses are based on the method described in the Terre Armee Internationale (1990) (TAI) Design Guide.
In the TAI internal stability analysis procedure, the vertical pressures on the maximum tension line at each strip level using factored ultimate limit state (ULS) design loads from the soil mass and surface live loads, are calculated by moment equilibrium and assuming a Meyerhof vertical stress distribution. Vertical pressures from the factored bridge and abutment ULS loads are calculated using a simplified theory of elasticity solution. The combined vertical stresses on the maximum tension line are converted to horizontal pressures by pressure coefficients that vary from at-rest at the surface to active at a depth of 6 m. The strip resisting length is taken as the length behind the maximum tension line. Three possible maximum tension lines are investigated; one behind the facing panels, one running to the rear of the abutment footing and one intermediate line that intersects the abutment footing.
Often the strip density over most of the height of the abutment walls is more critical under earthquake loading than under the G + Q load case. If there are high live load pressures on the top panels the density in several of the top rows of strips may be critical under G + Q.

2.1.3 Strip Length and Density

Results of analyses of RB’s with various strip lengths and densities to determine the critical earthquake acceleration coefficient, kc to initiate failure are shown in Figure 31. The analyses were based on a uniform RB with uniform strip densities and length over the height. Inertia loading was from the block alone with bridge loads neglected. The ground surface behind the wall was assumed to be horizontal.
The input parameters used in the analysis were values typically used in design and are summarised in Table 3. The results show that for a 10 m high wall to achieve a typical design level critical acceleration of 0.35 g the total strip lengths per unit area of facing would be 23.3, 21.3 and 20 m/m2 for strip densities of 10/3m, 8/3m and 6/3m respectively. (A width length of 3 m is a convenient unit as the standard facing panels are 1.5 m wide.) This indicates that for high walls relatively long blocks with lower strip densities may often be more economical than shorter blocks with higher densities.

Fig 31
Figure 31. Critical accelerations for various strip geometries

 

Table 3. Parameters for Critical Acceleration Analysis

Tab 3

2.1.4 Analysis Example

The Seddon Railway Overbridge (Figures 27 and 28) was subjected to very strong ground shaking in the magnitude 6.6, August 2013, Lake Grassmere earthquake. A PGA of 0.75 g was recorded 200 m from the bridge site. The bridge and walls were undamaged although displacements of up to 15 mm were observed at one of the abutment footings and small movements in the backfill caused minor misalignment of wall panels at one abutment (Wood and Asbey-Palmer, 2016).
To illustrate the application of the LE analysis method to an abutment wall a summary of a back-analysis of the performance in the Lake Grassmere Earthquake is presented in this section. Input parameters for the back-analysis and the main results are summarized in Table 4.
The earthquake inertia forces from the bridge superstructure and abutment spread footings (including the seating beams) were assumed to be distributed equally to the RBs at either end of the bridge, and were applied uniformly across the RB’s with the line of action at the top of the abutment seating beams.
Figure 32 shows the failure surface predicted by the LE analysis. The critical acceleration to initiate outward failure through the RB was estimated to be 0.41 g by both the LE and STARES analyses. Because the critical acceleration was less than the PGA’s in both components of the recorded horizontal accelerations at Seddon, significant outward sliding was considered likely. An upper limit to outward displacement on the RB failure plane was calculated using the recorded acceleration time-histories as inputs to a Newmark sliding block analysis (Newmark, 1960) carried out with a special purpose software program (DISPLMT, Houston et al, 1987).
The Newmark analysis gave a sliding displacement on the RB failure plane of 5.6 and 7.8 mm for the Seddon N00E and N90W time-history components respectively. The N00E component with a PGA of 0.62 g is directed along the axis of the bridge so the smaller of the two displacement values is the best estimate. A displacement sliding response curve is shown in Figure 33 for the N00E component together with the acceleration time-history input and illustrates the sliding steps that occur at the points when the input acceleration exceeds the critical acceleration.

Fig 32

Figure 32. Failure plane from LE earthquake load analysis.

Table 4. Seddon Railway Overbridge: Back-Analysis

Tab 4

 

Fig 33
Figure 33. Sliding displacement on LE failure plane. Seddon N00E EQ component.

The passive stiffness and capacity of the 1.97 m mean height abutment backwall (upstand, footing depth and footing key depth) when loaded against the backfill was calculated using empirical equations developed by Kahalili-Tehrani et al (2010) for the backbone response of vertical walls with homogenous backfills. This method does not account for the stiffening and increased resistance of the friction against the base of the footing or the softening effect of inertia loads in the backfill. To assess the significance of the resistance from the combined wall and footing and backfill inertia forces a typical abutment section was analysed using the LimitState:GEO software. LimitState:GEO carries out limit analyses using a discontinuity layout optimization technique (Smith and Cubrinovski, 2011).
The backbone curve from Kahalili-Tehrani et al is shown in Figure 34 and indicates a passive resistance of the abutment walls of approximately 300 kN/m (for unit width of wall).

Fig 34

Figure 34. Passive resistance of abutment using Kahalili-Tehrani et al, 2010

The LimitState:GEO analysis gave a total resistance for the combined wall and footing of 340 kN/m. Failure slip-lines from this analysis are shown in Figure 35. With a 0.62 g inertia force acting in the backfill and taken to be in phase with the abutment and superstructure inertia forces this reduced to approximately 150 kN/m. It is unlikely that all the inertia forces would be in phase for a significant time period so the average resistance would be greater than 250 kN/m. The combined abutment and superstructure inertia force at 0.62 g response acceleration was estimated to be 120 kN/m and the backbone curve gave a displacement of 10 mm at this load level.

Fig 35
Figure 35. Passive resistance slip-lines for bridge and abutment inertia loads

Under loads directed away from the abutment backfill (abutment pull loads) the inertia forces acting on the abutment are resisted by sliding friction on the base of the wall. Forces transferred to the top of the RB are then transferred to the facing panels which resist these outward pressures by load transfer to the reinforcing strips anchoring the panels to the top of the RB (typically the top two layers of strips).
A factor of safety of 0.85 against outward sliding of the abutment footing was calculated for a response acceleration of 0.62 g (PGA in Seddon N00E component). Newmark sliding block theory indicated that any sliding movement would be less than 2 mm. The RB facing panels and top strips were found to have adequate strength to transfer the abutment inertia forces into the RB’s.

2.2 Integral Abutments

2.2.1 Structural Form

In an integral bridge there are no movement joints in the superstructure between spans and between spans and abutments. If the superstructure and substructure are constructed monolithically the bridge is usually referred to as fully integral. An integral bridge which has bearings at the abutment and/or piers and as a result, the superstructure and substructure do not necessarily have to move together to accommodate the required translations and rotations, is usually referred to semi-integral. Figure 36 shows the main elements of an integral bridge abutment system, which consist of a bridge deck, girders, integral cast abutments and approach slabs. The bridge movement is accommodated at the ends of the approach slabs. Sleeper slabs are commonly used to provide vertical support for the ends of the approach slab where the slabs abut the roadway pavement. The abutments are often supported on piles but spread footing type abutments can be used in suitable ground conditions or on top of reinforced earth walls (see section 2.1).
Both in New Zealand and internationally, there is an increasing interest in the design and construction of integral and semi-integral bridges which have some marked advantages over other bridge construction forms, such as reduced maintenance. However, there are a number of design issues mainly related to soil-structure action effects at the abutments that arise from concrete creep, shrinkage and temperature movements and passive resistance effects under earthquake loads (Wood, 2016).

Fig 36
Figure 36. Elements of integral bridge system

2.2.2 Performance in Earthquakes

A large number of integral abutment type bridges have been subjected to strong ground shaking in earthquakes that have occurred over the past 45 years in New Zealand (NZ) and California. The performance of the bridges in California is of particular interest since approximately half the concrete bridges on the state highway system, constructed after about 1960, have integral abutments (or monolithic end-diaphragm abutments in Caltrans terminology). In contrast to California, the number of bridges constructed with integral abutments in New Zealand is only a small percentage (perhaps 5%) of the total and most of these were constructed prior to 1960.
Three State Highway (SH) bridges with integral abutments were subjected to very strong shaking (PGA ≥ 0.3 g) in the 4 September 2010 magnitude 7.1 Darfield and 22 February 2013 magnitude 6.3 Christchurch earthquakes, and a number of significant aftershocks following these main events. Two bridges with integral abutments were subjected to strong shaking (PGA > 0.13 g) in the 21 July 2013, magnitude 6.5 Cook Strait earthquake and the same two bridges together with three other bridges were subjected to strong shaking (PGA > 0.13 g) in the 16 August 2013 magnitude 6.6 Lake Grassmere earthquake.
A performance review of integral abutments on SH bridges in NZ and California identified the following design considerations:
Integral abutments are much stiffer than adjacent piers and therefore attract a large part of both the longitudinal and transverse inertia loads from the superstructure.
Earthquake forces on the abutments of long and wide bridges can be large and special detailing of the backwall and their foundations is required to resist these forces.
Approach slabs but approach or friction slabs should be used and well anchored to avoid separation from the abutment.
To reduce longitudinal displacements of straight bridges and horizontal rotations of skewed bridges, backfilling with densely compacted cohesionless soil is important.
Damping from abutment soil-structure interaction can increase the overall damping to a much higher value than the 5% often assumed in bridge design.

2.2.3 Seismic Design

The shape and height of the abutment walls will generally be determined by considerations other than seismic performance. The small initial stiffness of the backwall passive load response is almost independent of the wall height but the ultimate passive resistance is proportional to the square of the wall height so generally there are advantages in having the backwall as high as practicable.
The displacement based design method (DBD, Priestley et al, 2007) should be used for the analysis of bridges with integral abutments. This method provides a more satisfactory way of allowing for the relative stiffness of the piers and abutments and the effects of soil-structure interaction (damping and stiffness) from the sub-structure components than force based design procedures. (DBD methods of bridge analysis are covered in Part 3.)
The stiffness of the soil against the abutment walls can be determined using the hyperbolic force-displacement (HFD) relationship presented by Khalili-Tehrani et al, 2010. This relationship has been calibrated against earlier Log-Spiral Hyperbolic force-displacement models (Shamsabadi et al 2005, 2006 and 2007) which in turn were calibrated against several small-scale and full-scale tests of abutment walls and pile caps. The form of the HFD equation is:
F(y)= (a_r y)/(H+b_r y) H^n (2)

Where F and y are the lateral force per unit width of the backwall and the deflection respectively. The parameters ar and br are height-independent parameters that depend only on the backfill internal friction, cohesion, unit weight and soil strain at 50% of the ultimate stress. H is the backwall height. The exponent n is dependent on the soil cohesion and internal friction (n = 2 for cohesion = 0). The exponent range is 1< n < 2.
Figure 37 compares displacement versus force curves calculated using the HFD equation with experimental results from Rollins et al, 2010 who undertook full-scale tests to quantify the effects of cyclic and dynamic loading on the force-displacement relations for typical pile caps and abutment walls. The axes in the figure are dimensionless with the force Pd = P/0.5H2B, where P is the total force on the wall, the unit weight of the soil and H and B the height and width of the wall respectively.

Fig 37

Figure 37. Passive resistance from HFD equation and test results from Rollins et al 2010.

The HFD equation is strictly only applicable for a wall that is uniformly translated against the backfill. In many applications the wall will rotate as well as translate against the backfill and for these cases it is best to represent the force against the backfill by a series of Winkler springs over the height of the wall. The force-displacement relationship for each spring should be based on assuming a linear increase of stiffness with depth and with the total stiffness of the springs adding to the stiffness represented by the HFD equation.
The passive resistance and stiffness of abutment walls is reduced by skew angles. Passive force-deflection curves for skewed abutments based on laboratory testing of model walls have been presented by Jessee and Rollins (2013) and compared with numerical studies undertaken on skewed abutments by Shamsabadi et al (2006).
Damping associated with abutment dynamic cyclic loading can be estimated from the test results of Rollins et al, 2010. The values that they measured during slow cyclic loading appear to be appropriate for most bridge abutment applications. Median values for densely compacted sand, fine gravel and coarse gravel were all about 18%.
Static cyclic load tests reported by Rollins and Cole (2006) on a full-scale pile cap indicate that gaps of 50% to 70% of the peak displacement may develop in a coarse gravel backfill. However, these tests do not simulate the inertia loads in the backfill that arise in strong shaking. Cyclic inertia loads in the backfill force the backfill material back against the backwall to potentially develop active pressures against the backwall.

2.2.4 Friction Slabs

The impact of gapping on the dynamic response can be reduced by using settlement or friction slabs (Wood, 2010). Settlement slabs should be anchored to the abutment and be located at a depth of approximately 1 m so that they effectively act as friction slabs and improve the bridge performance by reducing any gapping and increasing the damping. Sliding of settlement slabs and abutment footings is a very effective method of increasing damping.
Yeo, 1987 carried out what appears to be the only published experimental research on the earthquake performance of friction slab abutments. A friction slab was combined with a vertical abutment wall in the final stage of a more comprehensive study of the performance of model abutment walls subjected to cyclic loading. The model walls were 1.0 m high x 2.4 m wide and the friction slab was 2.5 m long buried at a depth of 0.68 m. Cyclic loading was applied to the wall with a hydraulic actuator and a loading system that restrained the motion of the wall to pure horizontal translation (without rotation). A moist medium dense sand backfill was used. Figure 38 shows the test arrangement.
Force-displacement results for both the wall with a friction slab and a similar model without a friction slab are shown in Figure 39. The axes in the figure are dimensionless with the dimensionless force as defined for Figure 37.

Fig 38

Figure 38. Model wall testing of abutment walls with friction slabs.

Fig 39
Figure 39. Experimental cyclic force-displacement curves for translation of model abutments.

Comparison of the cyclic force-displacement curves in Figure 39 shows clearly the benefits of a friction slab for abutment walls under cyclic loading. It increases the stiffness and failure loads and for a given load level reduces the permanent displacement or “gapping” effect. Damping represented by the areas within the force-displacement loops was estimated to be approximately 22% and 16% for the walls with and without the friction slab respectively.

2.2.5 Integral Abutment Example

To quantify the longitudinal resistance available from a typical bridge integral abutment results were calculated for a 3 m high abutment wall fitted with and without a 6 m long friction slab located at a depth of 1.5 m below the top of the wall. The backfill material was assumed to be cohesionless gravel with an internal friction angle of 35o and unit weight of 20 kN/m3. A force-displacement relationship for the abutment without the friction slab was calculated using the HFD equation (Section 2.2.3) and the influence of the slab on the response was investigated using the LimitState:Geo software.
The failure slip lines for the abutment with the friction slab are shown in Figure 40 for both the case when the force applied by the bridge is directed towards the backfill (push) and when it is directed away from the backfill (pull). Failure loads from LimitState:GEO were; 950 and 640 kN/m (of width) for the push direction with and without a friction slab respectively, and 105 kN/m for the pull direction. The HFD equation gave a load of 620 kN/m at large displacements (> 200 mm) – in good agreement with the LimitState:GEO result for no friction slab. The LimitState:GEO result for the pull case is approximately equal to 0.85Ws tan; where is the soil internal friction angle and Ws is the weight of soil above the friction slab.

Fig 40

Figure 40. Slip-lines from LimitState:GEO for abutment fitted with friction slab.

The force-displacement relationship from the HFD equation for the 3 m high wall is plotted in Figure 41 together with curves for the wall with the 6 m long friction added. The total force curve (Push + Pull) is for the case when abutments at both ends of the bridge provide combined resistance to longitudinal loads. The curves for the walls with friction slabs added are approximate as LimitState:GEO does not provide displacements.

Fig 41

Figure 41. Force versus displacement and earthquake demand curves.

The superstructure of a typical two-lane NZ highway bridge constructed from prestressed concrete has a weight of approximately 10 kN/m2. The force ordinate in Figure 41 can therefore be interpreted as the response acceleration acting on a 100 m long bridge (weight = 1 MN/m of width) subjected to a longitudinal static acceleration. To assess the performance of this hypothetical bridge a demand curve based on the NZS 1170.5 response spectrum for a 1000 year return period and Site Subsoil Category C is superimposed on Figure 41. The spectrum is scaled to 15% equivalent viscous damping to account for the high damping expected from soil-structure interaction at the abutments. Intersection of the response and demand curves indicates that the wall passive resistance and the frictional resistance from slabs at both ends of the bridge (ignoring any resistance from the piers) limits the longitudinal displacement to less than 50 mm under a design level event. Integral abutments enhanced with friction slabs can therefore be very effective in reducing the longitudinal displacement response to a level unlikely to cause significant damage to any of the substructure components.

 

3. Conclusions

3.1 Stiff Retaining Walls

The flexibility of a retaining wall has a significant influence on the earthquake-induced soil pressures. Methods of predicting earthquake pressures on rigid and flexible walls are well documented in the literature but there is limited published information on methods of analysis for stiff walls with flexibility intermediate between these limiting cases. Were the wall geometry is relatively simple, theory of elasticity methods can be used to estimate earthquake actions on these stiff walls with sufficient accuracy for design. These methods also give an indication of whether significant non-linear soil behavior is likely and whether a more sophisticated finite element analysis is justified.

3.2 Overpass Abutment Walls

High bridge abutment walls of RE construction have performed well in the strong ground shaking generated by the 2010-2011 Canterbury earthquake sequence (Wood and Asbey-Palmer, 2013) and the 2013 Lake Grassmere earthquake. Laboratory model testing and back-analysis of a number of RE walls subjected to strong shaking indicates that the LE analysis method is satisfactory for design. The walls can be designed for accelerations significantly less than the PGA providing a Newmark sliding block analysis is carried out to estimate the displacements on the failure plane.

3.3 Integral Bridge Abutments

The stiffness and damping available from bridge abutment structures may often have a large influence on the earthquake response of the bridge. To reliably predict the longitudinal response of integral or semi-integral bridges a displacement based design approach that considers soil-structure interaction at the abutment backwalls, span seating beams and piles should be used.

 

4. Acknowledgements

The analytical and finite element work on stiff cantilever walls was carried out at California Institute of Technology by the author during a period of study leave from the New Zealand Ministry of Works and Development (MWD). The assistance and guidance received during this period from Professors G. W. Housner and R. F. Scott is gratefully acknowledged.
Rotating and friction slab wall testing projects were carried out at MWD Central Laboratories under the supervision of Stuart Thurston. This work was financially supported by the Road Research Unit of the National Roads Board and the Civil Directorate of MWD.

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