# A Mathematical Model to Simulate Static Characteristics of T-Beam Bridge with Wide Flange

Authors: Xiaomeng Zang; Genhui Wang; Jianchang Li; Rongcheng Hou; Yanan Gan; Junjie Wang; Wei Zhang
Journal: Journal of Mathematics
 (2021)

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## Abstract

This study considers various factors, such as shear lag effect and shear deformation, and introduces the self-stress equilibrium for shear lag warping stress conditions to analyze the static characteristics of T-beam bridges accurately. In the mechanical analysis, three generalized displacement functions are applied, and the governing differential equations and natural boundary conditions of the static characteristics of T-beams are established on the basis of the energy variational principle. In the example, the influences of the shear lag effect, different load forms, and span ratio on the mechanical properties of T-beam bridges are analyzed. Therefore, the method of this study enriches and develops the theoretical analysis of T-beams, and it plays a certain guiding role in designing such a structure.


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## Body

## 1. Introduction

Reinforced concrete T-beam bridges were widely used in the 1970s and 1980s because of their simple force and convenient design and construction [1–5]. In recent years, fabricated concrete T-beam bridges have been widely utilized in the construction of expressways and railways in China, such as the approach bridge of Qingshan Yangtze River Highway Bridge in Wuhan and the bridge of Chongqing Chengkai Expressway. However, with the increase in highway and railway traffic, such structures can be seriously damaged, such as cracks on the flange and web, which reduce their overall stiffness and further deteriorate their mechanical properties [6–9]. For example, in 773 overpasses and river crossing bridges in Beijing, most T-beam bridges have cracks, water seepage corrosion, and concrete spalling. Therefore, the refined mechanical analysis of T-beam bridges has theoretical and engineering significance.Currently, scholars have conducted intensive research on T-beam bridges, but structural diseases remain serious, which shows that bridge workers still lack understanding of the mechanical properties of the T-beam bridge. New mechanical conditions should be considered when studying the mechanical properties of such structures [10–15]. Referring to the existing references, it is found that the shear lag warpage stress, moment self-balance condition, and Timoshenko shear deformation are not considered simultaneously in the analysis of mechanical properties of this kind of structure, which leads to the limitation of the calculation results. Therefore, in the present study, various factors, such as shear lag warping self-stress equilibrium, shear lag, and shear deformation, are considered comprehensively [16–20]. The governing differential equations and natural boundary conditions of the T-beam bridge are established via the energy variational method. The effects of shear lag, span ratio, and boundary conditions on mechanical properties are analyzed using examples. According to the current external environmental conditions and bridge disease characteristics, the method adopted in this paper is to enrich and develop the existing theory of thin-walled structure analysis [21–24] and has certain guiding significance for such structural design.


## 2. Control Differential Equation and Natural Boundary Conditions of T-Beam Bridge

### 2.1. Setting of the Longitudinal Warping Displacement Function of T-Beam Flange

The force system shown in Figure1 acts on the T-beam shown in Figure 2. If the structure span is L, then the symmetrical bending state wz,θz represents the vertical deflection and vertical angle of the T-beam section of elementary beam theory, respectively, and vjz is the vertical deflection caused by the shear lag effect of the T-beam. Then, the longitudinal displacement of the T-beam flange uz is the sum of the theoretical value of the elementary beam and the longitudinal hysteresis displacement of the wing caused by the shear hysteresis effect.Figure 1
Coordinate and load system.Figure 2
Cross section of T-beams.The longitudinal displacement of the T-beam flange plate is as follows:(1)uzx,y,z=yθ+y−αφx−βvj′,where φx=1−x−tw/22/b2 is the nonuniform distribution function of the T-beam flange and α and β are correction coefficients of the flange satisfying the self-stress equilibrium for the shear lag warping stress. Among them, tw/2≤x≤b+tw/2.Then, the shear lag stress of the T-beam flange is as follows:(2)σYB=Ey−αφxvj″−Eβvj″,where α,β are the constant coefficients obtained to satisfy ∫AσYBdA=0 and ∫AσYBydA=0: α=3I/4h1bt,β=−I/h1A.The longitudinal displacement of the T-beam web is as follows:(3)ufx,y,z=yθ+y−βvj′.The shear stress of the T-beam flange is used as an independent stress system that satisfies the self-stress equilibrium for the shear lag warping stress. In other words, the T-shaped section will simultaneously satisfy the balance of shear lag and warpage self-stress and the bending moment.


### 2.2. Total Potential Energy of a T-Beam Bridge

The normal and shear stresses of the T-beam flange are expressed as(4)σZY=Eyθ′+Ey−αφvj″−Eβvj″,τj=G∂uz∂x=−Gα∂φ∂xvj′.The normal stress of the T-beam web is obtained as(5)σZF=Eyθ′+Ey−βvj″.The deformation potential energy of T-beam bridges is calculated as follows:The strain energies of the flange and web are obtained as(6)Uz1=12∬σZY2E+τj2G+σZF2EdAdz=12∫0lEIθ′2dz+12∫0l2EI1θ′vj″dz+12∫0lEI2vj″2dz+12∫0lGIGvj′2dz,where I=∫A1y2dA1+∫Afy2dAf;I1=0;I2=I+β2A+16/15α2bt1+8/3αbt1β−h1;IG=8α2t1/3b.Timoshenko’s shear strain energy is expressed as(7)UG=12∫0lkGAw′−θ2dz.The load potential energy is measured as(8)Up=−∫0lqyw+vjdz−Qw+vj|0l−M1vj′+Mzθ|0l.Then, the total potential of the system is as follows:(9)U=Uz1+UG+Up,where M1z is the bending moment of the x-axis generated by the shear lag effect of the T-beam flange; Mzz is the bending moment of the x-axis when the beam end produces a vertical corner of θz; Qz,qyz are the vertical shear force at the beam end and the vertical distribution force on the T-beam, respectively; E,G are Young’s and shear moduli of the T-beam material, respectively; A1,Af are the cross-sectional areas of the T-beam flange and web, respectively, in which A=A1+Af; and I is the moment of inertia of the T-beam of the x-axis.


### 2.3. Control Differential Equations and Natural Boundary Conditions of T-Beams

On the basis of the variational principle, the governing differential equation of the T-beam can be derived as(10)EIθ″+kGAw′−θ=0,(11)kGAw″−θ′+qy=0,(12)EI2vj4−GIGvj″−qy=0.The natural boundary conditions are as follows:(13)EIθ′−Mz|0lδθ=0,(14)kGAw′−θ−Q|0lδw=0,(15)EI2vj″−M1|0lδvj′=0,(16)EI2vj3+GIGvj′−Q|0lδvj=0.By substituting differential equations (10) and (11), wz and θz are calculated as(17)wz=c1z3+c2z2+c3z+c4+qy24EIz4,θz=c13z2+6EIkGA+c22z+c3+qykGAz+qy6EIz3,where c1, c2, c3, andc4 are the constant coefficients obtained in accordance with the corresponding boundary conditions of wz and θz.Similarly, by substituting differential equation (12),(18)vj4+−GIGEI2vj″+−1EI2qy=0.The solution of its characteristic equation isr1,2=±η.Therefore, differential equation (18) is calculated as(19)vjz=s1chηz+s2shηz+s3z+s4+−qy2GIGz2,where s1, s2, s3, and s4 are the constant coefficients obtained in accordance with the corresponding boundary conditions of vjz.On the basis of differential equations (10)–(12), the T-beam mechanical properties are composed of two independent mechanical systems, namely, the superposition of elementary beam theory and the shear lag theory system.


## 2.1. Setting of the Longitudinal Warping Displacement Function of T-Beam Flange

The force system shown in Figure1 acts on the T-beam shown in Figure 2. If the structure span is L, then the symmetrical bending state wz,θz represents the vertical deflection and vertical angle of the T-beam section of elementary beam theory, respectively, and vjz is the vertical deflection caused by the shear lag effect of the T-beam. Then, the longitudinal displacement of the T-beam flange uz is the sum of the theoretical value of the elementary beam and the longitudinal hysteresis displacement of the wing caused by the shear hysteresis effect.Figure 1
Coordinate and load system.Figure 2
Cross section of T-beams.The longitudinal displacement of the T-beam flange plate is as follows:(1)uzx,y,z=yθ+y−αφx−βvj′,where φx=1−x−tw/22/b2 is the nonuniform distribution function of the T-beam flange and α and β are correction coefficients of the flange satisfying the self-stress equilibrium for the shear lag warping stress. Among them, tw/2≤x≤b+tw/2.Then, the shear lag stress of the T-beam flange is as follows:(2)σYB=Ey−αφxvj″−Eβvj″,where α,β are the constant coefficients obtained to satisfy ∫AσYBdA=0 and ∫AσYBydA=0: α=3I/4h1bt,β=−I/h1A.The longitudinal displacement of the T-beam web is as follows:(3)ufx,y,z=yθ+y−βvj′.The shear stress of the T-beam flange is used as an independent stress system that satisfies the self-stress equilibrium for the shear lag warping stress. In other words, the T-shaped section will simultaneously satisfy the balance of shear lag and warpage self-stress and the bending moment.


## 2.2. Total Potential Energy of a T-Beam Bridge

The normal and shear stresses of the T-beam flange are expressed as(4)σZY=Eyθ′+Ey−αφvj″−Eβvj″,τj=G∂uz∂x=−Gα∂φ∂xvj′.The normal stress of the T-beam web is obtained as(5)σZF=Eyθ′+Ey−βvj″.The deformation potential energy of T-beam bridges is calculated as follows:The strain energies of the flange and web are obtained as(6)Uz1=12∬σZY2E+τj2G+σZF2EdAdz=12∫0lEIθ′2dz+12∫0l2EI1θ′vj″dz+12∫0lEI2vj″2dz+12∫0lGIGvj′2dz,where I=∫A1y2dA1+∫Afy2dAf;I1=0;I2=I+β2A+16/15α2bt1+8/3αbt1β−h1;IG=8α2t1/3b.Timoshenko’s shear strain energy is expressed as(7)UG=12∫0lkGAw′−θ2dz.The load potential energy is measured as(8)Up=−∫0lqyw+vjdz−Qw+vj|0l−M1vj′+Mzθ|0l.Then, the total potential of the system is as follows:(9)U=Uz1+UG+Up,where M1z is the bending moment of the x-axis generated by the shear lag effect of the T-beam flange; Mzz is the bending moment of the x-axis when the beam end produces a vertical corner of θz; Qz,qyz are the vertical shear force at the beam end and the vertical distribution force on the T-beam, respectively; E,G are Young’s and shear moduli of the T-beam material, respectively; A1,Af are the cross-sectional areas of the T-beam flange and web, respectively, in which A=A1+Af; and I is the moment of inertia of the T-beam of the x-axis.


## 2.3. Control Differential Equations and Natural Boundary Conditions of T-Beams

On the basis of the variational principle, the governing differential equation of the T-beam can be derived as(10)EIθ″+kGAw′−θ=0,(11)kGAw″−θ′+qy=0,(12)EI2vj4−GIGvj″−qy=0.The natural boundary conditions are as follows:(13)EIθ′−Mz|0lδθ=0,(14)kGAw′−θ−Q|0lδw=0,(15)EI2vj″−M1|0lδvj′=0,(16)EI2vj3+GIGvj′−Q|0lδvj=0.By substituting differential equations (10) and (11), wz and θz are calculated as(17)wz=c1z3+c2z2+c3z+c4+qy24EIz4,θz=c13z2+6EIkGA+c22z+c3+qykGAz+qy6EIz3,where c1, c2, c3, andc4 are the constant coefficients obtained in accordance with the corresponding boundary conditions of wz and θz.Similarly, by substituting differential equation (12),(18)vj4+−GIGEI2vj″+−1EI2qy=0.The solution of its characteristic equation isr1,2=±η.Therefore, differential equation (18) is calculated as(19)vjz=s1chηz+s2shηz+s3z+s4+−qy2GIGz2,where s1, s2, s3, and s4 are the constant coefficients obtained in accordance with the corresponding boundary conditions of vjz.On the basis of differential equations (10)–(12), the T-beam mechanical properties are composed of two independent mechanical systems, namely, the superposition of elementary beam theory and the shear lag theory system.


## 3. Natural Boundary Conditions Commonly Used for T-Beams

The specific boundary conditions of elementary beam theory can be obtained using equations (13) and (14). The commonly used boundary conditions are as follows.


### 3.1. Boundary Conditions forwz and θz

①
Uniform load:(20)wz|0l=0;θ′z|0l=0.②
Concentrated load:For a simple-supported T-beam, if the force applied between the spans includes one or more concentrated forces (Figure3) and the distances between adjacent forces of the concentrated force Pk are Lk1 and Lk2, the subscripts of wz and θz represent z1 or z2 coordinates. Therefore, the following continuous boundary conditions should be introduced at the k-point:(21)w1Lk1=w20,w1′Lk1=w2′0,θ1′Lk1=θ2′0,θ1Lk1−θ10=PkkGA.Figure 3
Coordinate and load system of T-beams.Similarly, the boundary conditions of shear lag theory can be obtained using equations (15) and (16). The commonly used boundary conditions are as follows.


### 3.2. Boundary Conditions forvjz

①
Uniform force:(22)vjz|0l=0;vj″z|0l=0.②
Concentrated force:If the force applied between the spans includes one or more concentrated forces (Figure3) and the subscript number of vj indicates whether it is in the z1 or z2 coordinate system, then the k-point should also introduce the boundary conditions as(23)vj10=0,vj1Lk1=vj20,vj1′Lk1=vj2′0,vj1″Lk1=vj2″0,vj2Lk2=0,vj1″0=0,vj2″Lk2=0,vj13Lk1−vj230=−PkEI2.


## 3.1. Boundary Conditions forwz and θz

①
Uniform load:(20)wz|0l=0;θ′z|0l=0.②
Concentrated load:For a simple-supported T-beam, if the force applied between the spans includes one or more concentrated forces (Figure3) and the distances between adjacent forces of the concentrated force Pk are Lk1 and Lk2, the subscripts of wz and θz represent z1 or z2 coordinates. Therefore, the following continuous boundary conditions should be introduced at the k-point:(21)w1Lk1=w20,w1′Lk1=w2′0,θ1′Lk1=θ2′0,θ1Lk1−θ10=PkkGA.Figure 3
Coordinate and load system of T-beams.Similarly, the boundary conditions of shear lag theory can be obtained using equations (15) and (16). The commonly used boundary conditions are as follows.


## 3.2. Boundary Conditions forvjz

①
Uniform force:(22)vjz|0l=0;vj″z|0l=0.②
Concentrated force:If the force applied between the spans includes one or more concentrated forces (Figure3) and the subscript number of vj indicates whether it is in the z1 or z2 coordinate system, then the k-point should also introduce the boundary conditions as(23)vj10=0,vj1Lk1=vj20,vj1′Lk1=vj2′0,vj1″Lk1=vj2″0,vj2Lk2=0,vj1″0=0,vj2″Lk2=0,vj13Lk1−vj230=−PkEI2.


## 4. Analysis of the Example of T-Beam Bridge

The material and geometric parameters for C50 reinforced concrete T-beam are as follows: elastic modulusE=3.5×104MPa, shear modulus G=1.5×104MPa, thickness of web tw=0.2m, thickness of wing plate t=0.1m, length of wing plate b=2.4m, and T-beam height h=1.2m. In the mechanical analysis, the uniform force qyz=9800N/m and the concentrated force Pkz=L×9800N, where L represents the simply supported beam spans or the sum of continuous beam spans. The concentrated force is applied to the simply supported beam midspan or the midspan of one of the continuous beams (Figure 4). Furthermore, the stress of the T-beam flange and the longitudinal deflection of the T-beam are calculated using the derivation formula in this study and other algorithms (note: in the calculation of ANSYS finite element method, the T-beam section is drawn in accordance with the intersection coordinates of the T-beam in Figure 2; then, the extrude function of ANSYS finite element is used to form the body; after dividing the cell grid, the simple or continuous boundary conditions are simulated to impose the corresponding constraints on the positions of T-beams for forming simple or continuous boundary conditions).Figure 4
Boundary conditions and load systems for the mechanical analysis of T-beams: (a) two-span continuous T-beam with concentrated load; (b) simply supported T-beam with concentrated load; (c) two-span continuous T-beam with uniform load; (d) simply supported T-beam with uniform load.
(a)(b)(c)(d)Tables1 and 2 and Figure 5 illustrate the following:(1)
For the simply supported T-beam bridge, the effect of the concentrated load is greater than that of uniform load under the same span, and the smaller the span width ratio, the stronger the shear lag effect. The shear lag effect of continuous T-beam bridges is more prominent than that of simply supported T-beam bridges. Moreover, the shear lag effect of continuous T-beam bridges with uniform load is greater than that with concentrated loads. Therefore, bridge experts should pay attention to this aspect.(2)
With the introduction of self-stress equilibrium for the shear lag warping stress, the mechanical properties of T-beams are decomposed into independent elementary beam theory and shear lag theory system, which is the innovation of this study. The example further shows that the mechanical properties of T-beam bridges are superimposed by the theoretical values of the elementary beam and the calculated values of shear lag theory.Table 1
Stress of the flange for two-span continuous T-beams (L1=L2=12m) (uniform load).
Horizontal coordinate of flange (m)00.40.81.21.62.02.4Elementary beam theoretical value (104 pa)42.72642.72642.72642.72642.72642.72642.726Shear lag effect value (104 pa)−28.565−26.381−19.978−9.7923.48218.94135.529Total stress value (104 pa)14.16116.34522.74832.93446.20861.66778.255Finite element value (104 pa)13.65715.57921.69131.28743.94558.79874.882Shear lag coefficient0.330.380.530.771.081.441.83Note: the uniform load calculation section is the fulcrum.Table 2
Stress of the flange for two-span continuous T-beams (L1=12m;L2=L3=6m) (concentrated load).
Horizontal coordinate of flange (m)00.40.81.21.62.02.4Elementary beam theoretical value (104 pa)59.68359.68359.68359.68359.68359.68359.683Shear lag effect value (104 pa)−35.569−32.850−24.877−12.1934.33623.58544.241Total stress value (104 pa)24.11426.83334.80647.49164.01983.268103.924Finite element value (104 pa)22.84225.51533.04745.17461.12379.35499.317Shear lag coefficient0.400.450.580.801.071.401.74Note: the calculated section is the fulcrum.Figure 5
Shear lag coefficients for simply supported T-beams.Tables3 and 4 and Figure 6 illustrate the following:(1)
The normal stress of the T-beam web is composed of two parts, namely, the sum of the theoretical value of the elementary beam and the influence of the shear lag effect. As the shear lag effect increases, its influence on the normal stress of the web increases.(2)
Existing research suggests that the normal stress of the T-beam web is only affected by elementary beam theory. The influence of the shear lag effect on the normal stress of the web cannot be ignored. As shown in Table3, the shear lag effect increases the compressive stress of the simply supported T-shaped web by 33.34%. In Table 4, the shear lag effect increases the tensile stress of the continuous T-shaped web by 42.57%. Ignoring the impact will likely cause structural damage; thus, this feature should be of great concern to the architect of the structure.(3)
A stress acts on the neutral axis of T-beam bridge elementary beam theory due to the shear lag effect. As shown in Table4, the stress value is 3.341×105pa. Hence, the neutral axis is not a neutral axis in the actual sense. The axis moves down.Table 3
Stress on webs for simply supported T-beams (L1=L2=6m) (concentrated load).
Vertical coordinates of web (m)0.20.10−0.2−0.4−0.6−0.8−1Elementary beam theoretical value (105 pa)−8.082−4.04108.08216.16524.24732.32940.412Shear lag effect value (105 pa)−4.043−3.542−3.041−2.040−1.038−0.0370.9651.966Total stress value (105 pa)−12.125−7.583−3.0416.04215.12724.21033.29442.378Finite element value (105 pa)−11.434−7.217−2.8625.74814.34523.04831.39639.895Effect of shear lag effect (%)33.3446.71100−33.75−6.86−0.152.904.64Table 4
Stress on webs for two-span continuous T-beams (L1=12m;L2=L3=6m) (concentrated load).
Vertical coordinates of web (m)0.20.10−0.2−0.4−0.6−0.8−1Elementary beam theoretical value (105 pa)5.9683.0310−5.968−12.124−18.185−24.247−30.309Shear lag effect value (105 pa)4.4243.8913.3412.2411.1400.040−1.060−2.160Total stress value (105 pa)10.3926.9223.341−3.727−10.984−18.145−25.307−32.469Finite element value (105 pa)9.7996.4573.143−3.485−10.297−17.058−23.614−30.491Effect of shear lag effect (%)42.5756.21100−60.13−10.38−0.224.196.65Note: the calculated section is the fulcrum.Figure 6
Stress on T-beam webs near the bridge bearing for continuous T-beams (L1=12m,L2=L3=6m, concentrated load).Table5 and Figure 7 illustrate the following:(1)
The vertical deflection of the T-beam bridge increases due to the influence of shear lag effect. Thus, the shear lag effect reduces the vertical stiffness of the T-beam bridge. Therefore, this study has theoretical and engineering practical significance.(2)
Similarly, the vertical deflection of the T-beam bridge is still the superposition of the theoretical value of the elementary beam and the influence of the shear lag effect, and no coupling relationship exists between the two.Table 5
Deflection for simply supported T-beams (z=L/2,L=12m) (uniform load).
Longitudinal coordinates of T-beam (m)0123456Elementary beam theoretical value (10−5m)024.34746.67365.53579.80988.69491.710Shear lag effect value (10−5m)01.3962.6353.6384.3684.8114.959Total deflection value (10−5m)025.74349.30869.17384.17793.50596.669Finite element value (10−5m)024.27147.35666.88581.22690.57993.716Effect of shear lag effect (%)05.425.345.265.195.155.13Figure 7
Deflections for simply supported T-beams (L=12m, uniform load).


## 5. Conclusion

The mechanical concept of this method is clear, and the theoretical foundation is reliable due to the introduction of the self-stress equilibrium for the shear lag warping stress. The method provides a great guiding role for designing simply supported beam or continuous T-beam bridges. In particular, the mechanical properties of the web of T-beam bridges are analyzed using the method of this study. The results show that the webs of such structure are affected by the shear lag effect, which is the main innovation of this research.The method in this paper has certain theoretical significance and engineering practical value. Therefore, it is hoped that the scholars can consider the new mechanical conditions in this paper and improve the calculation accuracy when they design or study the mechanical properties of T-beam bridge in the future.



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