# The Evaluation of Torsional Strength in Reinforced Concrete Beam

Mohammad Rashidi, Hana Takhtfiroozeh

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Mohammad Rashidi, Hana Takhtfiroozeh. The Evaluation of Torsional Strength in Reinforced Concrete Beam. Mechanics, Materials Science & Engineering Journal, 2017, 7, 10.13140/RG.2.2.16568.75521. hal-01508635

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# The Evaluation of Torsional Strength in Reinforced Concrete Beam

Mohammad Rashidi¹, Hana Takhtfiroozeh²

1. Department of Civil Engineering, Sharif University of Technology, Tehran, Iran

2. Department of Civil Engineering, Building and Housing Research Center, Tehran, Iran

<figure><img src="image_2.png" /></figure>

DOI 10.13140/RG.2.2.16568.75521

**Keywords:** torsional strength, concrete beam, transverse and longitudinal bars, reinforcement.

**ABSTRACT.** Many structural elements in building and bridge construction are subjected to significant torsional moments that affect the design. A simple experiment for the evaluation of the torsional strength of reinforced concrete beams as a one of this structural elements is presented in this research. The objective of this experiments would be the role of transverse and longitudinal reinforcement on torsion strength. Four beam test samples has been tested with the same length and concrete mix design. Due to the fact, that the goal of this experiment is to determine the effect of reinforcement type on torsion strength of concrete beams; therefore, bars with different types in each beam have been applied. It was observed that the ductility factor increases with increasing percentage reinforcement from the test results. It should be also noted that transverse bars or longitudinal bars lonely would not able to increase the torsional strength of RC beams and both of them can be essential for having a good torsional behaviour in reinforced concrete beams.

**Introduction.** The interest in gaining better understanding of the torsional behaviour of reinforced concrete (RC) members has grown in the past decades. This may be due to the increasing use of structural members in which torsion is a central feature of behaviour such as curved bridge girders and helical slabs. The achievements, however, have not been as much as those made in the areas of shear and bending. Dealing with torsion in today's codes of practice is also very primitive and does not contain the more elaborate techniques. Predictions of current standards for the ultimate torsional capacity of RC beams are found to be either too conservative or slightly risky for certain geometry, dimensions and steel bar sizes and arrangements.

Torsional moments in reinforced concrete are typically accompanied by bending moments and shearing forces. However, simplified methods in design codes are based on a simple combination of the pure shear methods and pure torsion methods. In the ACI code [1], the effects of the torsional moment are accounted for by superimposing the amount of transverse and longitudinal steel and the intensity of the shearing stresses required for torsion resistance to those required for shear resistance. The Canadian code [2] assumes a similar interaction and further superimposes the effects of torsion and shear on the longitudinal strain indicator required in the design solution. Moreover, interaction surfaces between shearing and axial forces and bending moment such as those suggested by Elfren et al. [3] and Ewida and McMullen [4] are still of practical importance. The use of such interaction surfaces and the use and development of the code equations require knowledge of the pure torsional strength of reinforced concrete.

Rahal and Collins [5] assigned the methods available for computing the torsional capacities to two main categories. Methods in the first category use semi-empirical equations chosen to fit available experimental data. The strength of these methods comes generally from their simplicity.

Methods in the second category use procedures based on more rational models such as the space truss model. These models are generally more time demanding, but their strength comes from their rationality and their ability to give the engineer a feel for the behavior of the structural member designed.
A recently developed simplified model [6] was shown to be an accurate and rational tool for calculating the shear strength of membrane elements subjected to shear. Similar to the General Method [7], this model is based on the equations of the MCFT. The MCFT is a powerful rational model capable of calculating the full response of sections subjected to shear, axial load, and bending and torsional moments [8, 9, and 10]. The new model was able to cast the results of the rational MCFT into a simple procedure. The applicability of the model was extended [11] to cover beams subjected to shearing and axial forces and bending moments. The effects of axial forces and bending moments on the shear strength were accounted for by a simplified superposition procedure.

This paper extends the effect of reinforcement type on torsion strength of concrete beams. The objective of this experiments would be the role of stirrups and longitudinal reinforcement on torsion strength. Four beam test samples has been tested with the same length and concrete mix design. The reinforcement of this samples has been different ranging from without reinforcement to complete reinforcement.

**Materials and methods.** Four experimental beam samples, without reinforcement, with just transverse reinforcement, with just longitudinal reinforcement, and both transverse and longitudinal reinforcement, has been tested to gain bending moment, cracking moment and ultimate bending moment. Appropriate torsional results originated from this experiment give us an information about the effect of reinforcement on Reinforced Concrete Beams.

The considered mix for the samples has been shown in table 1 below. According to the instructions, coarse aggregates have been sieved via a 2-cm sieve. Also, the samples considered in construction are three cylindrical samples in 30×15 cm dimensions and four beams samples in 60×10×10 cm dimensions.

<table><caption>Table 1. The considered mix for the samples.</caption><thead><tr><td>Part</td><td>Weight Ratio (kg/m³)</td></tr></thead><tbody><tr><td>Cement</td><td>500</td></tr><tr><td>Sand</td><td>800</td></tr><tr><td>Gravel</td><td>800</td></tr><tr><td>Water</td><td>220</td></tr><tr><td>Total</td><td>2320</td></tr></tbody></table>

Due to the fact, that the goal of this experiment is to determine the effect of reinforcement type on torsion strength of concrete beams; therefore, bars with different types in each beam have been applied. The ends of the beam has been used metal cube to avoid crunch of beams end [12]. In addition, in ends of beam, longitudinal as well as transverse reinforcement has been used to a distance of 10 cm. After reinforcement of samples according to figures 1 to 4, the stages of concreting and curing of concrete shall be conducted and then the samples shall be examined after 28 days of curing. Dimensions of cylindrical samples and beam samples are also shown in table 2 and 3 respectively.
<figure><img src="image_3.png" /><figcaption>Fig.1. Samples No. 1, without longitudinal and transverse reinforcement.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig.2. Samples No. 2, just longitudinal reinforcement.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig.3. Samples No. 3, just transverse reinforcement.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig.4. Samples No. 4, both longitudinal and transverse reinforcement.</figcaption></figure>

<table><caption>Table 2. Dimensions of Cylindrical Samples.</caption><thead><tr><td>Sample No.</td><td>The Average Diameter (Cm)</td><td>The Average Height (Cm)</td></tr></thead><tbody><tr><td>1</td><td>15.1</td><td>30.3</td></tr><tr><td>2</td><td>15.2</td><td>30.1</td></tr><tr><td>3</td><td>15.0</td><td>30.2</td></tr></tbody></table>
<table><caption>Table 3. Dimensions of the Beam Samples.</caption><thead><tr><td>Sample No.</td><td>Length (Cm)</td><td>Width (Cm)</td><td>Height (Cm)</td></tr></thead><tbody><tr><td>1</td><td>60.10</td><td>9.99</td><td>10.11</td></tr><tr><td>2</td><td>60.05</td><td>10.02</td><td>10.03</td></tr><tr><td>3</td><td>60.30</td><td>10.01</td><td>10.02</td></tr><tr><td>4</td><td>60.25</td><td>9.98</td><td>10.08</td></tr></tbody></table>

It should be noted, that the compressive strength test of the samples shall be conducted after capping the goal of which is to create a flat surface on the sample.

All the beams, which were experimented on, were 60 centimetres long, they were placed on a 55-centimetre-wide support and were loaded and tested. Two concentrated symmetrical loads, which were 25 centimetres away from each other were used for loading purposes. The weight of the rods, which are placed on the beam, was 37.8 kg. The used bars in this experiment are of type A2 and the current strength of 300 MPa. The loading model of the beams can be seen in figure 5.

<figure><img src="image_7.png" /><figcaption>Fig. 5. The loading model of the beam.</figcaption></figure>

## Torsion in the international Standards.

Provisions for torsional design of reinforced concrete members appear in majority of international standards of concrete design. While these provisions are conceptually similar, they contain variations that produce different results. Provisions of some of the more well-known standards are reviewed here in this section.

**Australian Standard (AS3600).** According to the Australian standard for concrete structures, AS3600, the ultimate strength in pure torsion, $T_{uc}$, for a beam without closed ties can be calculated as

$$T_{uc} = J_t (0.3\sqrt{f'_c}) \quad (1)$$
where f'c – is the compressive strength of concrete at 28 days;

$J_t$ – is the torsional rigidity of the cross-section.

This torsional rigidity for a rectangular cross-section with dimensions x×y (where x<y) can be determined as 0.4 x² y. For beams with closed ties, the ultimate torsional strength, $T_{us}$, is

$$T_{us} = f'_{ys} (A_{sw} / s) 2 A_t \cot\theta_t \quad (2)$$

where $A_t$ – is the area enclosed by the centre lines of longitudinal bars Figure 6;

s – is the centre-to-centre spacing of stirrups,

$f_{ys}$ – is the yield strength of stirrups,

$A_{sw}$ – is the cross-sectional area of stirrups,

$t_\theta$ – is the crack angle which can be taken as 45° or can vary linearly between 30° when $T^*=\phi T_{uc}$ and 45° when $T^*=\phi T_{u.max}$.

There are $T^*$ – is the factored design torque, $T_{uc}$ – is the ultimate torsional strength of a beam without torsional reinforcement, and $\phi$ is equal to 0.7.

The term $T_{u.max}$ is the ultimate torsional strength of a beam limited by web crushing failure and can be obtained from $T_{u.max}=0.2f'_cJ_t$. This is a simple equation to evaluate $T_{u.max}$. Other more complicated equations have been presented in the literature but not adapted by the standard. For example, Warner et al. [13] present $T_{u.max}$ as

<figure><img src="image_8.png" /><figcaption>Fig. 6. The cross-section of a rectangular reinforced concrete beam</figcaption></figure>

$$T_{u.max} = f'_c (A_{oh}^2 / p_h) \left( \frac{\sin \theta_t \cos \theta_t}{1.14 + 0.68 \cot^2 \theta_t} \right) \quad (3)$$
Where $A_{oh}$ is the area enclosed by the centre line of the exterior closed ties and $p_h$ is the perimeter. AS3600 suggests that the total longitudinal steel area, As, shall be obtained by

$$A_s = (f_{ys} / f_y) (A_{sw} / s) u_t \cot^2 \theta_t \quad (4)$$

Where $u_t$ – is the perimeter of $A_t$ (in Eq. (4));

$f_y$ – is the yield strength of longitudinal reinforcement.

Furthermore, according to this standard, the spacing of stirrups shall not be greater than the lesser 0.12$u_t$ and 300 mm.

**British Standard (BS8110).** The British standard for reinforced concrete structures, BS8110, indicates that the additional stirrups required to resist torsion in addition to what is required for shear shall be calculated from

$$A_{sv} / s > T_{us} / 0.8 \times y_1 \times y_1 (0.87 f_{ys}) \quad (5)$$

Where $A_{sv}$ – is the area of the two legs of stirrups at a section;

$x_1$ and $y_1$ – are the centre to center of the shorter and longer legs of stirrups, Figure 1. Moreover, BS8110 suggests that additional longitudinal reinforcement $A_s$ due to torsion should be provided as calculated by

$$A_s > A_{sw} f_{ys} (x_1 + y_1) / s f_y \quad (6)$$

This standard emphasises that the spacing of stirrups should not exceed the smallest of $x_1$, $y_1$ / 2 or 200mm. BS8110 only allows the use of its provisions for torsional design when the yield stress of reinforcement is not more than 460MPa.

**ACI Standard (ACI318-02).** ACI318-02 calculates the ultimate torsional strength of reinforced concrete beams as

$$T_{us} = f_{ys} (A_{sw} / s) 2 A_o \cot \theta_t \quad (7)$$

Where $A_o$ – is the gross area enclosed by the shear flow path, which can be taken equal to $0.85A_{oh}$, where $A_{oh}$ – is the area enclosed by the centre of stirrups.

ACI allows the crack angle $\theta_t$ of non-prestressed or low-prestressed members to be taken as 45°. Eq. (8) is based on the assumptions that all of the external torque is resisted by reinforcement and concrete resistance is negligible; that the concrete carries no tension; that the reinforcement yields, and that
the concrete outside the stirrups is relatively ineffective. The standard also indicates that the additional longitudinal reinforcement ($A_s$) required for torsion shall not be less than the value obtained from the following equation

$$A_s = (f_{ys} / f_y) (A_{sw} / s) u_t \cot^2 \theta_t \quad (8)$$

ACI318-02 recommends that the transverse torsional reinforcement (stirrup) shall be anchored by a 135° standard hook around a longitudinal bar and the spacing of transverse torsion reinforcement shall not exceed the smaller of $p_h/8/8$ or 12" ($\approx$304mm).

**European Standard.** According to the European Standard (Eurocode 2), three different ultimate values should be calculated and the minimum chosen. The first value is related to the stirrups contribution to the torsional resistance which can be calculated as

$$T_{u(1)} = f_{ys} (A_{sw} / s) 2 A_k \cot \theta_t \quad (9)$$

Where $A_k$ is the area enclosed by the centre-lines of the effective wall thickness. The effective wall thickness, $t_{ef}$, can be calculated as $A/u$ where A is the total area and u is the perimeter of the cross-section. The second value of the torsional strength corresponds to the longitudinal bars as

$$T_{u(2)} = f_y (As / u_k) 2 A_k \tan \theta_t \quad (10)$$

Where $u_k$ is the perimeter of the area $A_k$.

Torsional capacity of the concrete struts is the third value. It can be derived from

$$T_{u(3)} = 2v f_{ck} A_k t_{ef} \sin \theta_t \cos \theta_t \quad (11)$$

Where $f_{ck}$ – is the compressive strength of concrete, and v can be taken as 0.6(1 – $f_{ck}$ / 250).

The least of these three values is the torsional strength of the member. The European Standard also indicates that the variation of crack angle is in the order of $2.5 \le \cot \theta_t \le 1$ but can be taken as $\theta_t$=45°.

**Canadian Standard.** The method of calculating torsional strength of reinforced concrete beams in the Canadian Standard, CSA, is similar to ACI. In addition, CSA advises that the stirrups must be anchored by 135° hooks, the nominal diameter of the bar or tendon shall not be less than $s/16$, and the total area of longitudinal bars required around the section, $A_l$, (with a spacing not exceeding 300 mm) shall be calculated from $A_l p_h / s$, where $A_l$ is the area of a stirrup, $p_h$ is the perimeter of the centre line of the stirrups, and $s$ is the spacing of stirrups.
In the above mentioned standards, the method of evaluating the ultimate torsional capacity of reinforced concrete beams is similar. ACI standard for this experiment which is more prevalent in the vast majority of countries has been used.

**Discussion of test results.** A simple test for calculating the torsional strength of reinforced concrete beams was experimented with two concentrated symmetrical loads presented in figure 5. As can be seen in figure 6 the failure of beams is shown and subsequently the results of tests including sample rotation, momentum of the cross-section, crack momentum and ultimate momentum of the cross section is presented in table 5.

<figure><img src="image_9.png" /><figcaption>Fig. 6. Failure of beams.</figcaption></figure>

Table 5 indicates that the crack momentum of all samples was 5870 Kg.cm according to equation 12.

$$T_{cr} = 4 \sqrt{f'c A_c^2 / P_c} \qquad (12)$$

Where $A_c$ – is the area of beam cross-section;

$P_c$ – is the perimeter of the beam.

Ultimate momentum of the cross-section in sample No. 1 is equal to its crack momentum because this sample was not reinforced by longitudinal and transverse bars. However, this amount has been increased with the enhancement of reinforcement especially in the sample of 4. Moreover, the ductility of beams, if the rotation of samples increase in results, will grow. It was observed that the ductility factor increases with increasing percentage reinforcement. As can be seen in the test results in sample No. 4 with transverse and longitudinal bars the torsional strength and ductility of beam have been increased 95% and 50% respectively in comparison with sample No. 1.

In addition, it was noticed that sample No. 3 with just transverse bars had a more torsional strength compared to sample No. 2 with just longitudinal strength and it was concluded that transverse bars play an important role in torsional strength of Reinforced Concrete Beams. The results of experiment shows that the momentum of cross-section in sample No. 3 is 11500 Kg.cm, while this amount for sample No.2 would be 8500 Kg.cm.

It should be also noted that transverse bars or longitudinal bars lonely would not able to increase enough the torsional strength of RC beams and both of them can be essential for having a good torsional behaviour in reinforced concrete beams.
<table><caption>Table 5. The Results of the Experiment</caption><thead><tr><th>Sample No.</th><th>Sample rotation (Degree)</th><th>Momentum of the Cross-Section (Kg.cm)</th><th>Crack Momentum (Kg.cm)</th><th>Ultimate Momentum of the Cross-Section (Kg.cm)</th></tr></thead><tbody><tr><td>1</td><td>8.16</td><td>7850</td><td>5870</td><td>5870</td></tr><tr><td>2</td><td>8.78</td><td>8500</td><td>5870</td><td>6500</td></tr><tr><td>3</td><td>9.16</td><td>11500</td><td>5870</td><td>8320</td></tr><tr><td>4</td><td>12.20</td><td>15250</td><td>5870</td><td>10200</td></tr></tbody></table>

**Summary.** A simple experiment for the evaluation of the torsional strength of reinforced concrete beams is presented in this research. The following conclusions were drawn from the studies on reinforced concrete beams:

* It was observed that the ductility factor increases with increasing percentage reinforcement. The torsional strength and ductility of the sample with transverse and longitudinal bars have been increased 95% and 50% respectively in comparison with sample without reinforcement.

* The transverse bars play an important role in torsional strength of Reinforced Concrete Beams compared to longitudinal bars.

* It should be also noted that transverse bars or longitudinal bars lonely would not able to increase the torsional strength of RC beams and both of them can be essential for having a good torsional behaviour in reinforced concrete beams.

References

[1] ACI. Building code requirements for reinforced concrete (ACI 318-95) and commentary (ACI 318 R-95). Committee 318, American Concrete Institute (ACI), Detroit, Mich. 1995.

[2] CSA. Design of concrete structures for buildings. Standard A23.3-94, Canadian Standards Association (CSA), Rexdale, Ont. 1994.

[3] Elfren, L., Karlsson, I., and Losberg, A. Torsion-bending-shear interaction for concrete beams. ASCE Journal of the Structural Division, 100(8): 1657-1676, 1974.

[4] Ewida, A.A., and McMullen, A.E. Torsion-shear-flexure interaction in reinforced concrete members. Magazine of Concrete Research, 23(115): 113-122, 1981.

[5] Rahal, K.N., and Collins, M.P. Simple model for predicting torsional strength of reinforced and prestressed concrete sections. ACI Structural Journal, 93(6): 658-666, 1996.

[6] Rahal, K.N. Shear strength of reinforced concrete: Part I: Membrane elements subjected to pure shear. ACI Structural Journal, 97(1): 86-93, 2000a.

[7] AASHTO. AASHTO LRFD bridge design specifications, SI units, first edition, American Association of State Highway and Transportation Officials (AASHTO), Washington D.C. 1994.

[8] Vecchio, F.J., and Collins, M.P. The modified compression field theory for reinforced concrete elements subjected to shear. ACI Journal, 83(2): 219-231, 1986.

[9] Collins, M.P., and Mitchell, D. Prestressed concrete structures. Prentice Hall, Inc., Englewood Cliffs, N.J. 1986.
[10] Rahal, K.N., and Collins, M.P. The effect of cover thickness on the shear and torsion interaction — An experimental investigation. ACI Structural Journal, 92(3): 334–342, 1995a.

[11] Rahal, K.N. Shear strength of reinforced concrete Part II: Beams subjected to shear, bending moment and axial load. ACI Structural Journal, 97(2), 2000.

[12] Mohammad Rashidi & Hana Takhtfiroozeh. Determination of Bond Capacity in Reinforced Concrete Beam and Its Influence on the Flexural Strength. Mechanics, Materials Science & Engineering Vol.6, 2016. doi: 10.13140/RG.2.2.18300.95361

[13] Warner, R.F., Rangan BV, Hall AS, Faulkes KA. Concrete structures. Longman, South Melbourne, 1998.