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Amulu C. P., Ezeagu C. A. Page 45

Combined Torsion, Bending and Shear Analysis in Reinforced

Concrete beams

1Amulu C.P, and

2Ezeagu C. A.

[email protected]: or

[email protected]

Department of Civil Engineering, Faculty of Engineering Nnamdi Azikiwe University Awka.

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Abstract: This study investigates the effect of combined actions of torsion, bending and shear stresses on

rectangular cross-sectional plain and reinforced concrete beams at a given concrete mix design. The ultimate

torsional strengths and angle of cracks of the beams were determined experimentally through a simple test

arrangement set-up on fifteen beam specimens grouped into five groups; BC1 to BC5. The combined loading was

achieved through by loading the test beams at an eccentric distance away from the beam’s principal axis at the mid-

span, through a system of Universal Testing Machine at the Engineering laboratory of the Standard Organization of

Nigeria (Son), Enugu. Three international standard codes (BS 8110, Euro code 2 and ACI 318) were used to

calculate the ultimate torsional strengths predicted for torsional resistance provided by longitudinal and transverse

reinforcements and the strut of concrete cross-section. The values obtained from the codes were compared with that

of experimental results observation for validation. The ultimate torsional moments predicted by different classical

theories (elastic, plastic, skew-bending and Cowan’s equations) were also used to calculate the ultimate torsional

moments and compared with the experimental results. It was observed that the values predicted by the codes were

not the same with each other and the experimental results in some cases while in other cases, the predicted values

matches that of experimental results. This can be attributed to difference in certain assumptions, parameter and

constants taken during the formulation of their respective equations which influenced the results. It was also

observed from this study that ultimate torsional moments predicted by the classical theories are significantly

difference from that of experimental results. It was concluded from the study that the angle of cracks used in design

by different codes should not be fixed, but to be varied in other to obtain better economic and stable structures in

terms of longitudinal and transverse reinforcements together with high concrete section that can sustain the applied

loads.

Keywords: Bending, Flexural, Load, Moment and Torsion

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mailto:[email protected]

mailto:[email protected]


Introduction: Beams are the simplest structural elements that are used extensively to support

loads. On several situations, beams and slabs are subjected to torsion in addition to bending

moment and shear force. Loads acting normal to the plane of bending will cause bending

moment and shear force. However, loads acting away from the plane of bending will induce

torsional moment along with bending moment and shear (Davison and Owens, 2005),

(Kharagpur, 2008) and the interaction among these forces is important. Torsion is a load that

subjects a member to couples/moments that twists the member spirally. This implies that torsion

occurs when applied loads are eccentric from the centroid and the resultant does not pass through

the beam’s shear centre (Amulu, Ezeagu and Obiorah, 2016), (Onouye and Kane, 2007). Torsion

may be induced in a member in various ways during the process of load transfer in a structural

system. Torsion happens because of integrity and continuity of members and also under the

effect of external loads in concrete structure (Barghlame and Lotfollahi-Yaghin, 2011).Torsion

generated in a member can be classified into two main types of situations based on necessity of

analysis and design for torsion (Sengupta and Menon).Primary or equilibrium torsion and

Secondary or compatibility torsion: If a member is not properly reinforced for torsion with

closed stirrups and longitudinal steels in case of reinforced concrete beam, a sudden brittle

fracture may occur, leading to failure of the beam at torsional cracking loads. Shear stresses due

to torsion create diagonal-tension stresses that produce diagonal cracking (Leet and Bernal,

1997). In structural analysis, the effects of torsion are usually neglected and only bending, shear

and axial forces are taken into consideration. This is because torsion was considered to be a

secondary effect that will be covered in the factor of safety (Kamara and Rabbat, 2007). When

torsion is encountered in reinforced member, it usually occurs in combination with flexure and

transverse shear. The interactive behavior of torsion with bending moment and flexural shear in

reinforced concrete beam is fairly complex, owing to non-homogeneous, non-linear and

composite nature of the material and presence of cracks (Pillai and Menon, 2009).

(Panchacharam and Belarbi, 2002) and (Kharagpur 2008).

Statement of Problem: The design provisions in the areas of combined analysis for shear and

torsion are not of the same level of rationality and general applicability. The absence of rational

models has resulted in highly empirical design procedures characterized by large difference in

values evaluated from different design codes when compared to experimental test results. The


lack of fundamental behavioral models for reinforced concrete members subjected to shear and

torsional loading seems to be prime reason for the unsatisfactory nature of the highly empirical

design procedures used in design. This study provides the ideal of conceptual model to properly

represent the behavior of reinforced concrete member subjected to torsion and to compare results

obtained from different approaches in the standard design codes of practices with the ones

obtained experimentally. The study furthermore seeks to compare the response of reinforced

concrete and steel beams to torsional loading.

Aim of the Study: To optimize the effect of torsion on Reinforced concrete beams at a designed

mix ratio.

Objectives of study: The objectives of this study include:

1) To examine and compare the torsional design provisions/ procedures in BS 8110, Euro

code 2, and ACI 318 building code

2) To evaluate the torsional parameters (torsional crack load and failure load, ultimate

torsional capacity and angle of twist) of a reinforced concrete beams subjected to

combined shear, bending and torsional loading using standard design codes and

experimental work.

3) To validate the provisions of the codes by comparing the calculated design code values

with that of experimental results.

4) To evaluate the ultimate torsional strength of reinforced concrete beams obtained

experimentally, compare it with those predicted by theories: elasticity, plasticity, skew-

bending and space truss analogy.

Scope and limitations of the study: In this study, analysis and experimental work will be

carried out. In the analysis, torsional provisions of Euro code 2, BS 8110 and ACI 318 building

codes are evaluated against the experimental data for reinforced concrete beams under the

actions of combined shear, bending and torsion only. The experimental result will also be

evaluated with ultimate torsional strengths predicted analytically by different theories on torsion

in prismatic object.

Significance of the work: This study provides an evaluation of the design provisions for

combined actions of shear, bending and torsion given by the ACI 318 building code, BS 8110

and Euro Code 2 to experimental values. This study will make structural designers to be familiar

with the provisions of the BS 8110, Euro Code 2 and ACI 318 building code provisions to

torsional design and to determine the more conservative and consistent code among the three

codes studied. Specification recommendations are given for improving the design codes

Amulu C. P., Ezeagu C. A. 8

provisions. Structural standards and codes of practice are reviewed continuously and

improvements are implemented as research findings reveal more accurate methods of design.

LITERATURE REVIEW: Torsion may be induced in a member (reinforced concrete, timber

or steel beam) in various ways during the process of load transfer in a structural system. When a

beam is subjected to transverse loading such that the resultant force passes through the

longitudinal shear axis, the beam will not twist, but bends. However, when the resultant force

acts away from the shear center axis, moment is induced into the system (Davison and Owens,

2005). This moment causes a body to have tendency to rotate and if the system tries to resist

such rotational tendency, bending and /or torsion results (Onouye and Kane, 2007). This implies

that if applied loads are “eccentric” from the centroid and the resultant forces do not pass through

the member’s centroid, then torsion occurs. The distance from the centroid that the load passes

through is the “eccentricity” and the product of the eccentricity and the load gives the torsional

moment. This torsional moment tends to twist the beam to give a rotational displacement. There

are some other cases where beams are subjected to torsion as a result of external load acting

outside the ‘shear center’ axis of a given cross-section, or deformation resulting from the

continuity of beams or similar members that are joint at an angle to each other. In more general

cases, the determination of the torsional moment for any loading in torsion and shear requires

knowledge of location of the center of shear in addition to the geometric centroid (Ziegler,

1995), (Amulu, Ezeagu and Obiorah, 2016). Structural members subjected to torsion are of

different shapes such as T-shaped, inverted L-shaped, double T-shaped and box section. These

different configurations make the understanding of torsion in reinforced concrete members a

complex task (Panchacharam and Belarbi, 2002). Analysis of the torsional behavior of a

reinforced concrete member can be divided into three different phases namely i. pre-cracking

stage. ii. post-cracking stage.iii. the time of yielding of steel. (Csiko’s and Hegedus, 1998) and

(Barghlame and Lotfollahi-Yaghin, 2011).Prior to cracking, a torsional moment applied to a

concrete member is resisted by internal shear stresses. (Kamara and Rabbat, 2007). When a

concrete member is loaded in pure torsion, shear stresses develop. One or more cracks (inclined)

develop when the maximum principal tensile stress reaches the tensile strength of concrete.

(Pillai and Menon, 2009).From the uniqueness of solutions of the elasticity equations, it follows

that the assumed forms for the displacements are the exact solution to the torsional problems


(Wolfram mathematical documentation centre), (Namiq, 2012), (Ezeagu, Osadebe and Anyata,

2014). The properties of a material, undergoing ‘elastic process’ are not the same in all

directions, it is called anisotropic (Sadd, 2005) and (Ezeagu and Nwokoye, 2008). The problem

with this equation is that the obtained results are unconservative by approximately 50% (Csikos

and Hegedus, 1998). However, Reinforced concrete is combined with two kinds of materials

namely: steel and concrete. These two materials have two different kinds of characteristics. Steel

is a kind of material which is homogeneous, whereas concrete is not a homogeneous material.

The homogeneity of steel versus concrete mechanical attribute of these materials depends on

time and environment (Barghlame and Yaghin, 2011). In other words, limitation of the above

model is that the failure of reinforced concrete to torsion is not elastic.

METHODOLOGY: The methodology used in this study will be based on two phases: the first

phase is on experimental design of fifteen rectangular Reinforced concrete beams all of these

specimens loaded eccentrically away from beams principal axis by a combined action of

bending, shear and torsional stresses, applied through a system of Universal Testing machine.

The second phase is on evaluation and comparison of torsional design provisions of the

international standards codes examined; BS 8110, Euro Code 2 and ACI 318 building code. The

calculated values from these codes will be analyzed and validated against the experimental

values measured and the response of reinforced concrete beams to torsional crack load will be

compared with those of classical theories: elasticity, plasticity, skew-bending theories and space

truss analogy.

EXPERIMENTAL DESIGN: Designs to be carried out:

I. Design of reinforced concrete beams to be used for the experimental work (which is an

eccentric loaded beam). The ultimate Torsional moment predicted by codes under this

study will be compared with the values to be observed from Experimental work as

presented in table 3.2.

II. Concrete mix Design for compressive strength of concrete cubes at 28 days

III. The design examples of reinforced concrete beams against torsion, bending and shear

stresses using BS 8110, Euro code 2 and ACI 318 codes for equilibrium and

compatibility Torsional cases, the flow chart in figure 3.4 will be used as guide.


The torsional ultimate moments predicted by different theories (Elasticity, plasticity and skew-

bending theories) on torsion will be calculated and compared with the experimental values to be

observed using table 3.3 below as guide.

Choice of test assembles: The objective of the design of rectangular beam was to provide a

simplified test arrangement and to ensure a failure of the test beams in combined actions of

bending, shear and torsion without causing a failure in other elements. The test beam are seated

on two steel supports resting on the laboratory floor and could be loaded at any desired

eccentricity from the beam principal axis, through a system of Universal Testing machine with a

point-loading head. Thus, a known combination of torsional moment, bending moment and

shearing forces could be applied on the test beam simultaneously. The test beam is restricted at

both ends on the steel support in other to avoid rotation. To avoid failure in the steel support, it

was designed to have a higher compressive strength than the loaded test beam. The beam was

designed for stresses of 30𝑁/𝑚𝑚2 in concrete and 460𝑁/𝑚𝑚2 in reinforcement. The designed

test beam specimens are to carry a maximum load of 25𝑘𝑁/𝑚 at an eccentricity of 60𝑚𝑚,

required 8𝑚𝑚 diameter of stirrups at 150𝑚𝑚 centre to center and 10𝑚𝑚 diameter of

longitudinal reinforcement to resist calculated shear and torsion. However, the spacing of the

stirrups was varied for the test beam specimens to study the effect of transverse reinforcement on

the ultimate strength of the test beam.


Specimen: Five groups of specimens, each group contains three 150*150*1300𝑚𝑚 long

rectangular reinforced concrete; all the samples have effective span of 1200𝑚𝑚 designed, to be

constructed and tested in the Standard Organization of Nigeria (SON) Engineering laboratory,

PRODA junction Enugu-Abakaliki Express Road, Emene Industrial Layout Emene, Enugu State,

under the combined actions of bending, torsion and shear forces simultaneously applied by

loading the test beams at an eccentric distance from the beams principal axis through a system of

Universal Testing machine. A group of control beams are design to fail in flexure, shear and

torsion. The cross-section of the beams is 150 ∗ 150𝑚𝑚. The groups of reinforced concrete

beams consist of one with control beam and four beams designed with varied ratio of both

longitudinal and transverse reinforcements (see table 3.1).

Details of assemblies: The arrangement of test beam specimens is detailed in figures 3.1 and

3.2. The test beam specimens have the same cross-section of 150 ∗ 150 ∗ 1300𝑚𝑚 with beams

BC2, BC3, BC4 and BC5 reinforced transversely and longitudinally, while BC1 is a plain

concrete beam without both transverse and longitudinal reinforcements. The spacing of

transverse reinforcement and number of longitudinal steels of Beams BC2, BC3, BC4 and BC5

are varied (see table 3.1).

Experimental set-up: The test beams are supported over an effective span of 1300𝑚𝑚 and

subjected to an eccentric loading through a system of Universal Testing machine with a point

load at the middle of the beam (figures 3.1 and 3.2). A series of tests are to be carried out with

plain and reinforced concrete beam specimens loaded to a combined action of twisting moment,

flexure and shearing forces, simultaneously. The reinforcement ratios are varied both in the

transverse and longitudinal directions, in order to study its effect on the resistance of the test

beams to all applied load. The differences in the reinforcements can be seen in Table 3.1. The

test is a destructive one and the equipment to be used includes; Universal Testing machine and

compressive testing machine at Standard Organization of Nigeria (SON) Engineering laboratory

Enugu State.The actual loading of the test beam will be done by a system of Universal Testing

machine at an eccentric distance away from beams axis. The load is applied at an increased rate

of 20 percent. The ultimate torsional moment is to be determined. The load at which there is a

crack on the test beam is determined and the load that leads to failure of the test beam is to be


read from the digital measuring gauge on the machine. The beam is considered to have collapsed

when it could resist no more loads. Two methods are used to detect the cracking torsional

moment of the test beam; the first method is the visual observation of the test beam surfaces of

every load step. As the cracks began at the middle of the test beam, they are easily observable.

The second method is the observation of the sound of the cracking. The ultimate torsional

moment is taken as the maximum moment reached throughout the test. At every load step, the

cracks are marked. After the tests, these marks were drawn on paper so that the angle between

the cracks and the axis of the beam could be measured.

Table 3.1Summary of Reinforced Concrete Beams Tested (all the samples are 1500mm long).

Beam notation longitudinal bars (mm) Stirrups bars (mm) Spacing of links (mm)

BC1 Not reinforced

BC2 4T10 R8 200

BC3 4T10 R8 100

BC4 6T10 R10 200

BC5 6T10 R10 100

Figure 3.3 Loading and geometry details of reinforced concrete control beam

X 1200

1500

150 150

P

X e

150

P

150

X-X section


Table 3.2: Ultimate Torsional moment predicted by codes.

Torsional resistance Euro Codec 2 ACI 318-11 BS 8110

Longitudinal

reinforcement 𝑇𝑅𝑑1 = 2𝐴𝑘

𝐴𝑠𝑙𝑓𝑦𝑙

𝑈𝑘

𝑡𝑎𝑛𝜃 𝑇𝑈1 =

2𝐴0𝐴𝑙𝑓𝑦𝑙

𝑃ℎ𝑡𝑎𝑛𝜃

𝑇𝑈𝑆1 =𝐴𝑙(0.8𝑥1𝑦1)(0.87𝑓𝑦𝑙)

(𝑥1+𝑦1)

Transverse

reinforcement

𝑇𝑅𝑑2 = 2𝐴𝑘(𝐴𝑠𝑤

𝑆) 𝑓𝑦𝑡𝐶𝑜𝑡𝜃 𝑇𝑈2 = 2𝐴0(

𝐴𝑡

𝑆) 𝑓𝑦𝑡𝐶𝑜𝑡𝜃

𝑇𝑈𝑆2 = 𝐴𝑠𝑣

𝑆(0.8𝑥1𝑦1)(0.87𝑓𝑦𝑡)

Concrete section 𝑇𝑅𝑑3

= 2𝑣𝑓𝑐𝑑𝐴𝑘𝑡𝑒𝑓𝑆𝑖𝑛𝜃𝐶𝑜𝑠𝜃

𝑣 = 0.6(1 − 𝑓𝑐𝑘/250)

Taken as:

𝑇𝑈3

= ∅𝜆√𝑓𝑐′

𝐴𝑐𝑝2

𝑃𝑐𝑝

Taken as:

𝑇𝑈𝑠3 =1

2𝜏𝑚𝑎𝑥ℎ𝑚𝑖𝑛

2 (ℎ𝑚𝑎𝑥 −ℎ𝑚𝑖𝑛

3)

𝜏𝑚𝑎𝑥 ≈ 0.2√𝑓𝑐𝑢

The minimum of this value from each code should be adopted as the ultimate Torsional

resistance.

Table 3.3: Torsional moments predicted by different theories

Elasticity Plasticity Skew-bending

Torsional moment 𝑇𝑐𝑟 = 𝛼𝑏2𝐷𝜏𝑚𝑎𝑥 𝑇 = 0.5𝜏𝑡ℎ𝑚𝑖𝑛2 (ℎ𝑚𝑎𝑥 − ℎ𝑚𝑖𝑛/3)

𝑇𝑈𝑆 = 𝑇𝑐 + 𝛼𝑡

𝑥1𝑦1 𝐴𝑡 𝑓𝑡𝑦

𝑆

Presentation of Results : To validate the experimental results obtained the three standard design

codes under study; BS 8110, Euro Code 2 and ACI 318 will be evaluated for verification of the

results. In order to provide the accuracy and efficiency of the experimental results, the results

obtained will be compared with those from design codes and theoretical predictions (elastic,

plastic and skew-bending theories).


Table 4.1 the characteristics of tested beams

Sample test

Beam

Cross-

section

Length

(𝑚𝑚)

𝑓𝑐𝑢

(𝑀𝑝𝑎)

Dia. of

stirrup(𝑚𝑚)

Dia. of

main-bar

(𝑚𝑚)

𝑓𝑦𝑡

(𝑀𝑝𝑎)

𝑓𝑦𝑙

(𝑀𝑝𝑎)

S

(𝑚𝑚)

𝐴𝑡𝑆⁄

(𝑚𝑚2

/𝑚𝑚)

𝐴𝐿

(𝑚𝑚2)

BC1 150*150 1300 30 - - - - - - -

BC2 150*150 1300 30 8 10 434 445 200 0.503 314

BC3 150*150 1300 30 8 10 434 445 100 1.006 314

BC4 150*150 1300 30 8 10 434 445 200 0.503 471

BC5 150*150 1300 30 8 10 434 445 100 1.006 471

Table 4.2: Experimental results obtained from the tested beam samples

Sample Eccentricity e (𝑚𝑚)

Load at failure (𝑘𝑁)

Cracking Angle 𝜃 (𝑖𝑛 𝑑𝑒𝑔𝑟𝑒𝑒)

BC1

BC1

BC1

65

10.690

10.760

10.480

AVE 10.637

BC2

BC2

BC2

65

40.200

39.835

43.020

43

44

40

AVE 41.018 42.333

BC3

BC3

BC3

65

50.870

49.785

49.050

37

38

37

AVE 49.902 37.333

BC4

BC4

BC4

65

43.290

44.285

43.590

37

39

37

AVE 43.722 37.667

BC5

BC5

BC5

65

51.335

50.255

51.500

39

40

39

AVE 51.030 39.333


Table 4.3A: Torsional strength/ultimate torsional moment of beams according to standard codes’

prediction and experimental test beams.

Beam

sample

BS 8110

(𝜃 = 450)

(𝑘𝑁𝑚)

Euro Code 2

(𝜃 = 450)

(𝑘𝑁𝑚)

ACI 318-11

(𝜃 = 450)

(𝑘𝑁𝑚)

Experimental

Test Result

(𝑘𝑁𝑚)

Element failed/

mode of failure

𝑻𝑩𝑺,𝑨𝑳 𝑻𝑩𝑺,𝑨𝒕 𝑻𝑩𝑺,𝒄𝒐𝒏𝒄 𝑻𝑬𝑪,𝑨𝑳 𝑻𝑬𝑪,𝑨𝒕 𝑻𝑬𝑪,𝒄𝒐𝒏𝒄 𝑻𝑨𝑪𝑰,𝑨𝑳 𝑻𝑨𝑪𝑰,𝑨𝒕 𝑻𝑨𝑪𝑰,𝒄𝒐𝒏𝒄 𝑻𝑬𝑹

BS EC2 ACI

BC1 - - 1.234 - - 4.2223 - - 1.3121 0.6914 - - -

BC2 5.0265 1.6104 1.234 6.5647 6.3096 4.2223 5.6097 3.652 1.3121 2.6662 T,C - C

BC3 7.5398 3.2208 1.234 6.5647 12.6193 4.2223 5.6097 7.3047 1.3121 3.2436 T,C - C

BC4 5.0265 1.6104 1.234 9.8470 6.3096 4.2223 8.4151 3.652 1.3121 2.8419 T,C - C

BC5 7.5398 3.2208 1.234 9.8470 12.6193 4.2223 8.4151 7.3047 1.3121 3.31695 T,C - C

Table 4.3B; Comparison of the ratio of ultimate torsional strength between standards’ prediction

and experimental result

Test

Beam 𝑻𝑩𝑺,𝑨𝑳

𝑻𝑬𝑹

𝑻𝑩𝑺,𝑨𝒕

𝑻𝑬𝑹

𝑻𝑩𝑺,𝒄𝒐𝒏𝒄

𝑻𝑬𝑹

𝑻𝑬𝑪,𝑨𝑳

𝑻𝑬𝑹

𝑻𝑬𝑪,𝑨𝒕

𝑻𝑬𝑹

𝑻𝑬𝑪,𝒄𝒐𝒏𝒄

𝑻𝑬𝑹

𝑻𝑨𝑪𝑰,𝑨𝑳

𝑻𝑬𝑹

𝑻𝑨𝑪𝑰,𝑨𝒕

𝑻𝑬𝑹

𝑻𝑨𝑪𝑰,𝒄𝒐𝒏𝒄

𝑻𝑬𝑹

BC1 - - 1.7848 - - 6.1042 - - 1.8977

BC2 1.8853 0.6040 0.4628 2.4622 2.3665 1.5836 2.1040 1.3697 0.4921

BC3 2.3245 0.9930 0.3804 2.0239 3.8905 1.3017 1.7295 2.2520 0.4045

BC4 1.7687 0.5667 0.4342 3.4649 2.2202 1.4857 2.9611 1.2851 0.4617

BC5 2.2731 0.9710 0.3720 2.9687 3.8045 1.2729 2.5407 2.2055 0.3962

Torsional shear strength of reinforced concrete beam according to standards’ code prediction and

experimental test result (𝑇𝐸𝑅). The torsional resistance provided by the longitudinal

reinforcement (𝑇𝐴𝐿), shear/transverse reinforcement(𝑇𝐴𝑡), and concrete cross-section ( 𝑇𝑐𝑜𝑛𝑐). L

–longitudinal reinforcement, T-transverse/vertical reinforcement, C-concrete section.

NB: an element is considered to have failed when the experimental value from the tested beam is

greater than that predicted by standard codes for the three variables under consideration.


Table 4.4A: Torsional strength/ultimate torsional moment of beams according to standard codes’

prediction and experimental test beams.

Beam

sample

Euro Code 2

(𝜃 = 220)

(𝑘𝑁𝑚)

ACI 318-11

Experimental

Test Result

(𝑘𝑁𝑚)

Element failed

ACI (1) (𝜃 = 300)

(𝑘𝑁𝑚)

ACI (2) (𝜃 = 600)

(𝐾𝑁𝑚)

𝑇𝐸𝐶,𝐴𝐿 𝑇𝐸𝐶,𝐴𝑡 𝑇𝐸𝐶,𝑐𝑜𝑛𝑐 𝑇𝐴𝐶𝐼,𝐴𝐿 𝑇𝐴𝐶𝐼,𝐴𝑡 𝑇𝐴𝐶𝐼,𝑐𝑜𝑛𝑐 𝑇𝐴𝐶𝐼,𝐴𝐿 𝑇𝐴𝐶𝐼,𝐴𝑡 𝑇𝐴𝐶𝐼,𝑐𝑜𝑛𝑐 𝑇𝐸𝑅

EC2 ACI1 ACI2

BC1 - - 2.933 - - 1.3121 - - 1.3121 0.6914 - - -

BC2 2.6521 15.774 2.933 3.2390 6.3256 1.3121 9.7166 2.108 1.3121 2.6662 L C T,C

BC3 2.6521 31.5482 2.933 3.2390 12.652 1.3121 9.7166 4.217 1.3121 3.2436 L,C L,C C

BC4 3.9782 15.774 2.933 4.8589 6.3256 1.3121 14.575 2.108 1.3121 2.8419 - C T,C

BC5 3.9782 31.5482 2.933 4.8589 12.652 1.3121 14.575 4.217 1.3121 3.31695 C C C

Table 4.4B; Comparison of the ratio of ultimate torsional strength between standards’ prediction

and experimental result

Test

Beam 𝑇𝐸𝐶,𝐴𝐿

𝑇𝐸𝑅

𝑇𝐸𝐶,𝐴𝑡

𝑇𝐸𝑅

𝑇𝐸𝐶,𝑐𝑜𝑛𝑐

𝑇𝐸𝑅

𝑇𝐴𝐶𝐼1,𝐴𝐿

𝑇𝐸𝑅

𝑇𝐴𝐶𝐼1,𝐴𝑡

𝑇𝐸𝑅

𝑇𝐴𝐶𝐼1,𝑐𝑜𝑛𝑐

𝑇𝐸𝑅

𝑇𝐴𝐶𝐼2,𝐴𝐿

𝑇𝐸𝑅

𝑇𝐴𝐶𝐼2,𝐴𝑡

𝑇𝐸𝑅

𝑇𝐴𝐶𝐼2,𝑐𝑜𝑛𝑐

𝑇𝐸𝑅

BC1 - - 4.2421 - - 1.8977 - - 1.8977

BC2 0.9947 5.9163 1.10 1.2148 2.3725 0.4921 3.644 0.7906 0.4921

BC3 0.8176 9.7263 0.9042 0.9986 3.9006 0.4617 2.9956 1.300 0.4045

BC4 1.3998 5.5505 1.032 1.7097 2.2258 0.4617 5.1286 0.7418 0.4617

BC5 1.1994 9.5112 0.8842 1.4649 3.8143 0.3956 4.3941 1.2713 0.3956

Continuation of Tables 4.3 and 4.4 Experimental verification of ultimate capacity

Experimental verification of ultimate capacity

Test

Beam 𝑇𝐸𝑅

(𝑘𝑁𝑚)

𝑀𝐸𝑅

(𝑘𝑁𝑚)

𝑉𝐸𝑅

(𝑘𝑁)

BC1 0.6914 3.3271 5.7720

BC2 2.6662 12.4416 20.9628

BC3 3.2436 15.2066 25.4045

BC4 2.8419 13.2526 22.3145

BC5 3.31695 15.4451 25.9686


The measured and calculated values of ultimate torsional moment (KN-m) obtained from

experiment and predicted by different theories on torsion.

Table 4.5 values of theoretical predictions and experimental measures of ultimate torsional

moment of prismatic rectangular cross-section.

Beam sample

Elasticity

(𝐾𝑁𝑚)

Plasticity

(𝐾𝑁𝑚)

Skew-

Bending (𝐾𝑁𝑚)

Cowan’s

equation (𝐾𝑁𝑚)

Experimental

Result (𝐾𝑁𝑚)

BC1 0.769 1.2324 1.235 0.769 0.6914

BC2 0.769 1.2324 2.3878 4.4711 2.6662

BC3 0.769 1.2324 2.8652 8.1732 3.2436

BC4 0.769 1.2324 2.6469 4.4711 2.8419

BC5 0.769 1.2324 3.2317 8.1732 3.3169

0

1

2

3

4

5

6

7

8

9

0 2 4 6

ult

imat

e t

ors

ion

al m

om

en

t (k

Nm

)

Elasticity

Plasticity

Skew-bending

Cowan's Equation

ExperimentalResult

Figure 4.1 scatter graph of ultimate torsional moment obtained from classical theoretical predictions

and experimental measures of ultimate torsional moment


Figure 4.2. Crack pattern for BC1 beam

specimen

Figure 4.3.crack pattern for BC2 beam specimen

Figure 4.4. Crack pattern for BC3 beam

specimen

Figure 4.5.Crack pattern for BC4 beam

specimen

Figure 4.6. Crack pattern for BC5 beam specimen


Table 4.6a calculations from simply supported beam subjected to combined actions of torsion,

shear and bending stresses. (example 1)

𝐴𝑠𝑟𝑒𝑞,𝑏𝑒𝑛𝑑𝑖𝑛𝑔

(𝑚𝑚2)

𝐴𝑡 𝑠,𝑠ℎ𝑒𝑎𝑟

⁄

(𝑚𝑚2/𝑚𝑚)

𝐴𝑠𝑟𝑒𝑞,𝑡𝑜𝑟𝑠𝑖𝑜𝑛

(𝑚𝑚2)

𝐴𝑡 𝑠,𝑡𝑜𝑟𝑠𝑖𝑜𝑛

⁄


Area related

to shear

flow

(𝑚𝑚2)

Combined

transverse:

shear +

torsion


Combined

longitudinal:

bending +

torsion

(𝑚𝑚2)

BS

8110

156 0.253 85 0.7845 8000 1.0375 241

ACI

318

273 0.272 90 0.442 8774 0.714 363

Euro

code 2

144 0.496 57 0.25 0.746 1.10 201

Table 4.6b calculations from precast footbridge slab (example 2)

𝐴𝑠𝑟𝑒𝑞,𝑏𝑒𝑛𝑑𝑖𝑛𝑔

(𝑚𝑚2)

𝐴𝑡 𝑠,𝑠ℎ𝑒𝑎𝑟

⁄

(𝑚𝑚2/

𝑚𝑚)

𝐴𝑠𝑟𝑒𝑞,𝑡𝑜𝑟𝑠𝑖𝑜𝑛

(𝑚𝑚2)

𝐴𝑡 𝑠,𝑡𝑜𝑟𝑠𝑖𝑜𝑛

⁄


Area related

to shear

flow

(𝑚𝑚2)

Combined

transverse:

shear +

torsion

(𝑚𝑚2/

𝑚𝑚)

Combined

longitudinal:

bending +

torsion

(𝑚𝑚2)

BS

8110

929 0.63 401 0.78 51120 1.41 1330

ACI

318

571 0.6934 547 0.8861 88838.53 1.5795 1118

Euro

code 2

751 0.600 385 0.5 59166 1.10 1136

Summary of Result: In validating the ultimate torsional strengths observed from the experiment

on combined actions of torsion, bending and shear on 15 reinforced and non-reinforced concrete

beam samples, current international design codes’ predictions on the ultimate torsional strength

are used to calculate the corresponding values and compared with the experimentally observed

ones and values tabulated for comparison. Also, the ultimate torsional strengths predicted by

different classical theories on torsion: elastic, plastic, cowan and skew-bending theories are


calculated using their respective equations and values tabulated for comparison with the

experimental observed values. Another result presented was on two design examples of

structures subjected to combined actions of torsion, shear and bending; using the three design

codes under this study to determine the ultimate torsional strengths, areas of longitudinal and

transverse reinforcements required to resist the applied loads and the values tabulated for

comparison. The next result is on the use of Ms-excel to predict the ultimate torsional strengths

proposed by the three codes under consideration, when the areas of both longitudinal and

transverse reinforcements are given and their values are tabulated for comparison. The crack

patterns are also displayed.

Discussion: The ultimate torsional strength for rectangular cross-section of 15 reinforced

concrete beams subjected to combined actions of torsion, shear and bending were determined

using experiment, BS 8110, ACI 318 and Euro code 2 design code equations. Also the classical

theories of elastic, plastic, Cowan and skew-bending equations on ultimate torsional strength

were determined. All the calculated values from the codes and theories are compared with the

experimental observed values. Table 4.1 present the characteristics/ properties of the tested

reinforced and non-reinforced concrete beams. These characteristics are used also in the design.

Table 4.2 presents the loads at failure and cracking angles of the 15 reinforced and non-

reinforced concrete beams specimens subjected to combined actions of torsion, shear and

bending, applied at an eccentric distance at the mid-span of the beam specimens. From this table,

the ultimate torsional strength, bending moment and shear forces are determined. It can also be

seen that angle of crack is not (𝜃 = 450) , as generally taken, but varies. Table 4.3A and B

presents for comparison the ultimate torsional strength of tested beams and observed from

experiment and values calculated from the equations predicted by BS 8110, ACI 318 and Euro

code 2 for the general accepted angle of cracking of (𝜃 = 450). The ultimate torsional strength

predicted by the codes under this study with respect to the torsional resistance provided by the

transverse reinforcement (𝑇𝐴𝑡), longitudinal reinforcement 𝑇𝐴𝐿 , and the concrete cross-section

( 𝑇𝑐𝑜𝑛𝑐), the mode of failure / element failed are also presented. The B part of the table presents

the ratios of the calculated values to the experimental results. From these tables, it can be

indicated that an element has failed when the experimental result is greater than any of those

predicted by the codes for longitudinal reinforcement L, transverse reinforcement T, or concrete


section C. It can be seen from the table that some values predicted by the codes matches those

observed from experimental result. However, ultimate torsional strengths calculated for the

longitudinal reinforcement resistance from the three codes are higher than that observed from the

experiment. This indicates that the beams specimens failed before the longitudinal reinforcement

yield strength was reached. It can also be seen that the ultimate torsional strengths predicted by

the codes for Transverse reinforcement resistance are statistically the same to that obtained

experimentally except in Euro code 2 that over estimated its values when compared with

experimental results and the other two codes. The concrete section failed in the prediction by BS

8110 and ACI 318 codes while Euro code 2 predicted a higher value than that observed from

experiment. In other words, values calculated from Euro code 2 for the torsional resistance for

longitudinal and transverse reinforcements together with that of concrete section are highest,

followed by ACI 318, then BS 8110. Table 4.4 A and B presents for comparison, the ultimate

torsional strength predicted by the ACI 318 and Euro code 2, using the specified values of 𝜃

apart from 𝜃 = 450 . The B part of the table presents the ratios of the calculated values to the

experimental results. From the table, it can be observed that the ultimate torsional strength

predicted by Euro code 2 for longitudinal reinforcement resistance to torsion is statistically the

same with the experimental results while there was an over estimation by the torsional resistance

for transverse reinforcement. When the Euro code 2 values for 𝜃 = 450 𝑎𝑛𝑑 220 are compared,

it can be observed that the code was more conservative and consistence for 𝜃 = 220 for torsional

resistance provided by longitudinal reinforcement and concrete section while that for transverse

reinforcement was over estimated and non-conservative when compared with observed values

from experiment. This should be used where a higher torsional stresses are expected. The ACI

318 code using 𝜃 = 300 for torsional resistance provided by longitudinal reinforcement was

more conservative and consistent than those values gotten when 𝜃 = 450 while that provided by

the concrete section remain the same. The values provided by transverse reinforcement are over

estimated for 𝜃 = 300 than that of 𝜃 = 450. However, the ACI 318 code also gave a limit of

high value of 𝜃 = 600; the torsional resistance provided by longitudinal reinforcement is over

estimated when compared with the values gotten when 𝜃 = 450 𝑎𝑛𝑑 300. 𝜃 𝑣𝑎𝑙𝑢𝑒 for BS 8110

cannot be varied because the 𝜃 = 450 was built into the formulated equation for predicting

ultimate torsional moments. There is a continuation of tables 4.3 and 4.4 for Experimental


verification of ultimate capacity of the tested beams and its interaction with torsion plus shear,

and torsion plus bending. Table 4.5 presents for comparison the experimental observed and

calculated ultimate torsional moments predicted by different classical theories; elastic, plastic,

Cowan and Skew-bending. The table indicated that values predicted by elastic and plastic

theories are the same through-out the beam specimens because of constant dimension of the

beams (fixed X and Y dimensions) respectively. The values predicted by Cowan’s equation are

over estimated by almost twice and are not conservative and consistent. The values observed

from experimental result are slightly higher than those predicted by skew-bending theory. In

other words, it is only skew-bending theory values that come closer to that obtained

experimentally. The values predicted by elastic and plastic theories are under estimated and over

conservative when compared with experimental result. This indicates that the failure of

reinforced concrete beam subjected to combined actions of torsion, shear and bending cannot be

attributed to the two theories. Tables 4.6 a and b presents for comparison the calculations from

(a) example1( the design of simply supported beam loaded eccentrically at a distance away from

the beams principal axis, but at the mid-span and (b) for precast footbridge slab (example 2). A

detailed calculation was done on a footbridge beam to determine the areas of reinforcement

required by BS 8110, ACI 318 and Euro code 2 to sustain torsional force.

From the table a, Euro code 2 gave values that are more conservative and consistence in

providing for the areas required for torsional resistance for both longitudinal and transverse

reinforcements. From table b, it can be observed that areas of longitudinal reinforcement

required for torsion is more conservative by the Euro code 2 predictions while ACI 318 code

value is higher. The area of transverse reinforcement required for torsion only is more

conservative in Euro code 2 than other two codes. In the combined action of torsion and shear,

Euro code 2 values are more conservative while ACI 318 has the highest value. However, for

combined torsion and bending, ACI 318 becomes more conservative while BS 8110 is the

highest. It is clear that using Euro code 2, the design will be more economical and conservative

than using ACI 318 and BS 8110. Figure 4.1 shows a graph of the ultimate torsional strengths

predicted by classical theories and that of the experimental result. From the figure, it can be

observed that Cowan’s predictions are over estimated while elastic and plastic theories gave the

over conservative values. The skew-bending theory values are the closest to the experimental


result than any other theory. Figures 4.2 to 4.6 show a typical crack pattern for the various beam

samples from groups 1 to 5. The crack pattern in figure 4.2 indicated that the beam failed with

high cracking angle indicating no reinforcement in both longitudinal and transverse directions.

Figure 4.3 shows a crack pattern that has no provision for torsional reinforcement that is it

yielded as a result of bending stresses. While figures 4.4 to 4.6 show are crack pattern with

helical cracks inclined at an angle at an average of 𝜃 = 370.

Conclusions: The experimental investigation on the ultimate torsional strength of rectangular

cross-section reinforced concrete beams with plain, transverse and longitudinal reinforcement

subjected to combined actions of bending, shear and torsional stresses by using simple test

method was carried out. In this study, combined loading was achieved by applying an eccentric

load to the simply supported beams at the mid-span using universal testing machine. The

parameters in this study are eccentricity of the load which represents the magnitude of torsional

moment, variations in the amount of longitudinal and transverse reinforcements and given

compressive strength of concrete. The experimental results indicated that all beam specimens

failed by yielding of reinforcements or concrete crushing, or both after diagonal cracking which

is induced by the effect of combined actions of bending, shear and torsion. It can be concluded

that increase in the magnitude of ultimate torsional strength were as a result of increase in the

amount of longitudinal and transverse reinforcements. The comparison of the ultimate torsional

strength from the experimental results observed with the calculated values predicted by equations

of BS 8110, ACI 318 and Euro code 2 indicated that the codes studied in this work gave quite

consistent and conservative values in many cases while in few cases, uneconomic and non-

conservative results are obtained. This can be attributed to different values of 𝜃 specified by the

codes. Hence, it can be concluded that the value of angle of crack 𝜃 is not supposed to be fixed

as specified by the codes (𝜃 = 450), but a value that will ensure conservative and consistency in

providing the areas of reinforcement for both longitudinal and transverse direction. These values

should be between 300 𝑡𝑜 430 while lower values of 𝜃 = 220 should be used for highly stressed

reinforced concrete beams. 450 Value for 𝜃 was accepted generally based on the symmetric

arrangement of transverse reinforcements throughout the beam length. From the comparison of

ultimate torsional strengths predicted by different classical theories and experimental result, it

can be concluded that the classical theories were able to predict values that matches that of


experimental result, except in Cowan that over estimated its values while elastic theory under

estimated its values and over economical/conservative. Therefore failure of reinforced concrete

beams subjected to combined actions of torsion, shear and bending cannot be attributed to elastic

theory. The plastic theory should also not be used as it gave a value smaller than that observed

experimentally.

Recommendations: This study recommends that the value of angle of cracks should not be fixed

at 𝜃 = 450, but a value that will ensure a better economic and stable design that is neither too

conservative nor over estimation. This 𝜃 value should be between 300 𝑡𝑜 430 while a lower

value may be used where applied stresses are expected to be high.

It also recommends that inclusion of torsional design for members that are subjected to

its effects, in design as increase in reinforcement due to torsional stresses in both longitudinal

and transverse directs of the beam increases the torsional strength and capacity of such beam.The

assumption taken in formulating the codes equations should be revisited, especially in the

improvement of strength capacity of concrete strut to resist torsion.

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