The article presents parametric analysis regarding the impact of subgrade and backfill stiffness on values and distribution of bending moments in the structural elements of a small integral box bridge made of cast in situ reinforced concrete. The analyzed parameters are the modulus of subgrade reaction under and behind the bridge structure (k_{v} k_{h}). At the beginning, the author presents the integral box bridge and selected parts of the bridge design. In particular, the author focuses on the method of modeling of the subgrade stiffness parameters under and behind the bridge structure, as well as their impact on the values and distribution of bending moments in the bridge structural elements. The bridge was designed by the author and built on the M9 motorway between the towns of Waterford and Kilcullen in Ireland. In conclusions, the author shares his knowledge and experience relating to the design of small integral bridges and culverts and puts forward recommendations as to further research on these type of structures in Poland.

#### Keywords

- precast box bridge
- integral bridge
- design
- single-span bridge

An integral bridge can be defined as a bridge whose span is monolithically connected with the abutment walls and whose structure interacts with the surrounding soil due to thermal effects as well as various dead and live loads. Such elements as bridge bearings, mechanical expansion joints, and approach slabs are not required in this case, whereby the construction and maintenance of integral bridge are less expensive. Integral bridge structures have been widely used in the world since the 1930s. This paper presents parametric analysis regarding the impact of subgrade and backfill stiffness on values and distribution of bending moments in the integral box bridge structure elements made of cast in situ reinforced concrete. In the analysis is used one of the diagrams of load cases proposed in standard [8]. The design and construction of this bridge is described more thoroughly in publication [5]. Other types of integral bridges and viaducts, both single-span and multi-span ones, and arch bridges are described in [1, 2, 3, 4]. It is worth noting that the implementation of integral bridges on this section of the motorway contributed to a significant reduction in the time and cost of construction of the motorway.

The integral box bridge structure located in County Kilkenny in Ireland on the M9 motorway connecting the towns of Kilcullen and Waterford (Figs. 1–3) is described. The main purpose of this bridge is to pass agricultural traffic, agricultural machinery, and livestock to the pastures separated by the motorway. The bridge was designed in accordance with the guidelines for bridges and road culverts. The traffic loads and the load configuration conformed to the Irish standard for bridges and culverts. The bridge was designed as integral with the surrounding soil. The parameters of the subgrade and the backfill used to build the bridge were specified by a geotechnical engineer. These parameters are presented in the impact analysis below. The grading requirement for the 6N/6P class materials used to backfill the structure is given in Table 2 [14]. The bridge carries a motorway with two one-way carriageways, each 7 m wide, separated by a 2.6 m wide median strip with a concrete Jersey barrier. At the outer edge of each of the two carriageways, there is a 2.5 m wide shoulder limited by a steel safety barrier. A 4.0 m wide road with two 1.0 m wide sidewalks adjacent to it runs under the bridge. The embankment over the bride wing walls are protected with timber post and rail fencing with wire mesh. The basic parameters of the bridge are specified in Table 1.

Basic bridge parameters.

Effective span length | L_{t}=6.45 [m] |

Overall span length | L_{p}=6.9 [m] |

Skew angle | a=90° |

Wall, upper floor slab, and bottom slab thickness | h=0.45 [m] |

Minimal soil surcharge height over bridge structure | H_{n}=1.1 [m] |

Length of bridge without wing walls | L_{o}=30.6 [m] |

Overall length of wing walls | L_{s}=8.49 [m] |

Angle of rotation of wing walls relative to bridge length | b=45° |

Span height to length ratio | 1:15 |

Embankment height | 6.0 [m] |

Bottom slab and wing wall strip footing concrete class | C32/40 |

Bridge wall, upper floor slab, wing wall, and string course concrete class | C40/50 |

Live load type | HA and HB45 |

Grading of 6N and 6P class backfills [14].

125 | 100 | |

100 | 100 | |

75 | 65–100 | |

37,5 | 45–100 | |

10 | 15–75 | |

5 | 10–60 | |

0.6 | 0–30 | |

0.063 | 0–15 |

The design documentation was prepared in the Fehily Timoney and Company consulting office in Cork [13]. Working for Fehily Timoney and Company, the author designed this bridge.

The following standards, among others, were used to design the bridge:

BD31/01 The Design of Buried Concrete Box and Portal Frame Structures [8],

BA42/96 The Design of Integral Bridges [9],

BD37/01 Loads for Highway Bridges [10],

BS5400-04 Code of practice for design of concrete bridges [11].

In addition, the Irish Manual of Contract Documents for Road Works [14] and the project owner's (National Road Authority [12]) latest recommendations were used for the bridge design. Considering the interaction between the bridge and the surrounding soil, components, such as the backfill behind the bridge walls and the subgrade stiffness calculation, are described in the following section.

The class of material used to bury integral bridges and the way of built-in it has have a significant impact on the distribution of internal forces in the bridge structure elements. For this purpose, class 6N and 6P backfill materials are used on the British Isles. These materials usually consist of crushed rock, crushed concrete, natural gravel, crushed gravel, or combination of both, excluding argillaceous gravel aggregate. Detailed information about the backfills is given in the Manual of Contract Documents for Road Works [14] (Table 2). During the construction of the considered bridge, the effective angle of shearing resistance of the backfill ranged from f=35° to 40°.

It is important that before a bridge structure design begins, geotechnical investigations are carried out in the location where the future supports will be located, the modulus of subgrade reaction is determined, and the settlement of the supports is calculated. On the basis of this information, the designer can create a numerical model of the bridge structure, which will most accurately describe the actual foundation conditions. The model represents a structure on elastic supports, which behave flexibly, influenced by applied permanent and live loads. Moreover, if the stiffness of the structural members is low, the structure is flexible and better interacts with the surrounding soil, whereby the stresses in the structure elements are reduced and evenly distributed. For this reason, the cross-sections of the structural members of integral bridges can be smaller, whereby such bridges are less expensive to build than other types of bridges.

Only permanent loads were used in the parametric analysis. Considering the permanent character of the load, the shape of the bridge was assumed to be invariable along its length, and to simplify the calculations a two-dimensional structure with beam elements was adopted as the model of the bridge structure. The loads were applied to the bridge model according to one of the load cases diagrams given in standard [8]. The standard provides seven diagrams showing different load cases, which need to be considered in the design. The partial load factors for these loads are given in Figure 4. The calculated values of the loads and their denotations are shown in Table 3 and Figure 5. It is worth noting that loads such as load effects due to temperature, live load associated with traction, horizontal live loads, and other loads in the standards [8,10] as well as their combinations were not taken into account in the analysis. These loads additionally affected the values and distribution of internal forces in the bridge structure. Therefore, only for clarity of the analysis, the permanent loads were selected for the analysis.

Permanent load.

Road pavement | V_{1} | 2.3 • 0.2 • 9.81 • 1.75 • 1.1 = 8.7 |

Surcharge over bridge | V_{2} | 2.0 • 1.1 • 9.81 • 1.2 • 1.1 = 28.5 |

Earth pressure behind left abutment wall | H_{L1} | (2.3 • 0.2+2.0 • 1.1) • 9.81 • 0.33 • 1.5 • 1.1 = 14.2 |

H_{L2} | (2.3 • 0.2+2.0 • (1.1 + 5.9)) • 9.81 • 0.33 • 1.5 • 1.1 = 77.2 | |

Earth pressure behind right abutment wall | H_{P1} | (2.3 • 0.2+2.0 • 1.1) • 9.81 • 0.6 • 1.0 • 1.0 = 15.7 |

H_{P2} | (2.3 • 0.2+2.0 • (1.1 + 5.9)) • 9.81 • 0.6 • 1.0 • 1.0 = 85.1 | |

Self-weight of concrete | CW | 2.4 • 1 • 0.45 • 9.81 • 1.2 • 1.1 = 13.99 |

Concrete modulus of elasticity and Poisson's ratios [11].

Bottom slab, | 33.34 [GPa] | |

concrete C32/40 | E_{cm}, | 0.2 |

Abutment walls, upper floor slab, | ν | 35.22 [GPa] |

concrete C40/50 | 0.2 |

In the calculations, the structure was assumed to be founded on Winkler's unidirectional subgrade model. The elastic constraints connecting the bottom slab and abutment walls with the soil are only compression-loaded. This means that parts of the structure can detach from the surrounding soil. The superposition principle cannot be used in the calculations because of the nonlinear character of the bridge model supports. For this reason, all the loads involved were scaled up by applying a partial load factor and incorporated into a single load case. Ten numerical models were built. For clarity of the analysis, it is assumed that the effective angle of shearing resistance f’ is constant in all considered models. The variable parameters in the analyses are the modulus of subgrade reaction _{v}_{h}

Model and parameters analyzed.

_{h} [kN/m^{3}] | _{v} [kN/m^{3}] | |
---|---|---|

M-1 | 10,000 | 10,000 |

M-2 | 37,000 | 10,000 |

M-3 | 120,000 | 10,000 |

M-4 | 10,000 | 80,000 |

M-6 | 120,000 | 80,000 |

M-7 | 10,000 | 120,000 |

M-8 | 37,000 | 120,000 |

M-9 | 120,000 | 120,000 |

M-10 | 0 | ∞ |

Abaqus FEA software was used for the structure analysis [16]. The calculations of the modulus of the horizontal reaction of the subgrade behind the bridge abutment walls were carried out according to Ménard's empirical formula given in [17]. An example of calculations for the model M-5 is given below. The remaining values in Table 5 were adopted for the extreme values of medium dense and dense sand given in the publication [7].

_{h} | Modulus of horizontal subgrade reaction (backfill material) |

Coefficients dependent on soil type and consistency, e.g. gravel | |

_{p} | Pressuremeter modulus of soil, _{p}_{c} |

_{c} | Cone soil penetration resistance determined by cone penetration test (CPT) |

_{0} | Reference radius, r_{0} = 0.3m |

Bridge abutment wall height, D = 5.9 m, | |

Radius, a half of abutment wall height r = D / 2 = 5.9 / 2 = 2.95 m |

Geotechnical investigation carried out using the CPT probe showed that the cone resistance for the backfill used on the construction site amounted to _{c}^{3}. Hence, the calculated pressuremeter modulus of the soil was:

Substituting the above data into the Ménard formula for modulus _{h}

The above is the modulus of the horizontal reaction of the subgrade behind the bridge wall. The modulus of the vertical reaction of the subgrade on which the bridge was to be built amounted to:

After these moduli had been determined, their values were proportionally distributed on the bridge model's abutment walls (_{h}_{v}_{h}

Stiffness of elastic constraints for M-5 model.

^{3}] | ^{2}] | ||
---|---|---|---|

k_{1} | 0.3375 | k_{h} = 80,000 | 0.3375 • 80,000 = 27,000 |

k_{2} | 0.3625 | 0.3625 • 80,000 = 29,000 | |

k_{3} | 0.5 | 0.5 • 80,000 = 40,000 | |

k_{4} | 0.3375 | k_{v} = 37,068 | 0.3375 • 37,068 = 12,510 |

k_{5} | 0.3625 | 0.3625 • 37,068 = 13,437 | |

k_{6} | 0.5 | 0.5 • 37,068 = 18,536 |

Many factors, such as the soil classification its properties, initial soil state, the load type (short/long term), and its intensity and the shape and the size of the foundation, have influence on the value of the modulus of subgrade reaction. Short-term loading and unloading of the soil without occurrence of the consolidation and deconsolidation usually cause less settlement in it compared to when long-term static loading is acting on the soil. For this reason, the value of the soil dynamic modulus _{D}_{0} of the same soil subjected only to static loads [6]. Wider and larger area loadings will involve consolidation of the deeper soil layers. The foundation's rigidity affects the stress distribution under the foundation. It should be emphasized that to properly determine the soil parameters for the structure design, close cooperation is necessary between the geotechnical and the structural engineer.

The bending moment values _{y}

The parametric analysis shows that the values of bending moments _{y}

Bending moment values and their distribution in the abutment walls of the analyzed bridge model are mainly influenced by subgrade stiffness under the bridge. Backfill stiffness has small impact on bending moment values and their distribution in the abutment walls. Lower value of the modulus of subgrade reaction _{v}_{v}_{y}_{y}

Over the entire length of the bridge's bottom slab, bending moment values _{y}_{v}_{v}

Bending moment values used for bridge design.

Upper floor slab, midspan | 362 |

Upper floor slab, at support | 333 |

Bottom slab, midspan | 196 |

Bottom slab, at support | 287 |

Abutment wall at midspan | 182 |

The bridge was put into service in the first half of 2010. After three years of bridge exploitation, no major cracks were found on the bridge structure than those assumed in the calculations as well as uneven settlement of the structure.

The parametric analysis shows that when the bridge is designed with subgrade stiffness taken into account, one gets different values and distribution of bending moment than for the model on a rigid supports. Due to foundation of the bridge on a flexible subgrade, it was proper to include it in this bridge design. If the elasticity of the subgrade had not been taken into account in the numerical model, this could have led to excessive deflections and surface cracking in such bridge elements as the bottom surface of the upper floor slab, both surfaces of the bottom slab, and the inner surfaces of the abutment walls from the embankment side. It should be added that excessive cracking in the structure elements may appear on invisible surfaces such as the inner surfaces of the abutment walls or bottom slab from the embankment side and on the upper surface of the bottom slab, on which, for instance, a road surface may be built. Therefore, cracking of these elements may not be visible during bridge exploitation or during periodical inspections. This can lead to an unexpected structure failure or ultimately to construction disaster.

It should be emphasized that prior to design calculations that take subgrade stiffness into account the proper soil parameters for both the subgrade and the backfill must be determined. On the basis of such data the designer can build a numerical model of the structure founded on the specific subgrade stiffness. Therefore, close cooperation is required between the geotechnical and the structural engineer when designing this type of bridge. If one designs a bridge founded on a different subgrade than the target one (e.g., on a rigid subgrade), this can result in the over-reinforcement of some of the structural members and in the under-reinforcement of other structural members. After the inspection of the bridge and the other bridges on this motorway, it was concluded that it had been proper to take into account subgrade and backfill stiffness in the bridge calculations. It should be noted that the integral bridge presented here very well interacts with the surrounding soil, under applied permanent and live loads. Bridges of this type can have structural members with a smaller cross-section in comparison with conventional design solutions in which the structure–soil interaction is not taken into account. Consequently, they are cheaper to build than conventionally built bridges owing to the reduced quantity of the materials used. In the author's opinion, integral bridges can and should be built in Poland because they are less expensive and take less time to build. One should take into account the fact that the ambient temperatures in Poland are different than on the British Isles, and therefore, it is necessary to investigate integral bridge structures in our climate conditions. Such research would give bridge engineers a deep insight into the behavior of this type of bridges, whereby their span could be gradually increased. It is worth noting that increasingly more valuable publications on integral bridges and viaducts appear in Poland [15]. This indicates a growing interest in such structures on the part of bridge engineers in Poland.

_{h} | Modulus of horizontal subgrade reaction (backfill material) |

Coefficients dependent on soil type and consistency, e.g. gravel | |

_{p} | Pressuremeter modulus of soil, _{p}_{c} |

_{c} | Cone soil penetration resistance determined by cone penetration test (CPT) |

_{0} | Reference radius, r_{0} = 0.3m |

Bridge abutment wall height, D = 5.9 m, | |

Radius, a half of abutment wall height r = D / 2 = 5.9 / 2 = 2.95 m |

#### Bending moment values used for bridge design.

Upper floor slab, midspan | 362 |

Upper floor slab, at support | 333 |

Bottom slab, midspan | 196 |

Bottom slab, at support | 287 |

Abutment wall at midspan | 182 |

#### Basic bridge parameters.

Effective span length | L_{t}=6.45 [m] |

Overall span length | L_{p}=6.9 [m] |

Skew angle | a=90° |

Wall, upper floor slab, and bottom slab thickness | h=0.45 [m] |

Minimal soil surcharge height over bridge structure | H_{n}=1.1 [m] |

Length of bridge without wing walls | L_{o}=30.6 [m] |

Overall length of wing walls | L_{s}=8.49 [m] |

Angle of rotation of wing walls relative to bridge length | b=45° |

Span height to length ratio | 1:15 |

Embankment height | 6.0 [m] |

Bottom slab and wing wall strip footing concrete class | C32/40 |

Bridge wall, upper floor slab, wing wall, and string course concrete class | C40/50 |

Live load type | HA and HB45 |

#### Grading of 6N and 6P class backfills [14].

125 | 100 | |

100 | 100 | |

75 | 65–100 | |

37,5 | 45–100 | |

10 | 15–75 | |

5 | 10–60 | |

0.6 | 0–30 | |

0.063 | 0–15 |

#### Stiffness of elastic constraints for M-5 model.

^{3}] | ^{2}] | ||
---|---|---|---|

k_{1} | 0.3375 | k_{h} = 80,000 | 0.3375 • 80,000 = 27,000 |

k_{2} | 0.3625 | 0.3625 • 80,000 = 29,000 | |

k_{3} | 0.5 | 0.5 • 80,000 = 40,000 | |

k_{4} | 0.3375 | k_{v} = 37,068 | 0.3375 • 37,068 = 12,510 |

k_{5} | 0.3625 | 0.3625 • 37,068 = 13,437 | |

k_{6} | 0.5 | 0.5 • 37,068 = 18,536 |

#### Concrete modulus of elasticity and Poisson's ratios [11].

Bottom slab, | 33.34 [GPa] | |

concrete C32/40 | E_{cm}, | 0.2 |

Abutment walls, upper floor slab, | ν | 35.22 [GPa] |

concrete C40/50 | 0.2 |

#### Model and parameters analyzed.

_{h} [kN/m^{3}] | _{v} [kN/m^{3}] | |
---|---|---|

M-1 | 10,000 | 10,000 |

M-2 | 37,000 | 10,000 |

M-3 | 120,000 | 10,000 |

M-4 | 10,000 | 80,000 |

M-6 | 120,000 | 80,000 |

M-7 | 10,000 | 120,000 |

M-8 | 37,000 | 120,000 |

M-9 | 120,000 | 120,000 |

M-10 | 0 | ∞ |

#### Permanent load.

Road pavement | V_{1} | 2.3 • 0.2 • 9.81 • 1.75 • 1.1 = 8.7 |

Surcharge over bridge | V_{2} | 2.0 • 1.1 • 9.81 • 1.2 • 1.1 = 28.5 |

Earth pressure behind left abutment wall | H_{L1} | (2.3 • 0.2+2.0 • 1.1) • 9.81 • 0.33 • 1.5 • 1.1 = 14.2 |

H_{L2} | (2.3 • 0.2+2.0 • (1.1 + 5.9)) • 9.81 • 0.33 • 1.5 • 1.1 = 77.2 | |

Earth pressure behind right abutment wall | H_{P1} | (2.3 • 0.2+2.0 • 1.1) • 9.81 • 0.6 • 1.0 • 1.0 = 15.7 |

H_{P2} | (2.3 • 0.2+2.0 • (1.1 + 5.9)) • 9.81 • 0.6 • 1.0 • 1.0 = 85.1 | |

Self-weight of concrete | CW | 2.4 • 1 • 0.45 • 9.81 • 1.2 • 1.1 = 13.99 |

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