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

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.


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][2][3][4][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][7][8][9]. For example, in 773 overpasses and river crossing bridges in Beijing, most T-beam bridges have cracks, water seepage corrosion, and concrete spalling. erefore, 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][11][12][13][14][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. erefore, in the present study, various factors, such as shear lag warping self-stress equilibrium, shear lag, and shear deformation, are considered comprehensively [16][17][18][19][20]. e governing differential equations and natural boundary conditions of the T-beam bridge are established via the energy variational method. e 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][22][23][24] and has certain guiding significance for such structural design.

Setting of the Longitudinal Warping Displacement
Function of T-Beam Flange. e force system shown in Figure 1 acts on the T-beam shown in Figure 2. If the structure span is L, then the symmetrical bending state w(z), θ(z) represents the vertical deflection and vertical angle of the T-beam section of elementary beam theory, respectively, and v j (z) is the vertical deflection caused by the shear lag effect of the T-beam. en, the longitudinal displacement of the T-beam flange u z 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. e longitudinal displacement of the T-beam flange plate is as follows: 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, (t w /2) ≤ x ≤ (b + (t w /2)).
en, the shear lag stress of the T-beam flange is as follows: where α, β are the constant coefficients obtained to satisfy e longitudinal displacement of the T-beam web is as follows: e 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.

Total Potential Energy of a T-Beam Bridge.
e normal and shear stresses of the T-beam flange are expressed as e normal stress of the T-beam web is obtained as e deformation potential energy of T-beam bridges is calculated as follows: e strain energies of the flange and web are obtained as where Timoshenko's shear strain energy is expressed as e load potential energy is measured as en, the total potential of the system is as follows: where M 1 (z) is the bending moment of the x-axis generated by the shear lag effect of the T-beam flange; M z (z) is the bending moment of the x-axis when the beam end produces a vertical corner of θ(z); Q(z), q y (z) 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; A 1 , A f are the crosssectional areas of the T-beam flange and web, respectively, in which A � A 1 + A f ; and I is the moment of inertia of the T-beam of the x-axis.

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 e natural boundary conditions are as follows: By substituting differential equations (10) and (11), w(z) and θ(z) are calculated as where c 1 , c 2 , c 3 , andc 4 are the constant coefficients obtained in accordance with the corresponding boundary conditions of w(z) and θ(z). Similarly, by substituting differential equation (12), e solution of its characteristic equation is r 1,2 � ± η. erefore, differential equation (18) is calculated as where s 1 , s 2 , s 3 , and s 4 are the constant coefficients obtained in accordance with the corresponding boundary conditions of v j (z).
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.

Natural Boundary Conditions Commonly
Used for T-Beams e specific boundary conditions of elementary beam theory can be obtained using equations (13) and (14). e commonly used boundary conditions are as follows.

Journal of Mathematics
For a simple-supported T-beam, if the force applied between the spans includes one or more concentrated forces ( Figure 3) and the distances between adjacent forces of the concentrated force P k are L k1 and L k2 , the subscripts of w(z) and θ(z) represent z 1 or z 2 coordinates. erefore, the following continuous boundary conditions should be introduced at the k-point: Similarly, the boundary conditions of shear lag theory can be obtained using equations (15) and (16). e commonly used boundary conditions are as follows.

Boundary Conditions for
② Concentrated force: If the force applied between the spans includes one or more concentrated forces ( Figure 3) and the subscript number of v j indicates whether it is in the z 1 or z 2 coordinate system, then the k-point should also introduce the boundary conditions as

Analysis of the Example of T-Beam Bridge
e material and geometric parameters for C50 reinforced concrete T-beam are as follows: elastic modulus E � 3.5 × 10 4 MPa, shear modulus G � 1.5 × 10 4 MPa, thickness of web t w � 0.2 m, thickness of wing plate t � 0.1 m, length of wing plate b � 2.4 m, and T-beam height h � 1.2 m. In the mechanical analysis, the uniform force q y (z) � 9800 (N/m) and the concentrated force P k (z) � L × 9800 N, where L represents the simply supported beam spans or the sum of continuous beam spans. e 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).
Tables 1 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. e 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. erefore, 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. e 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. Figure 6 illustrate the following:

Tables 3 and 4 and
(1) e 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. e influence of the shear lag effect on the normal stress of the web cannot be ignored. As shown in Table 3, 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   Figure 3: Coordinate and load system of T-beams. 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 Table 4, the stress value is 3.341 × 10 5 pa. Hence, the neutral axis is not a neutral axis in the actual sense. e axis moves down. Table 5 and Figure 7 illustrate the following: (1) e vertical deflection of the T-beam bridge increases due to the influence of shear lag effect. us, the shear lag effect reduces the vertical stiffness of the T-beam bridge. erefore, 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

Conclusion
e 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.
e 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. e results show that the webs of such structure are affected by the shear lag effect, which is the main innovation of this research. e method in this paper has certain theoretical significance and engineering practical value. erefore, 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.

Data Availability
All data generated or used during the study appear in the submitted article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.