# Concrete Open-Wall Systems Wrapped with FRP under Torsional Loads

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Structural Model

**Figure 1.**B

^{(1)}: Reinforced concrete (RC) member; B

^{(2)}: fiber-reinforced composite material (FRP) strengthening; $\text{B}={\text{B}}^{(1)}\cup {\text{B}}^{(2)}$: strengthened member.

^{(1)}) or the FRP strengthening (B

^{(2)}) (Figure 1).

#### 2.1. Kinematics

^{(1)}or B

^{(2)}, has been developed previously in [13,14].

^{(i)}(i = 1, 2); ${\xi}_{\text{c}}\left(z\right)$, ${\eta}_{\text{c}}\left(z\right)$ are the displacement components of the point C along the x and y axes, respectively; $\alpha \left(z\right)$ and $\beta \left(z\right)$ are the cross-section flexural rotations; ${\zeta}_{\text{c}}\left(z\right)={\zeta}_{\text{M}}+\beta \left(z\right)\cdot {x}_{\text{M}}-\alpha \left(z\right)\cdot {y}_{\text{M}}$ with ${\zeta}_{\text{M}}={\zeta |}_{\text{s}=0}$ and $M({x}_{\text{M}},{y}_{\text{M}})$ denotes the origin of the curvilinear abscissa introduced over the mid-line of the cross-section of B

^{(i)}; $\dot{\rho}\left(z\right)$ is the derivative of the twisting rotation $\rho \left(z\right)$ with respect to the z coordinate, while $\omega \left(s\right)$ is the current sectorial area as in the classical theory of thin-walled beams.

^{(2)}(composite overlay) due to its low shear moduli of elasticity.

^{(i)}(i = 1, 2) depicted in Figure 2 to exhibit: (i) A rigid transformation in its own plane; (ii) A warping out of the same plane; (iii) The following angular sliding along the mid-line:

_{i}(s) denote specific polynomials which have been defined in a general manner in [14].

#### 2.2. Interface Model

^{(1)}and B

^{(2)}; furthermore, let $\text{\kappa}$ be the intersection between $\text{\Omega}$ and the plane that contains the current cross-section of $\text{B}={\text{B}}^{(1)}\cup {\text{B}}^{(2)}$.

^{(1)}and B

^{(2)}is simulated by means of bilateral, independent elastic springs, which cover the entire surface $\text{\Omega}$ and are arranged continuously along the local axes {

**n, t, k**}, the unit vector

**n**being oriented from B

^{(2)}to B

^{(1)}[16,17].

_{n}, C

_{t}and C

_{k}, in order, the compliance coefficients (per unit area) of the above mentioned springs along the directions

**n, t**and

**k**. The reactions of these interfacial springs ideally interposed between B

^{(1)}and B

^{(2)}are provided by the following relationships:

^{(1)}. They are expressed in the local reference system. Similarly, ${u}_{\text{n}}^{(2)},{u}_{\text{t}}^{(2)}$ and ${u}_{\text{k}}^{(2)}$ indicate the displacements of the same point $P\in \text{\Omega}$ if it accords, instead, to the kinematics of the beam B

^{(2)}.

^{(1)}and B

^{(2)}exhibits a progressively increasing stiffness as the quantities C

_{n}, C

_{t}and C

_{k}decrease. This provides an appropriate approximation of perfect contact interactions between B

^{(1)}and B

^{(2)}when the quantities C

_{n}, C

_{t}and C

_{k}tend towards zero.

#### 2.3. Constitutive Assumptions

## 3. Case-Study

_{(0)}, all displacements for both the beam B

^{(1)}and the beam B

^{(2)}are constrained to be zero; warping is not allowed too. A unit torsional couple is applied on the free end Σ

_{(l)}of the concrete wall-system.

_{p}: length of the strengthened region (L

_{p}= L); B, H: cross-sectional sizes of

**B**in reference to its mid-line; t

^{(1)}_{p}: overall thickness of the composite wrapping (t

_{p}= 20 mm); Σ

_{(0)}, Σ

_{(l)}: bottom/top end; C: unit torsional couple applied on the free end Σ

_{(l)}of

**B**.

^{(1)}**t**and

**k**of the local reference system, as indicated in the same figure.

**t**,

**k**) by an angle equal to 45° with respect to the previous case.

**k**axis.

Material | Young modulus [N/mm^{2}] | Shear modulus [N/mm^{2}] |
---|---|---|

Concrete (C20/25) | E_{c} = 28,460 | G_{c} = 14,230 |

Carbon fibres | E_{f} = 235,000 | – |

Epoxy resin | E_{r} = 3800 | G_{r} = 1380 |

Axis | D1 | D2 | U1 |
---|---|---|---|

n | 0.00 | 0.00 | 0.00 |

t | 0.25 | 0.25 | 0.00 |

k | 0.25 | 0.25 | 0.50 |

## 4. Numerical Results

^{(1)}as well as the warping torsional moment (${M}_{\omega}^{(2)}$) within the beam B

^{(2)}.

^{(2)}does not exhibit appreciable values of the Saint Venant torsional moment, due to the low thickness of the composite.

^{(2)}is almost completely extinguished if is removed the hypothesis of zero warping displacements at the bottom end ${\Sigma}_{(0)}$.

**Figure 6.**(

**a**) Saint Venant and warping torsional moments versus z/L—(FRP type: “D1”); (

**b**) Saint Venant and warping torsional moments versus z/L—(FRP type: “D2”); (

**c**) Saint Venant and warping torsional moments versus z/L—(FRP type: “U1”).

**k**axis: ${M}_{\omega}^{(2)}/C$. It emerges that the type “U1” makes it possible to achieve the greatest benefits in terms of reduction of internal torsional moment (${M}_{t}^{(1)}$ + ${M}_{\omega}^{(1)}$) within the concrete wall-system. The minimum and maximum values are summarized in Table 3.

D1 | D2 | U1 | |
---|---|---|---|

min | 0.08 | 0.03 | 0.15 |

max | 0.16 | 0.06 | 0.28 |

**n, t**and

**k**are presented in order.

_{(l)}($0.75\le z/L$), due to the well-known singularity of the contact between two elastic bodies, the result obtained via the interface model adopted are no longer reliable.

_{(0)}.

^{(1)}and B

^{(2)}. The three diagrams refer to case “U1” and have been evaluated at the cross-sections where the peak value is reached, as it can be argued by the previous Figures 8 a–c.

**Figure 8.**(

**a**) Normal interactions, t

**, versus z/L; (**

_{n}**b**) Tangential interactions along the

**t**direction, t

**, versus z/L; (**

_{t}**c**) Tangential interactions along the

**k**direction, t

**, versus z/L.**

_{k}**Figure 9.**(

**a**) Normal interactions, t

**, versus the s coordinate; (**

_{n}**b**) Tangential interactions along the

**t**direction, t

**, versus the s coordinate; (**

_{t}**c**) Tangential interactions along the

**k**direction, t

**, versus the s coordinate.**

_{k}## 5. Conclusion

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**MDPI and ACS Style**

Mancusi, G.; Feo, L.; Berardi, V.P.
Concrete Open-Wall Systems Wrapped with FRP under Torsional Loads. *Materials* **2012**, *5*, 2055-2068.
https://doi.org/10.3390/ma5112055

**AMA Style**

Mancusi G, Feo L, Berardi VP.
Concrete Open-Wall Systems Wrapped with FRP under Torsional Loads. *Materials*. 2012; 5(11):2055-2068.
https://doi.org/10.3390/ma5112055

**Chicago/Turabian Style**

Mancusi, Geminiano, Luciano Feo, and Valentino P. Berardi.
2012. "Concrete Open-Wall Systems Wrapped with FRP under Torsional Loads" *Materials* 5, no. 11: 2055-2068.
https://doi.org/10.3390/ma5112055