Integral Bridge Design - Derivation of the Spring Constant for Modelling the Soil-Structure Interaction

Transcription

1 Integral Bridge Design - Derivation of the Spring Constant for Modelling the Soil-Structure Interaction Sergei Terzaghi BE(Hons), MIPENZ Gillian Sisk BEng PhD MIEI CEng Synopsis Integral bridges present a challenge for load distribution calculations because the bridge deck, piers, abutments, embankments and soil must all be considered as a single compliant system. The thermal deck movements are accommodated by soil structure interaction between the supporting piles and the surrounding strata. Deck loading is also affected by the soil which acts as both load and support system to the piles upon which the structures are founded. Specifying a series of spring supports behind the abutments and adjacent to the foundation piles to approximate soil behaviour is a commonly used modelling method. The main difficulty with the Winkler spring type model is the derivation of an appropriate spring constant. A common practice is to use the subgrade stiffness for the soil type. In this paper, factors that influence the magnitude of spring stiffness are investigated, such as the overall range of movement expected, the soil type or select fill parameters (modulus and Poisson ratio), and the area of the structure resisting the movement. Relationships between the spring constant and each of these parameters are then determined and are used to derive the spring constant appropriate for the ground conditions and loading combinations. Introduction An integral bridge may be defined as having no expansion joints or sliding bearings, the deck is continuous across the length of the bridge. Integral bridges are alternatively referred to as integral abutment bridges, jointless bridges and rigid frame bridge. Semi-integral bridges typically have sliding bearings but no expansion joints. An important aspect of integral bridge design is the modelling of the pile-soil interaction. For conventional abutments with raked piles, the longitudinal movement of the abutment is relatively small and therefore the interaction between the piles and the soil does not have a significant influence of the design of the piles. For abutments with vertical piles and particularly those with only a single row, shortening of the superstructure due to creep and shrinkage and movements arising from traction forces, induce bending moments which are directly related to the combined stiffness of the pile and the surrounding soil. In order to calculate bending moments in the piles, the stiffness of the soil must be included. Integral Bridge Design Terzaghi/Sisk 1

2 This paper focuses on the determination of the soil stiffness and considers the factors that influence this value. Various Methods to Determine Modulus of Subgrade Reaction Table 1 below summarises a number of commonly used methods used to determine the modulus of horizontal subgrade reaction. These are used to compare to the approach adopted in this paper. Methods/References: Based on constants of horizontal subgrade reaction, m h or n h (Geoguide 1, 1993) Equation for wall analysis, k hj = m h z j /d for pile analysis, k hj = n h z j /B where: m h = 0.64K p γ ( y' / d) for loose sand = 1.09K p γ ( y' / d) for dense sand where: y = lateral deflection of wall at mid-depth embedment in soil K p = coefficient of passive earth pressure d = depth of wall embedment in soil γ = soil unit weight n h = 0.19 D r 1.16 where: D r = relative density refer to Schmertmann (1978) Based on E s (Geoguide 1, 1993) k hj = 0.8 to 1.8 E sj Based on Vesic (Poulos, 1980) 0.65 Esd khj = 12 d E I p 4 p E s 2 1 υs Based on Skempton for cohesive soil only (Poulos, 1980) k hj = (80 to 320) c u /d ; where: c u = undrained shear strength; d = pile diameter Table 1 Integral Bridge Design Terzaghi/Sisk 2

3 Research In order to assess the influence of the different factors, and validate a given analytical approach, a 3-d finite element model was set up using the program Plaxis, and the results used to calculate the equivalent spring stiffness. Three different soil models were used to also examine the influence of the way the soil was modeled on the results. The three soil models used were the Linear Elastic Model, the Mohr Coulomb Model, and the Hardening Soil Model. The essential characteristics of these models are: Linear Elastic model is a classic elastic model governed by an E value and poisson s ratio. This model requires only two stiffness/strength parameters The Mohr Coulomb Model is a linear elastic- perfectly plastic model, that is, it behaves in a linear elastic fashion up to the point of failure, and thereafter, behaves in perfectly plastic fashion, with non- recoverable deformation. This model requires 5 stiffness/strength parameters The hardening soil model is a non-linear elasto-plastic soil model incorporating strain dependent hardening (change in stiffness), development of plasticity, pressure dependent stiffness, and memory of previous soil history. As such, it is able to better capture elements of real soil behaviour. It requires 7 stiffness/strength key parameters with 5 secondary parameters. Whilst this sound more complex, in fact, it is often easier to capture the 7 key parameters with routine testing than it is to guesstimate the 5 required in the Mohr Coulomb model. As most analytic representations are based around Linear Elastic solutions, it is quite important to include in any analysis the Linear Elastic Model as a basis for comparison, similarly with the Mohr coulomb model as most people are familiar with its characteristics. The Hardening soil model was selected as it (properly calibrated) more closely represents the real soil behavior In selecting a spring stiffness for modeling soil behavior, one needs to consider the original definition of the modulus of subgrade reaction, from which the soil spring is typically derived, the loaded area, and the zone of influence of the loaded area, as well as such apparently secondary factors like total strain induced, strain rate, confining pressure, and cyclic loading (from both high (vibrations) and low frequency (eg thermal)). It is often difficult to capture all of these effects in a single spring value, or even in a sensitivity check. In order to evaluate the sensitivity of the selected spring constant to these factors a series of runs were carried out using the three model types above, with parameters selected to simulate the target soil, such that one could better select an appropriate spring constant for a given situation. A selection of the findings is given in the graphs below. A number of findings are evident straight away, with significantly different results by using the different soil models. This should not be a surprise, as each model Integral Bridge Design Terzaghi/Sisk 3

4 incorporates a different amount of plasticity and different response to strain. Also, there is more sensitivity to the loaded area than is commonly recognized. This again is function of the differing levels of strain as the loaded area becomes larger. The sensitivity of the chosen spring stiffness is surprisingly sensitive to the poisson s ratio, something not reflected elsewhere. Integral Bridge Design Terzaghi/Sisk 4

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6 Case Study - Pacific Highway Karuah to Bulahdelah Sections 2 and 3 Upgrade The use of integral construction was adopted for all the bridge structures on the Karuah to Bulahdelah Upgrade. This form of construction allowed the gland type expansion joints to be replaced by small movement deck joints, it allowed simplification of the elastomeric bearings by eliminating shear capacity requirements and it eliminated forward raking piles. The elimination of the gland type expansion joints at the abutments also results in a long term maintenance advantage. Figure 1 Typical Abutment Detail The bridges are single span structures supporting the carriageway varying in span from 18m to 33m. The superstructures comprise prestressed concrete T beams with a compositely acting reinforced concrete deck slab. The beams are supported on laminated elastomeric bearings which accommodated the rotation due to live loads. A small movement joint as per the standard RTA detail is located at each abutment. The spill-through abutments consist of reinforced concrete headstocks supported on 400mm square precast reinforced concrete driven piles. Stainless steel dowels resist the design longitudinal loads in shear (refer to Figure 1 for typical abutment detail) The site is generally underlain by alluvial clays of firm to stiff consistency over siltstone/sandstone bedrock Integral Bridge Design Terzaghi/Sisk 6

7 The approach to structural design of the bridges was considered with reference to BA 42/96, and requirements of the Project Scope of Work and Technical Criteria (SWTC), and specifically AS The Advice Note BA 42/96 is a British Technical Advice Note developed to give design requirements for integral abutments. It was written specifically to provide guidance on horizontal earth pressures, and soil-structure interaction and is based largely on research carried out by Springman et al (1996). At the time of the design, the RTA Bridge Policy Circular BPC 2007/05 had not been issued. This circular makes reference to the Advice Note BA42/96. It stipulates design requirements and limitations to the design of integral bridges in NSW which is keeping with current practice in Europe and the USA. Based on these references, the following methodology was used. The superstructure design was modelled as a grillage using either ACES to obtain design actions and deflections in the T beams and deck due to live load and superimposed dead loads such as parapets. The abutment, pile group and soil interaction was then modelled as a 2-dimensional frame in MicroStran incorporating the superstructure. Horizontal springs were incorporated at the interface of the piles and the existing material and between the abutments and the select fill. The spring stiffness values adopted were evaluated using the modulus of subgrade reaction as described in this paper. The modulus of subgrade reaction for the soil took into consideration the height of the abutment or pile diameter as appropriate, the overall range of bridge movement, the soil type or select fill and elastic modulus. Upper and lower bounds were provided and used as appropriate for each combination load case to determine the worst effect. The soil spring reactions were checked to ensure the soil was within its limiting stress. The load cases comprising permanent effects, temperature expansion or contraction and live load effects were applied to the integral bridge frame. The two critical ULS load cases were as follows: Dead, superimposed dead, creep, shrinkage, temperature contraction and SLS live and longitudinal braking all acting simultaneously. Active earth pressure was applied to the abutment moving in the direction of the longitudinal braking force and horizontal springs applied to the back of the abutment wall resisting this movement. Dead, superimposed dead, temperature expansion and SLS live and longitudinal braking all acting simultaneously. To determine the effects of the longitudinal braking, horizontal springs were applied to the abutment resisting the movement due to the longitudinal braking force and active earth pressure applied to the back of the other abutment wall. To determine the effects of the thermal expansion, horizontal springs were applied to both abutments. The stiffness of the springs were adjusted accordingly to ensure that the K* earth pressure was Integral Bridge Design Terzaghi/Sisk 7

8 not exceeded (refer to BA 42/96 earth pressure corresponding to the retained height and thermal displacement). Discussion For abutments with a single row of vertical piles, shortening of the superstructure due to creep and shrinkage and movements arising from earthquake and braking forces, induce bending moments which are directly related to the combined stiffness of the piles and the surrounding soil. In order to calculate bending moments in the piles, the stiffness of the soil must be included in the modeling of the structure. As an example on the Karuah to Bulahdelah bridges, increasing the spring stiffness by 100% resulted in an increase of approximately 30% in the bending moment in the piles. It is important to note that the modulus of subgrade reaction and the stiffness of a soil are not soil properties and for a given soil type will vary depending on the loaded area and the magnitude and rate of loading along with the stress history of the soil and also current confining pressure. As a consequence, it is important the selected distribution of spring values reflect that, and in fact may need to change depending on the load case. The likely range of spring values is also somewhat dependent on soil type, and in any event should not be relied upon in soft soils, as the concept of spring stiffness relies on the soil remaining within the elastic (0.01% strain) or pseudo-elastic range (0.1% strain) of soil behavior. Below Table 2 tabulates values of spring stiffness calculated using three different methods. This table would suggest that current practice could lead the piles being under or over designed, especially for the smaller piles and where the pile loads are as a result of prescribed displacements. Item B Material ν E (kpa) Based on E s (Geoguide 1, 1993) Based on Vesic (Poulos, 1980) Based on Terzaghi (2007) ks (kpa/m) ks (kpa/m) ks (kpa/m) M-C Model Abutment 1.9 Fill ,000 48, ,000 15,470 30,153 Abutment 1.45 Fill ,000 48, ,000 20,270 37,198 Pile 0.4 Fill ,000 48, ,000 73, ,156 Pile 0.4 alluvium ,000 8,000 18,000 11,128 16,859 Pile 0.4 Clay ,000 16,000 36,000 23,578 33,719 Pile 0.6 Fill ,000 48, ,000 51,195 73,825 Pile 0.6 alluvium ,000 8,000 18,000 7,753 12,304 Pile 0.6 Clay ,000 16,000 36,000 16,428 24,608 Pile 0.9 Fill ,000 48, ,000 34,130 53,879 Integral Bridge Design Terzaghi/Sisk 8

9 Based on E s (Geoguide 1, 1993) Based on Vesic (Poulos, 1980) Based on Terzaghi (2007) Item B Material ν E (kpa) Pile 0.9 alluvium ,000 8,000 18,000 5,169 8,980 Pile 0.9 Clay ,000 16,000 36,000 10,952 17,960 Pile 1.2 Fill ,000 48, ,000 25,600 43,089 Pile 1.2 alluvium ,000 8,000 18,000 3,877 7,181 Pile 1.2 Clay ,000 16,000 36,000 8,214 14,363 Table 2 Variation in values from different methods highlights the risk of under or overdesigning piles. Conventional use of springs is discouraged and the use of combined structural geotechnical models is encouraged. References Guide to Retaining Wall Design Geoguide 1 by Geotechnical Engineering Office, Civil Engineering Department, The Government of the Hong Kong Special Administrative Region, H.G. Poulos and E.H. Davis, 1980, Pile Foundation Analysis and Design Springman S M, A R M Norrish and C W W Ng (1996), Cyclic Loading of Sand Integral Bridge Abutments TRL Project Report 146. Crawthorne: Transport Research Laboratory Integral Bridge Design Terzaghi/Sisk 9