**2. Theoretical analysis**

**1. Introduction**

80 New Trends in Structural Engineering

External prestressing is a technique originally developed for reinforcing bridge structures and now has applications in architectural structures [1]. It has gone through three stages of development. In the early stage, external tendons were installed with curvature at the bottom and sides of a beam and held by deviators. Prestressing forces were applied by transverse tensioning. In the second stage, external tendons were installed with curvature only at the vertical sides of a beam, and prestressing forces were applied by a tensioning jack. Multiple weaknesses have been identified in this tensioning technique during practical applications. First, prestressing forces had to be applied to an independent working surface, but the working surface was usually obstructed by columns. Second, the friction between a deviator and an external tendon could weaken the effects of prestressing forces. To avoid the aforementioned problem with working surface, external tendons were usually continuously and axially installed along the full length of a beam. However, this would complicate the stress states of the columns and beam-column joints and thereby undermine the structure's seismic performance [2]. To overcome these disadvantages, retractable web members were introduced to apply prestressing forces in the later stage. Web members can be installed vertically or diagonally, and DWMs are an improvement on vertical web members (patent number: ZL 03134360.0). In DWM prestressing, external tendons are anchored to the upper parts of both beam sides, and retractable DWMs are used to stretch the tendons, which then transfer the prestressing forces to the beam. Compared to the two earlier external prestressing techniques, DWM prestressing has two main advantages: (1) The way in which the prestressing force is applied allows for easier and safer construction and enhances the effects of prestressing forces; (2) As external tendons are not continuously installed along the full beam length and do not span any column, the installation process will neither cause mechanical disturbance to the floor, columns, and other vertical elements nor occupy the space required to reinforce vertical elements; (3) external ten-

dons running through beam ends can increase the shear strength of beam ends [3, 4].

At present, stress increments in external tendons can be calculated mainly by the reduction factor method, regression analysis of section reinforcement ratio, deformation analysis, and so on. However, there is a lack of unified standard and the standard parameter values for an external tendon in the stiffness of a beam is infinitely great an ultimate state differ between standards from different countries [5–8]. Based on the assumption of infinite beam stiffness, a study [9] examined the force distribution in an external tendon that was subjected to a uniformly distributed load applied by DWM, with the increase in tendon length being used as the parameter. The relationship between tendon extension and load was derived. When DWM external prestressing is applied to a simply supported beam (SSB), the beam does not have infinite stiffness and tends to deform under the prestressing force [10, 11]. The load on tendons applied by web members was not uniformly distributed. Therefore, the mechanical behavior of a beam reinforced by this technique remains unknown. For this reason, the present study investigated the behavior of a SSB reinforced by external prestressing with three DWMs using a combination of theoretical derivation, numerical analysis, and experimental verification. Three variables, including initial tendon force, tendon cross-sectional area, and initial tendon sag, were considered and the influences of beam-end rotation and beam

#### **2.1. Increment in tendon stress**

#### *2.1.1. Fundamental assumptions*

The theoretical analysis was based on the following fundamental assumptions:


#### *2.1.2. Computing model of tendon*

**Figure 1** shows the curves describing deformation of an external tendon in a SSB reinforced by DWM prestressing. The dotted line L1 shown in the figure is the elliptic curve describing the deformation of a tendon under a uniformly distributed load provided in [9]. The solid line, L2 , is a broken line for a tendon stabilized by three web members; it is a polygon incised in L1 .

#### *2.1.3. Solving for external tendon force*

According to Song Yu [9], the shape function for a tendon under a uniformly distributed load has the following form:

$$\frac{\left(y - \frac{l^2 - 8f^2}{16f}\right)^2}{\left(\frac{l^2 + 8f^2}{16f}\right)^2} + \frac{\left(x - \frac{l}{2}\right)^2}{2\left(\frac{l^2 + 8f^2}{16f}\right)^2} = 1\tag{1}$$

where *l* and *f* are the tendon span and sag, respectively, after application of the initial prestressing force.

Let *<sup>m</sup>* <sup>=</sup> *<sup>l</sup>* <sup>2</sup> + 8 *f* <sup>2</sup> \_\_\_\_\_ 16*f* and compute the derivative of the shape function with respect to *x*. Then performing integration on the arc-length formula will yield the expression for initial tendon length, *s*o:

$$s\_0 = \frac{3l}{4} + \frac{\sqrt[3]{2}}{4} + \ln \frac{2}{2} \frac{\sqrt[3]{2}}{\sqrt{2}} \frac{m+l}{m-l} \tag{2}$$

When the SSB undergoes a beam-end rotation of *θ* under the action of an external load, the resulting span and sag of each tendon can be expressed as follows:

$$L' = l - 2 \cdot e \cdot \sin \theta \tag{3}$$

$$f = f + y\tag{4}$$

**2.2. Calculation of bearing capacity of a RC beam during normal service**

stressing force on the beam exerted by tendons, *ps*

After a SSB is reinforced by DWM external prestressing, the main forces acting on it include constant and live loads (e.g., concentrated force, *ki·Q*, and uniformly distributed load, *q*), pre-

Mechanical Performance of Simple Supported Concrete Beam-Cable Composite Element...

concentrated forces exerted on the beam by web members. This analysis focused on two randomly selected concentrated forces. The force analysis and the coordinate system used are

Let EI denote the flexural rigidity of a beam within its elastic range and *e* be the initial eccentricity of tendon. The force applied by a DWM can be decomposed into two components. This analysis did not consider the effect of the horizontal component on the beam's mechanical behavior in order to simplify the calculation. The analytical results proved reliable. Normally, the use of *n* web members will divide a beam into four segments. Thus three web members divide the beam into four segments. Computing the moment at an arbitrary cross-section of

> *i*=1 *n*

*x*(*l* − *bi* )/*l*

 −(*x* − *a*)(sin*γ* − sin*β*) + *x*sin*γ* (*a* ≤ *x* ≤ *a* + *l*

−(*l* − *x* − *a*)(sin*γ* − sin*β*) + (*l* − *x*)sin*γ* (*l* − *a* − *l*

'

−(*<sup>x</sup>* <sup>−</sup> *<sup>a</sup>*)(sin*<sup>γ</sup>* <sup>−</sup> sin*β*) <sup>+</sup> *<sup>x</sup>*sin*<sup>β</sup>* (*<sup>a</sup>* <sup>+</sup> *<sup>l</sup>*

−(*<sup>l</sup>* <sup>−</sup> *<sup>x</sup>* <sup>−</sup> *<sup>a</sup>*)(sin*<sup>γ</sup>* <sup>−</sup> sin*β*) <sup>+</sup> (*<sup>l</sup>* <sup>−</sup> *<sup>x</sup>*)sin*<sup>β</sup>* (*l*/<sup>2</sup> <sup>≤</sup> *<sup>x</sup>* <sup>≤</sup> *<sup>l</sup>* <sup>−</sup> *<sup>a</sup>* <sup>−</sup> *<sup>l</sup>*

⋅ sin*ϕ* ≤ *x* ≤ *l*)

(0 ≤ *x* ≤ *bi*

'

)

'⋅ sin*ϕ*)

⋅ sin*ϕ* ≤ *x* ≤ *l*/2)

'

*bi* (*l* − *x*)/*l* , reaction forces from the supports, and

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83

*f*(*x*) − *ps* ⋅ *m*(*x*) − *ps* ⋅ (*e* + *y*) ⋅ cos*γ* (9)

(*bi* <sup>≤</sup> *<sup>x</sup>* <sup>≤</sup> *<sup>l</sup>*) (10)

'⋅ sin*ϕ*)

⋅ sin*ϕ* ≤ *x* ≤ *l* − *a*)

(11)

*2.2.1. Computing model*

presented in **Figure 2**.

the beam gives

*m*(*x*) =

where

*2.2.2. Load-deformation relationship*

**Figure 2.** Analysis diagram of beam-cable element.

*EI* <sup>⋅</sup> *<sup>y</sup>*'' <sup>=</sup> *qx*(*<sup>l</sup>* <sup>−</sup> *<sup>x</sup>*) \_\_\_\_\_\_ <sup>2</sup> <sup>+</sup> *ki <sup>Q</sup>* <sup>⋅</sup> <sup>∑</sup>

*<sup>f</sup>*(*x*) <sup>=</sup> {

⎧

*x*sin*γ* (0 ≤ *x* ≤ *a*)

(*l* − *x*)sin*γ* (*l* − *a* − *l*

⎪ ⎨

⎪ ⎩

where *e* is the initial eccentricity of external tendon and *y* is the deflection of the beam under the action of an external load.

Change in tendon length is associated with tendon span and sag. After the SSB deforms, both the tendon span and sag will change. Substituting Eqs. (3) and (4) into Eq. (2) will give the tendon length for a given beam deflection.

$$s = \frac{3}{4} \frac{L}{4} + \frac{\sqrt{2} \, m^{\circ}}{4} + \ln \frac{2 \sqrt{2} \, m^{\circ} + L}{2 \sqrt{2} \, m^{\circ} - L} \tag{5}$$

where *m*' <sup>=</sup> *<sup>L</sup>* ' <sup>2</sup> + 8*f* ' 2 \_\_\_\_\_\_ 16 *f* '.

Since the behavior of external tendons is always elastic, according to the Hooke's law,

$$
\Delta S = \mathcal{S} - \mathcal{S}\_0 \tag{6}
$$

$$
\Delta S = \frac{\Delta F}{EA\_\bullet} S\_0 \tag{7}
$$

$$p\_s = P\_0 + \Delta \mathbf{F} \tag{8}$$

where ∆*s* is tendon extension, Δ*F* is the increment in tendon force, *E* is tendon's elastic modulus, *A*<sup>s</sup> is tendon cross-sectional area, *s0* is initial tendon length, and *pS* is the tendon force for a given beam deflection.

**Figure 1.** The external cable diagram of deformation.

Mechanical Performance of Simple Supported Concrete Beam-Cable Composite Element... http://dx.doi.org/10.5772/intechopen.76517 83

**Figure 2.** Analysis diagram of beam-cable element.

#### **2.2. Calculation of bearing capacity of a RC beam during normal service**

#### *2.2.1. Computing model*

Let *<sup>m</sup>* <sup>=</sup> *<sup>l</sup>*

<sup>2</sup> + 8 *f* <sup>2</sup> \_\_\_\_\_

82 New Trends in Structural Engineering

*<sup>s</sup>*<sup>0</sup> <sup>=</sup> \_\_3*<sup>l</sup>*

*f*

tendon length for a given beam deflection.

*s* = <sup>3</sup> *<sup>L</sup>*' \_\_\_

Δ*S* = \_\_\_\_ <sup>Δ</sup>*<sup>F</sup>*

is tendon cross-sectional area, *s0*

**Figure 1.** The external cable diagram of deformation.

the action of an external load.

<sup>2</sup> + 8*f* ' 2 \_\_\_\_\_\_ 16 *f* '.

where *m*' <sup>=</sup> *<sup>L</sup>* '

lus, *A*<sup>s</sup>

a given beam deflection.

16*f* and compute the derivative of the shape function with respect to *x*. Then perform-

\_\_ \_\_\_\_\_ <sup>2</sup> *<sup>m</sup>* <sup>+</sup> *<sup>l</sup>* 2 √ \_\_

<sup>2</sup> *<sup>m</sup>* <sup>−</sup> *<sup>l</sup>* (2)

<sup>2</sup> *<sup>m</sup>*' <sup>−</sup> *<sup>L</sup>*' (5)

.*S*<sup>0</sup> (7)

is the tendon force for

= *f* + *y* (4)

ing integration on the arc-length formula will yield the expression for initial tendon length, *s*o:

When the SSB undergoes a beam-end rotation of *θ* under the action of an external load, the

*L*' = *l* − 2 ⋅ *e* ⋅ sin*θ* (3)

where *e* is the initial eccentricity of external tendon and *y* is the deflection of the beam under

Change in tendon length is associated with tendon span and sag. After the SSB deforms, both the tendon span and sag will change. Substituting Eqs. (3) and (4) into Eq. (2) will give the

> \_\_ 2 *m*' + *L*' \_\_\_\_\_\_\_ 2 √ \_\_

'

<sup>4</sup> <sup>+</sup> <sup>√</sup> \_\_ <sup>2</sup> *<sup>m</sup>*' \_\_\_ <sup>4</sup> <sup>+</sup> ln <sup>2</sup> <sup>√</sup>

Since the behavior of external tendons is always elastic, according to the Hooke's law,

Δ*S* = *S* − *S*<sup>0</sup> (6)

*EAs*

*ps* = *P*<sup>0</sup> + *F* (8)

where ∆*s* is tendon extension, Δ*F* is the increment in tendon force, *E* is tendon's elastic modu-

is initial tendon length, and *pS*

<sup>4</sup> <sup>+</sup> <sup>√</sup> \_\_ \_\_\_<sup>2</sup> *<sup>m</sup>* <sup>4</sup> <sup>+</sup> ln <sup>2</sup> <sup>√</sup>

resulting span and sag of each tendon can be expressed as follows:

After a SSB is reinforced by DWM external prestressing, the main forces acting on it include constant and live loads (e.g., concentrated force, *ki·Q*, and uniformly distributed load, *q*), prestressing force on the beam exerted by tendons, *ps* , reaction forces from the supports, and concentrated forces exerted on the beam by web members. This analysis focused on two randomly selected concentrated forces. The force analysis and the coordinate system used are presented in **Figure 2**.

#### *2.2.2. Load-deformation relationship*

Let EI denote the flexural rigidity of a beam within its elastic range and *e* be the initial eccentricity of tendon. The force applied by a DWM can be decomposed into two components. This analysis did not consider the effect of the horizontal component on the beam's mechanical behavior in order to simplify the calculation. The analytical results proved reliable. Normally, the use of *n* web members will divide a beam into four segments. Thus three web members divide the beam into four segments. Computing the moment at an arbitrary cross-section of the beam gives

$$EI \cdot y^{\top} = \frac{qx(l-x)}{2} + k\_i Q \cdot \sum\_{i=1}^{n} \text{f(x)} - p\_s \cdot m(\mathbf{x}) - p\_s \cdot (e + y) \cdot \cos \gamma \tag{9}$$

where

$$f(\mathbf{x}) = \begin{cases} \mathbf{x}(l - b\_i) / l & (0 \le \mathbf{x} \le b\_i) \\ b\_i(l - \mathbf{x}) / l & (b\_i \le \mathbf{x} \le l) \end{cases} \tag{10}$$

$$m(\mathbf{x}) = \begin{cases} \mathbf{x}\text{siny} & \text{( $0 \le x \le a$ )} \\ -(\mathbf{x} - a)(\sin\gamma - \sin\beta) + \mathbf{x}\sin\gamma & \text{( $a \le x \le a + l \cdot \sin\alpha$ )} \\ -(\mathbf{x} - a)(\sin\gamma - \sin\beta) + \mathbf{x}\sin\beta & \text{( $a + l' \cdot \sin\alpha$  \le x \le l/2 $)} \\ -(l - \mathbf{x} - a)(\sin\gamma - \sin\beta) + (l - \mathbf{x})\sin\beta & \text{($ l/2 \le x \le l - a - l' \cdot \sin\alpha $)} \\ -(l - \mathbf{x} - a)(\sin\gamma - \sin\beta) + (l - \mathbf{x})\sin\gamma & \text{($ l - a - l' \cdot \sin\alpha $ \le x \le l - a)} \\ (l - \mathbf{x})\sin\gamma & \text{($ l - a - l' \cdot \sin\alpha $ \le x \le l$ )} \end{cases} \tag{11}$$

where *l'* is the length of a DWM, *Φ* is the angle between the DWM and the *y*-axis, *ki* is a load factor, *bi* is the distance from the concentrated load to the support, *r* is the angle between the midspan web member and the horizontal plane at its intersection with the tendon, and *β* is the angle between the DWM and the horizontal plane at its intersection with the tendon.

By taking the partial derivative of formula (9), we obtain

\*\*Lemma\*\*
and the horizontal plane at its intersection with the tendon, and  $\beta$  is the line of the DWM and the horizontal plane at its intersection with the tendon.

 $\text{we partial derivative of formula (9), we obtain}$ 
 $y(\mathbf{x}) = e\left[\frac{\cos\omega(\mathbf{x}-\frac{l}{2})}{\cos\left(\frac{\omega l}{2}\right)} - 1\right] + \frac{q}{p\_\ast \cdot \cos\gamma} \cdot \left\{\frac{\mathbf{x}}{2}(l-\mathbf{x}) - \frac{1}{\omega^2} \left[\frac{\cos\omega(\mathbf{x}-\frac{l}{2})}{\cos\left(\frac{\omega l}{2}\right)} - 1\right]\right\}$ 
 $+ \frac{Q}{p\_\ast \cdot \cos\gamma} \sum\_{i=1}^{Q} k\_i \cdot f(\mathbf{x}) - \frac{1}{\cos\gamma} \cdot m(\mathbf{x})$ 

where

$$
\omega = \sqrt{\frac{p\_\ast}{EI}} \cos \gamma \tag{13}
$$

**Figure 4.**

The material stress–strain curve.

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85

Eq. (12) describes the relationship between external load and RC beam deflection, which can be used as a theoretical basis for design of SSB reinforced by external prestressing with three web members.

#### **3. Finite element analysis**

#### **3.1. Constitutive relations**

A model of an RC beam reinforced by external prestressing was constructed by separate modeling using ABAQUS, a software suite for finite element analysis. The concrete was modeled using C3D8R, a linear reduced integration element with eight nodes. The rebars, web members, and external tendons were constructed of T3D2, a three-dimensional, two-node linear truss element. The slip between rebars and concrete was neglected. The constraints between the concrete and reinforcement cage were applied via the Embedded Region command. The constraint relationships between external tendon and concrete beam and between web members and concrete beam were achieved via the Kinematic coupling command. External prestressing force was applied by decreasing the temperature and solved by an implicit solver

**Figure 3.** Finite element model. (a) Concrete, (b) steel reinforcement, (c) web member.

Mechanical Performance of Simple Supported Concrete Beam-Cable Composite Element... http://dx.doi.org/10.5772/intechopen.76517 85

**Figure 3.** Finite element model. (a) Concrete, (b) steel reinforcement, (c) web member.

where *l'* is the length of a DWM, *Φ* is the angle between the DWM and the *y*-axis, *ki* is a load factor, *bi* is the distance from the concentrated load to the support, *r* is the angle between the midspan web member and the horizontal plane at its intersection with the tendon, and *β* is the

> { \_*x*

\_\_\_\_\_\_\_ *p* \_\_*<sup>s</sup>*

Eq. (12) describes the relationship between external load and RC beam deflection, which can be used as a theoretical basis for design of SSB reinforced by external prestressing with three

A model of an RC beam reinforced by external prestressing was constructed by separate modeling using ABAQUS, a software suite for finite element analysis. The concrete was modeled using C3D8R, a linear reduced integration element with eight nodes. The rebars, web members, and external tendons were constructed of T3D2, a three-dimensional, two-node linear truss element. The slip between rebars and concrete was neglected. The constraints between the concrete and reinforcement cage were applied via the Embedded Region command. The constraint relationships between external tendon and concrete beam and between web members and concrete beam were achieved via the Kinematic coupling command. External prestressing force was applied by decreasing the temperature and solved by an implicit solver

<sup>2</sup>(*<sup>l</sup>* <sup>−</sup> *<sup>x</sup>* ) <sup>−</sup>\_<sup>1</sup>

*ω*<sup>2</sup> [

cos*ω*(*<sup>x</sup>* <sup>−</sup> \_*<sup>l</sup>* 2 \_

cos( \_ *ωl* <sup>2</sup> ) <sup>−</sup> <sup>1</sup>

cos*<sup>γ</sup>* <sup>⋅</sup> *<sup>m</sup>*(*<sup>x</sup>* ) (12)

)

*EI* cos*γ* (13)

]}

angle between the DWM and the horizontal plane at its intersection with the tendon.

] <sup>+</sup> *<sup>q</sup>* \_\_\_\_\_\_\_\_ *ps* <sup>⋅</sup> cos*<sup>γ</sup>* ⋅

By taking the partial derivative of formula (9), we obtain

)

*ki* <sup>⋅</sup> *<sup>f</sup>*(*<sup>x</sup>* ) <sup>−</sup> \_\_\_\_\_ <sup>1</sup>

cos*ω*(*<sup>x</sup>* <sup>−</sup> \_*<sup>l</sup>* 2 \_

cos( \_ *ωl* <sup>2</sup> ) <sup>−</sup> <sup>1</sup>

*ω* = √

84 New Trends in Structural Engineering

where

web members.

*y*(*x* ) = *e* [

+\_\_\_\_\_\_\_\_ *<sup>Q</sup> ps* <sup>⋅</sup> cos*<sup>γ</sup>* ∑ *i*=1 2

**3. Finite element analysis**

**3.1. Constitutive relations**

**Figure 4.** The material stress–strain curve.

(**Figure 3**). The concrete was modeled with the plastic damage constitutive model provided in ABAQUS, the rebars were modeled with the improved rebar model developed by Esmaeilyxiao, and the web members were simulated using an ideally elastic-plastic model. The constitutive relations for the model materials are presented in **Figure 4**.

**Beam number**

\*

Indicates the test specimen to be tested.

**Table 2.** Simply supported beam's design parameters and change of bearing capacity.

**Cable sectional area (mm2 )** **Sag of cable (mm)**

**Cable internal force (kN)** **Percent (%)**

**Initial Comparison of bearing capacity of simulated specimens**

Mechanical Performance of Simple Supported Concrete Beam-Cable Composite Element...

**Percentage increase (%)** **L/200 hour (kN)**

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**Percentage increase (%)** 87

**L/300 hour (kN)**

#### **3.2. Parameter design**

The beam had dimensions of 3000 × 300 × 180 mm. It was constructed of C25 concrete, three Φ14 rebars in the concrete under tension, two vertical Φ14 rebars in the concrete under compression, Φ6 stirrups @100, and Φ14 web members (threaded). Design levels of the tendon parameters considered are as follows: initial tendon force, Level 3; tendon cross-sectional area, Level 3 (**Table 1**); and tendon sag, Level 4. Design values of other parameters are given in the first five lines of **Table 2**. FJGL denotes a control specimen, which was unreinforced. \* denotes a specimen to be tested in the experimental verification.

#### **3.3. Numerical results**

#### *3.3.1. Common features*

The load-deflection curve for a RC beam generally splits into four portions. The characteristics of the four portions and the corresponding stages of the beam's behavior are summarized below:



**Table 1.** Design parameters of pre-stressed cables.

Mechanical Performance of Simple Supported Concrete Beam-Cable Composite Element... http://dx.doi.org/10.5772/intechopen.76517 87


**Table 2.** Simply supported beam's design parameters and change of bearing capacity.

(**Figure 3**). The concrete was modeled with the plastic damage constitutive model provided in ABAQUS, the rebars were modeled with the improved rebar model developed by Esmaeilyxiao, and the web members were simulated using an ideally elastic-plastic model. The consti-

The beam had dimensions of 3000 × 300 × 180 mm. It was constructed of C25 concrete, three Φ14 rebars in the concrete under tension, two vertical Φ14 rebars in the concrete under compression, Φ6 stirrups @100, and Φ14 web members (threaded). Design levels of the tendon parameters considered are as follows: initial tendon force, Level 3; tendon cross-sectional area, Level 3 (**Table 1**); and tendon sag, Level 4. Design values of other parameters are given in the first five lines of **Table 2**. FJGL denotes a control specimen, which was unreinforced.

The load-deflection curve for a RC beam generally splits into four portions. The characteristics of the four portions and the corresponding stages of the beam's behavior are summarized

**1.** Elastic deformation. In this stage, the concrete at the beam bottom slightly deflected without fracturing and the corresponding portion of the load-deflection curve is nearly linear; **2.** Yielding. As the load increased, the bottom concrete showed increased deflection as a result of cracking. The rebars in the tensioned region reached the yield point earlier than the external tendons. The corresponding portion of the load-deflection curve contained a

**3.** Hardening. The neutral axis of a cross-section shifted downward and the external tendons

**4.** Failure. As the load continued increasing, the external prestressing tendon or the concrete under compression would fail after the tendon stress exceeded its ultimate strength or the compressive stress in the concrete exceeded its compressive strength. Their failure modes are different: the tendons failed via brittle fracture, while the concrete failed by

**) Breaking force (kN)**

tutive relations for the model materials are presented in **Figure 4**.

denotes a specimen to be tested in the experimental verification.

**3.2. Parameter design**

86 New Trends in Structural Engineering

**3.3. Numerical results**

*3.3.1. Common features*

noticeable turning point;

ductile fracture.

were fully engaged in the work;

**Table 1.** Design parameters of pre-stressed cables.

**No. Cable diameter (mm) Sectional area of cable (mm2**

1 Ф9 48 75 2 Ф12 86 133 3 Ф15 134 208

\*

below:

#### *3.3.2. Relationship between beam deflection and increment in tendon force*

1. The greater the initial tendon force, the smaller the maximum tendon deflection (or a tendon's energy dissipation capacity). **Figure 5a** shows the tendon force-deflection curves of a tendon with a cross-sectional area of 48 mm2 and sag of 200 mm for different initial tendon forces (20, 40, and 60 kN). It is clear that the maximum tendon deflection at the fracture point gradually decreased. This implies that it is unreasonable to optimize reinforcement design simply by increasing the initial tendon force.

**3.** The rate of tendon force growth increased with increasing tendon sag. **Figure 5c** shows the tendon force deflection curves for different tendon sags (200, 300, and 400 mm) when the

Mechanical Performance of Simple Supported Concrete Beam-Cable Composite Element...

Initial tendon force, tendon cross-sectional area, and tendon sag have different effects on the

**1.** Effect of initial tendon force. **Figure 6a** shows the load-deflection curves for four different

**Figure 6.** Effect of different factors on strengthening beam. (a) Initial internal force of cable, (b) sectional area of cable,

200 mm. An analysis of the curves reveals that an increase in the initial tendon force can increase the RC beam's ultimate bearing capacity and reduce the duration of the hardening stage. Decreasing initial tendon force has the opposite effects. As shortened duration of hardening is not expected for structural performance, a greater initial tendon force does

in this figure, a large tendon sag can ensure more effective reinforcement.

initial tendon forces when the tendon cross-sectional area was 48 mm2

and initial tendon force was 20 kN. As can be seen

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89

and tendon sag was

tendon cross-sectional area was 48 mm2

load-deflection curve for the RC beam:

and (c) sag of cable.

*3.3.3. Effects of different parameters on the load-deflection curve*

**2.** As the tendon cross-sectional area increased, the rate of growth in tendon force increased, and thus the contribution by the tendon became more significant. Conversely, a smaller tendon cross-sectional area is associated with a slower rate of increase in tendon force. **Figure 5b** illustrates the tendon force-deflection curves for different tendon cross-sectional areas (48, 86, and 134 mm2 ) when the tendon sag was 200 mm and initial tendon force was 20 kN. This figure demonstrates that a larger tendon cross-sectional area is better in reinforcement design.

**Figure 5.** Cable internal force deflection curve. (a) The initial tendon force, (b) sectional area of cable, and (c) sag of cable. \*The horizontal dotted line represents the tensile force at which the external prestressing tendon begins to fracture.

**3.** The rate of tendon force growth increased with increasing tendon sag. **Figure 5c** shows the tendon force deflection curves for different tendon sags (200, 300, and 400 mm) when the tendon cross-sectional area was 48 mm2 and initial tendon force was 20 kN. As can be seen in this figure, a large tendon sag can ensure more effective reinforcement.

#### *3.3.3. Effects of different parameters on the load-deflection curve*

*3.3.2. Relationship between beam deflection and increment in tendon force*

tendon with a cross-sectional area of 48 mm2

areas (48, 86, and 134 mm2

forcement design.

88 New Trends in Structural Engineering

design simply by increasing the initial tendon force.

1. The greater the initial tendon force, the smaller the maximum tendon deflection (or a tendon's energy dissipation capacity). **Figure 5a** shows the tendon force-deflection curves of a

forces (20, 40, and 60 kN). It is clear that the maximum tendon deflection at the fracture point gradually decreased. This implies that it is unreasonable to optimize reinforcement

**2.** As the tendon cross-sectional area increased, the rate of growth in tendon force increased, and thus the contribution by the tendon became more significant. Conversely, a smaller tendon cross-sectional area is associated with a slower rate of increase in tendon force. **Figure 5b** illustrates the tendon force-deflection curves for different tendon cross-sectional

20 kN. This figure demonstrates that a larger tendon cross-sectional area is better in rein-

**Figure 5.** Cable internal force deflection curve. (a) The initial tendon force, (b) sectional area of cable, and (c) sag of cable. \*The horizontal dotted line represents the tensile force at which the external prestressing tendon begins to fracture.

and sag of 200 mm for different initial tendon

) when the tendon sag was 200 mm and initial tendon force was

Initial tendon force, tendon cross-sectional area, and tendon sag have different effects on the load-deflection curve for the RC beam:

**1.** Effect of initial tendon force. **Figure 6a** shows the load-deflection curves for four different initial tendon forces when the tendon cross-sectional area was 48 mm2 and tendon sag was 200 mm. An analysis of the curves reveals that an increase in the initial tendon force can increase the RC beam's ultimate bearing capacity and reduce the duration of the hardening stage. Decreasing initial tendon force has the opposite effects. As shortened duration of hardening is not expected for structural performance, a greater initial tendon force does

**Figure 6.** Effect of different factors on strengthening beam. (a) Initial internal force of cable, (b) sectional area of cable, and (c) sag of cable.

not necessarily mean more effective reinforcement. After the initial tendon force exceeded a threshold (53% in this study), the external tendon will yield and fracture in advance and the RC beam becomes more likely to fail by brittle fracture.


increased, the plastic zone slowly extended toward the top and ends and reached the highest point between web members. The plastic zone area was significantly smaller than that observed in the unreinforced beam. A plastic zone developed at the RC beam top, which is characteristic of deformation of continuous beams. This suggests that after reinforcement, the stress in the beam was redistributed and the properties of the material were used to a greater extent.

Mechanical Performance of Simple Supported Concrete Beam-Cable Composite Element...

http://dx.doi.org/10.5772/intechopen.76517

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**Table 1** shows the number of test specimens and their materials. Two parts were prepared for each of the specimens indicated by \*. The specimens were divided into three groups. Then

**Figure 8** shows the design of the specimens (reinforcement ratio 1.27%). The properties of the rebars, steel wire ropes, and other materials used in the RC beam are presented in **Table 3**. The

Loading scheme: A three-point bend test was performed on the specimens using a hydraulic servo jack (**Figures 9** and **10**). Each test process was first controlled by load, which increased 10 kN for each stage. After the load reached 100 kN, displacement control was applied, and the displacement increased 5 mm each stage. The loading time was 3 min and the period of

Observation scheme: (1) observed variables: load, midspan displacement, beam-end displacement, and stress in concrete, wire ropes, rebars, and web members; (2) observation method: measurements by load transducer, displacement meter, and resistance strain gauge and manual measurement record using coordinate paper (rope length change was measured as strain in rope) and calculation using the Hooke's law; and (3) test devices: static strain gauge, ruler, and so on [12].

"1-" and "2-" were added to the original specimen numbers.

**Figure 7.** The plastic strain distribution of beam. (a) FJGL and (b) FGL-3-3.

concrete strength, at 28 MPa, was measured using rebound hammer.

**4. Tests and results**

**4.1. Experimental design**

sustained load was 30 min.

#### *3.3.4. Characteristics of plastic zone development*

The analysis above shows that the application of DWM external prestressing not only improved the bearing capacity of the SSB but also altered the plastic zone developed in the beam. **Figure 7** shows the contours of stress in the plastic zone throughout the deformation processes of the unreinforced beam and the RC beam. When the unreinforced beam was subjected to an external load, a plastic zone arose first in the beam segment in the stage of pure bending. As the load increased, the plastic zone tended to expand toward the two ends symmetrically about the midspan position. The height of the plastic zone at midspan gradually increased and always peaked around the midspan. The plastic zone within the segment in shear bending gradually expanded from the loading point to the supports.

In the RC beam, the plastic zone in the region corresponding to the pure-bending segment of the unreinforced beam expanded at a slower rate due to the presence of web members. The plastic zone's height decreased compared to that in the unreinforced beam. Along the beam bottom, it was symmetrically distributed about the midspan web member. As the load Mechanical Performance of Simple Supported Concrete Beam-Cable Composite Element... http://dx.doi.org/10.5772/intechopen.76517 91

**Figure 7.** The plastic strain distribution of beam. (a) FJGL and (b) FGL-3-3.

increased, the plastic zone slowly extended toward the top and ends and reached the highest point between web members. The plastic zone area was significantly smaller than that observed in the unreinforced beam. A plastic zone developed at the RC beam top, which is characteristic of deformation of continuous beams. This suggests that after reinforcement, the stress in the beam was redistributed and the properties of the material were used to a greater extent.
