Calculation methods of bridge deck for prestressed short rib T-beam bridges

Abstract

Because of the low height of the prestressed short rib T-beam bridge and the poor torsion resistance of the main beam, the positive moment in the middle span of the bridge deck will increase correspondingly compared with the normal rib beam bridge. At present, there is little research on the calculation method of the bridge deck of the prestressed short rib T-beam bridge. In this paper, the space finite element method and the continuous one-way slab method are used to calculate the forces on the bridge deck, based on the space finite element method, a finite element elastic supported continuous beam method is proposed to calculate the forces on the bridge deck. By comparing the calculation results of the three methods with the test results, the reasonable calculation method of the bridge deck is studied. The results show that the spatial finite element analysis method can simulate the mechanical performance of the deck of the bridge of the prestressed short rib T-beam bridge well, the stress calculation results are consistent with the test results, and the calculation accuracy is high, which can be used in the actual engineering design; The finite element analysis method of elastic support continuous beam can also simulate the mechanical performance of the deck of the bridge of the prestressed short rib T-beam bridge. The concept of the method is clear, the calculation is convenient, and it is more suitable for the application of engineering design; The calculation results of the continuous one-way slab method are too large to be safe for design.

Keywords

Bridge engineering short rib T-beam lane slab space finite element analysis elastic support continuous beam analysis

1. Introduction

With the rapid development of high-grade highways, it has become an inevitable trend to require bridges with lightweight structure and beautiful appearance [1]. Therefore, in order to reduce the average filling height of roadbed, reduce the number of earthwork, reduce land occupation and save project cost, etc. [2, 3, 4], The prestressed concrete short rib T-beam bridge is put forward. The prestressed concrete short rib T-beam has the advantages of convenient construction, reasonable stress, good durability and beautiful appearance [5, 6, 7], however, due to the small height of the structure, the torsional stiffness of the main girder is poor [8, 9], The poisitive mid-span moment of the bridge deck is correspondingly increase [10, 11, 12, 13]. At present, many theoretical discussions have been made on the calculation of bridge deck of different types of bridges [14, 15, 16, 17, 18, 19, 20], however, there is still a lack of theoretical support for the calculation of the bridge deck of this short rib T-beam bridge structure [21, 22]. In this paper, the spatial finite element analysis method, the finite element elastic support continuous beam method and the continuous one-way slab method are used to analyze the mechanical performance of the bridge deck of the prestressed concrete short rib T-beam bridge, and the calculation results are compared with the experimental results to explore the reasonable algorithm of the bridge deck of the prestressed concrete short rib T-beam bridge [23, 24, 25].

2. Experimental research

Taking Keshan 4 $\times$ 16 m simple-supported prestressed concrete short T-beam bridge as the test object, The cross section of the bridge is composed of five T-beams with short ribs, the calculated span is 14.96 m. A diaphragm beam is set at the end and middle of the bridge,the width of the diaphragm beam is 0.3 m, the concrete material is C50 concrete, $\Phi^{j}$ 15.2 (7 $\Phi$ 5) 1860 steel strand is used for prestressed reinforcement. See Table 1 for performance indexes of materials and Fig. 1 for main beam size. Under the condition that the loading position and the loading force are consistent with the actual loading, the bridge is tested under static load with 55T heavy vehicle, and the distance between the wheel centers on both sides is 1.8 m.

Table 1
Properties of materials

Material performance parameters
Concrete	Modulus of elasticity $E_{c}$ (Mpa)	Standard value of tensile strength $f_{tk}$ (Mpa)	Design value of tensile strength $f_{td}$ (Mpa)
	34500	2.65	1.83
	Standard value of compressive strength $f_{ck}$ (Mpa)	Design value of compressive strength $f_{cd}$ (Mpa)
	32.4	22.4
Steel strand	Diameter $d$ (mm)	Modulus of elasticity $E_{p}$ (Mpa)	Standard value of tensile strength $f_{pk}$ (Mpa)
	15.2	195000	1860
	Design value of tensile strength $f_{pd}$ (Mpa)
	1260

Figure 1.

Main beam dimensions.

2.1 Test loading condition

This test is divided into two working conditions for loading. Working condition 1: load the positive bending moment at the middle position of the side span of the bridge deck. When loading, the rear axle of the vehicle is 4 m away from the middle position of the bridge span, the front of the vehicle is facing the pier direction, and the rear of the vehicle is facing the middle position of the bridge span, so that the wheel center at one side of the rear axle of the vehicle is pressed at the middle position of the side span of the bridge deck. Working condition 2: load the positive bending moment at the middle position of the secondary side span of the bridge deck. When loading, the rear axle of the vehicle is 4 m away from the middle position of the bridge span, the front of the vehicle is facing the pier direction, and the rear of the vehicle is facing the middle position of the bridge span, so that the wheel center at one side of the rear axle of the vehicle is pressed at the middle position of the secondary side span of the bridge deck. As the effective working width a $=$ 0.97 m, the effective working width does not overlap, so the influence of the front axle and the middle axle of the vehicle on the calculation of the internal force of the bridge deck is ignored, so the loading diagram only shows the loading position of the rear wheel of the vehicle, as shown in Figs 2–4.

Figure 2.

Longitudinal loading position of test (m).

Figure 3.

Transverse loading position of vehicle under condition 1 (mm).

Figure 4.

Transverse loading position of vehicle under condition 2 (mm).

Figure 5.

Layout of measuring points (mm).

2.2 Measuring point arrangement

Under the action of vehicle load, when testing the stress in the middle of the bridge deck span, a strain gauge shall be arranged at the bottom of the slab in the middle of the side span and the secondary side span of the bridge deck. When testing the stress at the fulcrum of the bridge deck, a strain gauge shall be arranged at the top of the slab at the No. 2 and No. 3 fulcrums of the carriageway slab, as shown in Fig. 5.

3. Analytical method

3.1 Space finite element analysis

3.1.1 Model building

Considering that the bridge deck is a slab structure, it is more reasonable and accurate to use the plate element in the modeling. According to the actual structure, the beam element is used for the beam rib and diaphragm beam, and the slab element is used for the bridge deck, the beam rib and the bridge deck are rigidly connected, set the diaphragm according to the actual size at the fulcrum and midspan position, and the diaphragm is orthogonal to the main beam, the boundary condition is general support, and the whole bridge is divided into 252 elements, the space finite element diagram and model diagram are shown in Figs 6 and 7. The calculated load is input in the form of surface load. The side length of rectangular surface load is b1 $=$ 0.86 m, a1 $=$ 0.46 m, and the area load concentration is 110 kN/(0.86 m $\times$ 0.46 m) $=$ 278 kN/m ${}^{2}$ , see Figs 8 and 9 for the corresponding calculated load position of each working condition.

Figure 6.

Schematic diagram of plate-beam composite unit.

Figure 7.

Profile of plate-beam composite unit model.

3.1.2 Comparative analysis of models

In order to ensure the correctness and accuracy of the established model, the theoretical analysis results of side beams under self weight are calculated according to the design size of structural members and the gravity density of materials, and the results are compared with the finite element analysis results, as shown in Tables 2 and 3.

Table 2
Comparison of internal forces under dead load

Method	Midspan		Quarter point
	$M$ /KN $\cdot$ m	$Q$ /KN	$M$ /KN $\cdot$ m	$Q$ /KN
Finite element method	515.8	0.09	372.6	58.3
Theoretical method	539.9	0	394.8	56.6

Table 3

Deflection comparison under dead load

Method	Midspan		Quarter point
	Deflection/mm	Difference value/%	Deflection/mm	Difference value/%
Finite element method	13.90	1.42	6.93	3.75
Theoretical method	14.1		7.20

Figure 8.

Calculation load position diagram of condition 1 m.

Figure 9.

Calculation load position diagram of condition 2 m.

It can be seen from Tables 2 and 3 that the finite element analysis results of the composite element of the mining plate beam are in good agreement with the calculation results of the theoretical formula, which verifies the correctness of the finite element model and can be used for the mechanical performance analysis of the completed bridge.

3.2 Finite element elastic support continuous beam method

The bridge deck is an elastic support slab supported on the diaphragms and main girder ribs. In order to facilitate the engineering design, a finite element method of elastic support continuous beam is innovatively proposed, which is used for the stress calculation of the bridge deck. The key of this method is the correct calculation of the vertical and torsional stiffness of the elastic support. Due to the huge workload of theoretical methods, it is difficult to apply them to the actual production. Therefore, the following describes the method of calculating spring stiffness by using the finite element model and the method of establishing the finite element elastic support continuous beam model.

3.2.1 Calculation of vertical stiffness of spring by finite element method

In this paper, the calculation location of the carriageway slab is 4 m away from the mid span. Using the spatial finite element model established in Chapter 2.1, add the vertical force of $F=$ 100 kN on the bridge deck node corresponding to the beam rib, respectively calculate the vertical deflection $N_{i}$ of the corresponding point when each vertical force acts independently, and obtain the vertical stiffness of each spring through $F/N_{i}$ . As shown in Fig. 10.

$\displaystyle K_{V}=\frac{F}{N_{i}}$ (1)

Figure 10.

Calculation of vertical stiffness of the spring.

Figure 11.

A sketch of spring torsional stiffness calculation.

3.2.2 Calculation of torsion stiffness of spring by finite element method

Using the spatial finite element model established in Chapter 2.1, a torque of 100 kN $\cdot$ m is applied on the bridge deck node corresponding to the beam rib, and the bending moment value $M_{i}$ of one linear meter plate width at the acting position when each torque acts independently is calculated by using the model, then calculate the rotation angle $R_{i}$ of the corresponding action point around the longitudinal direction of the bridge when each torque acts alone, $M_{i}/R_{i}$ to obtain the torsional stiffness of each spring. The calculation diagram is shown in Fig. 11.

$\displaystyle K_{T}=\frac{M_{i}}{R_{i}}$ (2)

3.2.3 Theoretical calculation method of spring stiffness

Using arch support can more accurately simulate the elastic support effect of the main beam on the bridge deck. The thrust stiffness of the arch support is used to simulate the vertical spring stiffness of the main beam, and the bending stiffness is used to simulate the torsional spring stiffness. Take the calculation formula shown in Fig. 12a, Then there is a reaction $R_{i}$ and a reaction moment $T_{i}$ at any arch support, as shown in Fig. 12b, they are the vertical load and torque load on the main beam $i$ . $EI_{Q}$ in Fig. 12a is the flexural rigidity of the transverse structure (flange plate) within the effective longitudinal width of the beam bridge.

Figure 12.

Calculation diagram of continuous beam method with elastic support.

Figure 13.

Beam element calculation drawing.

Figure 13 shows the beam unit I. for beam unit I, if the coordinate system shown in Fig. 13 is used, the virtual work principle is used, and the internal displacement is eliminated by condensation method, then the

$\displaystyle\{F\}^{e}=[{{k}_{o}}]^{e}\{\delta\}^{e}$ (3)

$\{F\}^{e}=[{Q_{i},M_{i},Q_{j},M_{j}}]^{\text{T}}$ is the bar end force matrix; $\{\delta\}^{e}$ is the element node displacement matrix; $[{{k}_{o}}]^{e}$ is the element stiffness matrix, its expression is

$\displaystyle[{{k}_{o}}]^{e}=\frac{1}{E_{1}}\left[{{\begin{array}[]{*{20}c}{% \frac{2}{{b}}}&1&{-\frac{2}{{b}}}&1\\ {\textit{Symmetry}}&{E_{3}}&{-1}&{E_{2}}\\ &{\textit{Symmetry}}&{\frac{2}{{b}}}&{-1}\\ &&{E_{3}}&\\ \end{array}}}\right]$ (4)

Among

$\displaystyle E_{1}=\frac{{b}^{2}({1+12\mu})}{6EI_{o}},E_{2}=\frac{({1-6\mu}){% b}}{3}$ $\displaystyle E_{3}=\frac{2{b}({1+3\mu})}{3},\mu=\frac{EI_{o}}{GA_{s}{b}^{2}}$

In the above formulas, $EI_{o}$ is the bending rigidity of the continuous beam with generalized elastic support; $G$ is the shear modulus; As is the effective shear area of the section, As $=$ Af (Af is the web area); $b$ is the element length; $\mu$ is the parameter considering the shear deformation, if the shear deformation is not considered, then $\mu=$ 0. The element matrix is superposed into the total stiffness matrix [ $k$ ] in the order of elements, and the regular equation of matrix displacement method is obtained as follows

$\displaystyle[K]_{2{n}\times 2{n}}[\Delta]_{2{n}\times 2{n}}=[P]_{2{n}\times{n}}$ (5)

In the formula, $[P]_{2n\times n}$ is the total load matrix, and the boundary condition treatment is the same as the general matrix displacement method. From the above formula, it can be found that the displacement matrix $[\Delta]_{2n\times n}$ of the summary point is

$\displaystyle[\Delta]=[K]^{-1}[P]$ (6)

Therefore, when $p=$ 1 acts on any main beam $i$ , the vertical load and torque distributed by main beam $i$ can be obtained from the following two formulas respectively

$\displaystyle R_{{ki}}={k}_{\textit{pwk}}{w}_{{ki}}+{k}_{{p}\theta{k}}\theta_{% {ki}}(k,i=1,2,\ldots,n)$ (7) $\displaystyle T_{{ki}}={k}_{T\theta{k}}\theta_{{ki}}+{k}_{\textit{Twk}}{w}_{{% ki}}(k,i=1,2,\ldots,n)$ (8)

Thus, the vertical stiffness and torsional stiffness of the spring can be calculated.

3.2.4 Comparative analysis of spring stiffness

In order to ensure the correctness and accuracy of the spring stiffness calculated by the finite element method, the finite element calculation results are compared with the theoretical calculation results, as shown in Tables 4 and 5.

Table 4
Vertical stiffness of spring

Main beam number	Finite element method (kN/mm)	Theoretical method (kN/mm)
1#	50.3	49.3
2#	80.1	78.2
3#	94.6	91.6
4#	80.1	78.2
5#	50.3	49.3

Table 5

Torsional stiffness of spring

Main beam number	Finite element method (kN $\cdot$ m/rad)	Theoretical method (kN $\cdot$ m/rad)
1#	17000	16679
2#	26664	25966
3#	27907	26877
4#	26664	25966
5#	17000	16679

From Tables 4 and 5, it can be seen that the analysis results of the finite element method are in good agreement with the theoretical calculation results, which verifies the correctness of the calculation of spring stiffness by the finite element method, and can be used to calculate the spring stiffness of the elastic supported continuous beam.

3.2.5 Establishment of continuous beam model with elastic support

The beam element is used to establish the finite element model of the continuous beam with elastic support. The boundary condition is the form of point spring, and the spring stiffness is calculated according to the results in Sections 2.2.1 and 2.2.2. The calculated load is applied to the model in the form of uniform load of beam element, and the distribution length of uniform load is $b_{1}=$ 0.86 m. The load concentration is 110 kN/0.97 m $=$ 113 kN/m, The effective working width $a_{1}=$ 0.97 m and the height $h=$ 0.16 m of the section beam are taken as the width of the section beam. See Fig. 14 for the schematic diagram of the elastic support continuous beam and Fig. 15 for the calculation load application position of each working condition.

Figure 14.

Schematic diagram of elastically supported continuous beam.

Figure 15.

Calculation load application position diagram of condition 1 and 2.

3.3 Continuous one-way slab method

The length width ratio of the bridge deck is La/Lb $=$ 7.48. According to the reference [2], the torsional capacity of the main beam with t/h $=$ 0.35 is poor. The maximum design bending moment in the span under the load is calculated according to the following formula.

Pivot negative moment:

$\displaystyle M=-0.7M_{{op}}$ (9)

Medium-span bending moment:

$\displaystyle M=+0.7M_{{op}}$ (10) $\displaystyle M_{0{p}}=({1+\mu})\cdot\frac{P}{8{a}}\cdot\left({{L-}\frac{{b}_{% 1}}{2}}\right)$ (11)

among them $P=$ 220 kN, $a=$ 0.97 m, $b_{1}=$ 0.86 m, $L=$ 1.6 m, $\mu=$ 0.3. $P$ : Automotive axle load; $a$ : Effective working width of lane board; $b_{1}$ : Transverse side length of rectangular load acting on the top of concrete bridge deck under wheel load; $L$ : Distance between girders; $\mu$ : impact coefficient.

4. Calculation results and comparative analysis

In order to verify the correctness of the analysis method of the bridge deck of the prestressed short rib T-beam bridge, the theoretical calculation results are compared with the experimental results under the same technical standards and loading conditions.

See Tables 6 and 7 for the theoretical calculation and test results of stress at each measuring point under the load of condition 1 and 2.

Table 6
Stress comparison of working condition 1

Measuring point	Experimental study		Continuous beam method with finite element elastic support		Space finite element method		Continuous one-way slab method
	Stress (MPa)	Mean square deviation	Stress (MPa)	Error (%)	Stress (MPa)	Error (%)	Stress (MPa)	Error (%)
BY1	3.42	0.021	3.70	8.2	3.61	5.5	7.0	104.7
	3.45			7.2		4.6		102.9
	3.40			8.8		6.2		105.9
BY2	$-$ 2.70	0.026	$-$ 2.90	7.4	$-$ 2.48	8.1	$-$ 7.0	160.3
	$-$ 2.65			9.4		7.5		164.2
	$-$ 2.65			9.4		6.4		164.2

Note: the negative number in the table represents the tensile stress.

It can be seen from Table 6 that the average value of BY1 measurement result is $-$ 3.423, the range value is 0.05, the mean square difference is 0.021, the dispersion coefficient is 0.006, the average value of BY2 measurement result is $-$ 2.67, the range value is 0.05, the mean square difference is 0.026, and the dispersion coefficient is 0.009. The range value of the two measuring points is less than 0.05, the mean square difference is less than 0.026, and the dispersion coefficient is less than 0.0097. The analysis of the above errors can show that the test results of BY1 and BY2 measuring points are small in dispersion and accurate in data, and the test research of condition 1 is repeatable. At this time, the carriageway slab at BY1 section bears positive bending moment, and the carriageway slab at BY2 section bears negative bending moment, this shows that the carriageway slab is elastically supported on the main beam, and the loading position is on both sides of the No. 2 fulcrum under the loading condition 1, the vertical displacement of the carriageway slab on both sides of the No. 2 fulcrum is large, and the vertical displacement of the carriageway slab at the No. 2 fulcrum is small, which makes the carriageway slab at the No. 2 fulcrum bear the negative bending moment.

Table 7

Stress comparison of working condition 2

Measuring point	Experimental study		Continuous beam method with finite element elastic support		Space finite element method		Continuous one-way slab method
	Stress (MPa)	Mean square deviation	Stress (MPa)	Error (%)	Stress (MPa)	Error (%)	Stress (MPa)	Error (%)
BY3	3.55	0.017	3.36	5.3	3.66	3.1	7.0	97.2
	3.51			4.3		4.3		99.4
	3.54			5.1		3.4		97.7
BY4	$-$ 3.59	0.026	$-$ 3.44	4.2	$-$ 3.53	1.7	$-$ 7	95.0
	$-$ 3.60			4.4		1.9		94.4
	$-$ 3.54			2.8		0.3		97.7

It can be seen from Table 7 that the average value of BY3 measurement result is $-$ 3.53, the range value is 0.04, the mean square difference is 0.017, the dispersion coefficient is 0.0048, the average value of BY4 measurement result is $-$ 3.58, the range value is 0.06, the mean square difference is 0.026, and the dispersion coefficient is 0.007. The range value of the two measuring points is less than 0.06, the mean square difference is less than 0.026, and the dispersion coefficient is less than 0.007. The analysis of the above errors can show that the test results of BY3 and BY4 measuring points are small in dispersion and accurate in data, and the test research of condition 2 is repeatable. The test results can be used for comparative analysis with the theoretical calculation results. Because the loading position of condition 2 is close to the symmetrical loading, the bridge width is small, and the vertical displacement of each main beam is almost the same, the positive bending moment value in the middle of the span is very close to the negative bending moment value of the fulcrum when the middle of the span is loaded on the carriageway slab, so the test results of BY3 and BY4 are close.

Figure 16.

Comparative analysis of stress value of each measuring point.

It can be seen from Tables 6 and 7 and Fig. 16 that there is no significant difference between the results calculated by the finite element analysis method and the spatial finite element analysis method. The error mean of the calculation results of all measuring points by the spatial finite element analysis method is 4.42, and the error mean of the calculation results of all measuring points by the finite element analysis method is 6.38, it shows that the spatial finite element analysis method is more close to the test results, and the calculation results are more consistent with the actual. Theoretically, the spring stiffness of the elastic supported continuous beam is calculated by using the spatial finite element model, and the calculation results of the two should be completely unified. However, there are inevitably some systematic errors in the simulation process. It is precisely because of these errors that the calculation accuracy of the elastic supported continuous beam method is slightly lower than that of the spatial finite element analysis method. Comparing the calculation results of the finite element method with the test results, we can see that the spring stiffness is reasonable, and the method used to calculate the spring stiffness in this paper is feasible. The calculation results of continuous one-way slab method are much larger than the test results. The reason for this phenomenon is that the analysis method is partial to safety consideration, and a large safety factor is set in the calculation.

5. Innovation

1.
Theoretical research and innovation are made on the calculation method of the bridge deck of the prestressed short rib T-beam bridge, and the rationality of the method is verified by the completion test.
2.
The bridge deck is simplified as a continuous beam with elastic support on the main beam rib, and the finite element method of elastic support continuous beam is innovatively proposed for the stress calculation of the bridge deck.

6. Conclusion

Through the comparative analysis of theoretical calculation results and test results, the following conclusions are drawn:

The new finite element analysis method of elastic support continuous beam can accurately calculate the mechanical properties of the deck of the bridge of the prestressed short rib T-beam bridge. The concept of the method is clear, the calculation is convenient, and it is more suitable for the application of engineering design.

The method of using space finite element model to calculate the spring stiffness of continuous beam with elastic support is put forward. The result is accurate and practical.

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Calculation methods of bridge deck for prestressed short rib T-beam bridges

Abstract

Keywords

1. Introduction

2. Experimental research

Table 1 Properties of materials

3. Analytical method

3.1 Space finite element analysis

3.1.1 Model building

Table 2 Comparison of internal forces under dead load

3.2.1 Calculation of vertical stiffness of spring by finite element method

Table 4 Vertical stiffness of spring

Table 6 Stress comparison of working condition 1

References

Table 1
Properties of materials

Table 2
Comparison of internal forces under dead load

Table 4
Vertical stiffness of spring

Table 6
Stress comparison of working condition 1