Components of the laminated floor slab are made in the factory, prefabricated slabs are required to have lapped steel bars to be spliced on-site to form a force-bearing whole. The current conventional method is "beard rib" connection, but because of its "reinforced" structure that it has caused an increase in the process of opening holes, pouring and leak-proof grout sealing, etc. which is not convenient for industrial production. The "beard tendons" are also easy to bend and deform during transportation and hoisting, and even cause precast concrete damage and missing corners. Affecting the appearance and quality of the laminated board [9]. Formwork needs to be installed at the "beard ribs" of adjacent floor slabs on the construction site, and collisions also bring a lot of complicated procedures to the construction. Therefore, this project adopts the method of "no ribs"[10] in the joints between the laminated slabs (Fig. 3). At the same time, the composite wire mesh technology is used to improve the crack resistance of the slabs, so that the thickness of the precast laminated slabs is reduced to 30 mm, which is better than ordinary concrete. The laminated board is reduced by half, saving costs, and can be free of formwork and plastering, which greatly improves the efficiency of construction and installation.

### **3.2 U-Shaped Steel Bar Ring Buckle Connection Technology**

On the third floor of the project, the ➅ axis y direction AB and BC span reinforcements are 4 25, 2 25+2 22 respectively, and the left and right x direction beams are 2 25 reinforcements. At the intersection of the ➅ axis and the B axis, the lower part of the joint is thick, and the amount is large, which is easy to produce Rebar collision problem. In order to reduce the collision problem of steel bars in the node area and ensure the

**Fig. 3.** Closely assembled laminated floor slab

effective transmission of node force, this project adopts U-shaped steel bar ring buckle connection technology. As shown in Fig. 4, the upper part of the laminated beam is the post-cast part, and the upper longitudinal steel bars of the beam can penetrate the nodes or supports, because the node reinforcement is too dense, the lower beams and the column reinforcements collide, and the lower longitudinal reinforcements of the beam undergo U-shaped bending. After folding, anchor into the beam concrete without extending into nodes or supports. Then arrange the closed circular steel bars and the U-shaped steel bars at the beam end between the longitudinal bars in the core area of the beam-column node to stagger each other, thereby forming a pair of U-shaped ring buckles at the beam end, and place four ring buckles at the four corners of the ring buckle. Root short-inserted steel bars, U-shaped steel bar ring buckle connection section stirrups are encrypted. The U-shaped steel bar ring buckle connection technology forms a "pin type" connection through the core area concrete surrounded by the ring buckle steel bar. The ring buckle steel bar can effectively transmit the tension and pressure required for the connection, ensuring the safety of the connection and effectively improving the on-site construction Efficiency [11].

The U-shaped steel ring buckle connection technology is simple in structure and convenient in construction. It can avoid the collision of the prefabricated beam steel bars with the joints or the steel bars in the supports, and solves the problem of complex and difficult anchoring of the steel bars in the core area of the beam-column of the traditional post-cast integral connection node. At the same time, the length of the postcast section at the beam end can be shortened, and the standardization of prefabricated beam components can be realized.

**Fig. 4.** U-shaped node connection technology

### **3.3 Overlap Connection of Primary and Secondary Beams and Steel Slot Connection Technology**

The assembled monolithic concrete prefabricated secondary beam adopts the same superimposed form as the main beam, and the upper longitudinal steel bars are tied together on-site and poured together with the upper steel bars of the floor. For the connection of the lower reinforcement, that is, the connection of the primary and secondary beam nodes, the integral cast or shelving connection is often used [12]. Partially vacant sections are reserved in the prefabricated main girder using the cast-in-place method, which causes the main girder to be discontinuous and increases the difficulty of prefabrication and hoisting. However, the shelving type is not connected to the lower secondary beam steel bar, which belongs to the "hinged connection" and is not suitable for structures that bear dynamic loads and large spans. It can be seen that no matter whether the cast-in-place or shelving connection method is used, there are construction drawbacks. The design of this project The primary and secondary beam connections not only have no gaps in the primary beam but also connect the lower longitudinal ribs of the secondary beam. In addition to the U-shaped connection technology described above, there are also primary and secondary beam lap connections and steel slot connections.

The lap connection of the primary and secondary beams is to set a keyway on the side of the main beam and the part where the secondary beam is connected, and at the same time reserve the steel bars that overlap with the lower part of the secondary beam, and connect with the secondary beam reinforcement through the post-cast belt, as shown in Fig. 5. This lap method can not only avoid gaps in the main beam, but also realize the complete force of the primary and secondary beams, but the reserved lap steel bars for the main beam are not convenient for transportation and consume more steel. At the same time, the reserved post-pouring belt is longer and there are more wet operations.

In order to reduce the wet work of the post-cast belt, the project also adopted a new type of steel slot connection technology, that is, the steel slot is pre-embedded on the side of the main beam at the junction of the primary and secondary beams, and the stressed steel bars at the bottom of the secondary beams can be directly extended into the steel. The slot is anchored, mortar is poured, and concrete is finally poured to form a whole, as shown in Fig. 6. The construction difficulty of the steel slot connection is greatly reduced, and because of its special structure, when the connecting steel bar is stressed, the mortar or concrete poured in the gap between the steel slot and the connecting steel anchor head is in a three-way compression state. Its bearing capacity and ductility have been greatly improved to ensure that damage does not occur in the steel slot [13].

### **3.4 Bellows Through Hole Prefabricated Column**

In prefabricated concrete buildings, in order to ensure the reliability of vertical force transmission, longitudinal prefabricated column reinforcement is generally connected by grouting sleeves, but the grouting sleeve connection not only requires high manufacturing precision of the prefabricated columns, but also detects the compactness of the mortar inside the sleeves. The difficulty is also higher, and the construction cost is also higher [14]. Therefore, this project uses a bellows through-hole precast column, that is, a metal corrugated pipe with the same height as the column is inserted at the position of the

**Fig. 5.** Lap connection of main-secondary beams

**Fig. 6.** Connection of main-secondary beam steel slots

steel bar inside the precast column, and the steel wire rope is tied and fixed with the column stirrup skeleton, and then concrete is poured, as shown in Fig. 7. When the upper and lower columns are connected, the longitudinal ribs of the upper and lower columns are connected by mechanical connection or welding at the nodes first, and then the prefabricated columns with the through-hole of the bellows are hoisted to the installation position, and finally mortar or special purpose is poured along the through-holes of the bellows. Grout until it is below the height of the bellows.

**Fig. 7.** Bellows through hole prefabricated column

The bellows through-hole prefabricated column has a large bellows cavity, which is much larger than the diameter of the column longitudinal ribs. When the beam-column steel bars collide, the longitudinal steel bars in the corrugated pipe have a larger space that can be moved, which greatly increases the fault tolerance rate, thereby reducing the precision requirements of the prefabricated column components and improving the construction efficiency. The longitudinal ribs of the upper and lower columns are welded or mechanically connected, and the integrity of the longitudinal ribs is good, which improves the load-bearing capacity of the precast column, and does not require the use of sleeves and high-strength mortar, which further reduces the cost.

# **4 Bim Collaborative Forward Design**

The "language" of traditional building structure communication is mainly in the form of two-dimensional plan drawings, which has certain requirements for the space imagination ability of design and construction personnel. In addition, many problems cannot be seen on the plan drawings, but they cannot be carried out in actual construction, resulting in more on-site changes. In order to adapt to the increasing requirements of the construction industry for building structure design, BIM technology came into being. As a new comprehensive technology for building model design, BIM technology abstracts the specific data of architectural design through software Data processing is expressed in the form of 3D models [15]. Especially in prefabricated buildings, the components are prefabricated in the factory, and the joints are poured on site. The reinforcing bars of various components are overlapped or anchored at the joints, resulting in dense reinforcing bars at the joints and significant collision problems. If you continue to display with flat drawings, it will be difficult to deal with the collision problems Therefore, this project uses 3D modeling Rhino software to build 3D models of all components used in this project, especially for the node areas with dense steel bars and new connection technology. The modeling structure shows that all nodes using the new connection technology have not occurred. Joint collision problem, construction is convenient, and the joint steel bar with collision problem is adjusted, the steel bar is anchored by the combination of hook and anchor plate, and the colliding steel bar is bent, and the bending ratio is not more than 1:6. A structural reinforcement is added to the bent steel bar. The collision example is shown in Fig. 8. The two front and rear beams in figure a collide with the two outermost steel reinforcement bars as shown in Fig. 8 (a) (Beam End). The two ends of the steel bars are bent to the middle but collide with the column reinforcement as shown in Fig. 8 (b) (Middle bend). Therefore, the final solution is to bend upwards and add structural steel bars to the bending parts as shown in Fig. 8 (c)(Bends upwards).

**Fig. 8.** BIM technology application examples

# **5 Conclusion**

With the advancement of industrialization, the development of prefabricated buildings is an inevitable trend. However, there are still some problems that restrict development, mainly including insufficient standardization in design, various conflicts and collision problems in construction affect the integrity of structural connections, in order to promote the industrialization of prefabricated buildings, a standardized design system must be established, and research and development Various new connection technologies and new components optimize the structural system.

The design of prefabricated concrete buildings is integrated with the structural splitting. Under the premise of meeting the building functions, the principle of "less specifications, more combinations" and the modular coordination standard are followed to carry out standardized, modular and integrated architectural design. And the structure can also reduce the types of components through reinforcement merging, etc., and carry out standardized design of node connections.

The use of new connection technologies, such as U-shaped buckle steel bar connection technology and steel card slot connection technology, can effectively improve the problems of steel bar collision and construction inconvenience in the node area, and improve construction efficiency; at the same time, the use of new components can not only solve some of the current construction Problems, superior performance, low cost, easy to popularize and use, and meet the requirements of building industrialization.

**Acknowledgments.** Foundation Items: Research Foundation for Young and Middle-aged Teachers in Guangxi Universities(2019KY0730); Science Research and Technology Development Project of Hezhou City, (Hezhou technology 20001); Innovation and Entrepreneurship Training Program for College Students<PHONE_NUMBER>65).

# **References**

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# **Symplectic Elastic Solution of Multi-layer Thick-Walled Cylinder Under Different Interlayer Constraints**

Zhongyu Jiang(B) , Yajun Zhang, Huaqing Liu, and Xuanxuan Li

Key Laboratory of Mechanics, School of Architecture and Civil Engineering, Anhui Polytechnic University, Wuhu 241000, China

<EMAIL_ADDRESS>
**Abstract.** Multi-layer thick-walled cylinder is a common supporting structure in engineering, which is widely used in various engineering fields. Considering the complex boundary conditions and the different interlayer constraints, it is difficult to solve the theoretical solution of multi-layer thick walled cylinder. In this paper, the general solution expressions of displacement and stress of multi-layer thick-walled cylinder are derived in Hamiltonian mechanics system. The complex boundary conditions are transformed into the form of algebraic sum by Fourier series expansion, and the complex boundary problems are solved by superimposing the special solutions of each order expansion term. At the same time, according to the characteristics of different interlayer constraints, the corresponding conditions of interlayer continuous smooth are proposed. Combined with the boundary conditions of thick-walled cylinder, the linear equations with undetermined coefficients are established. By solving the equations, the mechanical problems of multi-layer thick-walled cylinder are finally solved. By comparing the mechanical responses of multi-layer thick-walled cylinder under different constraint conditions, it is concluded that the overall mechanical performance of the tight interlayer connection is better, and the circumfluence stress component is more prominent than other stress components. Finally, the influence of lateral pressure coefficient and elastic modulus ratio on the circumferential stress of multi-layer thick-walled cylinder is discussed. These research results provide the necessary theoretical basis for solving the mechanical problems of multi-layer thick-walled cylinders.

**Keywords:** Multi-layer thick-walled cylinder · Hamiltonian mechanics · Smooth interlayer contact · Tight interlayer connection · Complex boundary conditions

# **1 Introduction**

Thick-walled cylinder is a common engineering structure, which is widely used in mining, hydropower, chemical, military and other fields [1]. Thick-walled concrete cylinder is often used in mine engineering to ensure that the wellbore is still in elastic state under large load conditions. However, when the thickness of the wellbore is already large, simply relying on increasing the thickness of the wellbore will not only increase the time and labor cost, but also fail to significantly improve the elastic ultimate bearing capacity of the wellbore [2, 3]. It is a common economic and reasonable method to adopt multi-layer composite cylinder, which can effectively reduce the wall thickness and improve the ultimate bearing capacity of the structure [4].

On the basis of considering the different tensile and compression elastic modulus of the material, Wang Su et al. [5] established the stress expression of double-layer thickwalled cylinder under uniform internal pressure. The elastic limit solution corresponding to the internal pressure is obtained, and the influence of the cylinder parameters on the elastic limit is discussed. Lu, A. et al. [6] discussed the optimal design method of double-layer thick-walled cylinder by using the mixed penalty function method. Under the known uniform external load conditions, the minimum wall thickness of the cylinder, and the optimum thickness ratio and the elastic modulus ratio of the inner and outer layers are calculated. The disadvantage of this method is that it can only solve the problem of multi-layer thick-walled cylinder under uniform load with the help of classical elastic mechanics solution. Qiu J et al. [7] proposed an analytical method to solve the pressure and stress between multiple contact pairs by using the theory of multi-layer thick-walled cylinder, and applied it to the design of interference fit for engine crankshaft bearing. It mainly uses the finite element method to analyze the stress of the multi-layer thickwalled cylinder, which can not give the relationship expression between the variables accurately. Wu, Q. et al. [8] studied the plane strain problem of double-layer thickwalled cylinder by using the power series method of complex variable function, and obtained the analytical solution of stress of two-layer cylinder under the condition of complete contact. It is difficult to determine the complex potential function during the derivation of this method, and it needs to be recalculated for different external boundary conditions, which lacks generality. Abbas Loghman et al. [9] studied the magnetothermo-elastic response of a double-layer thick-walled cylinder. The minimum effective stress distribution and the minimum radial displacement can be obtained by selecting the appropriate uniformity parameters under thermal-magnetic mechanical load.

With the development of modern mathematical and physical methods, Zhong [10] and Yao et al. [11] introduced the concept of symplectic geometry into Hamiltonian mechanical system, and established a new solution system of elasticity. It has shown great advantages in dealing with complex elastic mechanics problems. Zhou Jianfang et al. [12] extended the method of separating variables in Hamilton mechanics system, and applied it to the elasticity problem of non-homogeneous boundaries in polar coordinates. This method successfully solves the problem of thick-walled cylinder subjected to nonuniform hydrostatic pressure, which fully shows the superiority of Hamilton mechanical system. Based on the Hamiltonian state space method, Tseng and Tarn [13] discussed the theoretical solution of the stress field around a circular hole when the elastic plate with a hole is subjected to unidirectional tension by the method of variable separation and symplectic eigenfunction expansion. It can be seen that the symplectic elastic mechanics method has a wide application prospect in the field of basic research. It avoids the subjectivity of stress function selection, and uses rational logical derivation to solve the problem. Therefore, it is of great significance to study the theoretical solution of multi-layer thick-walled cylinder structure by symplectic elasticity in polar coordinate system.

## **2 Hamiltonian Mechanics in Polar Coordinates**

### **2.1 The Mixed State Equation of Sector in Polar Coordinates**

Define new variables: *S*<sup>ρ</sup> = ρσρ, *S*<sup>ϕ</sup> = ρσϕ, *S*ρϕ = ρτρϕ, in a typical sector region (Fig. 1) *R*1≤ρ≤R2, α≤ϕ≤β. Then perform variable substitution ξ = ln ρ, that is ρ = *<sup>e</sup>*<sup>ξ</sup> ; and <sup>ξ</sup><sup>1</sup> <sup>=</sup> ln *<sup>R</sup>*1, <sup>ξ</sup><sup>2</sup> <sup>=</sup> ln *<sup>R</sup>*2. In the variational principle, <sup>ϕ</sup> is simulated as time coordinates, ξ is horizontal, and the *S*<sup>ρ</sup> lateral force factors is eliminated. Then the mixed energy variational principle of the Hamiltonian system under polar coordinates is obtained [14].

$$\begin{split} \delta \int\_{\alpha}^{\beta} \int\_{\xi\_1}^{\xi\_2} \left\{ S\_{\rho\varphi} \frac{\partial u\_{\rho}}{\partial \varphi} + S\_{\varphi} \frac{\partial u\_{\varphi}}{\partial \varphi} + S\_{\varphi} \left( u\_{\rho} + \nu \frac{\partial u\_{\rho}}{\partial \xi} \right) - S\_{\rho\varphi} \left( u\_{\varphi} - \frac{\partial u\_{\varphi}}{\partial \xi} \right) + \\ \frac{1}{2} E \left( \frac{\partial u\_{\rho}}{\partial \xi} \right)^2 - \frac{1}{2E} \left[ \left( 1 - \nu^2 \right) S\_{\varphi}^2 + 2(1 + \nu) S\_{\rho\varphi}^2 \right] \right\} d\xi d\varphi = 0 \end{split} \tag{1}$$

The mixed energy variational principle is expanded, and the sign (·) = ∂/∂ϕ is used to represent the derivative of ϕ, then the Hamiltonian regular equations can be written as

$$
\begin{pmatrix}
\dot{u}\_{\rho} \\
\dot{u}\_{\varphi} \\
\dot{S}\_{\rho\varphi} \\
\dot{S}\_{\varphi}
\end{pmatrix} = \begin{bmatrix}
0 & 1 - \frac{\partial}{\partial\xi} & \frac{2(1+\nu)}{E} & 0 \\
0 & 0 & -1 - \frac{\partial}{\partial\xi} & 0
\end{bmatrix} \begin{pmatrix} u\_{\rho} \\ u\_{\varphi} \\ S\_{\rho\varphi} \\ S\_{\varphi} \end{pmatrix} \tag{2}
$$

The total-state vector *υ* = *u*<sup>ρ</sup> *u*<sup>ϕ</sup> *S*ρϕ *S*<sup>ϕ</sup> *<sup>T</sup>* and operator matrix *<sup>H</sup>* are introduced, and the Hamiltonian regular equations Eq. (2) become

$$
\dot{\upsilon} = H\upsilon\tag{3}
$$

It can be proved that the operator matrix *H* is a Hamiltonian operator matrix in symplectic geometric space [15]. The Hamiltonian operator matrix is written in block form.

$$H = \begin{bmatrix} A & D \\ B - A^\* \end{bmatrix} \tag{4}$$

where

$$\begin{aligned} \mathbf{A} &= \begin{bmatrix} 0 & 1 - \frac{\partial}{\partial \xi} \\ -1 - \nu \frac{\partial}{\partial \xi} & 0 \end{bmatrix}, \mathbf{B} = \begin{bmatrix} -E \frac{\partial^2}{\partial \xi^2} & 0 \\ 0 & 0 \end{bmatrix}, \mathbf{D} = \begin{bmatrix} \frac{2(1+\nu)}{E} & 0 \\ 0 & \frac{1-\nu^2}{E} \end{bmatrix}, \\\ \mathbf{A}^\* &= \begin{bmatrix} 0 & -1 + \nu \frac{\partial}{\partial \xi} \\ 1 + \frac{\partial}{\partial \xi} & 0 \end{bmatrix}. \end{aligned}$$

The homogeneous boundary conditions on both sides ξ = ξ<sup>1</sup> and ξ = ξ<sup>2</sup> are:

$$E\frac{\partial\mu\_{\rho}}{\partial\xi} + \nu \mathbf{S}\_{\varphi} = 0, \mathbf{S}\_{\rho\varphi} = 0,\tag{5}$$

**Fig. 1.** Mechanical problems of sector in polar coordinates

### **2.2 Basic Eigensolutions of Homogeneous Equations**

Solving the Hamilton regular equations Eq. (2) under the homogeneous boundary condition Eq. (5), the general solution of the equation is only related to the properties of the Hamilton operator *H* [16]. Therefore, the eigenvalues are μ = 0, ±*i* respectively.

When the eigenvalue μ = 0, the eigensolution and its Jordan-type eigensolution are expressed as

$$\begin{aligned} \Psi\_0^0 &= \begin{pmatrix} 0\\ e^{\xi} \\ 0\\ 0 \end{pmatrix};\\ \psi\_0^1 &= \begin{pmatrix} c\_1 e^{\xi} + c\_2 e^{-\xi} + \frac{1-v}{2} \xi e^{\xi} \\ 0 \\ 0 \\ \frac{E}{1-v} c\_1 e^{\xi} + \frac{E}{1+v} c\_2 e^{-\xi} + \frac{E}{2} e^{\xi} \left(\xi + \frac{2-v}{1-v}\right) \end{pmatrix} \end{aligned} \tag{6}$$

where *<sup>c</sup>*<sup>1</sup> = −<sup>1</sup> <sup>2</sup> <sup>−</sup> <sup>1</sup>−*<sup>v</sup>* 2 *R*2 <sup>2</sup> ln *<sup>R</sup>*2−*R*<sup>2</sup> <sup>1</sup> ln *R*<sup>1</sup> *R*2 2−*R*<sup>2</sup> 1 ; *<sup>c</sup>*<sup>2</sup> = −1+*<sup>v</sup>* 2 *R*2 2*R*2 1 *R*2 2−*R*<sup>2</sup> 1 ln *R*2 *R*1 .

When the eigenvalue μ = ±*i*, the eigensolution and its Jordan-type eigensolution are respectively expressed as

$$\begin{aligned} \boldsymbol{\Psi}\_{i}^{0} &= \begin{pmatrix} 1 \ i \ 0 \ 0 \end{pmatrix}^{T} \\ \boldsymbol{\Psi}\_{-i}^{0} &= \begin{pmatrix} 1 \ -i \ 0 \ 0 \end{pmatrix}^{T}; \\ \boldsymbol{\Psi}\_{i}^{1} &= \begin{pmatrix} \boldsymbol{u}\_{\rho}^{1} i \ \boldsymbol{u}\_{\varphi}^{1} \ \boldsymbol{S}\_{\rho\varphi}^{1} \ \boldsymbol{S}\_{\varphi}^{1} i \end{pmatrix}^{T} \\ \boldsymbol{\Psi}\_{-i}^{1} &= \begin{pmatrix} -\boldsymbol{u}\_{\rho}^{1} i \ \boldsymbol{u}\_{\varphi}^{1} \ \boldsymbol{S}\_{\rho\varphi}^{1} - \boldsymbol{S}\_{\varphi}^{1} i \end{pmatrix}^{T} \end{aligned} \tag{7}$$

where

$$\begin{cases} u\_{\rho}^{1} = \frac{1}{2}(1-\upsilon)\xi + \alpha(1-3\upsilon)e^{2\xi} + \beta(1+\upsilon)e^{-2\xi} \\ \qquad u\_{\varphi}^{1} = -\frac{1}{2}[1+\upsilon + (1-\upsilon)\xi] + \alpha(\xi+\upsilon)e^{2\xi} + \beta(1+\upsilon)e^{-2\xi} \\ \qquad S\_{\rho\varphi}^{1} = E\left(\frac{1}{2} + 2\alpha e^{2\xi} - 2\beta e^{-2\xi}\right) \\ \qquad S\_{\varphi}^{1} = E\left(\frac{1}{2} + 6\alpha e^{2\xi} + 2\beta e^{-2\xi}\right) \end{cases};$$

<sup>α</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> 4 *R*2 <sup>1</sup> <sup>+</sup> *<sup>R</sup>*<sup>2</sup> 2 <sup>β</sup> = −α*R*<sup>2</sup> 1*R*2 2

.

The original problem corresponding to the eigensolution and its Jordan eigensolution is solved as follow

$$\boldsymbol{\sigma}\_{\mu}^{n} = \sum\_{m=0}^{n} \frac{1}{m!} \boldsymbol{\varphi}^{m} \boldsymbol{\Psi}\_{\mu}^{n-m} (n=0,1) \tag{8}$$

### **2.3 Special Solutions of Non-homogeneous Boundary Conditions**

It is usually difficult to solve the elastic mechanics problem under complex boundary conditions. Considering that the geotechnical material is a small deformation elastic body, the principle of linear elastic superposition is applicable. Any complex boundary function can be Fourier transformed and decomposed into simple regular triangular series. Therefore, the complex boundary loads are decomposed into a series of triangular series loads, and the stress and deformation laws under each order of triangular series loads are obtained, then the original problems are solved by superposition of the calculation results of each order [17].

In polar coordinates, the boundary load is expanded by Fourier series as:

$$f(\varphi) = \frac{a\_0}{2} + \sum\_{k=1}^{\infty} (a\_k \cos k\varphi + b\_k \sin k\varphi) \tag{9}$$

where *<sup>a</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup> π " π <sup>−</sup><sup>π</sup> *<sup>f</sup>* (ϕ)*d*ϕ, *ak* <sup>=</sup> <sup>1</sup> π " π <sup>−</sup><sup>π</sup> *<sup>f</sup>* (ϕ) cos *<sup>k</sup>*ϕ*d*ϕ, *bk* <sup>=</sup> <sup>1</sup> π " π <sup>−</sup><sup>π</sup> *<sup>f</sup>* (ϕ)sin *<sup>k</sup>*ϕ*d*ϕ.

According to the property of the Hamiltonian operator matrix [18], the block operator *A* has an orthogonal eigenfunction system in the Hilbert space *X* ×*X*, and the eigenvalue and eigenfunction system of *A* can be expressed as:

$$U\_k = \begin{pmatrix} \cos k\varphi \\ \sin k\varphi \end{pmatrix}, (k = 0, \pm 1, \pm 2, \cdots) \tag{10}$$

Block operator −*A*<sup>∗</sup> has orthogonal eigenfunction systems in Hilbert space *X* × *X* and the eigenvalues and eigenfunction systems of −*A*<sup>∗</sup> can be expressed as:

$$
\hat{U}\_k = \begin{pmatrix} \sin k\varphi \\ \cos k\varphi \end{pmatrix}, \ (k0 \pm 1 \pm 2 \cdots) \tag{11}
$$

First, a simple case is considered, that is, the inner side is homogeneous boundary and the outer side is only subjected to *k*-order normal cosine boundary load. The specific form is:

$$\begin{cases} \text{When } \xi = \xi\_1, \, E \frac{\partial u\_\rho}{\partial \xi} + \nu \mathbf{S}\_\varphi = 0, & \mathbf{S}\_{\rho \varphi} = 0 \\ \text{When } \xi = \xi\_2, \, E \frac{\partial u\_\rho}{\partial \xi} + \nu \mathbf{S}\_\varphi = P\_k \cos k\varphi, \, \mathbf{S}\_{\rho \varphi} = 0 \end{cases} \tag{12}$$

where, *P*<sup>0</sup> is the *k*-order normal load coefficient, *k* is the coefficient of series term. Especially when *k* = 0, the boundary load is a constant.

According to the eigenfunctions Eqs. (10) and (11) of the block operators *A* and −*A*∗, the special solution of the equation satisfying the non-homogeneous boundary conditions Eq. (12) can be written as

$$\begin{cases} \tilde{u}\_{\rho} = P\_k \left( A\_1 e^{\lambda\_1 \xi} + A\_2 e^{\lambda\_2 \xi} + A\_3 e^{\lambda\_3 \xi} + A\_4 e^{\lambda\_4 \xi} \right) \cos k\varphi \\ \tilde{u}\_{\varphi} = P\_k \left( B\_1 e^{\lambda\_1 \xi} + B\_2 e^{\lambda\_2 \xi} + B\_3 e^{\lambda\_3 \xi} + B\_4 e^{\lambda\_4 \xi} \right) \sin k\varphi \\ \tilde{S}\_{\rho\varphi} = P\_k \left( C\_1 e^{\lambda\_1 \xi} + C\_2 e^{\lambda\_2 \xi} + C\_3 e^{\lambda\_3 \xi} + C\_4 e^{\lambda\_4 \xi} \right) \sin k\varphi \\ \tilde{S}\_{\varphi} = P\_k \left( D\_1 e^{\lambda\_1 \xi} + D\_2 e^{\lambda\_2 \xi} + D\_3 e^{\lambda\_3 \xi} + D\_4 e^{\lambda\_4 \xi} \right) \cos k\varphi \end{cases} (13)$$

where *Ai*, *Bi*, *Ci* and *Di* are undetermined coefficients.

These constants in the above formula are not completely independent, they should also satisfy equation Eq. (3). Based on the relationship between these constants, the quartic equation of the eigenvalue λ and the coefficient *k* of the series term can be derived.

$$(\lambda^4 - 2\left(k^2 + 1\right)\lambda^2 + \left(k^2 - 1\right)^2 = 0\tag{14}$$

Solve the equation and get

$$
\lambda\_{1,2} = \pm (k+1); \; \lambda\_{3,4} = \pm (k-1) \tag{15}
$$

Thus, the specific expression of the special solution is determined under *k*-order normal outward load. Then the specific values of undetermined coefficients *Ai*, *Bi*, *Ci* and *Di* are determined accosrding to the boundary condition Eq. (12).

Similarly, the inner side is considered as homogeneous boundary, while the outer side is only affected by *k*-order tangential sine boundary load. The specific form is as follows:

$$\begin{aligned} \text{When } \xi = \xi\_1, & E\frac{\partial u\_\rho}{\partial \xi} + \nu \mathbf{S}\_\varphi = 0, \mathbf{S}\_{\rho \varphi} = 0 \\ \text{When } \xi = \xi\_2, & E\frac{\partial u\_\rho}{\partial \xi} + \nu \mathbf{S}\_\varphi = 0, \mathbf{S}\_{\rho \varphi} = T\_k \sin k\varphi \end{aligned} \tag{16}$$

where, *T*<sup>0</sup> is the *k*-order tangential load coefficient, *k* is the coefficient of series term.

By the same method, a special solution of the equation that satisfies the boundary condition Eq. (16) can be calculated. Finally, according to the principle of linear elastic superposition, the theoretical solution which satisfies the complex boundary condition of the original problem can finally be obtained.

# **3 Stress Analysis of Multi-layer Thick-Walled Cylinder**

### **3.1 Problem Description**

Considering the multi-layer thick-walled cylinder, the inner and outer radii of the *i*-th layer are *Ri* and *Ri*+1, respectively, and the material constants are *Ei* and *vi*. The stresses in the *x* and *y* directions on the outside of the thick-walled cylinder are σ*<sup>x</sup>* and σ*<sup>y</sup>* (as shown in Fig. 2). The lateral pressure coefficient *K*<sup>0</sup> = σ*y*/σ*<sup>x</sup>* is defined according to the stress values in the *x* and *y* directions. According to the theory of stress state, the force on the outside of the structure is transferred to the outermost thick-walled cylinder, and the stress boundary conditions remain unchanged along the axis of the multi-layer thick-walled cylinder. In this way, the problem can be simplified as the plane strain problem.

**Fig. 2.** The plane force diagram of multi-layer thick-walled cylinder

### **3.2 Boundary Conditions and Continuous Smooth Conditions**

The inner and outer boundary conditions of the multi-layer thick-walled cylinder can be expressed as:when ρ = *R*1,

$$
\sigma^1\_{\rho}(\mathcal{R}\_{\mathbb{I}}) = 0, \ \mathfrak{r}^1\_{\rho\varphi}(\mathcal{R}\_{\mathbb{I}}) = 0; \tag{17}
$$

when ρ = *Rn*+1,

$$\begin{cases} \sigma\_{\rho}^{n}(\mathcal{R}\_{n+1}) = \frac{1+K\_{0}}{2}\sigma\_{x} + \frac{1-K\_{0}}{2}\sigma\_{x}\cos 2\varphi\\ \tau\_{\rho\varphi}^{n}(\mathcal{R}\_{n+1}) = \frac{1-K\_{0}}{2}\sigma\_{x}\sin 2\varphi \end{cases} \tag{18}$$

There are usually many constraint modes between layers of multi-layer thick-walled cylinder, among which two limit modes are smooth contact and tight connection, while the others are between them. The radial stress and displacement of the interlayer are continuous with different constraint modes, but the difference is the equilibrium condition between layers. The continuous smooth conditions of the two limit modes are listed below.

# • Smooth interlayer contact

When ρ = *Ri*,*i* = 2, ··· ,*n*, the continuous condition is:

$$\begin{cases} u\_{\rho}^{i-1}(\mathcal{R}\_{l}) = u\_{\rho}^{l}(\mathcal{R}\_{l}) \\ \sigma\_{\rho}^{i-1}(\mathcal{R}\_{l}) = \sigma\_{\rho}^{l}(\mathcal{R}\_{l}) \end{cases} \tag{19}$$

At the same time, the stress equilibrium condition is:

$$
\pi\_{\rho\varphi}^{l-1}(\mathcal{R}\_l) = \pi\_{\rho\varphi}^{l}(\mathcal{R}\_l) = 0 \tag{20}
$$

• Tight interlayer connection

The continuity condition between layers is the same as formula (19). The equilibrium condition of stress and displacement between layers is as follows:

$$\begin{cases} \boldsymbol{u}\_{\boldsymbol{\varphi}}^{i-1}(\mathcal{R}\_{l}) = \boldsymbol{u}\_{\boldsymbol{\varphi}}^{i}(\mathcal{R}\_{l}) \\ \boldsymbol{\tau}\_{\rho\boldsymbol{\varphi}}^{i-1}(\mathcal{R}\_{l}) = \boldsymbol{\tau}\_{\rho\boldsymbol{\varphi}}^{i}(\mathcal{R}\_{l}) \end{cases} \tag{21}$$

According to formula (13), the general solution of displacement and stress of thickwalled cylinder of each layer can be written:

$$\begin{cases} \boldsymbol{\mu}^{l}\_{\rho} = \left( A^{l}\_{1} e^{\lambda\_{1}\xi} + A^{l}\_{2} e^{\lambda\_{2}\xi} + A^{l}\_{3} e^{\lambda\_{3}\xi} + A^{l}\_{4} e^{\lambda\_{4}\xi} \right) \cos k\varphi\\ \boldsymbol{\mu}^{l}\_{\varphi} = \left( B^{l}\_{1} e^{\lambda\_{1}\xi} + B^{l}\_{2} e^{\lambda\_{2}\xi} + B^{l}\_{3} e^{\lambda\_{3}\xi} + B^{l}\_{4} e^{\lambda\_{4}\xi} \right) \sin k\varphi\\ \boldsymbol{S}^{l}\_{\rho\varphi} = \left( C^{l}\_{1} e^{\lambda\_{1}\xi} + C^{l}\_{2} e^{\lambda\_{2}\xi} + C^{l}\_{3} e^{\lambda\_{3}\xi} + C^{l}\_{4} e^{\lambda\_{4}\xi} \right) \sin k\varphi\\ \boldsymbol{S}\_{\varphi^{l}} = \left( D^{l}\_{1} e^{\lambda\_{1}\xi} + D^{l}\_{2} e^{\lambda\_{2}\xi} + D^{l}\_{3} e^{\lambda\_{3}\xi} + D^{l}\_{4} e^{\lambda\_{4}\xi} \right) \cos k\varphi \end{cases} \tag{22}$$

Where, the relationship between undetermined coefficients of each layer is

$$\begin{cases} A^i\_j = \frac{\left[ (1 - \nu \lambda\_i)(1 + \lambda\_i) - k^2 \right]}{Ek\lambda\_i^2} C^l\_j \\ B^i\_j = \frac{\lambda\_i^3 + \lambda\_i^2 + (\nu k^2 - 1)\lambda\_i + k^2 - 1}{Ek^2\lambda\_i^2} C^l\_j \ j = 1, 2, 3, 4 \\ D^i\_j = \frac{1 + \lambda\_i}{k} C^l\_j \end{cases}$$

According to the statistical data, the undetermined coefficients of multi-layer thickwalled cylinder are 4*n* in total, and the continuous smooth conditions between layers are 4 × (*n* − 1) equations, plus the 4 equations of the inner and outer boundary conditions, a total of 4*n* linear equations, from which all 4*n* undetermined coefficients can be calculated. By substituting the solved undetermined coefficients into formula (22), the stress and displacement fields of thick-walled cylinder of each layer can be calculated.

## **3.3 Example**

The following parameters are selected for analysis and discussion of the calculation example: the radius of double-layer thick-walled cylinder from inside to outside is *R*<sup>1</sup> = 3 *m*, *R*<sup>2</sup> = 4 *m* and *R*<sup>3</sup> = 5 *m* respectively. The outside of the thick-walled cylinder is subjected to σ*<sup>x</sup>* = 7 *MPa* and the lateral pressure coefficient is taken as *K*<sup>0</sup> = 0.6. The elastic modulus of the inner and outer layers are taken as 2*E*<sup>1</sup> = *E*<sup>2</sup> = 40 *GPa* respectively. Poisson's ratio is the same, take *v*<sup>1</sup> = *v*<sup>2</sup> = 0.25. According to the symmetry of the force on the multi-layer thick-walled cylinder, only a quarter model is taken for study, that is, θ ∈ [0°, 90°]. There are two kinds of constraint modes between thickwalled cylinder: 1. Smooth interlayer contact, i.e. zero shear stress between layers; 2. Tight interlayer connection. The effects of two kinds of constraint modes on the stress and displacement fields of multi-layer thick-walled cylinder are compared.

**Fig. 3.** Radial displacement *u*ρ nephogram with different interlayer constraint modes (unit: m)

Figures 3, Fig. 4, Fig. 5, Fig. 6 and Fig. 7 describe the nephogram of stress and displacement field under two different interlayer constraint modes. It can be seen from the comparison that the distribution patterns of radial displacement *u*<sup>ρ</sup> and radial stress σρ under two different interlayer constraints are basically the same, and smooth contact will cause the radial displacement value to be larger. However, the two constraint modes have great influence on the circumpolar displacement *u*ϕ, the circumpolar stress σϕ and the shear stress τρϕ. The distribution of nephogram under different constraints is completely

**Fig. 4.** Circumferential displacement *u*θ nephogram with different interlayer constraint modes (unit: m)

**Fig. 5.** Radial stress σρ nephogram with different interlayer constraint modes (Unit: MPa)

different. The thick-walled cylinder with smooth interlayer contact is bounded by layers, and the inner and outer layers nephogram are distributed independently with similar distribution rules; while for the thick-walled cylinder with tight interlayer connection, the inner and outer nephogram are obviously continuous distribution. In terms of numerical value, the extreme value of stress and displacement under tight interlayer connection is smaller than that under smooth interlayer contact. Comprehensive analysis shows that the tight interlayer connection of multi-layer thick-walled cylinder increases the interlayer constrained force, the deformation is more coordinated, and the anti-deformation ability of thick-walled cylinder is improved. The overall mechanical properties are better than that of the thick-walled cylinder with smooth interlayer contact.

**Fig. 6.** Circumferential stress σϕ nephogram with different interlayer constraint modes (Unit: MPa)

**Fig. 7.** Shear stress τρϕ nephogram with different interlayer constraint modes (Unit: MPa)

### **3.4 Analysis of Influencing Factors**

The influences of elastic modulus *E*, lateral pressure coefficient *K*<sup>0</sup> and other factors on the stress field distribution of thick-walled cylinder are discussed below. Because the mechanical properties of the tight interlayer connection of multi-layer cylinder are better than the smooth interlayer contact, only the tight interlayer connection of multi-layer thick-walled cylinder is considered. See the above example for other calculation parameters. In order to better study the relative relationship between the stress components of thick-walled cylinders and the influencing factors, and to obtain the general rule, each stress component is firstly dimensionless. The dimensionless values σρ/σ*x*, σϕ/σ*<sup>x</sup>* and τρϕ/σ*<sup>x</sup>* are defined, and the relationship between the dimensionless stress values and the elastic modulus ratio of the inner and outer cylinders *E*1/*E*<sup>2</sup> (0.1 ~ 10) and the lateral pressure coefficient *K*<sup>0</sup> (0 ~ 1) are analyzed. The variation rule of each stress field along the circumference direction of the cylinder and along the thickness direction of the cylinder is discussed.

# • Influence of elastic modulus ratio

The calculation model takes the lateral pressure coefficient *K*<sup>0</sup> = 0.6, and the elastic modulus ratio *E*1/*E*<sup>2</sup> of the inner and outer cylinders varies between (0.1 ~ 10). By comparing the nephogram of each stress component in the above example, it can be seen that the circumferential stress value is much larger than other stress components, and the extreme value of circumferential stress appears on the inner and outer circumference of the cylinder respectively. Therefore, the following focuses on the distribution of circumferential stress σϕ along the inner and outer sides of the two-layer cylinder.

**Fig. 8.** Variation of σ ϕ/σ*x* along the cylinder thickness 0° direction

**Fig. 9.** Variation of σ ϕ/σ*x* along the cylinder thickness 30° direction

Figure 8, Fig. 9, Fig. 10 and Fig. 11 respectively describes the variation rule of circumferential stress σϕ/σ*<sup>x</sup>* along different directions of the cylinder thickness with different elastic modulus ratio. It can be seen from the figure that when *E*<sup>1</sup> = *E*2, the circumferential stress is discontinuous at the interface of thick-walled cylinder, which is caused by boundary conditions; only when *E*<sup>1</sup> = *E*2, the circumferential stress is continuous at the interface of thick-walled cylinder. The closer the elastic modulus ratio *E*1/*E*<sup>2</sup> approaches 1, the smaller the difference of circumferential stress on both sides of the cylinder interface is; otherwise, the greater the difference is. The closer the ratio of elastic modulus *E* to 1, the smaller the difference of circumferential stress on both sides of the cylinder interface is, and vice versa. It can also be seen that when the direction angle is relatively small, the circumferential stress is an increasing function along the thickness of the cylinder, that is, for the same layer of cylinder, the stress of the outer cylinder is greater than that of the inner cylinder, and the stress extreme occurs in the outer cylinder. When the direction angle is relatively large, the circumferential stress is

**Fig. 10.** Variation of σ ϕ/σ*x* along the cylinder thickness 60° direction

3 3.5 4 4.5 **Fig. 11.** Variation of σ ϕ/σ*x* along the cylinder thickness 90° direction

a decreasing function along the thickness of the cylinder, and the stress extreme occurs in the inner cylinder.

• Influence of lateral pressure coefficient *K*<sup>0</sup>

The calculation model takes the elastic modulus ratio *E*1/*E*<sup>2</sup> = 0.5 of the inner and outer cylinders, and the lateral pressure coefficient *K*<sup>0</sup> varies between (0 ~ 1). The distribution of circumferential stress ratio σϕ/σ*<sup>x</sup>* along the thickness direction of thick wall cylinder is discussed.

4

K0 =0 K0

K0

3 3.5 4 4.5 -1 0 1 2 3

=0.1 K0

=0.7 K0

=0.3 K0

=1

=0.9 K0

=0.5

**Fig. 12.** Variation of σ ϕ/σ*x* along the cylinder thickness 0° direction

**Fig. 13.** Variation of σ ϕ/σ*x* along the cylinder thickness 30° direction

cylinder thickness 90° direction

Figure 12, Fig. 13, Fig. 14 and Fig. 15 respectively describes the variation rule of circumferential stress σϕ/σ*<sup>x</sup>* along different directions of the cylinder thickness with different lateral pressure coefficients. It can be seen from the figures that (1) The circumferential stress is discontinuous at the interface of the multi-layer thick-walled cylinder. (2) When the direction angle is relatively small, there is an intersection point of circumferential stress in the outer cylinder corresponding to different lateral pressure coefficient *K*0; while when the direction angle is relatively large, the intersection point of the circumferential stress appears in the inner cylinder. (3) When the direction angle and the lateral pressure coefficient *K*<sup>0</sup> are relatively small, the inside circumferential stress of the inner cylinder will be negative, while the outer cylinder will not be negative; when the direction angle is relatively large and the lateral pressure coefficient *K*<sup>0</sup> is relatively small, the negative circumferential stress value will appear on the outside of the outer cylinder. (4) In the 0° direction, there is a minimum value of circumferential stress inside the inner cylinder; there is a maximum value of circumferential stress outside the outer cylinder. In the 90° direction, there is a maximum value of circumferential stress inside the inner cylinder; there is a minimum value of circumferential stress outside the outer cylinder.

## **4 Conclusion**

cylinder thickness 60° direction

These conclusions provide specific theoretical guidance and technical support for solving the mechanical problems of multi-layer thick-walled cylinder.

**Funding.** This research was funded by the National Natural Science Foundation of China (Grant No. 42102082); and Key Research Program of Anhui Polytechnic University (Grant No. KZ42020043); National College Student Innovation and Entrepreneurship Training Program (Grant No.<PHONE_NUMBER>22).

**Conflicts of Interest.** On behalf of all authors, the corresponding author states that there is no conflict of interest.

# **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Research on Construction Process of Steel Beam Incremental Launching Based on Finite Element Method**

Yancai Xiao1, Kun Fu2, Zhuang Li1,3(B) , Zhiping Zeng1,4, Jian Bai5, Zhibin Huang1,6, Xudong Huang1, and Yu Yuan7

<sup>1</sup> School of Civil Engineering, Central South University, Changsha, Hunan, China<EMAIL_ADDRESS>
<sup>2</sup> Kunming Metro Construction Management Co., Ltd., Kunming, Yunnan, China <sup>3</sup> China Railway Electrification Engineering Group Co., Ltd., Beijing, China

<sup>4</sup> MOE Key Laboratory of Engineering Structures of Heavy Haul Railway (Central South University), Changsha, Hunan, China

<sup>5</sup> Luoyang Kebosi New Material Technology Co., Ltd., Luoyang, Henan, China <sup>6</sup> Southeast Coastal Railway (Fujian) Co., Ltd., Fuzhou, Fujian, China

<sup>7</sup> Information Engineering School, Nanchang University, Nanchang, Jiangxi, China

**Abstract.** In order to ensure the normal operation of the traffic under the bridge, reasonable calculation methods and construction techniques should be adopted for the construction of the newly added railway station. This paper establishes a structural calculation finite element model to calculate and analyze the various construction stages of the steel beam incremental launching construction of the newly-added Gaoping station on the Yichuang-Wanzhou Railway, and systematically study the mechanical properties of the steel beam in the process. The results show that: (1) The deflection of each rod can meet the requirements of the railway bridge steel structure construction specification. However, when the length of the front cantilever of the steel beam reaches 11.4 m, the maximum deflection of the upper and lower chord bars is close to the limit. (2) The load-bearing capacity of each member of the steel beam meets the requirements, which indicates that the structural design of the steel beam and the incremental launching construction plan are reasonable. (3) In view of the complexity and uncertainty of the incremental launching construction process, real-time monitoring of the construction process is required, and the beam should be dropped in time when abnormal conditions occur to ensure the safe operation of the existing line.

**Keywords:** Newly added station · Incremental launching construction · Steel beam structure · Finite element · Mechanical properties

# **1 Introduction**

Along with the demand of socio-economic and transportation development, the bridge construction technology has also been developed tremendously [1–5]. And a common construction methods called incremental launching has received wide attention from scholars. Lou Song et al. analyzed the mechanical behavior of large tonnage steel truss beam during step incremental launching and sliding construction based on MIDAS Civil software modeling [6]; Zhou Jianting et al. proposed a construction control method for incremental launching of large span rail steel box laminated beam bridge in the context of the north bank approach bridge of Nanjimen rail special bridge [7]; Liu Junhua used finite element method to study the facility construction method and incremental launching safety analysis method of a large span arch beam combination system bridge incremental launching construction [8]; Based on the law of lateral deflection of steel channel beam tangential incremental launching, Yang Zengquan proposed the construction process of steel channel beam step incremental launching without temporary pier and pier side support in the middle [9]; Shi Xiaoye et al. proposed a guide beam strengthening scheme with double rows of longitudinal stiffening ribs by analyzing the local instability and buckling deformation characteristics of the guide beam [10].

In this paper, a structural computational finite element model is established to calculate and analyze each construction stage of steel beam incremental launching construction by taking the newly added flyover at Gaoping station on Yichuang-Wanzhou Railway as a relying project, and calculating the steel beam displacement and stress respectively to verify the safety of steel beam structure during incremental launching construction.

# **2 Project Overview**

Gaoping station is a new station on the Yichuang-Wanzhou Railway, which is one of the main skeleton of China's "eight vertical and eight horizontal" high-speed railroad network, with up to 96 high-speed trains running every day. In order to reduce the impact on the operation of existing lines and ensure the safety of the entire construction process, the best construction method of the station flyover is incremental launching method [11–13].

The bridge is equipped with 4 sets of step incremental launching equipment with longitudinal center spacing of 12.6 + 14.4 + 12.8 + 7.9 + 5.5 m and transverse center spacing of 9 m. The steel beam push can be divided into three stages. The first stage is to assemble a 27 m steel beam, and then push 6 m integrally to the end of the steel beam to reach the top D7 and D8 of the walker; the second stage is to continue to assemble a 6 m steel beam. Then push it integrally for 10 m; the third stage is to continue to assemble the remaining 10.075 m steel beam, push it 9.6 m integrally until the steel beam is pushed to the design position.

# **3 Finite Element Modeling**

### **3.1 Calculation Parameters**

The steel used for the steel beam of the flyover at Gaoping station includes Q345qD and Q235. The allowable stress for compression and bending of Q345qD steel is 250 MPa, and the allowable stress for shear is 140 MPa; the allowable stress for compression and bending of Q235 steel is 170 MPa, and the allowable shear stress is 100 MPa [14].

# **3.2 Calculation Load**

• Self-weight of steel beam

Taking into account the self-weight of steel beams, detailed structures such as bridge decks, diaphragms and stiffeners are not included in the finite element model. Therefore, the self-weight coefficient is increased to balance the gravity of the detailed structure, and the self-weight coefficient is taken as 1.37.

• Self-weight of incremental launching temporary piers

The self-weight of the temporary pier support and the assembled support is automatically loaded by the program. Taking into account that the welds and stiff plates are not included in the model, the self-weight coefficient is taken as 1.2.

• Lncremental launching horizontal force

Horizontal loads in the transverse and longitudinal directions are applied to the top of the temporary piers, and the load is taken as 5% of the reaction force of the vertical fulcrum.

• Construction load

Consider the construction load of 2.0 kN/m<sup>2</sup> on the working platform on the top of the temporary pier.

# **3.3 Calculation Conditions**

The temporary piers supporting the steel beam structure at different positions during the entire incremental launching construction process will change accordingly, and the whole process will be divided into 17 working conditions (calculated once every 2 m of incremental launching). The reaction force of temporary pier support, steel beam deflection and stress are calculated under each working condition. The calculation models at different stages are shown in Figs. 1, 2 and 3, and the description of the controlled working conditions and the length of the front and rear cantilever during the pushing process of the steel beam are shown in Table 1.

**Fig. 1.** The first stage. **Fig. 2.** The second stage.

**Fig. 3.** The third stage.

**Table 1.** Controlled construction steps.

(*continued*)

**Table 1.** (*continued*)

# **4 Calculation and Analysis of Steel Beam Pushing Process**

### **4.1 Steel Beam Displacement Calculation**

The maximum displacement of each member of the steel beam under various controlled construction conditions is shown in Fig. 4. The deflection of the lower chord (LC) and the upper chord (UC) refers to the vertical displacement, and the web member deflection (WM) refers to the horizontal displacement.

**Fig. 4.** Maximum deflection of incremental launching steel beam member under various controlled construction conditions.

It can be seen from Fig. 4 that under different working conditions, the maximum deflection of the upper chord and the lower chord of the steel beam are basically the same, and both are greater than the deflection of the web. The maximum deflection of each member appears in working condition 7 (the front cantilever length is 11.4 m). At this time, the maximum vertical deflection of the lower chord is −27.4 mm, the maximum vertical deflection of the upper chord is −27.5 mm, and the maximum horizontal deflection of the web member is −2.5 mm. Figure 5 shows the displacement cloud picture for working condition 7.

**Fig. 5.** Steel beam deflection cloud map for working condition 7.

According to "Steel Structure Design Standard", the allowable value of deflection for working condition 7 is 28.5 mm [15]. From the above analysis, it can be seen that although the deflection of each member can meet the requirements of the railway bridge steel structure construction code, the maximum deflection of the upper chord and the lower chord is close to the limit value, indicating that the steel beam structure has made full use of some chords' flexural stiffness.

### **4.2 Steel Beam Stress Calculation**

Figure 6 shows the maximum normal stress of the upper chord (UC), lower chord (LC), vertical web member (VWM), and inclind web member (IWM) of steel beams under various controlled construction conditions.

As can be seen from the above figure, the maximum value of the normal stress appears in the upper chord during the whole process of the steel beam incremental launching construction. The phenomenon is particularly significant in working condition 7 and working condition 16. The maximum normal stress of the lower chords of the steel beam appears in the working condition 7, which is 107.9 MPa. The maximum normal stress of the upper chords and the vertical web members both appear in the working condition 16, which are 52.4 MPa and 65.7 MPa, respectively. And the maximum normal stress of the inclind web members appears in the working condition 17, which is 54.2 MPa. The steel used in the steel beam is marked with Q235, whose allowable stress is 250 MPa.

**Fig. 6.** Maximum normal stress of incremental launching steel beam member under various controlled construction conditions.

Therefore, the load-bearing capacity of each member of the steel beam can meet the requirements during the incremental launching process, and the structure is safe and controllable.

# **5 Conclusion**

Basing on the above researches, conclusions and suggestionscan be drawn as follows: (1) In the whole process of steel beam incremental launching construction, the deflection of each member can meet the requirements of the railway bridge steel structure construction code. But when the length of the front cantilever of the steel beam reaches 11.4 m (working condition 7), the maximum deflection of the upper chord and the lower chord is close to the limit. (2) During the construction of the steel beam incremental launching, the load-bearing capacity of each member of the steel beam meets the requirements. It shows that the design of steel flyover structure and the design of the incremental launching construction plan are reasonable, and the structure of the incremental launching process is safe and controllable. (3) In view of the complexity and uncertainty of the incremental launching construction process, real-time monitoring of the construction process is required, and the beam should be dropped in time when abnormal conditions occur to ensure the safe operation of the existing line under the bridge.

**Acknowledgments.** The research is financially supported by the Science and Technology Research Development Project of China Railway Nanchang Group Co., Ltd. (grant CRNG202007), Natural Science Foundation of Hunan Province, China (Grant 2019JJ40384), High-Speed Railway Foundation Joint Fund Project (Grant U1734208), Scientific Research Test Project of China Railway Corporation (Grant SY2016G001), which is gratefully acknowledged by the authors.

# **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Numerical Simulation and Experimental Study on Concrete Block Fabricated Foundation**

Huiyuan Liu1, Xibin Yang1, Jie Li1, Mintao Ding2(B) , Congyue Song3, and Yongping Li4

 Inner Mongolia Power (Group) Co., Ltd., Inner Mongolia, China China Electric Power Research Institute, Beijing, China<EMAIL_ADDRESS>Ordos Electric Power Bureau, Ordos, China Inner Mongolia Electric Power Survey and Design Institute Co., Ltd., Inner Mongolia, China

**Abstract.** The fabricated foundation can be prefabricated in the factory, so it is easy to control the size, weight and quality of components, and it's easy to transport since the size is controllable. The operation links such as on-site concrete pouring and maintenance, on-site formwork and steel bar binding are omitted, the opening time of foundation pit is reduced, and the construction time is greatly shortened. In this paper, two different types of concrete block fabricated foundation (Foundation-A and Foundation-B) are designed. The foundation is divided into concrete blocks one by one, which are connected by bolts and steel plates on site. In order to study the connection performance and bearing capacity of the two types of foundation, the in-situ test and the numerical simulation analysis of fabricated foundations is carried out. The research shows that both foundation types can meet the requirements of bearing capacity, but Foundation-B is better than Foundation-A in bearing performance, integrity, processing difficulty and construction difficulty.

**Keywords:** Fabricated foundation · Concrete block · Bearing capacity

# **1 Instruction**

The fabricated foundation has strong social and economic benefits in transmission line projects with severe natural conditions such as high altitude, lack of water, difficult sand and gravel collection, long transportation distance or urgent time limit [1–3]. Therefore, further research on fabricated foundation has important practical and engineering significance. Since the 1960s, scholars have studied the bearing mechanism of aeolian sand foundation by indoor model test, numerical analysis and field test, and deduced the theoretical calculation formula of ultimate bearing capacity [4–9]. The ultimate bearing capacity of foundation is influenced by the geometric dimension of foundation, buried depth and ground overload [10, 11]. Liu [12] studied the bearing mechanism of straight column extended bottom foundation on eolian sand foundation under uplift load based on field tests. Qian [13] carried out field test research on inclined column extended foundation of aeolian sand foundation under the combined load of uplift horizontal force. Lu [14] conducted a compressive test on the prefabricated concrete assembled foundation of Qinghai-Tibet AC750 kV/DC ±400 kV frozen soil power grid interconnection project. Under compression load, the displacement of the ring flange connecting the column, the bottom slab, and the joint seam of the two consolidated bottom slabs conformed with that of the top plane of the foundation. The displacement and the stress of the different connected parts redistributed until the new balance developed. As a result, the different connected parts of the precast concrete assembly foundation co-operated to carry the compression load. The joint working performance of foundation components under external load is the key and difficult point of design, as well as an important guarantee for the safety and stability of foundation. At present, there are few relevant specifications and experience at home and abroad [15–17]. Compression Behavior on Precast Concrete Assembly Spread Foundation for Qinghai-Tibet AC/DC Grid Interconnection Project.

# **2 Model Design**

In this paper, two types of fabricated foundation are designed, including Foundation-A and Foundation-B, both are formed by concrete blocks of different sizes. The two types of foundation are shown in Fig. 1.

**Fig. 1.** Structure diagram of concrete block fabricated foundation

Foundation-A is composed of special-shaped concrete blocks, and the concrete blocks are overlapped into a Z-shape. Foundation-B adopts laminated type, and the two layers of concrete blocks are staggered and tower overlapped with each other to form the whole foundation. The overall structure of the two foundation types is similar to the spread foundation with two steps, with a buried depth of 3.5 m, a bottom plate width of 3 m, and a square section with side length of 1 m in the middle column. In order to increase concrete area at the junction of the upper and lower layers of special-shaped blocks of Foundation-A, the step of Foundation-A is narrower than that of Foundation-B(The step of Foundation-A is 400 mm wide and 300 mm high, and the step of Foundation-B is 600 mm wide and 300 mm high).

The in-situ test is adopted in this research. The test foundation is prefabricated according to the above sizes. Uplift and compression static load test shall be conducted for Foundation-A and Foundation-B respectively.

# **3 In Situ Test**

In this test, two kinds of in-situ static load tests of foundations were carried out, including two test conditions of uplift and compression. In the uplift test, it can be observed that a circular soil failure surface is formed around the foundation, as shown in Fig. 2. After the test, dig out the foundation was digged out to observe the foundation connection, as shown in Fig. 3.

˄a˅Foundation-A ˄b˅Foundation-B

**Fig. 2.** Failure surface of uplift test

From the uplift failure mode of the test, it can be seen that the failure of Foundation-A and Foundation-B is mainly due to the shear failure of the soil. While observing the connection of the test foundation, it can be seen that the concrete blocks and connections are not damaged, which means that the two types of foundation structures can meet the requirements of bearing capacity. The load-displacement curve of two foundations is shown in Fig. 4.

**Fig. 4.** Load-displacement curve

The test results show that the bearing performance of Foundation-B is better than that of Foundation-A. In the in-situ test, displacement rods (Fig. 5, composed of steel bars welded on the upper surface of the foundation, the steel bars extend vertically out of the soil surface, and the steel bars and the soil are separated by PVC tubes. During the test, the displacement of the foundation at this position can be known by putting the displacement monitoring on top of the steel bar extended out of the surface). The displacement rods are respectively set at the four corners of two concrete steps, It is used to measure the displacement of each step in the test. The displacement of the displacement rods is shown in Fig. 6. In the figure, the displacement of the top of the foundation is measured by the sensor at the top of the foundation, the displacement of the upper step and the lower step are measured by the displacement rods, and the displacement rods is the average value of the displacement of the upper and lower steps.

It can be seen from the displacement rods:

(1) From the data of the compression load test, the displacement difference between two steps of Foundation-A is less than that of Foundation-B. In terms of mechanical structure, Foundation-A is a special-shaped structure, and the upper and lower steps belong to the same concrete block, so the displacement difference is less than that of

**Fig. 5.** Load-displacement curve

(c) Uplift of Foundation-A (d) Uplift of Foundation-B

**Fig. 6.** Load-displacement curve of displacement rods

Foundation-B. However, the displacement difference between the top and the lower part of Foundation-A is larger than Foundation-B, indicating that the connection between the surrounding concrete blocks and the middle column of Foundation-A is weak under the compression load, the punch-resistant capability is weak under the compression load. Therefore, the connection performance of Foundation-B is better than that of Foundation-A.

# **4 Numerical Simulation**

The calculation model of the foundation is divided into two steps, the first step is the global finite element model, which includes the foundation and nearby foundation soil. The joint reaction force of the contact surface between foundation and soil can be obtained according to the whole finite element model. The second step is the substructure element model, the foundation substructure model includes the concrete block of the foundation, the connecting bolt between the blocks and the equivalent connecting key. During the calculation, the reaction force of the foundation boundary node calculated by the global finite element model in the first step is applied as the input, and the force of the connecting bolt and the equivalent connecting key is calculated. The finite element model is shown in the Fig. 7.

The global finite element model adopts Solid185 element, with a total of 576875 nodes and 565104 elements. Materials are divided into soil and concrete. Concrete has a calculated density of 2400 kg/m3 and an elastic modulus of 28 GPa, while soil has a calculated density of 1600 kg/m3 and an elastic modulus of 200 MPa. Friction contact (TARGE170 element and CONTA174 element) was adopted for the foundation and foundation, and the friction coefficient was set as 0.5.

As shown in the Fig. 8, the Foundation-A substructure model adopts solid185 element (concrete) and Link180 element (simulating bolt and steel plate). The model consists of 65,674 nodes and 65,148 units. The Foundation-B consists of 67,263 nodes and 62,920 units. Friction contact was adopted between the blocks (TARGE170 element and CONTA174 element), and the friction coefficient was set as 0.5.

**Fig. 7.** Finite element model (overall foundation model)

**Fig. 8.** Finite element model (Foundation)

According to the calculation results, the stress of Foundation-A is smaller under the upward drawing condition and larger under the downward pressing condition. In order to maintain the stability of the foundation under the uplift condition, the stress should be greater than 0. Therefore, the uplift load that Foundation-A can bear is smaller. With the same foundation bearing capacity, Foundation-A can bear a smaller down load. Therefore, from the perspective of foundation bearing capacity, Foundation-B is a better choice.

Foundation-A is connected by 32 vertical bolts, Foundation-B is connected by 12 vertical bolts for layer 1 flange, and 128 bolts for layer 2 flange. Table 1 shows the maximum tension of vertical bolts and the corresponding bolt specifications. It can be seen from the table that the vertical connecting bolt is most dangerous for Foundation-A under the action of downward pressure, and the vertical connecting bolt is most dangerous for Foundation-B under the condition of upward pulling. According to the calculation of the corresponding bolt area based on the maximum bolt axial force, it can be seen that the vertical bolt of the 1st floor flange of Foundation-B has the highest dimension requirement, and M27 bolt is required at least. This is because the total number of bolts is small and the bending moment caused by transverse load is greatly affected under the pulling up condition. However, there are more bolts in the second layer, so the stress of the single bolt is smaller.

**Table 1.** Bearing capacity of Bolts

Link element is used for equivalent simulation of the steel plate. The sum of the axial forces of two adjacent LINK elements is the force of the steel plate. For the upper connecting steel plate, the steel plate is pulled under the pulling condition. For the lower side connecting steel plate, the steel plate is pulled under the lower pressing condition. Table 2 shows the maximum tensile stress of the connecting key, the corresponding specification of the connecting bolt and the thickness of the steel plate. The thickness of the steel plate is calculated according to the minimum bolt size. When the bolt size is larger, the thickness of the steel plate can be reduced appropriately, so the calculation results in the table are more safe and conservative. As can be seen from the table, although the steel plate in Foundation-B is subjected to Foundation-A larger tension, the bolt area and the corresponding steel plate thickness obtained in Foundation-B are smaller because the number of stressed bolts in basic scheme B is twice that in Foundation-A on the same steel plate.

**Table 2.** Bearing capacity of Connector

# **5 Conclusion**

In this paper, two types of fabricated foundation are designed. The connection performance and bearing capacity of the two types of foundation are verified by in-situ test, and the test results are analyzed by numerical simulation.

# **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

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# **Pavement Crack Detection and Quantification Based on Scanning Grid and Projection Method**

Zhaoyun Sun, Lili Pei(B) , Bo Yuan, Yaohui Du, Wei Li, and Yuxi Han

School of Information Engineering, Chang'an University, Xi'an, Shaanxi, China<EMAIL_ADDRESS>
**Abstract.** Pavement cracks are difficult to monitor and quantify due to their complex texture and easy to be disturbed by noise and illumination. To solve this problem, a road crack monitoring and quantification method based on vehicle video is proposed. First, a method for extracting morphological features of dynamic road cracks is proposed. Combine automated vehicle-mounted equipment with GPS signals to obtain crack images with location information. Then, a calculation algorithm of crack parameters based on the combination of UK scanning grid and projection method is proposed, which uses the reverse engineering principle of perspective transformation to correct the image and divides the entire image into grid blocks. Finally, based on the analysis of different crack grades, the crack distress evaluation method is improved. The experimental results show that the proposed method has strong reliability and adaptability and achieves high-frequency and wide-range road detection.

**Keywords:** Pavement cracks · Video image detection · Feature extraction · Crack quantification

# **1 Introduction**

Crack detection technology is becoming more and more mature, providing a lot of technical support for pavement maintenance. Scholars from all over the world have gradually developed a variety of pavement detection system based on intelligent detection technology [1], designed the double connectivity detection of pavement crack detection algorithm [2], image measurement method of asphalt pavement damage [3]. Asphalt pavement crack detection based on mathematical morphology [4] based on Prim pavement crack connection algorithm of minimum spanning tree [5]. The processing algorithm in HARRIS pavement damage detection system will be affected by environmental factors. Therefore, the elements of a sidewalk automatic measurement and 3D methods, based on 3D data of pavement crack detection algorithm [6, 7] is proposed.

The crack images are easily influenced by texture features and there's a lot of noise [8], so the next step is to filter the image, this method is often combined with neural network classifier, for distress image segmentation [9]. For early pavement management and repair, the circular Radon transform RGB camera image is used to propose a set of better road infrastructure evaluation indicators [10]. When using cameras and deep learning networks for crack detection, there is a problem of insufficient filtering of the road background. In order to improve the detection efficiency as a whole, a lightweight Tiny-Darknet is combined with YOLOv3, and open VINO is used for model optimization and reasoning to accelerate the detection of road cracks. But this method slows down the number of frames detected on the video image [11].

Traditional detection methods are time-consuming, while pavement detection vehicles are expensive [12]. In this paper, the improved UK scanner method divides video frame image into grid block, which can not only calculate the number of cracks, but also calculate the critical rate of cracks, and increase the accuracy of crack detection and statistics.

# **2 Crack Image Acquisition and Feature Extraction**

Perform image extraction and a series of image processing on the road video, and then correct the image and remove duplicate parts. The process of crack image acquisition and feature extraction is shown in the Fig. 1.

**Fig. 1.** The process of crack image acquisition and feature extraction.

# **2.1 Image Acquisition**

A video data acquisition system is designed to collect images of pavement cracks, including a lightweight waterproof high-resolution GARMIN motion camera and a vehicle. Set a specific angle and height to the camera fixed on the vehicle to ensure complete collection of road images. Use GPS locator to record the longitude, latitude, altitude and time of the vehicle during the video capture process. The camera acquires 20 to 40 frames per second. According to the principle of dynamic photography, a four-point calibration method is used to calibrate the detection area of the road video frame image.

# **2.2 Feature Extraction**

This paper uses the region growth algorithm to further process the image. First, enhancing non-scale filtering and adaptive threshold segmentation, the contour of the crack is obtained. However, this contour is not a complete crack but a part of the real crack.

**Fig. 2.** Crack growth algorithm implementation flow based on region growth.

Second, calculate the complete crack size according to the extracting cracks image features. The crack growth algorithm based on regional growth is shown in Fig. 2.

The traditional region growth algorithm can only grow adjacent pixels. If a crack is not a seed point and is not adjacent to other crack seeds, these cracks cannot be detected. The proposed crack growth algorithm optimizes the traditional region growth algorithm, which the search scope is extended to the n\*n region. The condition is that the absolute difference between pixel values is less than or equal to the threshold K. Experiments are carried out based on the optimization region of n\*n region convolution. Let n be 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31. The threshold K is set from 1 to 50. After comparative analysis, when n = 15, K = 15, the effect is best.

### **2.3 Frame Correction and the Overlapping Part Removal**

When acquiring video frame image of road surface, the detection area in the image is irregular quadrilateral due to the difference in Angle of view and acquisition distance. It leads to the errors of different pixel points corresponding to the actual coordinate system values, which directly affects the calculation accuracy of geometric parameters of crack distress. Therefore, the image detection area shown in Fig. 3(a) needs to be corrected by reverse-engineering method of perspective transformation to restore to the real rectangular road surface image shown in Fig. 3(b). Then, by using the image crack feature extraction algorithm given in Sect. 2.2, the obtained binary image is shown in Fig. 3(c).

There is inevitable overlap between the two frames in the detection area. By analyzing the longitude, latitude, altitude and time information recorded every second, the method of removing overlap in the detection area is shown in Fig. 4.

**Fig. 3.** Crack feature extraction.

**Fig. 4.** Remove overlap schematic diagram.

Figure 4 shows only the first frame image can be fully used, and the actual length of pavement cracks in other frame images can be calculated according to Eq. (1). *L*<sup>1</sup> is the total length of road surface, *l* is the corresponding length of road surface in the first video frame image, and *N* is the number of frames.

$$len = \frac{L\_1 - l}{N - 1} \tag{1}$$

# **3 Crack Image Distress Parameters Calculation and Assessment**

The geometrical parameters of crack include crack area, crack length and crack width. As long as the grids are small enough, the grids can be approximately regarded as external rectangles of cracks. For each sub-block crack image, the image is scanned. If there is crack pixel, the sub-block is marked as a crack region sub-block. Otherwise, the sub-block will be not marked.

Set the initial pixel area of the crack *Ax* = 0 and scan each pixel in the sub-block from left to right from top to bottom. If the pixel is a crack, increase the pixel area, *Ax* = *Ax* + 1. If the pixel is the background, continue to scan the next pixel point until all the pixels in the sub-block are scanned, and the pixel area of the sub-block can be obtained. According to the actual length and width of each pixel, the actual area of each pixel can be calculated. The crack area in this sub-block *A* can be obtained by multiplying the number of pixels *Ax*.

Initially set the length and width of crack projection *Tc* = 0, *Tk* = 0. Scan each column pixel in the sub-block from left to right. If a column pixel contains crack pixels, the projection width will increase, that is *Tk* = *Tk* + 1, otherwise no operation will be carried out and continue to scan the next column pixel until all the columns are scanned, then the projection width of crack *Tk* can be obtained. Similarly, scan each row pixel in the sub-block from top to bottom to calculate the projection length *Tc*.

The camera correction information for each pixel is calculated based on the actual length and width. Projection length *Tc* and projection width *Tk* become an actual length *Lc* and an actual width *Lk* . Since the sub-blocks of the whole pavement are small, the crack length can be approximated according to the Pythagorean theorem, as shown in Eq. (2).

$$L = \sqrt{L\_c^2 + L\_k^2} \tag{2}$$

The crack width can be approximately divided by the actual crack area by the actual length.

The UK Scanner pavement crack damage assessment method is shown in Fig. 5. The left picture is the original pavement image, and the right picture is the crack map measured after grid division.

**Fig. 5.** UK Scanner pavement crack damage assessment schematic.

The UK scanner method divides the pavement crack grade into three categories: light, medium and heavy. Different grades of cracks are different in the evaluation of pavement condition. The measurement method of DMI (distress measurement index) for relevant research and evaluation is shown in Eq. (3).

$$DMI = \frac{1 \ast N\_1 + 2 \ast N\_2 + 3 \ast N\_3}{(1 + 2 + 3) \ast N} \tag{3}$$

In Eq. (3), *N*<sup>1</sup> is the number of light crack distresses, *N*<sup>2</sup> is the number of medium crack distresses, *N*<sup>3</sup> is the number of heavy crack distresses, N is the total number of cracks, and 1, 2, 3 are the specified coefficients of cracks of different grades.

The crack width standard is *k*<sup>1</sup> mm, *k*<sup>2</sup> mm, where *k*<sup>1</sup> ≤ *k*2, the crack width *D* ≤ *k*<sup>1</sup> is the light crack, the crack width *k*<sup>1</sup> < *D* ≤ *k*<sup>2</sup> is the medium crack, the crack width *D* > *k*<sup>2</sup> is heavy cracks. Calculation of the pavement condition index (DMI): The different grades of cracks have different degrees of distress to the pavement. Set the light crack coefficient as, the medium crack coefficient as *kM* , and the heavy crack coefficient as *kL*. Calculate the test pavement index DMI as shown in Eq. (4), and calculate the pavement distress index. The crack data is processed, classified, calculated and stored in the database.

$$DMI = \frac{k\_S \* N\_S + k\_M \* N\_M + k\_L \* N\_L}{N\_S + N\_M + N\_L} \tag{4}$$

# **4 Test Results Analysis**

In order to test the accuracy of crack feature extraction algorithm, 20 image samples were extracted from different video samples collected, and two methods were used for processing. 1) The grid was divided with 15\*6 (pixel) as the side length, and introduce 'N1' as the pseudo-ground truth. 2) The image is detected by crack feature extraction algorithm. The grid is divided into 15\*6 (pixel) edges, and the number N2 of the grid containing cracks is counted in the background of the system. The relative error calculation formula obtained by comparing the detection results of the two methods is shown in Eq. (5). The statistical results are shown in Table 1.

$$\delta = \frac{|N\_1 - N\_2|}{N\_1} \times 100\% \tag{5}$$

**Table 1.** Statistical table of cracks relative errors.

The average relative error of the 20 samples is 31.57%, and the error between the manual detection value and the system extraction value is analyzed as follows: N1 > N2 means that the grid containing cracks has not been detected, and N1 < N2 means that many interferences have been mistakenly identified as cracks, such as water and shadows. As the upper half of the perspective image has less image information than the lower half, the incomplete and missing part of the corrected image leads to blurring, which affects the detection results.

The length of the segment taken from the collection video is 4 2552 s, and the detection distance is 6213 m. All cracks were divided into three degrees according to the principle that the crack width less than 3 mm was considered as mild crack, the crack width between 3 and 6 mm was considered as moderate crack, and the crack width greater than 6 mm was considered as severe crack.

The number of crack grids, the total number of crack grids, the total number of grids and the percentage of crack grids in each interval are respectively counted. The test data are counted every 1000 m, and the number of crack grids with different damage degrees in each interval is analyzed and compared, as shown in Fig. 6. The blue part is the number of mild crack grids, the orange part is the number of moderate crack grids, and the gray part is the number of severe crack grids.

**Fig. 6.** Crack level grid number statistics.

The actual damage degree of crack distress in test sections is shown in Fig. 7. It can be seen that the distress degree of the measured section is the most serious in the range of 3000–4000 m.

**Fig. 7.** Pavement distress assessment distribution.

# **5 Conclusions**

This paper proposes methods to solve the problem of crack feature extraction and parameter quantification in video. The main conclusions can be summarized in the following two aspects:

1) The image is segmented using automatic thresholds, and non-crack interferences are filtered according to the area and crack morphological characteristics. The traditional area growth algorithm is improved to solve the problem of the algorithm's inability to grow across areas and identify the characteristics of cracks.

2) A crack quantification method based on the combination of UK Scanner grid method and projection method is proposed. Count the crack information of all grid sub-blocks and use the improved UK Scanner pavement crack damage evaluation method to evaluate the detected road.

**Acknowledgments.** This work was supported by the National Natural Science Foundation of China (51978071); Project the Chang'an University Ph.D. Candidates' Innovative Capacity Development Grant Program (Grant No.<PHONE_NUMBER>41).

# **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Test and Evaluation for Performance of Composite Pavement Structure**

Zhihao Yang1, Linbing Wang2(B) , Dongwei Cao3, Rongxu Li3, and Hailu Yang1

<sup>1</sup> University of Science and Technology Beijing, National Center for Materials Service Safety, No. 30 Road Xueyuan, Haidian District, Beijing, China

<sup>2</sup> Joint USTB Virginia Tech Lab on Multifunctional Materials, Department Civil and Environmental Engineering, USTB, VA Tech, Blacksburg, Virginia 24061, USA<EMAIL_ADDRESS>
<sup>3</sup> China-Road Transportation Verification and Inspection Hi-Tech Co. Ltd., No. 8 Road Xitucheng, Haidian District, Beijing, China

**Abstract.** Perpetual pavement has become an important research field of highway development in China. Reasonable selection of pavement structure and ensuring the durability of the structure are one of the necessary measures to build perpetual pavements. The inverted asphalt pavement structure can not only provide high strength and good bearing capacity of semi-rigid base, but also make use of the graded crushed stones for restraining the reflection cracks of semi-rigid base. This paper presented a study on three pavement structures are, namely, a semirigid asphalt pavement and two inverted asphalt pavements. The performances of the three pavement structures after one million loading repetition are obtained. Taking rutting depth, deflection and dynamic response as evaluation indexes, the feasibility of inverted asphalt pavement structure as perpetual pavement structure is evaluated. It is found that the composite asphalt pavement structure with permeable asphalt mixture of large particle size as base and cement stabilized macadam as subbase has the best performance as perpetual pavement.

**Keywords:** Composite asphalt pavement structure · Accelerated pavement test · Rut · Subgrade settlement · Dynamic response

# **1 Introduction**

Perpetual asphalt pavement is one of the hotspots in the field of highway development in China. Semi-rigid base or flexible base structure is usually used in perpetual pavement in China. The pavement structure with flexible material between asphalt layer and semirigid subbase is called inverted asphalt pavement structure [1]. Semi-rigid base has high stiffness and stress diffusion capacity, but its bottom is prone to produce temperature shrinkage cracks and dry shrinkage cracks when the temperature and humidity of the surrounding environment changes. Such cracks may gradually develop into bottom-up reflection cracks on the surface, which leads to the cracking of the pavement [2–4]. The inverted structure not only has the advantages of high strength and good bearing capacity of semi-rigid base, but also has the advantages that the flexible base can reduce the reflective crack [5, 6]. The lower layer of graded broken stone can bear the vertical load from the top to the bottom of the road surface and dissipate the strain energy from the bottom to the upper cement stabilized macadam base, and it can also make full use of the characteristics of the high compressive strength and low tensile strength of graded macadam base to reduce the damage of pavement structure caused by water erosion and overloading of semi-rigid materials [7].

Researchers in the United States, Australia, South Africa and other countries found that flexible interlayer can significantly reduce the asphalt pavement structure in the use of structural reflection cracks [8]. However, the mix of composite structures varies from country to country. The flexible interlayer of composite pavement structures in South Africa is usually 15 cm of graded or asphalt stabilized macadam, while graded macadam is commonly used in Australia and France [9]. The composite pavement structure consisting of asphalt stabilized macadam base course and granular subbase or cement stabilized subbase course is usually adopted in Japanese highways [10]. Since the modulus of the flexible interlayer of graded macadam is smaller than that of water-stabilized macadam, the stress transmitted by the base of water-stabilized macadam can be reduced, the inverted base asphalt pavement structure can give full play to the graded macadam base of the mechanical characteristics and drainage characteristics, so that the pavement structure has good durability [11].

According to the summary, the inverted structure with graded crushed stone flexible interlayer is rarely used at home and abroad. Moreover, most foreign studies are aimed at the composite pavement formed by paving asphalt mixture layer on cement concrete pavement, and there is a lack of research on cement stabilized macadam subbase. Through material design and accelerated loading test, this paper systematically compares and studies the performance differences between two kinds of Inverted Asphalt Pavement and semi-rigid asphalt pavement. Lay the foundation for the inverted asphalt pavement structure to be used as a perpetual pavement.

# **2 Pavement Structure and Material Design**

### **2.1 Pavement Structure**

According to the perpetual asphalt pavement structure in Hebei province and the perpetual pavement structure in Shandong Binda Expressway, three pavement structures are selected. In order to study the durability of composite base course, the control variable method was used. That is, the surface layer and the roadbed structure are consistent, but the roadbed is different. The base of Structure A is Large Stone Porous asphalt Mixes (LSPM) and cement stabilized macadam. The base of structure B is graded crushed stone and cement stabilized crushed stone. The base of structure C is two layers of cement stabilized macadam. Structure A and structure B were the experimental group, and the structure C was the control group. The structure is shown in Table 1.

**Table 1.** The structural forms in the accelerated pavement test section.

### **2.2 Materials Design**

### **2.2.1 Mixture**

SBS modifier/crumb rubber composite modified asphalt (SRA) was used as binder for asphalt mixture. To ensure the rationality of gradation design, the maximum nominal size of gradation increases from the surface layer down. The grading is shown in Table 2. In addition, the performance of the four kinds of asphalt mixture was studied. As shown in Table 3. The process of splitting strength test are shown in Fig. 1.

**Table 2.** Mix proportion of asphalt mixtures.

As can be seen from Table 3, SMA-13 has the best high temperature stability and low temperature stability among all asphalt mixtures. And its dynamic modulus is 9777 MPa, which shows that its ability of resisting traffic load is the strongest. This also meets the functional requirements of SMA-13 as an anti-wear layer. However, according to the

**Fig. 1.** The process of splitting strength test.

**Table 3.** The performance of asphalt mixture.

fatigue test data, the fatigue life of SMA-13 is lower than AC-16C and AC-20C, but higher than LSPM-25. This is due to the discontinuous gradation of SMA-13 and its larger void ratio will reduce the bond strength between coarse aggregates, which reduces its fatigue life.

In summary, through the experimental verification, we can conclude that the asphalt mixture used in the test road has good road performance.

## **2.2.2 Unbound Aggregate**

Unbound aggregate is a mixture of aggregates of different sizes obtained by gradation design. Graded macadam obtains its load-bearing capacity through the force between its internal aggregates. Mixture gradation was shown in Table 4. Although the graded macadam base can prevent the reflection crack very well, but its bearing capacity is usually less than that of asphalt or cement stabilized base. In order to enhance the shear strength and rutting resistance of graded broken stone, 1‰ polypropylene fiber was added to the mixture. The properties of graded broken stone are shown in Table 5.

**Table 4.** Mix proportion of unbound aggregate.

**Table 5.** The performance of aggregate.

By comparing the performance of graded broken stone with and without fiber, we can see that the performance of graded broken stone with fiber is greatly improved. The standard CBR value is increased by 17.75%, and the flooding CBR value is increased by 19.66%. The most remarkable thing is that the shear strength and splitting strength of graded broken stone increased by 36.63% and 23.84% respectively after adding fiber. This shows that its ability to resist wheel load is greatly improved.

### **2.2.3 Cement Stabilized Macadam**

In order to obtain the best bearing capacity of cement-stabilized macadam, the influence of different cement-based binders on the performance of cement-stabilized macadam was studied during the design of cement-stabilized macadam. Three binders, 5% Portland cement, 5% super sulphur cement and 2% steel slag powder + 3% Portland cement, were compared in this test. The gradation and properties of cement stabilized macadam are shown in Tables 6 and 7. The process of splitting strength test are shown in Fig. 2.

From Table 7, it can be seen that the unconfined compression strength of supersulphur cement stabilized macadam is the highest at 7 days, reaching 6.11 MPa under the same curing condition. Its 14 days splitting strength is 1.21 MPa. The results show that it has reached the design requirements of the intensity of the base of the expressway and the

**Table 6.** Mix proportion of cement stabilized macadam.

**Fig. 2.** The process of splitting strength test.

Test projects Unit Results 5%portland cement 5% super sulfated cement 2% steel slag powder + 3% portland cement Optimum moisture content % 5.7 5.2 6.1 7d unconfined compressive strength MPa 3.73 6.11 1.98 14d splitting strength MPa 0.45 1.25 0.27 28d plitting strength MPa 0.49 1.33 0.31 14d flexural-tensile strength MPa 0.73 1.97 0.41

**Table 7.** The performance of cement stabilized macadam.

first-class highway under the extremely heavy traffic condition. The compressive strength of steel slag powder + Portland cement stabilized macadam is the lowest in 7 days, which is only 1.98 MPa. Its 14 days cleavage strength is also the lowest, only 0.27 MPa. When the curing period reached 28 days, the unconfined compressive strength of three kinds of cement stabilized macadam increased obviously. Among them, the unconfined compressive strength of super-sulfur cement stabilized macadam increased most significantly, reaching 10.73 MPa. However, compared with the unconfined compressive strength, the indirect tensile strength of three kinds of cement stabilized macadam did not increase obviously.

Super sulfated cement was selected as binder for cement stabilized macadam.

# **3 Experimental Design and Data Collection Scheme**

# **3.1 Experimental Design**

In this research, based on the above research on the structure and materials, a test road of perpetual asphalt pavement was built with an effective loading length of 12 m. Three kinds of pavement structures were tested by linear accelerated loading test equipment, as shown in Fig. 2. The test parameters are: axle load 100 kN and tire pressure 0.7 MPa. In addition, a constant temperature control device is used to control the test temperature at 25 °C. At the same time, through the sensors embedded in the test road in advance, the road surface ruts, deflections, the settlement of the subgrade, and the longitudinal and transverse tensile strain at the bottom of surface course and base course were monitored and studied for different loading amounts. The test road and accelerated loading equipment as shown in Figs. 3 and 4 shows the embedding position and mark number of each sensor.

**Fig. 3.** The test road and accelerated loading equipment.

**Fig. 4.** Layout of sensor embedding position**.**

# **3.2 Data Collection Scheme**

### **3.2.1 Rut Cross Section**

The rutting was caused by the road surface deformation under the action of different amounts of axle loading. The durability of the different structures could be obtained by studying the change law of the rutting section for different load times amounts. In this experiment, a three-dimensional laser section instrument was used to scan the rutting sections for different amounts of axle loading. Each structure needed to collect five groups of sections, and after processing, the average value was taken as the rutting depth of each structure.

### **3.2.2 Deflection**

In this experiment, a Falling Weight Deflectometer (FWD) was used to measure the deflection of the road surface under different amounts of axle loading. The deflection data of each structure were measured three points, and the deflection data of each measuring point were measured three times. The average value of three measured points was taken as the deflection data of the structure.

### **3.2.3 Strain**

Tensile strain is an important index to control pavement cracking. In order to measure the tensile strain of the base and the surface layer, the transverse strain sensor, the longitudinal strain sensor and the vertical strain sensor are embedded. Firstly, the dynamic response of three kinds of asphalt pavement under different axle load is studied. The dynamic response of the road surface under the action of 80 kN, 100 kN and 120 kN is measured by the acceleration loading equipment. Secondly, the dynamic response of asphalt layer under different loading times is studied. It is an important data to reflect the internal fatigue failure of pavement structure and to judge the service condition of pavement structure under the action of axle load.

# **4 Results and Analysis**

### **4.1 Rut Cross Section**

Figure 5 presents the results of the rutting depth data of the three structures after different loading amounts.

**Fig. 5.** Rut depth curve**.**

As can be seen from Fig. 5, the rutting depth curves of structure a and structure B are quite similar. Among them, the rutting depth of structure a increases evenly from 0 to 500,000 times and slows down from 500,000 to 800,000 times. But it began to grow rapidly after 800000 times. The rutting depth of C structure increases relatively slowly. The B structure is between the A and C structures. This is because the base of structure A and B structure is flexible, and the aggregate is further compressed and compacted by the wheel load at the early stage, which leads to the increase of rut depth. At this stage, the cement-stabilized macadam subbase is still in the state of cementation. This stage corresponds to the load number of 0 to 500000 times. At the end of the first stage, the cement stabilized macadam subbase with C structure will be broken gradually under the action of axle load. This is the transition phase, which corresponds to 500000 to 800000 times. After 800000 times, the cement stabilized macadam subbase of the C structure is treated as an equivalent granular state, leading to a further increase in rutting depth. However, the strength of B structure base is much higher than that of graded crushed stone. Therefore, the duration of the first phase is greater than that of the A structure. After one million times, the rutting depth of three pavement structures: structure A *>* structure B *>* structure C. In the composite structure, the rutting resistance of structure B is better than that of structure A.

### **4.2 Deflection**

Figure 6 presents the results of the deflection test data after different loading amounts.

According to the curve chart of the deflection value change in Fig. 6, the deflection value of structure A was the largest, followed by the values of structure B, and the value of

**Fig. 6.** Road surface deflection value change rule graph.

structure C was the smallest. In other words, the deflection value of the composite asphalt pavement with base layer of LSPM was generally lower than that of the composite asphalt pavement with base layer of unbound aggregate, but both were larger than the deflection value of the semi-rigid asphalt pavement. This showed that the bearing capacity of semirigid asphalt pavement is better than that of the composite asphalt pavement. In addition, in the composite asphalt pavement, the bearing capacity of LSPM base was better than that of the unbound aggregate base.

### **4.3 Dynamic Response of Pavement Structure Under Different Axle Load**

In order to study the dynamic response of the three pavement structures under different axle loads, the axle loads of 80 kN, 100 kN and 120 kN are adopted. The applied loading speed is 22 km/h. The strain-time curves of the three structures under different axial loads are shown in Figs. 7, 8, 9, 10, 11, 12, 13, 14 and 15.

**Fig. 7.** Transverse (left) and longitudinal (right) flexural strains of surface layer of A structure at different loading times.

It can be seen that the strain in the pavement structure increases with the increase of axle load and there is obvious correlation between them. And the shape of the strain curve is related to the material properties of the structure and the depth of the monitoring point.

**Fig. 8.** Transverse (left) and longitudinal (right) flexural strains of middle course of A structure at different loading times.

**Fig. 9.** Transverse (left) and longitudinal (right) flexural strains of lower layer of A structure at different loading times.

**Fig. 10.** Transverse (left) and longitudinal (right) flexural strains of surface layer of B structure at different loading times.

**Fig. 11.** Transverse (left) and longitudinal (right) flexural strains of middle course of B structure at different loading times.

**Fig. 12.** Transverse (left) and longitudinal (right) flexural strains of lower layer of B structure at different loading times.

**Fig. 13.** Transverse (left) and longitudinal (right) flexural strains of surface layer of C structure at different loading times.

**Fig. 14.** Transverse (left) and longitudinal (right) flexural strains of middle course of C structure at different loading times.

**Fig. 15.** Transverse (left) and longitudinal (right) flexural strains of lower layer of C structure at different loading times.

In the case of A structure, when the axial load is increased from 80 kN to 100 kN, the longitudinal and transverse tensile strains of the upper layer increase by 33.3% and 62% respectively. In addition, the transverse strain of the three structures is more sensitive to load magnitude than the longitudinal strain. This may imply that this asphalt pavement structure under heavy load may be sensitive to longitudinal cracks.

In structure C, the bearing capacity is strong because the base is made of semi-rigid materials. It is shown in the strain diagram that the strain of the lower layer of structure C is the smallest, indicating that its bending deformation is the smallest. It is found that the transverse and longitudinal strains of structure B are smaller than those of structure A. This is because the base of structure B using LSPM, has a stronger load-bearing capacity than graded crushed stone materials. The maximum transverse tensile strain of structure A and structure B appears in the lower layer, and the maximum longitudinal tensile strain appears in the middle layer.

### **4.4 Dynamic Response of Pavement Structure Under Different Loading Times**

Figures 16 and 17 show how the longitudinal and transverse flexural tensile strains in the three asphalt layers vary with loading times.

**Fig. 16.** Transverse bending strain at the bottom of asphalt layer at different loading times.

**Fig. 17.** Longitudinal bending strain at the bottom of asphalt layer at different loading times.

The longitudinal flexural-tensile strain of the three pavement structures is smaller than that of the transverse flexural-tensile strain at different loading times. In the course of the test, the strain value of the three pavement structures fluctuates up and down, which is mainly caused by the temperature change of the pavement layer during the loading. However, the bending and tensile strains of the three structures show a decreasing trend. Among them, the transverse and longitudinal flexural tensile strains appear at the bottom of the asphalt layer of structure C is the largest. This shows that the bottom of structure C asphalt layer is easy to incur Buttom-Up crack extending from bottom to top. In the inverted asphalt pavement structure, the bending strain of structure B is less than that of semi-rigid structure, but its changing trend is similar to that of structure C. However, the bending strain at the bottom of A structure asphalt does not increase during the loading. This indicates that the fatigue cracking resistance of structure A is the strongest. That is to say, the bending and tensile strain at the bottom of asphalt layer can be reduced obviously by setting graded broken stone base.

# **5 Results and Analysis**

Through material design and accelerated loading test, this paper studies the performance of composite asphalt pavement structure and its feasibility as a perpetual pavement structure. The following conclusions are obtained:

**Acknowledgements.** The research is supported by grants from the National Key Research and Development Program of China (No. 2017YFF0205600, 2017YFF0205602) and the National Key R&D Program of China (2019YFE0117600). The sponsorship is gratefully acknowledged. The contents of this paper only reflect the views of the authors and do not reflect the official views or policies of the sponsors.

# **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Parameter Sensitivity Analysis of Long Span PC Continuous Beam Bridge with Corrugated Steel Webs**

Yu Jiang(B) and Peiheng Long

Department of Civil and Traffic Engineering, Beijing University of Civil Engineering and Architecture, No. 15 Yongyuan Road, Daxing District, Beijing, China<EMAIL_ADDRESS>
**Abstract.** In order to explore the regularity of alignment and stress variation of long-span corrugated steel web (CSW) continuous beam bridge during cantilever casting construction was taken as the engineering background. Based on numerical simulation, the sensitivity of parameters of alignment and stress of the main beam is carried out on cast-in-place section weight, modulus of elasticity and temperature gradient. The results show that the obvious influence of cast-in-situ section weight and temperature gradient on the alignment and stress is the key control parameters, while modulus of elasticity is the secondary control parameters. Therefore, it is necessary to monitor concrete dense and environmental temperature change in real time during construction, closure at a suitable temperature. Correct construction errors in time, ensure structural safety and smooth alignment.

**Keywords:** Corrugated steel web · Parameter sensitivity · Cantilever casting construction · Deflection · Stress

# **1 Introduction**

With the continuous growth of China's economy, new materials and new technologies are also emerging. CSW replace concrete webs to form a new type of bridge structure, which has been widely used in China [1]. Since France built the world's first highway bridge with CSW-Cognac bridge, it have been deeply studied and developed because of their unique "accordion effect" [2], small axial stiffness, reduced constraints on the roof and floor, improved prestress efficiency, less cracks in the webs, great shear performance [3], good energy dissipation capacity and seismic performance [4]. However, there are many errors in the cantilever casting construction of long-span CSW bridge, and there are errors between the actual construction state and the design theoretical state. In order to reduce the influence of error, the parameter sensitivity of CSW has been studied. It is analyzed that the main beam quality, concrete elastic modulus, prestress loss, shrinkage and creep are linear sensitivity parameters [5]. It is recognized that concrete density, elastic modulus and prestress loss are the main parameters of linear control, and temporary load is the secondary parameter [6]. The least square method is used to identify the parameters, combined with the grey system theory to predict the construction deflection error, compare it with the measured value, and the feedback is timely corrected, so as to form an adaptive control [7]. The influence of temperature difference and temperature gradient on bridge deformation is studied by numerical simulation and actual measurement. The error between the measured value and the calculated value of the model is used to identify and correct the design parameters and guide the subsequent construction [8].

At present, the parameter sensitivity research of long-span CSW bridge mostly focuses on the main beam alignment at the completion stage, and there is a lack of sensitivity research on the construction process and main beam stress. This paper mainly analyzes the parameter sensitivity of the alignment and stress in the maximum cantilever stage and the completion stage, monitors the changes of stress and alignment, and corrects the deviation in time to ensure that the main beam reaches the ideal design state.

## **2 Project Overview**

A bridge has a total length of 319.0 m, crosses the river valley, with a span of 83 m + 153 m + 83 m. It is a three span PC composite continuous box girder bridge with CSW. The design standard is a two-way four lane expressway, the design reference period is 100 years, Fig. 1 show general layout of bridge.

**Fig. 1.** General layout of bridge (unit: m).

The upper structure of the bridge is single box single chamber variable section CSW box girder, the roof and floor are made of C50 concrete, the width of the roof is 13.1 m, the length of the cantilever on both sides is 3.3 m, the thickness of the roof is 0.3 m, the thickness of the cantilever plate end is 0.2 m, the width of the floor is 6.5 m, and the thickness of the floor is 0.3 m~1.1 m. The height of the fulcrum is 8.8 m, the height of the side span end and the middle beam are 3.5 m. The thickness of the box girder floor varies from the fulcrum to the middle of the span according to a parabola of 1.8 times, Fig. 2 show standard transverse section.

Figure 3 show dimension drawing of CSW. The type of CSWs used in the experiment is the BCSW-1600. The length of the flat subpanel and the horizontal projected length of the inclined are 430 mm and 370 mm, a height of 220 mm and the thickness is 10~22 mm.

**Fig. 2.** Standard transverse section (unit: m).

**Fig. 3.** Dimension drawing of CSW (unit: mm).

# **3 Establishment of Finite Element Model**

The bridge adopts Midas civil 2020 finite element software for numerical simulation calculation. Using the CSW section provided by the software, a total of 84 elements are established for the main beam. The single-side cantilever had 16 suspension casting sections with a length of 4.8 m, and the length of middle span and side span closure section was 3.2 m, adopting the method of side span first and then middle span closure, Fig. 4 show finite element model of the main bridge.

**Fig. 4.** Finite element model of the main bridge.

# **4 Parameter Sensitivity Analysis**

In order to reasonably control the alignment and stress and guide the design and construction, parameter sensitivity is vital to the alignment and stress control of the completed bridge. Therefore, the ±10% variation of the reference value of each parameter is taken to analyze its influence on the alignment and stress of the main beam, Table 1 show variation range of each parameter.

## **4.1 Cast-In-Situ Section Weight**

Due to the influence of concrete dense temperature variation, size error of formwork and other factors during cantilever pouring, the actual weight value of cast-in-place

**Table 1.** Reference value and variation interval of design variables.

section always deviates from the theoretical value, resulting in the variation of section dead weight. Change the concrete dense to simulate the change of section weight, and monitor the change of main beam alignment and stress.

**Fig. 5.** Stress variation of main beam in bridge completion and cantilever stage under different concrete dense.

**Fig. 6.** Deflection variation of main beam in bridge completion and cantilever stage under different concrete dense.

It can be seen from Figs. 5 and 6 that the variation range of roof and floor stress is almost the same in the maximum cantilever stage and bridge completion, and the maximum variation of 0# block is 1.5 Mpa. With the increase of concrete dense the tensile stress of roof and compressive stress of floor increase. The maximum deflection variation is 12.5 mm at the quarter point of side span and middle span.

### **4.2 Modulus of Elasticity**

The modulus of elasticity will increase with the pouring age, and the measured value of the modulus of elasticity is generally greater than the theoretical value after the completion of the bridge. it directly affects the stiffness of concrete, thus affecting the alignment and stress of the main beam, so the sensitivity analysis of the change of elastic modulus should be carried out.

It can be seen from Figs. 7 and 8 that the variation of modulus of elasticity in the maximum cantilever and bridge completion has no obvious effect on the stress, the maximum value is 0.33 Mpa, and the change of roof stress is greater than that of the floor. The maximum deflection in the maximum cantilever and bridge completion is only

**Fig. 7.** Stress variation of main beam in bridge completion and cantilever stages under different modulus of elasticity.

**Fig. 8.** Deflection variation of main beam in bridge completion and cantilever stages under different modulus of elasticity.

5.0 mm, and the modulus of elasticity has little effect on the alignment and stress of the main beam.

### **4.3 Temperature Gradient**

For the temperature load, seasonal temperature difference load and sunshine temperature difference load are mainly considered in the construction, but the seasonal temperature difference load has no obvious influence on the structural stress and deformation. This paper only studies the influence of temperature gradient change on the alignment and stress of main beam in the construction and bridge completion.

**Table 2.** Stress variation of roof and floor of 0# block under different temperature gradient in the maximum cantilever stage.

It can be seen from Table 2 that during the construction process, the cantilever structure is a static structure, the temperature secondary internal force does not change significantly, and only slight stress changes are generated in the cantilever 0# block [9].

It can be seen from Figs. 9 and 10 that the change of roof stress is greater than that of floor stress at bridge completion stage. The stress change of floor stress is within 1.2 Mpa, and the maximum change of roof stress can be up to 5.7 Mpa, the maximum deflection in the bridge completion stage is 9 mm. The change of temperature gradient has a significant impact on the stress and alignment of the main beam in the bridge completion stage.

**Fig. 9.** Stress variation of main beam in bridge completion stage under different temperature gradient.

**Fig. 10.** Deflection variation of main beam in bridge completion under different modulus of elasticity.

# **5 Sensitivity Calculation**

There is no clear functional relationship between design parameters and objective parameters, so it is difficult to obtain the analytical solution of objective function accurately. In order to facilitate parameter analysis and reasonably optimize the alignment and stress during construction. Parameter sensitivity is introduced to quantify the impact of each parameter on target parameters [10].

$$\eta = \lim\_{\Delta x \to \text{yields}} \frac{\Delta \text{y}/\text{y}}{\Delta x/\text{x}} \tag{1}$$

Where, *y* is the target control parameter, i.e. stress and deformation in cantilever state and bridge completion state. *x* is the design parameter.

According to Formula (1), take ±10% of the change of each parameter to calculate the sensitivity of it to alignment and stress. The calculation results are shown in the Table 3.

**Table 3.** Sensitivity of design parameters in bridge completion and maximum cantilever stages.

Note: Higher sensitivity value means higher sensitivity of parameters

It can be seen from Table 3 that the sensitivity of concrete dense and temperature gradient is strong, and the sensitivity of modulus of elasticity is weak.

# **6 Conclusion**

**Acknowledgments.** I sincerely thank my tutor for my careful guidance and all the members of the research group for their opinions and findings.

# **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Preparation and Impact Resistance of Carbon Fiber Reinforced Metal Laminates Modified by Carbon Nanotubes**

Zehui Jia1, Lingwei Xu2, Shuangkai Huang1, Haoran Xu1, Zhimo Zhang1, and Xu Cui1(B)

<sup>1</sup> College of Civil Aviation, Shenyang Aerospace University, Shenyang 110136, China<EMAIL_ADDRESS>
<sup>2</sup> College of Aeroengine Academy, Shenyang Aerospace University, Shenyang 110136, China

**Abstract.** Fiber reinforced metal laminates (FMLs) are a kind of interlaminar hybrid composites made of metal sheets and fibers alternately stacked and cured at a certain pressure and temperature. In this paper, through the simulation of ABAQUS finite element software and recording the change of projectile velocity, the energy loss of projectile is calculated and the impact resistance is judged. Through the comparison of three groups of simulation experimental results, the energy absorbed by carbon fiber reinforced metal laminate is about 300 times that of aluminum alloy plate, which fully shows that carbon fiber reinforced metal composite has excellent impact resistance compared with aluminum alloy. After adding 1 wt% carbon nanotubes to carbon fiber reinforced metal laminates, the absorbed energy is about 10 times that of the original, which shows that carbon nanotubes improve the ultimate yield stress of resin and materials in epoxy resin and enhance the weakness that the composites are easy to delamination under impact load.

**Keywords:** FMLs · Carbon fiber · Finite element · Carbon nanotubes · Impact load

# **1 Introduction**

Fiber reinforced metal laminates (FMLs) are formed by alternately stacking metal material layers and fiber reinforced composite layers, similar to sandwich structure. Each material layer is bonded by adhesive under certain temperature and vacuum conditions. It can show the excellent mechanical properties of fiber composites and metal materials, the high strength, high stiffness and fatigue resistance of carbon fiber reinforced composites, as well as the high toughness and excellent damage tolerance of metal materials [1–3]. As an excellent modifier and reinforcing material, carbon nanotubes have attracted extensive attention of scientists all over the world [4, 5]. After adding CNTs, the performance of FMLs is greatly improved, and the crack resistance, interlaminar fracture toughness and impact resistance of the matrix are improved [6].

Avila et al. [7] studied the effect of adding nano clay to glass reinforced composites under low velocity impact. The results show that adding 5 wt% nano clay can increase the energy absorption by 48%. Khoramishad et al. [8] studied the effect of multi walled carbon nanotubes on the impact resistance of fiber metal laminates and found that adding 0.5 wt% CNT can improve the impact resistance of fiber metal laminates. Xian Xingjuan [9] studied the static and fatigue tensile failure characteristics of CFRP (epoxy resin matrix) composite laminates with 0° and 45° ply without notch and with straight edge notch through the analysis of a series of test results of 120 groups of specimens. Lin Xiaohong and others used the finite element analysis software ABAQUS/explicit for low-speed impact analysis of carbon fiber epoxy resin composites in the 0°/90° paving direction. The Johnson cook model parameters were used for aluminum alloy, and the Hashin damage criterion was used for fiber layer to establish the bonding layer element, analyze the loss of impact energy and observe and analyze the stress distribution diagram. Therefore, ABAQUS is used to simulate the results of FMLs under high-speed impact, and to simulate the impact resistance of FMLs with CNTs.

# **2 Preparation and Testing of Carbon Fiber Reinforced Metal Laminates Modified by Carbon Nanotubes**

## **2.1 Preparation of Carbon Fiber Reinforced Metal Laminate**

The carbon fiber reinforced metal laminate used in the experiment is composed of two metal plates and a layer of orthogonal carbon fiber woven cloth. The metal aluminum plates are aluminum alloy 2024-T3 aluminum commonly used in aircraft, with a thickness of about 0.3 mm, and the thickness of orthogonal carbon fiber woven step is about 0.25 mm. E51 epoxy resin is used between layers. CNTs were added to epoxy resin by ultrasonic method. Finally, the method of autoclave curing is used for molding (Table 1).

**Table 1.** Shows the material parameters of 2024-T3 aluminum alloy.

### **2.2 High Speed Impact Test**