# Analytical Model for the Prediction of Instantaneous and Long-Term Behavior of RC Beams under Static Sustained Service Loads

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## Abstract

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## 1. Introduction

## 2. Instantaneous Flexural Behavior

#### 2.1. Material Models

#### 2.1.1. Concrete in Compression

_{cm}is the concrete compressive strength, η = ε/ε

_{c1}, k = (1.05·E

_{cm}· ε

_{c1})/f

_{cm}, ε

_{c1}the concrete compressive strain corresponding to the peak stress f

_{cm}, ε

_{cu}is the ultimate compressive strain in concrete as per EC2 [14], and E

_{cm}is the secant modulus of elasticity. Figure 1 shows the adopted compressive behavior of concrete. All the above mentioned parameters can be evaluated as functions of the compressive strength [14] (Equations (2)–(4)).

#### 2.1.2. Concrete in Tension

_{ctm}can be determined as a function of the compressive strength using Equation (5). The full tensile behavior is divided into two parts as shown in Figure 2. The first part is linear and can be described by the classical Hooke’s law up to the peak stress. The second part is nonlinear and can be expressed by a stress-crack opening curve (σ-w) where the stress decreases with the increase of the crack opening (Figure 2). The preferred model to explain the softening behavior is the power law model (Equation (6)) [16,17]. According to the established law, n stands for its power and w

_{u}for the critical crack width where zero tensile strength corresponds.

#### 2.1.3. Steel Reinforcement Behavior

_{y}and the second part represents the plastic phase which is taken here as an inclined line up to the ultimate steel strain ε

_{uk}and a maximum steel stress f

_{yk}= k · f

_{y}. Equation (7) [14] represents this behavior with k = 1.25, ε

_{y}= f

_{y}/E

_{s}and ε

_{uk}= 10%.

#### 2.2. Mechanical Model of RC Section Subjected to Bending

_{c}), the steel strain (ε

_{s}), and the height of the compression zone (x), are present in the non-ultimate state (also known as the serviceability state) versus two equilibrium equations.

_{0}at the beam axis. Equation (8) can be used to calculate the strain in the strip that is positioned at y distances from the beam’s bottom.

_{0}, which satisfy the equilibrium of the cross-section. The numerical solution of the equilibrium equations was performed using the Newton–Raphson method with two variables. The cross section has been divided into n + 1 strip defining n layers in order to the integrals of Equation (9). The force of each layer is computed by multiplying the average stress by the area of the layer. As a result, sums can be used in place of integrals. After reaching the equilibrium and determining the two variables (ε

_{0}and k) all strains values as well as stresses along the section could be determined, in addition to the height of the compression zone (Equation (10)). These results are essential for long-term analysis.

#### 2.3. Initial Loading Model Assessment

## 3. Long-Term Flexural Behavior

#### 3.1. Material Models

_{0}) is the creep coefficient according to EC2 [14] Annex B which represents the ratio between the long-term strain at time t due to a sustained load applied at time t

_{0}and the instantaneous strain at time t

_{0}while E

_{eff}(t,t

_{0}) is the effective modulus of elasticity. In uniaxial compression, Equation (14) describes the chosen stress–strain relationship, where ε

_{cu,LT}is the adjusted ultimate compressive strain at time t. This model is shown in Figure 7, where it can be seen that the stress–strain curve is dependent on the loading duration (t − t

_{0}) and the creep coefficient value φ (t, t

_{0}).

_{cm}was replaced by E

_{eff}(t, t

_{0}) (Equation (12)) as illustrated in Figure 8.

#### 3.2. The Mechanical Model of RC Section Subjected to Sustained Bending

#### 3.3. Long-Term Loading Model Assessment

## 4. Conclusions

- Any simply supported beam under any level of loading can be examined instantly and over time using the established model.
- The concrete stress–strain relationship can be modified using the EMM approach to produce reliable numerical results.
- The model may also be used to calculate the height and width of the cracks following creep advancement. Unfortunately, due to a lack of experimental studies, a comparison of crack evolution during creep was not conducted in the current work.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Jacques, B.; Jean-Pierre, O. Les Bétons: Bases et Données Pour Leur Formulation; Eyrolles: Paris, France, 1996. [Google Scholar]
- Ghali, A.; Favre, R.; Elbadry, M. Concrete Structures: Stresses and Deformations: Analysis and Design for Serviceability, 3rd ed.; Spon Press: London, UK, 2002. [Google Scholar]
- Torres, P.P.; Ghorbel, E.; Wardeh, G. Towards a New Analytical Creep Model for Cement-Based Concrete Using Design Standards Approach. Buildings
**2021**, 11, 155. [Google Scholar] [CrossRef] - De Vittorio, S. Time-Dependent Behaviour of Reinforced Concrete Slabs; University of Bologna: Bologna, Italy, 2011. [Google Scholar]
- Van Mullem, T. Analysis and Numerical Simulation of Long-Term Creep Tests on Concrete Beams; Ghent University: Ghent, Belgium, 2016. [Google Scholar]
- Shaaban, I.G.; Saidani, M.; Nuruddin, M.F.; Malkawi, A.B.; Mustafa, T.S. Service ability Behavior of Normal and High-Strength Reinforced Concrete T-Beams. Eur. J. Mater. Sci. Eng.
**2017**, 2, 99–110. [Google Scholar] - Tošić, N.; de la Fuente, A.; Marinković, S. Creep of recycled aggregate concrete: Experimental database and creep prediction model according to the fib Model Code 2010. Constr. Build. Mater.
**2019**, 195, 590–599. [Google Scholar] [CrossRef] - Mayfield, B. Creep and shrinkage in concrete structures, Edited by Z. P. Bazant and F. H. Wittman, Wiley, Chichester, 1982. No. of pages: 363. Price: £24.95. Earthq. Eng. Struct. Dyn.
**1983**, 11, 591–592. [Google Scholar] [CrossRef] - Gilbert, R.I.; Ranzi, G. Time-Dependent Behaviour of Concrete Structures; Spon Press: London, UK, 2010. [Google Scholar]
- Reybrouck, N.; Criel, P.; Van Mullem, T.; Caspeele, R. Long-term data of reinforced concrete beams subjected to high sustained loads and simplified prediction method. Struct. Concr.
**2017**, 18, 850–861. [Google Scholar] [CrossRef] - Tošić, N.; Marinković, S.; Pecić, N.; Ignjatović, I.; Dragaš, J. Long-term behaviour of reinforced beams made with natural or recycled aggregate concrete and high-volume fly ash concrete. Constr. Build. Mater.
**2018**, 176, 344–358. [Google Scholar] [CrossRef] - Sryh, L.; Forth, J. Long-Term Flexural Behaviour of Cracked Reinforced Concrete Beams with Recycled Aggregate. Int. J. Concr. Struct. Mater.
**2022**, 16, 19. [Google Scholar] [CrossRef] - Shallal, M.A. Prediction of Long-Term Deflection of Reinforced Concrete Beams Suitable for Iraqi Conditions. J. Babylon Univ. Sci.
**2013**, 21, 1328–1347. [Google Scholar] - EN 1992-1-1; Eurocode 2: Design of Concrete Structures: Part 1–1: General Rules and Rules for Buildings. European Committee for Standardization: Brussels, Belgium, 2004.
- Shaaban, I.G.; Mustafa, T.S. Towards efficient structural and serviceability design of high-strength concrete T-beams. Proc. Inst. Civ. Eng.—Struct. Build.
**2021**, 174, 836–848. [Google Scholar] [CrossRef] - Mohamad, R.; Hammadeh, H.; Kousa, M.; Wardeh, G. Fracture based non linear model for reinforced concrete beams. Geomate J.
**2020**, 18, 110–117. Available online: https://geomatejournal.com/geomate/article/view/408 (accessed on 16 December 2022). [CrossRef] - Ghoson, D.; Mayada, K.; Wardeh, G. The Effect of Paste Volume and Concrete Properties on Size Independent Fracture Energy. J. Mater. Eng. Struct.
**2014**, 1, 58–72. [Google Scholar] - Wardeh, G.; Ghorbel, E. Prediction of fracture parameters and strain-softening behavior of concrete: Effect of frost action. Mater. Struct.
**2015**, 48, 123–138. [Google Scholar] [CrossRef] - Ignjatović, I.S.; Marinković, S.B.; Mišković, Z.M.; Savić, A.R. Flexural behavior of reinforced recycled aggregate concrete beams under short-term loading. Mater. Struct.
**2013**, 46, 1045–1059. [Google Scholar] [CrossRef] - Seara-Paz, S.; González-Fonteboa, B.; Martínez-Abella, F.; Eiras-López, J. Flexural performance of reinforced concrete beams made with recycled concrete coarse aggregate. Eng. Struct.
**2018**, 156, 32–45. [Google Scholar] [CrossRef] - Lee, K.-W. Nonlinear Time Dependent Design and Analysis of Slender Reinforced Concrete Columns; University of Hamburg: Hamburg, Germany, 2004. [Google Scholar]

**Figure 1.**The adopted behavior of concrete in compression [14].

**Figure 3.**The adopted behavior of Steel Reinforcement [14].

Data | Reybrouck et al., 2017 [10] | Tošic et al., 2018 [11] | Sryh and Forth, 2022 [12] | |||||
---|---|---|---|---|---|---|---|---|

B2-L52 | NAC7 | NC | ||||||

b [mm] | 150 | 150 | 160 | 300 | ||||

h [mm] | 280 | 280 | 200 | 150 | ||||

Bot. Reinf. | 5 T 14 | 8 T 14 | 2 T 10 | 3 T 16 | ||||

L [mm] | 2800 | 2800 | 3200 | 4000 | ||||

fy [MPa] | 461 | 461 | 587 | 500 | ||||

Es [GPa] | 195.5 | 195.5 | 200 | 200 | ||||

t_{0} [day] | 28 | 28 | 7 | 28 | ||||

t_{1} [day] | 1426 | 1600 | 457 | 118 | ||||

f_{cm,t0} [MPa] | 35 | 40.3 | 32.9 | 41.5 | ||||

E_{c,t0} | 31,000 | 27,800 | 30,100 | 30,600 | ||||

F_{ctm} [MPa] | 4.12 | 4.34 | 5.6 | 5.3 | ||||

σ/f_{cm} | 0.62 | 0.59 | 0.46 | 0.46 | ||||

M_{external} [kN.m] | 34 | 42.2 | 7.628 | 17.2 | ||||

Results | Paper | Present Work | Paper | Present Work | Paper | Present Work | Paper | Present Work |

Δ,_{0} (t = t_{0}) [mm] | 7.27 | 6.64 | 7.08 | 6.42 | 9.17 | 12.31 | 30.49 | 31.15 |

Δ,_{LT} (t = t_{1}) [mm] | 13.49 | 12.54 | 14.51 | 12.93 | 18.79 | 19.65 | 48.79 | 48.97 |

Ratio | 1.86 | 1.89 | 2.05 | 2.01 | 2.05 | 1.60 | 1.60 | 1.57 |

Variance in Δ,_{0} | 0.09 | 0.09 | 0.34 | 0.02 | ||||

Variance in Δ,_{LT} | 0.07 | 0.11 | 0.05 | 0.00 | ||||

ε_{c,0} (t = t_{0}) [×10^{−4}] | −8.70 | −8.10 | −8.50 | −9.25 | −5.47 | −5.45 | NA | −7.87 |

ε_{c,LT} (t = t_{1}) [×10^{−3}] | −2.43 | −2.31 | −2.43 | −2.61 | −1.89 | −1.36 | NA | −1.43 |

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**MDPI and ACS Style**

Bakleh, B.; Hasan, H.; Wardeh, G.
Analytical Model for the Prediction of Instantaneous and Long-Term Behavior of RC Beams under Static Sustained Service Loads. *Appl. Mech.* **2023**, *4*, 31-43.
https://doi.org/10.3390/applmech4010003

**AMA Style**

Bakleh B, Hasan H, Wardeh G.
Analytical Model for the Prediction of Instantaneous and Long-Term Behavior of RC Beams under Static Sustained Service Loads. *Applied Mechanics*. 2023; 4(1):31-43.
https://doi.org/10.3390/applmech4010003

**Chicago/Turabian Style**

Bakleh, Bassel, Hala Hasan, and George Wardeh.
2023. "Analytical Model for the Prediction of Instantaneous and Long-Term Behavior of RC Beams under Static Sustained Service Loads" *Applied Mechanics* 4, no. 1: 31-43.
https://doi.org/10.3390/applmech4010003