# Flexural Analysis Model of Externally Prestressed Steel-Concrete Composite Beam with Nonlinear Interfacial Connection

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Finite Element Formulations and Developed Procedure

#### 2.1. Schemes of the Proposed Model

#### 2.2. Fiber Beam Element Model with Nonlinear Interfacial Connection

_{cs}(introduced DOF compared with the conventional fiber beam element). The cross-section is defined by discrete fibers, including concrete fibers, steel fibers and reinforcement fibers.

**u**and resistance vector

**F**of the element are defined in Equations (1) and (2), respectively. The axial force N, shear force V, moment M and interfacial shear force F

_{cs}are defined as the resistance force concerning u, v, θ and u

_{cs}. The subscript i and j denote the corresponding nodal number.

**u**

^{b}can be expressed as Equation (3). The transformation relation between

**u**

^{b}and

**u**is shown in Equation (4). c

_{x}= (L + ∆u

_{x})/L

_{n}, c

_{x}= ∆u

_{y}/L

_{n}. L is the initial element length and L

_{n}is the deformed element length. ∆u

_{x}and ∆u

_{y}denote relative displacement along the local x and y axes, respectively.

**u**

^{b}, the deformation field inside the element can be denoted by the Hermite polynomial interpolation method, as shown in Equation (5), in which ξ = x/L, x is the local coordinate in the element, and 0 ≤ ξ ≤ 1. The axial and slip deformations are interpolated by linear equations and the rotation deformation is interpolated by a quadratic polynomial.

**E**(x) of the section at location x can be expressed as Equation (6) by derivation. ε

_{s}(x) and ε

_{c}(x) denote the sectional axial strain of the steel and the concrete beam, respectively. φ(x) denotes the sectional curvature and u

_{cs}(x) denotes the interfacial slip displacement at location x.

_{k}, z

_{k}), fiber area A

_{k}, fiber region flag ${\phi}_{k}^{s}$ and ${\phi}_{k}^{\mathrm{c}}$, fiber stress σ

_{k}and fiber tangent modulus E

_{k}, as shown in Figure 2b. For fibers in the concrete slab ((y

_{k}, z

_{k}) ∈ Ω

_{c}), ${\phi}_{k}^{s}=0$ and ${\phi}_{k}^{\mathrm{c}}=1$; otherwise ((y

_{k}, z

_{k}) ∈ Ω

_{s}), ${\phi}_{k}^{s}=1$ and ${\phi}_{k}^{\mathrm{c}}=0$. Then, the axial strain of fiber k can be yielded by Equation (7). The slip displacement at the location x can be denoted as Equation (8).

_{cs}is the average interfacial shear force per unit length. The resisting force vector

**F**can be identified from Equation (10) and is expressed by Equation (11).

**q**is the element resisting vector in the basic coordinate system and

**r**

_{s}(x) is the sectional resisting force vector.

**K**can be obtained by the partial derivative of Equation (11) to

**u**, as shown in Equation (12).

**k**

^{g}is the geometric stiffness matrix and

**k**

^{m}is the material stiffness matrix in the basic coordinate system. The derived expression of

**k**

^{m}is shown in Equation (13), in which

**k**

_{s}is the sectional stiffness matrix. Based on the idea of the fiber section method, the integration style of

**r**

_{s}and

**k**

_{s}can be calculated by the algebraic summary of the total fibers’ contribution, as shown in Equations (14) and (15). The integrations of the

**k**

^{m}and

**q**of the element are implemented using the Gauss–Lobatto method.

_{cs}(x) can be calculated. Based on the constitutive model of material and interfacial connection, the force and the tangent modulus can be updated. With the summation of fibers in the cross-section at the integration point, the sectional stiffness matrix and resistance vector can be obtained from Equations (14) and (15). From the integration along the beam length, the stiffness matrix and resistance vectors are calculated and then assembled to form the whole structural matrix for the next iteration. In this way, the stiffness matrix and internal force vectors were updated from the fiber to the section and then to the element and structural level in turn. The material, geometric and interfacial nonlinear behaviors of the composite beam were all considered in the proposed element.

#### 2.3. Multi-Node Slipping Cable Element for External Tendon

_{p}nodes and n

_{p}-1 segments were defined. The element displacement vector

**U**

_{t}is defined in Equation (16), in which ${u}_{i,t}=\left[\begin{array}{cc}{u}_{i,t}& {v}_{i,t}\end{array}\right]$ was the displacement of node i at step t.

_{0}which is the sum length of all segments, after deformation, the deformed element length can be expressed by the displacement vector

**U**

_{t}, as shown in Equation (17),

_{p}is the area of the external tendon, σ

_{p}and E

_{p}denote the tendon stress and tangent modulus at time t, respectively, which is calculated according to the material constitute model. ${k}_{t}^{g}$ and ${k}_{t}^{m}$ denote the geometric and material stiffness matrix, respectively.

#### 2.4. Rigid Link between Beam and Tendon

**u**

_{c}and DOFs of tendon node named

**u**

_{t}) can be expressed as Equation (20). As shown in Figure 6, one end of the rigid link connected to the beam node is the rigid connection and the other end connected to the external tendon is the hinge connection. The translational and rotational displacements of beam nodes lead to the translation of tendon nodes.

#### 2.5. Constitutive Model

- (1)
- Concrete constitutive model

_{Fc}is the concrete crushing energy, G

_{F}is the concrete fracture energy, f

_{c}is the concrete compression strength, f

_{t}is the concrete tension strength, E

_{c}is the initial elastic modulus of concrete. The values of G

_{Fc}, G

_{F}and G

_{F}

_{0}are determined following the suggestion of CEB-FIP [38].

- (2)
- Prestressing steel tendon constitutive model

_{p}and f

_{py}denote the initial elastic modulus and yield strength of the prestressing tendons. b

_{p}represents the ratio of strain-hardening modulus to the initial elastic tangent. In this paper, we adopt b

_{p}= 0, R0 = 18 for the Steel02 model definition and the default values are used for the other parameters (see http://opensees.berkeley.edu/ (accessed on 1 March 2020)). Meanwhile, the initial prestress σ

_{pe}is defined in Steel02.

- (3)
- Constitutive model for steel beam and reinforcements

_{s}and f

_{y}denote the elastic modulus and yield of steel, respectively. b denotes the ratio of strain-hardening modulus to E

_{s}. For the steel beam, b = 0.005; for reinforcements, b = 0.0 is adopted in the simulation.

- (4)
- Interfacial shear-slip model

_{u}denotes average interfacial shear capacity per unit length. A

_{us}is the average area of the shear stud cross-section per unit length. f

_{s}represents the average shear force at the interface under u

_{cs}slip displacement. f

_{u}is the yield strength of steel studs. n and m are the curve shape calibration parameters according to the shear-slip tests of shear connector specimens. If there were no test data available, the default value m = 0.558 and n = 1 mm

^{−1}can be adopted. Figure 7d shows the interfacial slip-shear curves. In OpenSees, we use the MultiLinear material model to define the curve and the initial modulus is determined as the secant modulus at 0.1 mm slip displacement.

#### 2.6. Computational Procedure Development

#### 2.7. Highlights of the Proposed Model

- (1)
- A new modeling method is proposed by introducing two relative slip DOFs into the ends of the fiber beam element. An 8-DOF fiber beam element was built for the composite beam with material, geometric and interfacial nonlinearity.
- (2)
- Taking the external tendon as the whole member, a multi-node slipping cable element is proposed with a complete stiffness and resistance matrix. Compared with the equivalent load method, the numerical convergence was improved and can be widely used for external tendons with different profiles.
- (3)
- The methods and framework of the element classes developed in OpenSees are presented.

## 3. Experimental Verification

#### 3.1. Verification for the Nonlinear Interfacial Behaviors

_{u}can be calculated using Equation (23). The values of V

_{u}for NCB-4, NCB-5 and NCB-6 were 860 N/mm, 383 N/mm and 688 N/mm, respectively. Additionally, the shear connection degree η was 1, 0.45 and 0.69 for NCB-4, NCB-5 and NCB-6, respectively.

#### 3.2. Verification for the Case of Simply Supported EPCB

_{c}= 31.7 MPa and the elastic modulus E

_{c}= 28.6 GPa. The yield strength of the steel beam f

_{y}= 293 MPa and the yield stress of the cover-plate was 358 MPa. According to the push-out tests of the shear connector, the maximum interfacial shear capacity of a single connector was 75 kN and the authors recommended α = 0.3, β = 0.55. The reinforcements of 8 mm diameter were embedded into the concrete slab, and the yield stress of the rebar was 428 MPa. The external prestressing was implemented using two nominal diameters of 15.7 mm seven-wire strands, whose A

_{p}= 150 mm

^{2}, E

_{p}= 197.8 GPa and initial effective prestress σ

_{pe}= 950 MPa.

#### 3.3. Verification for the Case of Continuous EPCB

_{c}= 30 MPa, E

_{c}= 30 GPa and f

_{t}= 2.5 MPa. For the steel beam, the yield strength is 249.3 MPa for the top flange, 272.3 MPa for the bottom flange and 287.7 MPa for the web. The elastic modulus of the steel beam is 200 GPa. The composite beam is prestressed by two unbonded steel tendons in the steel box beam. The cross-section area of the tendon is 139 mm

^{2}and the initial effective prestress is 885.8 MPa. The yield strength of the external tendons is 1860 MPa and the elastic modulus is 200 GPa. The loading scheme is shown in Figure 15.

## 4. Parametric Analysis

_{c}= 30 MPa and the elastic modulus E

_{c}= 30 GPa; for the steel beam, the yield strength f

_{y}= 300 MPa and the elastic modulus E

_{s}= 200 GPa; for the external tendons, the yield strength f

_{py}= 1860 MPa, the elastic modulus E

_{p}= 195 GPa, and the initial effective prestress σ

_{pe}= 1000 MPa.

_{d}= 6 m, the effective height of external tendon d

_{p}= 550 mm, and the average interfacial shear capacity at the unit length was set V

_{u}= 900 N/mm (shear connector degree η = 1). During the parameter analysis, the parameters’ values were all the same as those of the reference beam except the discussed parameter. For each parameter analysis, two loading types including one-point loading at mid-span and uniform loading were applied and analyzed.

#### 4.1. Effects of Deviator Spacing

_{d}is set to range from 0.4 m to 19.6 m at intervals of 0.4 m. A total of 49 analysis models are built to discuss the effects of deviator spacing on the structural behaviors. The calculated results of tendon stress increments, effective height decrements d

_{p}and flexural capacity are presented in Figure 18. With an increase of deviator spacing, the stress increments are slightly decreased but the decrements of d

_{p}are increased. The results reveal that the second-order effects are significant for cases with large deviator spacing. Along with the variation in S

_{d}from 0.4 m to 19.6 m, the flexural capacity decreases by 6.7% under one-point loading states and 10.8% under uniform loading states. The increasing S

_{d}causes a decrease in flexural capacity.

#### 4.2. Effects of External Tendon Effective Height

_{p}ranges from 110 mm to 550 mm at intervals of 40 mm. A total of 12 analysis models are built to discuss the effects of d

_{p}on the structural behaviors. The results are shown in Figure 19. Interestingly, we find that the effects of d

_{p}on its decrements are small, which is different from the effects of S

_{d}. With an increase in d

_{p}, the stress increments and flexural capacity are increased. The stress increments and decrements of d

_{p}under uniform loading are all larger than the ones under one-point loading states. Along with the variation of d

_{p}from 110 mm to 550 mm, the flexural capacity increases by 20.0% under one-point loading states and 23.0% under uniform loading states. The increasing d

_{p}increases the flexural capacity.

#### 4.3. Effects of Interfacial Slip

_{u}ranges from 100 to 1500 N/mm (shear connection degree η ranges from 0.11 to 1.67) to analyze the effects of shear connection on the flexural behaviors. The load-displacement curves of composite beams with various V

_{u}under one-point loading and uniform loading are shown in Figure 20a,b, respectively. With the increase in V

_{u}, the structural stiffness and capacity are increased. Figure 21 shows the stress increments, decrements of d

_{p}and flexural capacity variation caused by an increase in V

_{u}. The results show that the variation of stress increments and d

_{p}decrements are not monotonic. For weak shear connection models (V

_{u}≤ 300 N/mm for one-point loading states, V

_{u}≤ 400 N/mm for uniform loading states), the structural failure is mainly caused by interfacial slip, and the increasing interfacial shear capacity leads to larger structural deformation capacity before peak loads. Then, the stress increments and decrements of d

_{p}increase with an increase in V

_{u}. For stronger shear connection models (V

_{u}> 300 N/mm for one-point loading states, V

_{u}> 400 N/mm for uniform loading states), the concrete slab crushing failure is replaced by interfacial slip failure. The stiffness contribution of the concrete slab decreases the structural ultimate deformation and then causes the decreasing stress increments and d

_{p}decrements. For the cases of V

_{u}reaching the full shear connection, the stress increments and d

_{p}decrements show almost no change.

## 5. Conclusions

- Interfacial slip effects are inevitable for steel-concrete composite beams during their service life, even when they are designed with full shear connections. The structural stiffness, capacity and failure modes are all affected. The proposed fiber beam element model considering interfacial slip effects can be used to predict the capacity, deformation and failure mode, which are all verified to agree well with the experimental results.
- Ignoring the friction at the deviators, the external tendon has equal traction along its whole length. The conventional truss element could not satisfy the strain-compatibility property in multiple segments, causing some overestimation of the stress increments and flexural capacity. The proposed slipping cable element is built considering the strain-compatibility property, which shows better agreement with the experimental results.
- The parameter analysis results reveal that the deviator spacing, external tendon effective height, interfacial shear capacity and loading type all affect the flexural capacity of EPCBs. An increase in the deviator spacing decreases the ultimate effective height of the external tendon, which then leads to a decrease in the flexural capacity. The increased external tendon effective height increases the ultimate stress increments and flexural capacity. With an increase in the interfacial shear capacity, the flexural capacity increases gradually but the stress increments and effective height of the tendon do not vary monotonously. The proposed model provides an effective method for predicting the flexural behaviors of EPCBs.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Components | Yield Strength (MPa) | Ultimate Strength (MPa) | Elastic Modulus (MPa) |
---|---|---|---|

Rebar φ6 | 440 | 524 | 2.23 × 10^{5} |

Steel Beam | 342 | 447 | 2.23 × 10^{5} |

Shear Studs | 488 | 552 | - |

Specimen | Cubic Compressive Strength (MPa) | Prismatic Compressive Strength (MPa) | Elastic Modulus (MPa) |
---|---|---|---|

NCB-4 | 30.0 | 22.8 | 22,800 |

NCB-5 | 29.4 | 22.3 | 22,300 |

NCB-6 | 35.6 | 27.1 | 27,100 |

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**MDPI and ACS Style**

Yan, W.; Chen, L.; Han, B.; Xie, H.; Sun, Y.
Flexural Analysis Model of Externally Prestressed Steel-Concrete Composite Beam with Nonlinear Interfacial Connection. *Appl. Sci.* **2022**, *12*, 4699.
https://doi.org/10.3390/app12094699

**AMA Style**

Yan W, Chen L, Han B, Xie H, Sun Y.
Flexural Analysis Model of Externally Prestressed Steel-Concrete Composite Beam with Nonlinear Interfacial Connection. *Applied Sciences*. 2022; 12(9):4699.
https://doi.org/10.3390/app12094699

**Chicago/Turabian Style**

Yan, Wutong, Liangjiang Chen, Bing Han, Huibing Xie, and Yue Sun.
2022. "Flexural Analysis Model of Externally Prestressed Steel-Concrete Composite Beam with Nonlinear Interfacial Connection" *Applied Sciences* 12, no. 9: 4699.
https://doi.org/10.3390/app12094699