Combined Torsion and Bending in Reinforced and Pre Stressed Concrete Beams Using Simplified Method...

402 ACI Structural Journal/July-August 2007

ACI Structural Journal, V. 104, No. 4, July-August 2007.MS No. S-2006-029.R1 received September 4, 2006, and reviewed under Institute

publication policies. Copyright © 2007, American Concrete Institute. All rights reserved,including the making of copies unless permission is obtained from the copyrightproprietors. Pertinent discussion including author’s closure, if any, will be published in theMay-June 2008 ACI Structural Journal if the discussion is received by January 1, 2008.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

This paper presents a simplified model for the design and analysisof reinforced and partially- and fully-prestressed concrete beamssubjected to combined torsional and bending moments. This modelis an extension of the existing simplified method for combinedstress-resultants (SMCS) model. The interaction between torsionand flexure is achieved by superposing the steel required for thetwo moments. The observed ultimate loads of 111 beams arecompared with the calculations of the proposed model and verygood agreement is obtained. This includes comparing interactiondiagrams and the effects of concrete strength, stirrups spacing, andT-beam flange width on the ultimate capacity. The calculations bythe ACI code equations are also evaluated and shown to givesatisfactory and, in some cases, overly conservative, results. Thesimplicity of the proposed model is illustrated using a design andan analysis example.

Keywords: beams; bending; prestressed concrete; reinforced concrete;shear; stress; torsion.

INTRODUCTIONMany structural elements such as spandrel beams, eccen-

trically loaded bridge girders, and beams curved in plan aresubjected to the effects of combined actions. Torsional andflexural moments (T and M, respectively) can be dominant inthe design of such members. Only longitudinal steel is requiredto resist the flexural moment, whereas both transverse andlongitudinal steel are required to resist the torsional moment.

Designing for the flexural moment is simple, and theflexure theory based on the assumption that plane sectionsremain plane has been used with satisfactory results. Thetreatment of pure torsion and torsion combined with otherstress resultants in design codes,1,2 however, is not unified.The literature reports advanced models for combinedtorsion.3-7 These models, however, require the use ofcomputers and are not readily suitable for implementation indesign codes. There is a lack of a simple model for the designand analysis of sections subjected to various combinations ofthe six possible stress resultants on a beam cross section.

The simplified method for combined stress-resultants(SMCS) is a simplification of the results by the modifiedcompression field theory (MCFT).8 The SMCS model wasoriginally developed for the case of thin reinforced concretemembrane elements subjected to in-plane shearing stresses,9

and was found to give very good results. Its application wasextended to apply to membrane elements subjected to in-plane shearing and normal stresses,10 to reinforced concretebeams subjected to pure torsion11 and to combined shear,bending moment, and axial loads.12 The main features of thismodel are its simplicity and generality, where it was appliedto both membrane elements and beam members undervarious loadings without loss of its simplicity. The generality

feature is not available in many other simple noniterativemethods for calculation of the torsional strength.1,13

RESEARCH SIGNIFICANCEThere is a lack of a simple model for the analysis and

design of membrane elements and beam elements subjectedto various combinations of stress resultants. This paperextends the application of the SMCS model to the case ofbeams subjected to combined bending and torsion. Thismodel is applicable to members with adequate amounts oflongitudinal and transverse reinforcement.

SMCS FOR PURE SHEARIN MEMBRANE ELEMENTS

This section gives a summary of the basic SMCS model.More details can be found elsewhere.9

Figure 1 shows a reinforced concrete membrane elementadequately reinforced in the x and y directions and subjectedto in-plane shearing stresses. The mechanical reinforcementratios in the x and y directions ωx and ωy are defined as

(1)ωxρx fyx

fc′-----------=

Title no. 104-S38

Combined Torsion and Bending in Reinforced and Prestressed Concrete Beams Using Simplified Methodfor Combined Stress-Resultantsby Khaldoun N. Rahal

Fig. 1—Reinforced concrete membrane element subjected toin-plane shearing stresses.

403ACI Structural Journal/July-August 2007

(2)

The ultimate strength of this element depends mainly onthe amount and strength of reinforcement in the x and ydirections and on the concrete strength. The equations of theMCFT8 were used to calculate the ultimate strength and thecorresponding strains of the elements for various cases.Figure 2 shows the increase in the normalized shear strengthv/fc′ as the reinforcement level ωy in the y direction isincreased while maintaining ωx constant. For example, a20 MPa (2900 psi) concrete panel reinforced with ρx fyx = ρy fyy= 2 MPa (290 psi) is analyzed, and the ultimate strength isfound to be 2 MPa (290 psi) with both x and y reinforcementyielding before concrete crushing. These results correspondto ωx = ωy = 0.1 and v/fc′ = 0.1, and plot as Point A in Fig. 2.Analyzing a similar panel but with ρy fyy = 10 MPa (1450 psi)results in an ultimate shear strength of 3.36 MPa (487 psi),with only the x reinforcement yielding before concretecrushing. These results correspond to ωx = 0.1, ωy = 0.4 andv/fc′ = 0.168, and plot as Point B in Fig. 2.

Repeating the analysis for various amounts ωy results inthe lower curve in Fig. 2. Nearly the same curve can beobtained if the analysis was based on ωx = 0.1 obtained bysetting fc′ = 35 MPa (5080 psi) and ρx fyx = 3.5 MPa (508 psi).

Increasing ρx fyx to 4.0 MPa (580 psi) to give ωx = 0.2 andrepeating the analyses at various values of ωy results in theupper curve in Fig. 2. For reinforcement levels below thosecorresponding to points marked C and D, the y reinforcementyields before crushing of the concrete, and the element isunder-reinforced. For larger reinforcement ratios, theconcrete crushes before yielding of the y reinforcement(partially or fully over-reinforced element) and the relativeincrease in strength is significantly lower.

The analysis was repeated for various values of ωx and ωyand the results are plotted in Fig. 3. Reinforcement levelscorresponding to those marked C and D in Fig. 2 are joinedtogether to form a balanced curve. Figure 3 shows two

ωyρy fyy

fc′-----------=

balanced curves, one corresponding to levels of ωx beyondwhich x reinforcement does not yield, and the other corre-sponding to levels of ωy beyond which y reinforcement doesnot yield. Due to symmetry, ωx and ωy can be interchanged.

The two balanced curves split the graph into four regionscorresponding to four modes of failures of the membranes.The first region is where both x and y reinforcement yieldbefore concrete crushing (Mode 1: fully under-reinforcedsection), the second region is where only x reinforcementyields before concrete crushing (Mode 2: partially under-reinforced section), the third region is where only the yreinforcement yields before concrete crushing (Mode 3:partially under-reinforced section), and the fourth region iswhere concrete crushing takes place before any yielding inthe reinforcement (Mode 4: fully over-reinforced section).Hence, Fig. 3 gives not only the maximum shear stress butalso the mode of failure at ultimate conditions.

Part of the behavior summarized in Fig. 2 and 3 can beexplained by studying the equations that govern the equilibriumof the membrane element shown in Fig. 1

σx = f2cos2θ + f1sin2θ + ρx fsx (3)

σy = f2sin2θ + f1cos2θ + ρy fsy (4)

v = (–f2 + f1)sinθcosθ (5)

In under-reinforced elements, both x and y reinforcementyield ( fsx = fyx and fsy = fyy), and the ability of the diagonalcracks to transmit the tensile stresses drops to zero ( f1 = 0).For pure shear, the normal stresses σx and σy are equal tozero, and Eq. (3) to (5) can be rearranged to give the ultimateshear stress of under-reinforced elements and the corre-sponding angle θ as follows

(6)

(7)

ν ρx fyxρy fyy= or v fc′⁄ ωxωy=;

θρy fyy

ρx fyx

--------------ωy

ωx

------= =

ACI member Khaldoun N. Rahal is a Professor in the Department of Civil Engineeringat Kuwait University, Kuwait City, Kuwait. He is a member of Joint ACI-ASCE Committee445, Shear and Torsion. He is Past President of the ACI Kuwait Chapter.

Fig. 2—Relationship between normalized shear strength v/fc′and mechanical reinforcement ratios.

Fig. 3—Shear strength curves for reinforced membraneelements.


Equation (6) is plotted in Fig. 2, and is shown to match withthe results of the MCFT up to Points C and D (that is, forMode 1, fully under-reinforced elements). It shows that forthese elements, the shear strength comes solely from thesteel contribution. For partially or fully over-reinforcedelements, there is a significant concrete contribution, whichis implicitly included in the total shear strength v.

It is noted that Eq. (6) is similar to the plastic solution forfully under-reinforced membranes presented by Braestrup.14

However, SMCS and the theory of plasticity are different inthree of the four regions in Fig. 3, and in the boundariesbetween these regions. A detailed comparison between theresults of the plastic theory and the SMCS for membraneelements subjected to in-plane shear stresses is given inReference 9 (closure to discussion).

Equal reinforcement in the x and y directions leads to thefollowing simplifications of Eq. (6)

v = ρx fyx = ρy fyy (8a)

(8b)

SMCS FOR TORSIONThe equations of the SMCS for torsion are based on the

hollow tube analogy, where the cross section subjected to atorque T is modeled as a hollow tube with constant thicknesstd (refer to Fig. 4). The torque causes a field of shearingstresses (nonuniform over td) that circulate around in thewalls of the tube. Similar to the use of the equivalentcompressive stress block in the theory of flexure, an equivalentfield of constant principal compressive stresses and shearflow q can be assumed over a thickness a0 of the tube. Thebasic relationship between T and q is given by

T = 2qA0 (9)

where A0 is the area enclosed by the shear flow path shownin Fig. 4. The shear flow is related to the shear stress v andthe equivalent thickness of the wall as follows

q = a0v (10)

The walls of the twisted beam (Fig. 4) are assumed to be thinmembrane elements similar to those shown in Fig. 1. Theirultimate shear strength can hence be obtained from Fig. 3.Consequently, the SMCS model can be applied to the case oftorsion if the torque is related to the shear strength v in the

vfc′----- ωx ωy= =

walls and if the reinforcement indexes (Eq. (1) and (2)) arerelated to the actual longitudinal and transverse reinforcementin the section.

Based on the results of a simplified model,13 the thicknessof the wall and the area and perimeter enclosed by the shearflow path can be taken as

(11)

A0 = 0.8Ac (12)

p0 = 0.9pc (13)

For normal-strength concrete where the concrete strength isbelow 50 MPa (7250 psi), the stress-strain relationship incompression can be represented by a parabola. If the peakcompressive strain equal to (1.5 × the strain at peak stress),the relationship between a0 and td can be taken as

a0 = 0.833td (14)

Substituting Eq. (10), (11), (12), and (14) into Eq. (9) givesthe following equation for the nominal torsional moment T

(15)

Equation (15) provides the relationship between thetorsional capacity of the cross section and the shear stresscapacity of the thin membrane walls.

The transverse steel ratio (taken as the y direction steel fora vertical wall) is calculated as

(16)

The total symmetrical longitudinal steel provides reinforcementfor a series of membrane elements of length p0 and thicknessa0. Hence, the longitudinal steel ratio is calculated as follows

ρx = (17)

Combining Eq. (1), (2), (11), (13), (14), (16), and (17) andaccounting for the prestressed reinforcement in the elementgives the following equations for the reinforcement indexesin the walls

(18)

(19)

Equations (18) and (19) apply to sections symmetricallyreinforced in the longitudinal direction.

td 0.5Ac

pc

-----=

T 0.67Ac

2

pc

--------v=

ρyAt

sa0

--------=

AL

p0a0

-----------

ωLAL fyL Aps fpy+

0.375Ac fc′--------------------------------------=

ωtAt fyt pc

0.42sAc fc′---------------------------=

Fig. 4—Hollow tube model for torsional strength.

ACI Structural Journal/July-August 2007 405

It is to be noted that the torsion equations are based on theouter dimensions of the cross sections, which implies that theconcrete outside the hoops does not spall at ultimate load. Ifspalling is expected due to a relatively large concrete clearcover, the terms pc and Ac in Eq. (15), (18), (19), and (21) canbe replaced with the ph and A0h, respectively.

Unsymmetrically reinforced sectionsFigure 5(a) shows an unsymmetrically reinforced section.

The membrane element in the top flange of the tube isweaker than that in the bottom flange, and its ultimate shearstrength is critical in calculating the ultimate torsionalcapacity. The additional strength of the stronger wall can notbe achieved, and the strength of the unsymmetrical sectioncan be accurately and conservatively taken as that of a sectionsymmetrically reinforced with the weaker reinforcement.15,16

Hence, the strength of the section shown in Fig. 5(a) is taken tobe the same as that shown in Fig. 5(b) where the strongerbottom steel is replaced with an amount equal to the weakertop steel.

FLEXURE BY SUPERPOSITIONOF REINFORCEMENT

Superposition of the longitudinal reinforcement requiredto resist M to that required to resist T is adopted to accountfor the interaction between the two moments. This is illustratedin the following procedures for the cases of design and analysis,and is verified in the following section.

Design procedure1. Design for M (say positive) using the flexure theory, and

calculate amount of tensile (bottom) steel.2. Calculate ν using Eq. (15).3. Select a reinforcement indexes (say, ωL) and obtain the

other index (ωt) using Fig. 3 (or using Eq. (6) if section isfully under-reinforced).

4. Calculate amounts of longitudinal and transverse steelfrom Eq. (18) and (19). Select stirrups size and spacing.Distribute longitudinal steel symmetrically to top and bottomflanges (and on sides if skin reinforcement is to be provided).

5. In the tension zone, combine (bottom) longitudinal steelfrom Steps 1 and 4 (to resist M and T, respectively).

6. In the compression zone, reduce the (top) longitudinalsteel (required to resist T) by the amount equivalent to thecompression force caused by bending, given approximately by

(20)

Step 6 is similar to the approach permitted in the ACI code1

(where jd = 0.9). General design requirements such asproviding a minimum of four longitudinal corner bars andlimiting the spacing of the transverse and longitudinal steelneed to be respected. The procedure is illustrated inAppendix A using a solved example.

Capacity calculationIf the cross section is not symmetrically reinforced or if a

bending moment is acting, either the top or the bottom flange(whichever is weaker in the longitudinal direction) can becritical in determining the beam strength. The flexuraltension flange typically has larger reinforcement, but isweakened by the flexural tensile force, while the flexural

MjdfyL

-----------

compression flange typically has smaller reinforcement, butis strengthened by the flexural compressive force.1,16 Thestrength in the longitudinal direction effective in resisting thetorsional moment is that from the actual reinforcement,modified by the flexural tensile or compressive force. Asshown in Fig. 5, the total amount of longitudinal reinforcementresisting torsion is twice the critical (modified) steel. Anyskin reinforcement that contributes to the resistance of thewall can be added to this longitudinal index. Accordingly,the longitudinal reinforcing index is taken as

(21)

where M is positive if it causes tension in the bottom of thecross section and negative otherwise, and steel includesnonprestressed and prestressed reinforcement, as well asskin reinforcement.

Capacity calculation procedure1. Select bending moment M at which co-existing

torsional moment is to be calculated.2. Calculate ωt based on Eq. (19) and ωL based on Eq. (21).3. Use Fig. 3 (or, if the section is under-reinforced, Eq. (6))

to obtain v/f ′c.4. Calculate T using Eq. (15).The procedure is illustrated in Appendix B using a

solved example.

ACI PROVISIONSThe basic ACI1 equilibrium equation that relates the

torsional strength to the amount of transverse reinforcementand is based on the hollow tube model

(22)

ACI permits the area enclosed by the shear flow A0 to be takenas 0.85A0h. A similar equilibrium equation relates the torsionalstrength to the amount of longitudinal reinforcement

(23)

ωL

2M jd⁄ 2 As fy( )top∑+

0.375Ac fc′---------------------------------------------------------

2– M jd⁄ 2 As fy( )bot∑+

0.375Ac fc′------------------------------------------------------------

⎩⎪⎪⎨⎪⎪⎧

≤

T 2A0At fyt

s---------- θcot=

T 2A0AL fyL

ph

------------- θtan=

Fig. 5—Strength of unsymmetrically reinforced sections.


Equating T from Eq. (22) and (23) results in the ACI equationfor the required amount of longitudinal reinforcement fortorsional resistance

(24)

ACI requires that the angle of inclination θ of the diagonalstruts of the truss model shall not be smaller than 30 degreesnor larger than 60 degrees. ACI further suggests that theangle to be taken as 45 degrees for reinforced members and37.5 degrees for prestressed members. The Commentary, on theother hand, suggests that the angle can be obtained by analysis.

ALAt

s-----= ph

fyt

fyL

------ θ2cot

To avoid concrete crushing before yielding of the reinforce-ment and to limit the crack width at service load, the ACIcode requires that

(25)

If the cross section is hollow and its wall thickness t issmaller than A0h/ph, then the left-hand side term for torsionalshearing stress is replaced with T/(1.7A0ht).

The steel required to resist the torsional moment issuperimposed on the steel required to resist the flexural moment.In the compression zone, the longitudinal steel required fortorsion can be reduced using Eq. (20) (with jd = 0.9d) due tothe favorable effect of the flexural compression force.

EXPERIMENTAL VERIFICATIONA total of 111 beam specimens4,17-23 are used to evaluate

the ability of the proposed model and of the ACI codeprovisions to calculate the strength of reinforced andpartially prestressed beams subjected to combined torsionand bending. The specimens tested in these series includehollow and solid, nonprestressed and partially prestressed,symmetrically and nonsymmetrically reinforced, and rectan-gular and T sections. These test results studied the effects ofT to M ratio, nonsymmetry in longitudinal reinforcement,amount of transverse reinforcement, concrete compressivestrength, and size of T-beam flanges. Thirty-eight of thesebeams are selected for detailed comparisons, and the crosssection geometry and reinforcement are given in Fig. 6 andTable 1. A summary of the results of the 111 test specimensis given in Table 2. The results from the ACI equations arealso listed. One set of results is based on an angle θ of 45 degreesfor reinforced members and 37.5 degrees for partiallyprestressed members, and the other set is based on calculatingan angle between 30 and 60 degrees that satisfies the trussmodel Eq. (22) to (24) is also shown.

Symmetrically reinforced nonprestressed beamsGroup 2 of the specimens tested by McMullen and

Warwaruk17,18 contained five nonprestressed solid

Tph

1.7A0h2

------------------ 0.83 fc′≤

Fig. 6—Details of beams used in detailed evaluation of SMCS model.

Table 1—Properties of reinforcement in beams used in verification

Bar size Area, mm2 (in.2) fy, MPa (ksi) Used in

Lon

gitu

dina

l ste

el

No. 3 71 (0.11) 366 (53.0) Groups 1 to 4

No. 3 71 (0.11) 376 (54.5) TB

No. 3 71 (0.11) 406 (58.9) TBS

No. 3 71 (0.11) 552 (80.0) TBU

No. 4 129 (0.20) 433 (62.8) TBS

No. 4 129 (0.20) 393 (57.0) TBU

No. 5 200 (0.31) 337 (48.9) Groups 1 to 4

No. 5 200 (0.31) 363 (52.6) TB

No. 6 283 (0.44) 323 (46.8) Groups 1 to 4

No. 8 510 (0.79) 436 (63.2) TBU, TBS

φ4.2 13.9 (0.022) 640 (92.8) A-2, B11, C17, D15 flanges

φ12 113 (0.175) 540 (78.3) A-2, B11, C17, D15 webs

Hoo

ps

No. 3 71 (0.11) 376 (54.5) TB

No. 3 71 (0.11) 379 (55.0) 1-1 to 1-5, Group 3

No. 3 71 (0.11) 370 (53.6) 1-6, Groups 2, 4

No. 4 129 (0.20) 379 (55.0) TBU

No. 4 129 (0.20) 443 (64.2) TBS

φ4.2 13.9 (0.022) 640 (92.8) A-2, B11, C17, D15 flanges

φ6.5 33.2 (0.051) 330 (47.8) A-2, B11, C17, D15 webs

Note: TB series prestressing steel: effective prestress 1145 MPa (166 ksi), ultimatestrength 1703 MPa (247 ksi).


specimens tested under various combinations of T to Mratios. The longitudinal reinforcement was symmetricallydistributed around the solid cross section as shown in Fig. 6.Figure 7(a) shows the experimentally observed and thecalculated T-M interaction curves. The model is capable ofaccurately modeling the interaction. For the five beams, theaverage ratio of experimental to calculated ultimate momentwas 1.00 and the coefficient of variation (COV) was 2.6%.These numbers were 1.32 and 15.2% for the ACI variable θanalysis and 1.36 and 11.7%, respectively, for the ACI45-degree analysis. Equation (25) (safeguard againstconcrete crushing) was critical in determining the strength ofmembers with significant torsion, and is shown to giverelatively more conservative results. Where bending wassignificant, the results based on θ = 60 degrees providedmore accurate results compared with the calculations basedon θ = 45 degrees.

Symmetrically reinforced-partiallyprestressed beams

Mardukhi19 tested five symmetrically reinforced, partiallyprestressed hollow members (Series TB) under variouscombinations of torsion and bending. Figure 7(b) shows thecomparison between the calculated and observed results and agood agreement is observed. For the five beams, the averageratio of experimental to calculated ultimate moment was 1.03and the COV was 5.5%. These values are relatively similar tothose of Group 2, pointing to consistency in the results of themethod for reinforced and partially prestressed concretebeams when symmetrically reinforced in the longitudinaldirection. In the zone of predominant bending, both the longi-tudinal and transverse reinforcement were below balancedvalues, and Eq. (6) was used instead of Fig. 3 to calculate thetorsional shear strength v.

The average and COV values were 1.13 and 10.5% for theACI variable θ analysis and 1.34 and 22.8%, respectively, forthe ACI 45-degree analysis. In pure torsion and predominanttorsion, the amount of transverse reinforcement was critical,and using a small θ of 30 degrees provided more accurateresults. In predominant bending, the amount of longitudinal

reinforcement was critical, and a larger value of the angle55 degrees provided more favorable results.

Unsymmetrically reinforced beamsThe six nonprestressed solid specimens of Group 117,18

were similar to those in Group 2, except that a smalleramount of longitudinal reinforcement was provided in the

Table 2—Experimental verification

ReferenceNumber and type of beam

specimens

Distribution of longitudinal

reinforcementNominal size,

mm (in.)

Concrete strength,MPa (psi)

Experimental/calculated

SMCS

ACI (θ = 450 degrees)

ACI(30 degrees ≤ θ ≤ 60 degrees)

Mean COV, % Mean COV, % Mean COV, %

McMullen and Warwaruk17-18

20 rectangular solidreinforced beams

Five symmetrical15 unsymmetrical

152 x 305(6 x 12)

30 to 40(4350 to 5800) 0.98 5.8 1.31 14.9 1.19 12.9

Mardukhi19 Five rectangular hollowpartially prestressed beams Symmetrical 305 x 432

(12 x 17)≈38

(5500) 1.03 5.5 1.34 22.8 1.13 10.5

Onsongo4 Five hollow and four solidrectangular reinforced beams Unsymmetrical 508 x 410

(20 x 16.1)15 to 46

(2200 to 6670) 1.15 13.1 1.38 20.9 1.35 21.1

Gesund et al.20 12 rectangular solidreinforced beams Unsymmetrical 203 x 203,152 x 305

(8 x 8, 6 x 12)27 to 40

(3900 to 5800) 0.91 14.1 1.34 13.6 1.26 14.9

Zararis and Penelis21

42 T- and four rectangularsolid reinforced beams Unsymmetrical 100 x 210*

(4 x 8.3)14 to 41

(2030 to 5950) 1.11 15.2 1.98 20.9 1.65 18.3

Pandit and Warwaruk22

14 rectangular solidreinforced beams

Three symmetrical 11 unsymmetrical

152 x 305(6 x 12)

32 to 40(4650 to 5800) 0.95 10.3 1.25 13.3 1.15 12.7

Lampert and Thurlimann23

Five square hollowreinforced beams Unsymmetrical 500 x 500

(19.7 x 19.7)26

(3770) 1.04 4.5 1.32 13.2 1.11 3.76

111 beam specimens 1.04 14.7 1.59 28.2 1.39 23.5*Flange dimensions of T beams: 152 to 203 mm (6 to 8 in.) thickness, and 400, 700, and 1000 mm (15.7, 27.6, and 39.4 in.) width.

Fig. 7—T-M interaction diagrams in symmetricallyreinforced and partially prestressed beams.


flexural compression flange. Figure 8(a) compares theexperimentally observed and the calculated T-M interactioncurves. Smaller levels of flexural moments increased thetorsional capacity due to the strengthening effect of the flexuralcompressive force on the weaker top flange. The SMCSmodel was capable of accurately modeling the interaction,including the increase in torsional strength at relatively lowflexural moments. The average ratio of the experimental tocalculated ultimate moment in the six specimens was 1.00and the COV was 4.1%.

The ACI equations were considerably conservative incalculating the torsional strength at relatively low flexuralmoment, but were more accurate at higher levels of M. In thecases where the longitudinal reinforcement in either thecompression or the tension flange was critical in determiningthe overall strength, using larger values of θ provided largerstrength and more accurate calculations. The average and COV

values were 1.21 and 8.8% for the ACI variable θ analysis and1.38 and 15.0%, respectively, for the ACI 45-degree analysis.

The specimens of Group 317,18 had smaller amounts oftransverse and bottom longitudinal reinforcement. Figure 8(b)shows the observed and the calculated T-M interactioncurves. The proposed model was unconservative for twospecimens. The average ratio of the experimental to calculatedultimate moment in the five specimens was 0.96 and theCOV was 10.0%. These values were 1.11 and 11.6% for theACI variable θ analysis and 1.21 and 21.2% respectively forthe ACI 45-degree analysis.

The under-reinforced TBU series tested by Onsongo4

consisted of five hollow beams unsymmetrically reinforcedin the longitudinal direction. Figure 8(c) shows the observedand calculated interaction diagrams. The proposed modelaccurately calculated the interaction, while the ACI codeprovisions were considerably conservative, except forSpecimen TBU2. This specimen, along with TBU4 sufferedfrom difficulties during casting, which led to a reduced wallthickness in the top flange and hence possibly a reducedcapacity. The average ratio of the experimental to calculatedultimate moments in the five specimens was 1.08 and theCOV was 8.3%, respectively.

Similar to the observation in Fig. 7(a) and 8(a), Eq. (25)under-estimated the maximum torsional strength where itwas critical (in pure torsion and at relatively low T/M). Also,larger values of the angle θ were obtained when the strengthin the longitudinal direction in the top or bottom flangeswas critical. The average and COV of the experimental tocalculated ultimate strength were 1.39 and 27.8% for thevariable θ analysis, and 1.44 and 26.8% for the θ = 45-degreeanalysis, respectively.

Effect of concrete strengthThe four specimens of the TBS4 series were tested to study

the effect of the fc′ on the strength at a T/M of approximately1.25. The specimens were solid and unsymmetricallyreinforced in the longitudinal direction, as shown in Fig. 6,and the concrete strength ranged from 15.5 to approximately46 MPa (2200 to 6670 psi). Figure 9(a) shows the observedand calculated results. The tests showed an increase in beamcapacity at higher concrete strength. The proposed SMCScaptured this trend, but over-estimated the increase forconcrete strength above 33 MPa (4800 psi). The average andCOV of the ratio of observed to calculated moment were1.24 and 14.2%, respectively, for the proposed SMCS model,and 1.31 and 9.1% for both ACI methods. The ACI calculatedstrength was limited by concrete crushing (Eq. (25)) and areshown again to be conservative.

Effect of stirrups spacingThe four specimens of Group 417,18 were tested to study

the effect of the stirrups spacing on the strength at a T/M ofapproximately 0.6. The cross sections of these specimenswere similar to that of Group 3, and Specimen 3-4 fromSeries 3 tested at the same T/M fits within the graph. Thespacing of the stirrups ranged from 76 to 230 mm (3 to 9 in.),and was larger than the ACI limit of ph/8 in four out of thefive specimens. The proposed SMCS model and the ACIvariable θ analysis accurately captured the decrease instrength at larger stirrups spacing even where the spacingcan be considered inadequately large. The average and COVof the ratio of observed to calculated strength were 0.97 and4.8% for the proposed SMCS model, 1.10 and 1.2% for the

Fig. 8—T-M interaction diagrams in unsymmetricallyreinforced beams.


ACI variable θ analysis, and 1.25 and 8.5% for the ACI45-degree analysis, respectively.

Effect of flange width in T-beamsFigure 9(c) shows the experimentally observed and the

calculated strength of a series of four specimens from anexperimental program21 designed to study the effect offlange size on the strength of T-beams subjected tocombined torsion and bending. Both the web and the flangewere reinforced with longitudinal and transverse steel, andthe flange width ranged from 100 mm (4 in.) (rectangularsection) to approximately 1000 mm (39.4 in.) (refer to Fig. 6).The four specimens were tested under T/M of approximately1.18. The proposed method captured the trend in increase instrength with an overhang width up to approximately fivetimes the flange thickness, but slightly under-estimated theincrease in strength at larger overhang size. The average andCOV of the ratio of observed to calculated moment were1.09 and 9.0%, respectively, for the proposed SMCS model;1.94 and 5.6%, respectively, for the ACI variable angleanalysis; and 2.00 and 6.5% for the ACI 45-degree analysis,respectively. The ACI results are shown to be undulyconservative.

Overall performance of proposed modelTable 2 shows the average and COV of the experimental

to calculated strength of the 111 specimens.4,17-23 The ACIresults were more conservative than those of the proposedmodel, mainly in members subjected to significant torsion asshown in the previous section. The conservatism in Eq. (25)is partially due to the assumption of spalling of the concretecover in torsion, a phenomenon that did not affect the resultsmost (if not all) of the 111 specimens because of the relativelysmall thickness of clear cover used. In addition, spallingdoes not affect all sides of the cross section subjected tocombined stresses24 as assumed by the ACI equation. Theproposed model resulted in a smaller COV, pointing to amore uniform calculation of the strength at the various levelsof T/M and variables affecting the results.

Table 3 compares the performance of the SMCS model forcombined torsion and bending with that for the case ofbeams subjected to pure torsion;11 membrane elementssubjected to in-plane shearing stresses;9 membrane elementssubjected to in-plane shearing and normal stresses;10 andbeam elements subjected to shear, bending, and axialloads.12 The results were slightly more conservative andwith slightly higher variation when shear was combined withbending. In general, however, the performance of the SMCSmodel can be considered consistent in both beam andmembrane elements subjected to the stress-resultants shown.

SUMMARY AND CONCLUSIONSA simple method for the design and capacity calculation of

strength of reinforced and prestressed concrete memberssubjected to combined torsion and bending was presented.The interaction between the two moments was achieved byadopting the concept of superposition of the longitudinalreinforcement for the two cases.

The calculations of the SMCS model were compared withthe experimental results from 111 nonprestressed andpartially prestressed rectangular and T-beam specimenssubjected to combined torsion and bending. Full interactioncurves were calculated using the proposed model, and wereshown to be in very good agreement with the observedresults. The model also captured the effect of the concretestrength, the amount of transverse reinforcement, the

Fig. 9—Effect of fc′ , stirrups spacing and T-beam flangewidth on strength of beams.

Table 3—Comparison with performance of SMCS in other studies (total 415 specimens)

Type of elements Stress-resultantsNo. of

specimens

Observed/calculated

Mean COV, %

Beams (this study) Torsion and bending 111 1.04 14.7

Membrane elements9 In-plane shear 46 1.01 12.5

Membrane elements10 In-plane shearand normal 14 1.17 12.2

Beams11 Pure torsion 83 1.03 11.1

Beams12 Shear, bending and axial load 161 1.28 18.8


distribution of the longitudinal reinforcement, and the size ofthe T-beam flanges on the beam strength. The performance ofthe model was consistent with that in previous studies on:1) pure torsion in beams; 2) combined shear, bending, and axialload in beams; 3) pure shear in membrane elements; and4) combined shear and normal stresses in membrane elements.

The equations of the ACI code were also compared withthe experimental results and were found to be satisfactory.They showed a significantly higher level of conservatism inbeams subjected to pure or predominant torsion, especiallywhen the upper limit set by ACI Eq. (11-18) (Eq. (25)) wascritical in determining the strength. This conservatism can bepartially attributed to the assumption of spalling of theconcrete outer cover in torsion calculation.

In using the ACI code for capacity calculations,calculating the angle θ (between 30 and 60 degrees) based onthe actual reinforcement was found to provide more accurateresults than simply using 45 degrees for nonprestressedmembers and 37.5 degrees for prestressed members. Thiscalculation typically provided larger torsional strength.

In general, the results of the proposed SMCS model weremore favorable than those of the ACI equations. Given thatthis proposed model can be applied not only to beam elementsbut also to membrane elements subjected to various stressresultants, it is suggested that the SMCS model can be the basisof a more general and unified treatment of shear and torsion inreinforcement and prestressed concrete structural elements.

NOTATIONA0 = area enclosed in shear flow resultantA0h = area enclosed in centerline of outermost closed stirrup or hoopAc = area enclosed in outer perimeter of cross sectionAL = total area of symmetrical non-prestressed longitudinal reinforcement

in section Aps = total area of symmetrical prestressed longitudinal reinforcement

in sectionAs = area of bottom or top longitudinal reinforcement in sectionAt = area of one leg of transverse closed stirrup or hoopa0 = depth of equivalent stress block in shear flow zoneb = width of flange in T-beambw = width of web in T-beamd = effective depth in bendingf ′c = compressive strength of concretef1, f2 = principal tensile and compressive stress in membrane elementfpy = yield strength of symmetrical prestressed longitudinal reinforcementfsx, fsy = stress in x and y direction reinforcement in membrane elementfy = yield stress in bottom or top longitudinal reinforcement in sectionfyL = yield stress in symmetrical non-prestressed longitudinal

reinforcementfyt = yield stress of stirrups or hoopsfyx, fyy = yield stress of x direction and y direction reinforcement in

membrane elementhf = depth of flange in T-beamjd = flexural lever arm, can be taken as 0.9dM = acting flexural momentp0 = perimeter of the shear flow resultantpc = outer perimeter of sectionph = perimeter of centerline of outermost closed stirrup or hoopq = shear flow in hollow tube models = spacing of stirrups or hoopsT = torsional momentt = thickness of walls in hollow sectionstd = depth of shear flow zonev = maximum shear stress in walls of tubeθ = angle of inclination of diagonal strut in truss model ρx, ρy = reinforcement ratio in x and y directionsσx,σy = membrane element stresses in x and y directions

REFERENCES1. ACI Committee 318, “Building Code Requirements for Structural

Concrete (ACI 318-05) and Commentary (318R-05),” American ConcreteInstitute, Farmington Hills, Mich., 2005, 430 pp.

2. American Association of State Highway and Transportation Officials,“AASHTO LRFD Bridge Design Specifications and Commentary,” SIUnits, 3rd Edition, Washington, D.C., 2004.

3. Ewida, A. A., and McMullen, A. E., “Torsion-Shear-FlexureInteraction in Reinforced Concrete Members,” Magazine of ConcreteResearch, V. 33, No. 115, 1981, pp. 113-122.

4. Onsongo, W. M., “The Diagonal Compression Field Theory forReinforced Concrete Beams Subjected to Combined Torsion, Flexure, andAxial Load,” PhD thesis, Department of Civil Engineering, University ofToronto, Toronto, Ontario, Canada, 1978, 246 pp.

5. Cocchi, G. M., and Volpi, M., “Inelastic Analysis of ReinforcedConcrete Beams Subjected to Combined Torsion, Flexural and AxialLoads,” Computers and Structures, V. 63, No. 3, 1996, pp. 479-494.

6. Rahal, K. N., and Collins, M. P., “Analysis of Sections Subjected toCombined Shear and Torsion—A Theoretical Model,” ACI StructuralJournal, V. 92, No. 4, July-Aug. 1995, pp. 459-469.

7. Karayannis, C. G., and Chalioris, C. E., “Strength of PrestressedConcrete Beams in Torsion,” Journal of Structural Engineering andMechanics, V. 10, No. 2, 2000, pp. 165-180.

8. Vecchio, F. J., and Collins, M. P., “Modified Compression FieldTheory for Reinforced Concrete Elements Subjected to Shear,” ACIJOURNAL, Proceedings V. 83, No. 2, Mar.-Apr. 1986, pp. 219-231.

9. Rahal, K. N., “Shear Strength of Reinforced Concrete, Part I—Membrane Elements Subjected To Pure Shear,” ACI Structural Journal,V. 97, No. 1, Jan.-Feb. 2000, pp. 86-93, and closure to discussion, V. 97,No. 6, Nov.-Dec. 2000, pp. 910-913.

10. Rahal, K. N., “Membrane Elements Subjected to In-Plane Shearingand Normal Stresses,” ASCE Structural Journal, V. 128, No. 8, 2002,pp. 1064-1072.

11. Rahal, K. N., “Analysis and Design for Torsion in Reinforced andPrestressed Concrete Beams,” Structural Engineering and Mechanics,V. 11, No. 6, 2001, pp. 575-590.

12. Rahal, K. N., “Shear Strength of Reinforced Concrete, Part II: BeamsSubjected to Shear, Bending Moment and Axial Loads,” ACI StructuralJournal, V. 97, No. 2, Mar.-Apr. 2000, pp. 219-224.

13. Rahal, K. N., and Collins, M. P., “Simple Model for Predicting theTorsional Strength of Reinforced and Prestressed Concrete Sections,” ACIStructural Journal, V. 93, No. 6, Nov.-Dec. 1996, pp. 658-666.

14. Braestrup, M. W., “Plastic Analysis of Shear in Reinforced Concrete,”Magazine of Concrete Research, V. 26, No. 89, Dec. 1974, pp. 221-228.

15. Mitchell, D., and Collins, M. P., “The Behaviour of StructuralConcrete in Pure Torsion,” Publication No. 74-06, Department of CivilEngineering, University of Toronto, Toronto, Ontario, Canada, 1974, 88 pp.

16. Lampert, P., and Collins, M. P., “Torsion, Bending, and Confusion—An Attempt to Establish the Facts,” ACI JOURNAL, Proceedings V. 69,No. 8, Aug. 1972, pp. 500-504.

17. McMullen, A. E., and Warwaruk, J., “Concrete Beams in Bending,Torsion and Shear,” Proceedings, ASCE, V. 96, 1970, pp. 885-903.

18. McMullen, A. E., and Warwaruk, J., “The Torsional Strength ofRectangular Reinforced Beams Subjected to Combined Loading,” ReportNo. 2, Civil Engineering Department, University of Alberta, Alberta,Canada, 1967, 162 pp.

19. Mardukhi, J., “The Behaviour of Uniformly Prestressed ConcreteBox Beams in Combined Torsion and Bending,” MASc thesis, Universityof Toronto, Toronto, Ontario, Canada, 1974, 73 pp.

20. Gesund, H.; Schuette, F. J.; Buchanan, G. R.; and Gray, G. A.,“Ultimate Strength in Combined Bending and Torsion of ConcreteBeams Containing Both Longitudinal and Transverse Reinforcement,”ACI JOURNAL, Proceedings, V. 61, No. 12, Dec. 1964, pp. 1509-1521.

21. Zararis, P. D., and Penelis, G. G., “Reinforced Concrete T-Beams inTorsion and Bending,” ACI JOURNAL, Proceedings V. 83, No. 1, Jan.-Feb.1986, pp. 145-155.

22. Pandit, G. S., and Warwaruk, J., “Reinforced Concrete Beams inCombined Bending and Torsion,” Torsion in Structural Concrete, SP-18,American Concrete Institute, Farmington Hills, Mich., 1968, pp. 133-163.

23. Lampert, P., and Thurlimann, B., “Torsions-Biege-Versuche anStahlbetonbalken,” Bericht Nr. 6506-3, Institut fur Baustatik, ETH Zurich,Germany, Jan. 1969.

24. Rahal, K. N., and Collins, M. P., “Effect of Cover Thickness onShear and Torsion Interaction-An Experimental Investigation,” ACIStructural Journal, V. 92, No. 3, May-June 1995, pp. 334-342.

APPENDIX A: DESIGN EXAMPLEDesign a reinforced concrete section for: M = 1500 kN·m

(1106 k·ft), T = 700 kN·m (516 k·ft). Use fc′ = 30 MPa(4350 psi), fyt = fyL = 400 MPa (58 ksi), cover to steel = 30 mm(1.18 in.). Preliminary analysis suggests the section shown in


Fig. A1 with geometric properties: pc = 3925 mm (154.5 in.),Ac = 945,000 mm2 (1464.8 in.2).

Step 1: Design for flexure: Assuming φ25 mm (1 in.)longitudinal bars and φ14 mm (0.55 in.) hoops are used, d =900 – 30 – 14 – 13 = 843 mm (33.2 in.). The required ratio ofreinforcement is 0.501% corresponding to a bottom steel areaof approximately 5150 mm2 (8 in.2) (within limits ofmaximum and minimum reinforcement).

Step 2: Equation (15) gives a shear stress v = 700 (106)(3925)/(945,0002)/(0.67) = 4.59 MPa (665 psi). The normalizedshear stress is v /fc′ = 4.59/30 = 0.153.

Step 3: Because the section is under-reinforced for torsionresistance, the most straight-forward design, though notnecessarily the most economical, is to use Eq. (8).

Step 4: With ωt = ωL = 0.153, the amounts of steel arecalculated using Eq. (18) and (19):

At/s = 0.153(0.42)(945,000)(30)/(400)/(3925) = 1.16 mm2/mm(0.0457 in.2/in.)

AL = 0.153(0.375)(945,000)(30)/400 = 4066 mm2 (6.3 in.2)The maximum spacing of φ14 mm (0.55 in.) stirrups is

132 mm (5.2 in.). Choose s = 130 mm (5 in.). This satisfiesthe upper limit of d/2 and 1/8 hoop perimeter usually consideredin building codes.

Step 5: The depth is relatively large, and hence 6φ14 barsare provided as skin reinforcement. The remaining area is4066 – 6(154) = 3142 mm 2 (4.87 in.2) is split in two halves(1571 mm 2 [2.44 in.2] each) in the top and bottom flange.Total bottom steel is that from M and that from T = 5150 +1571 = 6721 mm 2 (10.4 in.2). Fourteen φ25 (No. 8) barsprovide the required amount and are placed with clearspacing of approximately 60 mm (2.36 in.), which satisfiesthe code requirements of minimum spacing.

Step 6: The top steel can be reduced using Eq. (20) by:1500(106)/0.9/843/400 = 4942 mm2 (7.66 in.2). Hence, notop reinforcement is needed. However, 4φ14 are used toprovide minimum reinforcement in the top flange.

The results of the design are summarized in Fig. A-1.

APPENDIX B: CAPACITY CALCULATION EXAMPLECalculate the torsional capacity of a TBU4 specimen (Fig. 6

and Table 1) when a moment M = 277 kN·m (204.3 k·ft) isacting. The area and perimeter enclosed within the outerdimensions of the section are calculated as pc = 1836 mm(72.3 in.) and Ac = 208,280 mm2 (322.8 in.2). The stirrupsare No. 4: At = 129 mm2 (0.2 in.2), fyt = 379 MPa (55 ksi),spacing s = 76 mm (3 in.) The concrete compressive strengthfc′ is 34.8 MPa (5050 psi).

Step 1: M = 277 kN·m (204.3 k·ft).Step 2: Calculate the transverse reinforcement index using

Eq. (19)

Calculate critical ωL using Eq. (21). Top steel: As = 387 mm2

(0.6 in.2), fy = 393 MPa (57 ksi), skin steel As = 1/2(426)mm2 (0.33 in.2), fy = 552 MPa (80 ksi), d = 376 mm (14.8 in.)

=

0.80

Bottom flange steel: As = 3570 mm2 (5.53 in.2), fy = 436 MPa(63.2 ksi), skin steel As = 1/2 (426) mm2 (0.33 in.2), fy =552 MPa (80 ksi)

= 0.63

The bottom reinforcement is critical.Step 3: With ωL = 0.63, ωt = 0.388, Fig. 3 gives v/fc′ =

0.32, and v = 0.32(34.8) = 11.14 MPa (1616 psi).Step 4: The ultimate T is calculated using Eq. (15) as follows

= 176 kN·m (130 k·ft)

This point corresponds to the same T/M as Specimen TBU3.

ωt129 379( ) 1836( )

0.42 76( ) 208280( ) 34.8( )------------------------------------------------------------ 0.388= =

ωL top–2 277( ) 10

6( )( ) 0.9( ) 376( ) 2 387( ) 393( ) 426 552( )+ +⁄

0.375 208,280( ) 34.8( )--------------------------------------------------------------------------------------------------------------------------------------=

ωL bot–2 277( ) 10

6( )– 0.9( ) 376( ) 2 3570( ) 436( ) 426 552( )+ +⁄

0.375 208,280( ) 34.8( )--------------------------------------------------------------------------------------------------------------------------------=

T 0.67 208,208( )2

1836--------------------------11.14=

Fig. A1—Cross section in design example.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Combined Torsion and Bending in Reinforced and Pre Stressed Concrete Beams Using Simplified Method...

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