# Calculation of Reinforcement of a Tie with RF-CONCRETE Members

### Technical Article

This article deals with the determination of the concrete reinforcement for a beam stressed by tension only according to EN 1992-1-1. The aim is to show the tensile load of a member-type element (without imposed deformations) and to define the concrete reinforcement in accordance with the standard's construction rules and provisions using the RFEM structural analysis software.

#### What means tension for a concrete element?

A section of a structural element is stressed by simple tension when the forces acting on one side of the section are reduced at the section's center of gravity to a single force N. This normal force N is then perpendicular to the section and directed towards the side where the forces act. The self-weight is neglected in the concrete and the section is uniformly subjected to tension.

#### Tensile Stress in Steel

For steel with a σ - ε diagram showing an inclined graph, the equation to the right of the graph corresponding to the tensile behavior of steel is written according to the steel's characteristic values set forth in §3.2. 7 (2) of EN 1992-1-1.

${\mathrm{\sigma}}_{\mathrm{s}}={\mathrm{f}}_{\mathrm{yd}}+\frac{\mathrm{k}\xb7{\mathrm{f}}_{\mathrm{yd}}-{\mathrm{f}}_{\mathrm{yd}}}{{\mathrm{\epsilon}}_{\mathrm{uk}}-\frac{{\mathrm{f}}_{\mathrm{yd}}}{{\mathrm{E}}_{\mathrm{s}}}}\xb7\left[{\mathrm{\epsilon}}_{\mathrm{s}}-\frac{{\mathrm{f}}_{\mathrm{yd}}}{{\mathrm{E}}_{\mathrm{s}}}\right]$

σ_{s} |
Stress in reinforcement |

f_{yd} |
Design value of yield strength = f_{yk} / γ_{s} |

k |
Characteristic limits ratio = f_{tk} / f_{yk} |

ε_{uk} |
Limit deformation |

E_{s} |
Modulus of elasticity |

ε_{s} |
Deformation in reinforcement = ε_{ud} = 0.9 ⋅ ε_{uk} |

f_{yk} |
Characteristic yield strength |

γ_{s} |
Partial safety factor of steel |

f_{tk} |
Characteristic value of tensile strength |

ε_{ud} |
Design value of limit deformation |

#### Longitudinal Reinforcement

Please note: the concrete under tension is neglected for pure tension. Thus, only the steels fully balance the N_{ed} tensile forces. So, the necessary reinforcement area is determined according to the tensile force and the provided stress.

A_{s} = N_{Ed} / σ_{s}

A_{s} ... Reinforcement area

N_{ed} ... Ultimate normal force

#### Application of Theory Using RF-CONCRETE Members

We look at an example of an element subjected to simple tension by analyzing the results obtained for the longitudinal reinforcements. Below you find the input data:

- Permanent loads: N
_{g}= 100 kN - Variable loads: N
_{q}= 40 kN - Square section: 20/20 cm
- Concrete strength class: C25/30
- Steel: S 500 A for inclined graph
- Diameter of longitudinal reinforcement: ϕl = 12 mm
- Diameter of transverse reinforcement: ϕt = 6 mm
- Concrete cover: 3 cm
- Control of cracking not required.

In order to verify the material settings in RF-CONCRETE Members, image 02 describes the materials used for concrete and reinforcement.

#### Ultimate Limit State

Design loading in ultimate limit state:

N_{Ed} = 1.35 ⋅ 100 + 1.5 ⋅ 40 = 195.00 kN

#### Provided Tensile Stress

Ultimate limit state for a durable, transient design situation:

f_{yd} = 500 / 1.15 = 435 MPa

k = 525 / 500 = 1.05 according to Table C.1 of EN 1992-1-1

ε_{uk} = 25 ‰

ε_{ud} = 0.9 ⋅ 25 = 22.5 ‰

σ_{s} = 435 + (1.05 ⋅ 435 - 435) / (2.5 - 435 / (200 000)) ⋅ [2.25 - 435 / (200 000)] = 454 MPa

#### Required Longitudinal Reinforcement

Longitudinal reinforcements for the ultimate limit state:

A_{s} = 0.195 / 454 ⋅ 10^{4} = 4.30 cm²

#### Provided Longitudinal Reinforcement

Having configured the reinforcing steels with a diameter of 12 mm in RF-CONCRETE Members, the provided reinforcement determined automatically by the add-on module is 4 bars, with a symmetrical distribution on the lower and upper parts of the section, i.e. 2 x 2 HA12 which results in the following reinforcement area:

A_{s} = 4 ⋅ 1.13 = 4.52 cm²

#### Transverse Reinforcement

With the transverse reinforcement being also defined by the user, RF-CONCRETE Members can automatically determine the spacings according to the standard, and check whether the their arrangement is conform.

In our case, by imposing stirrups having a diameter of 6 mm, the program gives us a spacing of 0.122 m, but also displays the warning message n° 155) in the Notes column that can be seen in image 07.

The formula referring to §9.2.2 (8) of EN 1992-1-1 is defined below.

S_{l,max} = 0.75 ⋅ d

S_{l,max} ... Maximum transverse spacing of stirrups

d ... Effective height

d = h - e - ∅t - ∅_{l}/2

h ... Cross-section height

e ... Concrete cover

The previous formulas give us the following results:

d = 0.200 - 0.03 - 0.006 - 0.012 / 2 = 0.158 m

S_{l,max} = 0.75 ⋅ 0.158 = 0.12 m

By clicking "Modify reinforcement…" as shown in image 08, the stirrup spacing can be modified to 0.11 m, and the warning message disappears.

#### Conclusion

Having set the parameters beforehand, RF-CONCRETE Members gives the number of rebars required according to the defined arrangement, in order to verify the tensile load according to the internal forces coming from RFEM. Depending on the displayed warning messages, it is also possible for the user to modify the reinforcement and its arrangement after calculation.

#### Author

#### M.Eng. Milan Gérard

Sales & Technical Support

Milan Gérard works at the Paris site. He is responsible for sales and provides technical support to our French-speaking customers.

#### Keywords

Eurocodes Tension Reinforcement

#### Reference

#### Links

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