Research on transverse distribution coefficient of external prestressing and carbon fiber reinforced beam

1. Introduction

Due to the limitation of technology level and the increase of traffic load, and a certain degree of damage generated in a part of bridges, the bearing capacity is reduced, and a large number of dangerous old bridges emerge takes place [1]. Therefore, in order to protect people’s lives and property safety, a proper reinforcement for these damaged bridges structure to improve their bearing capacity becomes the primary task in bridge maintenance in our country at present. Joint reinforcement is a new method of reinforcement put forward on the basis of traditional strengthening methods such as externally prestressing reinforcement, carbon fiber reinforcement and stick steel reinforcement [2-4]. Joint reinforcement has higher reliability, security, and thus becomes the subject of active research of bridge repair and reinforcement [5].

Wen-Ping Xu [6] has had an experimental research of stick steel-external prestressing reinforcement, structural characteristics such as ultimate bearing capacity and crack width of concrete T beam were compared with those with the stick steel reinforcement, external prestressing and stick steel-external prestressing respectively. Test results were as follows: the strengthening effect of joint reinforcement is “1 + 1 > 2”; Wei Lu [7] has studied carbon fiber cloth-external prestressing reinforcement. Mechanics regularity of joint reinforcement beam was got by the research of T beam reinforced with carbon fiber, external prestressing and carbon fiber- external prestressing respectively. The results were as follows: When beam reinforced with carbon fiber- external prestressing is under the force, the utilization rate of carbon fiber is higher than the utilization rate of external prestressing tendons, the carbon fiber cloth makes a greater contribution in the early stage, and the external prestressing tendons will be studied more later. Their researches show that the joint reinforcement has a certain engineering application value. It is of great significance to study the joint reinforcement. And some studies on the forced performance of strengthened beams have been done.

Bolduc [8] has studied the influence of fatigue loading, prior cracking and patch materials on flexural performance of reinforced concrete members retrofitted with externally bonded CFRP plates. And many available test data are obtained through retrofitting and testing of a 18.3 m (60 ft) pre-stressed box girder retrofitted with CFRP composite plates with mechanical anchors. Choi [9] conducted an experiment of effective stresses of concrete beams strengthened using carbon-fiber-reinforced polymer and external prestressing tendons. The effective stress of the RC beams was estimated by analyzing the experimental reinforcing effect and the resulting behavior of each specimen. It was found that the strengthening effect of CFRP was affected by the initial damage in the RC beams. On the other hand, in external post-tensioning, application of strain induced by an initial effective stress in the RC beams. As a result, the effective stress of these RC beams was reduced, and their performance was improved. Han [10] has highlighted that the external prestressing method was limited for strengthening by that it is hard to make the external anchorage with big load carrying capacity, and the fiber strengthening method was also limited by the volume of reinforcement. Therefore, a new strengthening method of PSC girder using external prestressing and glass fiber was proposed. For this study, Han accomplished field tests which assess the load carrying capacity before and after strengthening using external prestressing and glass fiber reinforcement, also various values were measured when external tendons were pre-stressed. As a result, the ultimate load carrying capacity of the bridges, which were reinforced with external tendons and glass fiber, was higher than the original designed internal force of the bridges. Their experiments deeply studied the forced performance of external prestressing and carbon fiber reinforced beam and laid the foundation for a theoretical analysis.

The research results of the joint reinforcement methods mentioned above provided an important reference for a further research, nevertheless these studies were mainly concentrated on the experimental research and gave a little reliable spatial analysis and design theory of the bridge superstructure after reinforcement. Based on the above study background, the joint reinforcement theory of external prestressing and carbon fiber reinforcement was studied in this paper, the analytical solutions of beam after being reinforced with external prestressing and carbon fiber were derived, the transverse distribution coefficient of girder after joint reinforcement was solved, and the influence of pre-stressed steel and carbon fiber on the overall mechanical performance of the main girder was obtained, the design theory of this paper can be effectively used to calculate the displacement and stress state of the main girder of the bridge strengthened with the external pre-stress method. It can provide a theoretical basis for the engineering application.

2. Basic assumptions

According to the actual deformation and stress state of reinforced beam, the basic assumptions used in this article are as follows [11, 12]: (1) Concrete beam accords with flat section assumption; (2) Carbon fiber cloth bears axial force only; (3) There is slip strain difference at the interface of carbon fiber cloth and concrete beam, and the shear stress transferred at interface is proportional to the displacement difference of carbon fiber cloth and concrete beam; (4) External pre-stressed tendons have common deformation with a composite beam at the anchor point, and the stress and strain of prestressed tendons has a linear relationship; (5) The stiffness of crossbeam at the min-span is infinite, and the deformation of mid-span section is a straight line under load; (6) The load assigned to each beam is proportional to the deflection; (7) The torsional effect of the main girder is ignored.

3. Basic equation of composite beam after carbon fiber affixing

3.1. Displacement function of composite beam

Because the deformation of concrete and carbon fiber is incoordinated [13], there are three basic unknown functions of the beam displacement after reinforcement, namely: $u_c\left(x\right)$, $u_f\left(x\right)$, $w\left(x\right)$. Where $u_c\left(x\right)$, $u_f\left(x\right)$ are the axial displacements of concrete beam and carbon fiber cloth at section$x$, respectively. $w\left(x\right)$ is the vertical displacement of composite beam. The vertical lift effect of composite beam was not considered, so $w\left(x\right)=w_c\left(x\right)=w_f\left(x\right)$.

According to Fig. 1, the displacement function expressions of composite beam after pasting carbon fiber are given as:

1

$\left\{\begin{array}{l}{u}_c\left(x\right)=u_{c0}\left(x\right)+z_{cb}\theta \left(x\right),\\ w\left(x\right)=w_c\left(x\right)=w_f\left(x\right),\\ u_f\left(x\right)=u_f\left(x\right),\end{array}\right.$

where $u_{c0}\left(x\right)$ is the axial displacement of concrete beam at the centroid section; $\theta \left(x\right)=w\text{'}\left(x\right)$ is the deflection angle at section $x$; $z_{cb}$ is the distance from the centroid section of concrete beam to interface at the beam bottom, $z_{cb}$ is negative under the section centroid, otherwise it is positive.

$\mathrm{\Delta }{u}_{cf}$ is the sliding displacement difference between the concrete beam and carbon fiber cloth, according to Eq. (1):

2

$\mathrm{\Delta }{u}_{cf}\left(x\right)=u_c\left(x\right)-z_{cb}{w}^{\text{'}\left(x\right)}-u_f\left(x\right).$

Fig. 1Cross-sectional displacement of composite beam

3.4. Balance equations

Based on the relationship between the force and the distribution force in the literature [16], differentiating Eq. (5), then put Eq. (3) into the formula after being differentiated, is:

$q_{fc}\left(x\right)=\frac{d{N}_c}{dx}=E_c{A}_c{u\text{'}\text{'}}_c\left(x\right),$

6

$q_{fc}\left(x\right)=\frac{d{N}_f}{dx}=-E_f{A}_f{u^{\text{'}\text{'}}}_f\left(x\right).$

Fig. 2Force diagram of micro-unit at interface

4. Deflection equations of simply supported beam reinforced with external prestressing and carbon fiber cloth

4.1. Displacement formulas of composite beam reinforced with external prestressing

Composite beam reinforced with external prestressing can be considered as the internal statically indeterminate problem [17, 18] as shown in Fig. 3.

Fig. 3Simply supported beam after joint reinforcement

Axial force balance equation of composite beam, when external force is zero, is:

7

$x_1+N_c\left(x\right)+N_f\left(x\right)=0.$

Force method equation can be got:

8

${\delta }_{11}{x}_1+{\delta }_{1p}=0,$

where ${\delta }_{11}$ is the relative displacement of prestressing tendons’ left and right sides at the location of min-span when $x_1=$ 1, it is the constant displacement; ${\delta }_{1p}$ is the relative displacement of prestressing tendons’ left and right sides at the location of min-span under external load, it is the load displacement.

4.2. Solution of constant displacement

According to equilibrium conditions of the internal and external bending moments:

9

$M_p\left(x\right)=M_c\left(x\right)-N_f\left(x\right)z_{cb}.$

According to Eq. (2) and Eq. (4):

10

$q_{fc}\left(x\right)=k_{fc}\left[u_c\left(x\right)-z_{cb}w\text{'}\left(x\right)-u_f\left(x\right)\right].$

Substituting Eqs. (3) and (6) into Eq. (7):

11

$x_1+E_c{A}_c{u\text{'}}_c\left(x\right)=-E_f{A}_f{u^{\text{'}}}_f\left(x\right).$

Substituting Eqs. (3) and (5) into Eq. (9):

12

$M_p\left(x\right)=E_c{I}_{c}w\text{'}\text{'}\left(x\right)-E_f{A}_f{u\text{'}}_f\left(x\right)z_{cb}=\left\{\begin{array}{l}0,0\le x\le l_{c1},\\ x_1{e}_p,l_{c1}\le x\le l_{c1}+l_{c2}\\ 0,l_{c1}+l_{c2}\le x\le l.\end{array}\right.,$

According to Eqs. (6), (10) and (11):

13

$w^{\text{'}\text{'}\left(x\right)}=\frac{E_c{A}_c{E}_f{A}_f}{E_c{A}_c{z}_{cb}{k}_{fc}}{u^{\text{'}\text{'}\text{'}}}_f\left(x\right)-\frac{E_f{A}_f}{E_c{A}_c{z}_{cb}}{u^{\text{'}}}_f\left(x\right)-\frac{E_c{A}_c}{E_c{A}_c{z}_{cb}}{u^{\text{'}}}_f\left(x\right)-\frac{x_1}{E_c{A}_c{z}_{cb}}.$

Substituting Eq. (13) into Eq. (12):

14

$\alpha {u\text{'}\text{'}\text{'}}_f\left(x\right)-\beta {u^{\text{'}}}_f\left(x\right)=\left\{\begin{array}{l}{\gamma }_1,0\le x\le l_{c1},\\ {\gamma }_2,l_{c1}\le x\le l_{c1}+l_{c2}\\ {\gamma }_3,l_{c1}+l_{c2}\le x\le l,\end{array}\right.,$

where:

$\alpha =\frac{E_c{I}_c{E}_f{A}_f}{z_{cb}{k}_{fc}},\beta =\frac{E_c{I}_c{E}_f{A}_f}{E_c{A}_c{z}_{cb}}+\frac{E_c{I}_c}{z_{cb}}+E_f{A}_f{z}_{cb},$
${\gamma }_1=\frac{E_c{I}_c{x}_1}{E_c{A}_c{z}_{cb}},{\gamma }_2=x_1{e}_p+\frac{E_c{I}_c{x}_1}{E_c{A}_c{z}_{cb}},{\gamma }_3=\frac{E_c{I}_c{x}_1}{E_c{A}_c{z}_{cb}}.$

Solve the differential equations:

15

${u\text{'}}_f\left(x\right)=\left\{\begin{array}{l}{c}_1{e}^{R_{1}x}+c_2{e}^{-R_{1}x}-\frac{{\gamma }_1}{\beta },0\le x\le l_{c1},\\ c_3{e}^{R_{1}x}+c_4{e}^{-R_{1}x}-\frac{{\gamma }_2}{\beta },l_{c1}\le x\le l_{c1}+l_{c2},\\ c_5{e}^{R_{1}x}+c_6{e}^{-R_{1}x}-\frac{{\gamma }_3}{\beta },l_{c1}+l_{c2}\le x\le l,\end{array}\right.$

where $R_1=\sqrt{\beta /\alpha }$; $c_1$, $c_2$, $c_3$, $c_4$, $c_5$, $c_6$ are undetermined coefficients.

Fig. 4Force diagram of simply supported beam

Substituting Eq. (15) into Eq. (13) when $0\le x\le l_{c1}$:

16a

$w^{\text{'}\text{'}\left(x\right)}=\frac{E_f{A}_f}{z_{cb}{k}_{fc}}\left(c_1{R}_1^2{e}^{R_{1}x}+c_2{R}_1^2{e}^{-R_{1}x}\right)$
$-\left(\frac{E_f{A}_f}{E_c{A}_c{z}_{cb}}+\frac{1}{z_{cb}}\right)\left(c_1{e}^{R_{1}x}+c_2{e}^{-R_{1}x}-\frac{{\gamma }_1}{\beta }\right)-\frac{x_1}{E_c{A}_c{z}_{cb}}.$

When $l_{c1}\le x\le l_{c1}+l_{c2}$:

16b

$w^{\text{'}\text{'}\left(x\right)}=\frac{E_f{A}_f}{z_{cb}{k}_{fc}}\left(c_3{R}_1^2{e}^{R_{1}x}+c_4{R}_1^2{e}^{-R_{1}x}\right)$
$-\left(\frac{E_f{A}_f}{E_c{A}_c{z}_{cb}}+\frac{1}{z_{cb}}\right)\left(c_3{e}^{R_{1}x}+c_4{e}^{-R_{1}x}-\frac{{\gamma }_2}{\beta }\right)-\frac{x_1}{E_c{A}_c{z}_{cb}}.$

When $l_{c1}+l_{c2}\le x\le l$:

16c

$w^{\text{'}\text{'}\left(x\right)}=\frac{E_f{A}_f}{z_{cb}{k}_{fc}}\left(c_5{R}_1^2{e}^{R_{1}x}+c_6{R}_1^2{e}^{-R_{1}x}\right)$
$-\left(\frac{E_f{A}_f}{E_c{A}_c{z}_{cb}}+\frac{1}{z_{cb}}\right)\left(c_5{e}^{R_{1}x}+c_6{e}^{-R_{1}x}-\frac{{\gamma }_3}{\beta }\right)-\frac{x_1}{E_c{A}_c{z}_{cb}}.$

Integral Eq. (16) when $0\le x\le l_{c1}$:

17a

$w^{\text{'}\left(x\right)}=\frac{E_f{A}_f}{z_{cb}{k}_{fc}}\left(c_1{R}_1{e}^{R_{1}x}-c_2{R}_1{e}^{-R_{1}x}\right)$
$-\left(\frac{E_f{A}_f}{E_c{A}_c{z}_{cb}}+\frac{1}{z_{cb}}\right)\left(\frac{c_1}{R_1}{e}^{R_{1}x}-\frac{c_2}{R_1}{e}^{-R_{1}x}-\frac{{\gamma }_1}{\beta }x\right)-\frac{x_1}{E_c{A}_c{z}_{cb}}x+c_7.$

When $l_{c1}\le x\le l_{c1}+l_{c2}$:

17b

$w^{\text{'}\left(x\right)}=\frac{E_f{A}_f}{z_{cb}{k}_{fc}}\left(c_3{R}_1{e}^{R_{1}x}-c_4{R}_1{e}^{-R_{1}x}\right)$
$-\left(\frac{E_f{A}_f}{E_c{A}_c{z}_{cb}}+\frac{1}{z_{cb}}\right)\left(\frac{c_3}{R_1}{e}^{R_{1}x}-\frac{c_4}{R_1}{e}^{-R_{1}x}-\frac{{\gamma }_2}{\beta }x\right)-\frac{x_1}{E_c{A}_c{z}_{cb}}x+c_8.$

When $l_{c1}+l_{c2}\le x\le l$:

17c

$w^{\text{'}\left(x\right)}=\frac{E_f{A}_f}{z_{cb}{k}_{fc}}\left(c_5{R}_1{e}^{R_{1}x}-c_6{R}_1{e}^{-R_{1}x}\right)$
$-\left(\frac{E_f{A}_f}{E_c{A}_c{z}_{cb}}+\frac{1}{z_{cb}}\right)\left(\frac{c_5}{R_1}{e}^{R_{1}x}-\frac{c_6}{R_1}{e}^{-R_{1}x}-\frac{{\gamma }_3}{\beta }x\right)-\frac{x_1}{E_c{A}_c{z}_{cb}}x+c_9.$

Integral Eq. (17) when 0 $\le x\le l_{c1}$:

18a

$w\left(x\right)=\frac{E_f{A}_f}{z_{cb}{k}_{fc}}\left(c_1{e}^{R_{1}x}+c_2{e}^{-R_{1}x}\right)$
$-\left(\frac{E_f{A}_f}{E_c{A}_c{z}_{cb}}+\frac{1}{z_{cb}}\right)\left(\frac{c_1}{R_1^2}{e}^{R_{1}x}+\frac{c_2}{R_1^2}{e}^{-R_{1}x}-\frac{{\gamma }_1}{2\beta }{x}^2\right)-\frac{x_1}{2E_c{A}_c{z}_{cb}}{x}^2+c_{7}x+c_{10}.$

When $l_{c1}\le x\le l_{c1}+l_{c2}$:

18b

$w\left(x\right)=\frac{E_f{A}_f}{z_{cb}{k}_{fc}}\left(c_3{e}^{R_{1}x}+c_4{e}^{-R_{1}x}\right)-\left(\frac{E_f{A}_f}{E_c{A}_c{z}_{cb}}+\frac{1}{z_{cb}}\right)\left(\frac{c_3}{R_1^2}{e}^{R_{1}x}+\frac{c_4}{R_1^2}{e}^{-R_{1}x}-\frac{{\gamma }_2}{2\beta }{x}^2\right)$
$-\frac{x_1}{2E_c{A}_c{z}_{cb}}{x}^2+c_{8}x+c_{11}.$

When $l_{c1}+l_{c2}\le x\le l$:

18c

$w\left(x\right)=\frac{E_f{A}_f}{z_{cb}{k}_{fc}}\left(c_5{e}^{R_{1}x}+c_6{e}^{-R_{1}x}\right)-\left(\frac{E_f{A}_f}{E_c{A}_c{z}_{cb}}+\frac{1}{z_{cb}}\right)\left(\frac{c_5}{R_1^2}{e}^{R_{1}x}+\frac{c_6}{R_1^2}{e}^{-R_{1}x}-\frac{{\gamma }_3}{2\beta }{x}^2\right)$
$-\frac{x_1}{2E_c{A}_c{z}_{cb}}{x}^2+c_{9}x+c_{12}.$

Integral Eq. (15):

19

$u_f\left(x\right)=\left\{\begin{array}{l}\frac{c_1}{R_1}{e}^{R_{1}x}-\frac{c_2}{R_1}{e}^{-R_{1}x}-\frac{{\gamma }_1}{\beta }x+c_{13},0\le x\le l_{c1},\\ \frac{c_3}{R_1}{e}^{R_{1}x}-\frac{c_4}{R_1}{e}^{-R_{1}x}-\frac{{\gamma }_2}{\beta }x+c_{14},l_{c1}\le x\le l_{c1}+l_{c2}\\ \frac{c_5}{R_1}{e}^{R_{1}x}-\frac{c_6}{R_1}{e}^{-R_{1}x}-\frac{{\gamma }_3}{\beta }x+c_{15},l_{c1}+l_{c2}\le x\le l.\end{array}\right.,$

According to Eqs. (10), (17) and (19) when 0 $\le x\le l_{c1}$:

20a

$u_c\left(x\right)=-\frac{E_f{A}_f}{E_c{A}_c}\frac{1}{R_1}{e}^{R_{1}x}{c}_1+\frac{E_f{A}_f}{E_c{A}_c}\frac{1}{R_1}{e}^{-R_{1}x}{c}_2+z_{cf}{c}_7+c_{13}+\frac{E_f{A}_f}{E_c{A}_c}\frac{{\gamma }_1}{\beta }x-\frac{x_1}{E_c{A}_c}x.$

When $l_{c1}\le x\le l_{c1}+l_{c2}$:

20b

$u_c\left(x\right)=-\frac{E_f{A}_f}{E_c{A}_c}\frac{1}{R_1}{e}^{R_{1}x}{c}_3+\frac{E_f{A}_f}{E_c{A}_c}\frac{1}{R_1}{e}^{-R_{1}x}{c}_4+z_{cf}{c}_8+c_{14}+\frac{E_f{A}_f}{E_c{A}_c}\frac{{\gamma }_2}{\beta }x-\frac{x_1}{E_c{A}_c}x.$

When $l_{c1}+l_{c2}\le x\le l$:

20c

$u_c\left(x\right)=-\frac{E_f{A}_f}{E_c{A}_c}\frac{1}{R_1}{e}^{R_{1}x}{c}_5+\frac{E_f{A}_f}{E_c{A}_c}\frac{1}{R_1}{e}^{-R_{1}x}{c}_6+z_{cf}{c}_9+c_{15}+\frac{E_f{A}_f}{E_c{A}_c}\frac{{\gamma }_3}{\beta }x-\frac{x_1}{E_c{A}_c}x.$

According to the displacement continuous equations of composite beam and balance relation of axial force, boundary conditions can be got as follows:

$\omega \left(0\right)=0,\omega \left(l\right)=0,\omega \left(l_{c1L}\right)=\omega \left(l_{c1R}\right),\omega {\left(l_{c1}+l_{c2}\right)}_L=\omega {\left(l_{c1}+l_{c2}\right)}_R,u_f\left(0\right)=0,$
$u_c\left(0\right)=0{,u}_f\left(l_{c1L}\right)=u_f\left(l_{c1R}\right){,u}_f{\left(l_{c1}+l_{c2}\right)}_L=u_f{\left(l_{c1}+l_{c2}\right)}_R,{u^{\text{'}}}_f\left(0\right)=0,$
${u^{\text{'}}}_f\left(l\right)=0,{u^{\text{'}}}_f\left(l_{c1L}\right)={u^{\text{'}}}_f\left(l_{c1R}\right),\begin{array}{l}\\ {u\text{'}}_f{\left(l_{c1}+l_{c2}\right)}_L={u\text{'}}_f{\left(l_{c1}+l_{c2}\right)}_R,\end{array}$
${u^{\text{'}\text{'}}}_f{\left(l_{c1}+l_{c2}\right)}_L={u^{\text{'}\text{'}}}_f{\left(l_{c1}+l_{c2}\right)}_R,{u^{\text{'}\text{'}}}_f\left(l_{c1L}\right)={u^{\text{'}\text{'}}}_f\left(l_{c1R}\right),{\omega }^{\text{'}\left(l_{c1L}\right)}={\omega }^{\text{'}\left(l_{c1R}\right)},$
$\omega \text{'}{\left(l_{c1}+l_{c2}\right)}_L=\omega \text{'}{\left(l_{c1}+l_{c2}\right)}_R.$

Substituting Eqs. (15), (18), (19) and (20) into boundary conditions:

21

$\mathbf{C}=\left[\begin{array}{c}{c}_1\\ c_2\\ ⋮\\ c_{15}\end{array}\right]=\frac{\mathbf{B}}{\mathbf{A}},$

$\begin{array}{l}\mathbf{A}=\\ \left[\begin{array}{ccccccccccccccc}k& k& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& e^{R_{1}l}k& e^{-R_{1}l}k& 0& 0& l& 0& 0& 1& 0& 0& 0\\ e^{R_1{l}_{c1}}k& e^{-R_1{l}_{c1}}k& -e^{R_1{l}_{c1}}k& -e^{-R_1{l}_{c1}}k& 0& 0& l_{c1}& -l_{c1}& 0& 1& -1& 0& 0& 0& 0\\ 0& 0& e^{R_1\left(l_{c1}+l_{c2}\right)}k& e^{-R_1\left(l_{c1}+l_{c2}\right)}k& -e^{R_1\left(l_{c1}+l_{c2}\right)}k& -e^{-R_1\left(l_{c1}+l_{c2}\right)}k& 0& \left(l_{c1}+l_{c2}\right)& -\left(l_{c1}+l_{c2}\right)& 0& 1& -1& 0& 0& 0\\ \frac{1}{R_1}& -\frac{1}{R_1}& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0\\ \frac{e^{R_1{l}_{c1}}}{R_1}& -\frac{e^{-R_1{l}_{c1}}}{R_1}& -\frac{e^{R_1{l}_{c1}}}{R_1}& \frac{e^{-R_1{l}_{c1}}}{R_1}& 0& 0& 0& 0& 0& 0& 0& 0& 1& -1& 0\\ 0& 0& \frac{e^{R_1\left(l_{c1}+l_{c2}\right)}}{R_1}& -\frac{e^{-R_1\left(l_{c1}+l_{c2}\right)}}{R_1}& -\frac{e^{R_1\left(l_{c1}+l_{c2}\right)}}{R_1}& \frac{e^{-R_1\left(l_{c1}+l_{c2}\right)}}{R_1}& 0& 0& 0& 0& 0& 0& 0& 1& -1\\ -\frac{E_s{A}_s}{E_c{A}_c{R}_1}& \frac{E_s{A}_s}{E_c{A}_c{R}_1}& 0& 0& 0& 0& z_{cs}& 0& 0& 0& 0& 0& 1& 0& 0\\ 1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& e^{R_{1}l}& e^{-R_{1}l}& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ e^{R_1{l}_{c1}}& e^{-R_1{l}_{c1}}& -e^{R_1{l}_{c1}}& -e^{-R_1{l}_{c1}}& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& e^{R_1\left(l_{c1}+l_{c2}\right)}& e^{-R_1\left(l_{c1}+l_{c2}\right)}& -e^{R_1\left(l_{c1}+l_{c2}\right)}& -e^{-R_1\left(l_{c1}+l_{c2}\right)}& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ R_1{e}^{R_1\left(l_{c1}+l_{c2}\right)}& -R_1{e}^{-R_1\left(l_{c1}+l_{c2}\right)}& -R_1{e}^{R_1\left(l_{c1}+l_{c2}\right)}& R_1{e}^{-R_1\left(l_{c1}+l_{c2}\right)}& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ e^{R_1{l}_{c1}}k{R}_1& -e^{-R_1{l}_{c1}}k{R}_1& -e^{R_1{l}_{c1}}k{R}_1& e^{-R_1{l}_{c1}}k{R}_1& 0& 0& 1& -1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& e^{R_1\left(l_{c1}+l_{c2}\right)}k{R}_1& -e^{-R_1\left(l_{c1}+l_{c2}\right)}k{R}_1& -e^{R_1\left(l_{c1}+l_{c2}\right)}k{R}_1& e^{-R_1\left(l_{c1}+l_{c2}\right)}k{R}_1& 0& 1& -1& 0& 0& 0& 0& 0& 0\end{array}\right]\end{array},$

$\begin{array}{l}\mathbf{B}=\left[\begin{array}{lllllllll}0& -\frac{{\gamma }_3}{2\beta }{l}^{2}m+\frac{x_1}{2E_c{A}_c{z}_{cs}}{l}^2& \frac{{\gamma }_2}{2\beta }{l_{c1}}^{2}m-\frac{{\gamma }_1}{2\beta }{l_{c1}}^{2}m& \left(\frac{{\gamma }_3}{2\beta }-\frac{{\gamma }_2}{2\beta }\right){\left(l_{c1}+l_{c2}\right)}^{2}m& 0& -\frac{{\gamma }_2}{\beta }{l}_{c1}+\frac{{\gamma }_1}{\beta }{l}_{c1}& & & \end{array}\right.\\ \left.\begin{array}{llllll}-\frac{{\gamma }_3}{\beta }\left(l_{c1}+l_{c2}\right)+\frac{{\gamma }_2}{\beta }\left(l_{c1}+l_{c2}\right)0\frac{{\gamma }_1}{\beta }\frac{{\gamma }_3}{\beta }& -\frac{{\gamma }_2}{\beta }+\frac{{\gamma }_1}{\beta }& -\frac{{\gamma }_3}{\beta }+\frac{{\gamma }_2}{\beta }& 0& \frac{{\gamma }_2}{\beta }{l}_{c1}m-\frac{{\gamma }_1}{\beta }{l}_{c1}m& \left(\frac{{\gamma }_3-{\gamma }_2}{\beta }\right)\left(l_{c1}+l_{c2}\right)m\end{array}\right],\end{array}$

where:

$k=\frac{E_f{A}_f}{z_{cb}{k}_{fc}}-\frac{1}{R_1^2}\left(\frac{E_f{A}_f}{E_c{A}_c{z}_{cb}}+\frac{1}{z_{cb}}\right),m=\frac{E_f{A}_f}{E_c{A}_c{z}_{cb}}+\frac{1}{z_{cb}}.$

The constant displacement ${\delta }_{11}$ can be got by the relative displacement of prestressing tendons’ left and right sides at the location of min-span:

22

${\delta }_{11}=u_c\left(l_{c1}\right)-u_c\left(l_{c1}+l_{c2}\right)+{\theta }_c\left(l_{c1}\right)e_p-{\theta }_c\left(l_{c1}+l_{c2}\right)e_p+\frac{x_1{l}_{c2}}{E_p{A}_p}.$

Substituting Eqs. (17) and (20) into Eq. (22):

23

${\delta }_{11}=-\frac{E_f{A}_f}{E_c{A}_c}\frac{1}{R_1}{e}^{R_1{l}_{c1}}{c}_1+\frac{E_f{A}_f}{E_c{A}_c}\frac{1}{R_1}{e}^{-R_1{l}_{c1}}{c}_2+z_{cb}{c}_7+c_{13}+\frac{E_f{A}_f}{E_c{A}_c}\frac{{\gamma }_1}{\beta }{l}_{c1}-\frac{x_1}{E_c{A}_c}{l}_{c1}$
$-\frac{E_f{A}_f}{E_c{A}_c}\frac{1}{R_1}{e}^{R_1\left(l_{c1}+l_{c2}\right)}{c}_3-\frac{E_f{A}_f}{E_c{A}_c}\frac{1}{R_1}{e}^{-R_1\left(l_{c1}+l_{c2}\right)}{c}_4-z_{cb}{c}_8-c_{14}+\frac{x_1}{E_c{A}_c}\left(l_{c1}+l_{c2}\right)$
$-\frac{E_f{A}_f}{E_c{A}_c}\frac{{\gamma }_2}{\beta }\left(l_{c1}+l_{c2}\right)+\left[\frac{E_f{A}_f}{z_{cb}{k}_{fc}}\left(c_1{R}_1{e}^{R_1{l}_{c1}}-c_2{R}_1{e}^{-R_1{l}_{c1}}\right)\right.-\frac{x_1}{E_c{A}_c{z}_{cb}}{l}_{c1}$

$+c_7-\frac{E_f{A}_f}{z_{cb}{k}_{fc}}\left(c_3{R}_1{e}^{R_1\left(l_{c1}+l_{c2}\right)}-c_4{R}_1{e}^{-R_1\left(l_{c1}+l_{c2}\right)}\right)$
$-\left(\frac{E_f{A}_f}{E_c{A}_c{z}_{cb}}+\frac{1}{z_{cb}}\right)\left(\frac{c_1}{R_1}{e}^{R_1{l}_{c1}}-\frac{c_2}{R_1}{e}^{-R_1{l}_{c1}}-\frac{{\gamma }_1}{\beta }{l}_{c1}\right)$
$+\frac{x_1}{E_c{A}_c{z}_{cb}}\left(l_{c1}+l_{c2}\right)-c_8+\left(\frac{E_f{A}_f}{E_c{A}_c{z}_{cb}}+\frac{1}{z_{cb}}\right)$
$\times \left.\left(\frac{c_3}{R_1}{e}^{R_1\left(l_{c1}+l_{c2}\right)}-\frac{c_4}{R_1}{e}^{-R_1\left(l_{c1}+l_{c2}\right)}-\frac{{\gamma }_2}{\beta }\left(l_{c1}+l_{c2}\right)\right)\right]\times e_p+\frac{x_1{l}_{c2}}{E_p{A}_p}.$