Modiﬁed Variable Angle Truss-Model for torsion in ...¬ed Variable Angle Truss-Model for torsion in reinforced concrete beams L ... ler) and Compatibility Compression Field Theory

ORIGINAL ARTICLE

Modified Variable Angle Truss-Model for torsionin reinforced concrete beams

L. F. A. Bernardo • J. M. A. Andrade •

S. M. R. Lopes

Received: 8 February 2012 / Accepted: 21 May 2012 / Published online: 3 July 2012

� RILEM 2012

Abstract A new computation procedure is devel-

oped to predict the overall behaviour of reinforced

concrete beams under torsion. This procedure is based

on a modification of the classic Variable Angle Truss-

Model in order to make it capable of predicting the

behaviour of the beams under torsion for all loading

states. The theoretical predictions are compared with

the results from reported tests. Conclusions are

presented. The main conclusion is that the new

procedure described in this paper gives very good

predictions when compared with the actual overall

behaviour of the beams.

Keywords RC beams � Torsion � Truss-Model �Theoretical behaviour

1 Introduction

Since the beginning of last century, several studies have

been developed to lead to the current knowledge of the

torsional behaviour of Reinforced Concrete (RC)

beams. Such models can be divided into two principal

theories: Skew-Bending Theory, which was the basis of

the American Code between 1971 and 1995, and the

Space Truss Analogy, which is the base of the European

Model Code since 1978 and also the American Code

since 1995 and as a great historical value.

One of the latest advanced theoretical models based

on the Space-Truss Analogy, is the Variable Angle

Truss-Model (VATM). This model was presented by

Hsu and Mo [22]. The authors used a r–e relationship

for the concrete in the struts which takes into account

for the softening effect instead of a conventional r–erelationship for uniaxial compression. The VATM

resulted from the successive developments of the

original Spatial-Truss Analogy of Rausch [33],

namely by Andersen [2], Cowan [15], Walsh et al.

[36], Lampert and Thurlimann [24], Elfgren [16, 17],

Muller [29] and Collins and Mitchell [28] and Mitchell

and Collins [14].

The several versions of the Space Truss Analogy

can be classified into Plasticity Compression Field

Theory (Lampert and Thurlimann, Elfgren and Mul-

ler) and Compatibility Compression Field Theory

(Collins, Hsu and Mo).

Several authors also proposed simplified versions

of VATM, namely: Collins and Mitchell [14] (model

called as Space Truss Theory with Concrete Cover

Spalling, which is the basis of the Canadian Model

Code), Rahal and Collins [31], Bhatti and Almughrabi

[12] and Wang and Hsu [37], among others. However,

these models can only give the ultimate torsional

L. F. A. Bernardo (&) � J. M. A. Andrade

Department of Civil Engineering and Architecture,

C-MADE, University of Beira Interior, Edifıcio II das

Engenharias, Calcada Fonte do Lameiro, 6201-001

Covilha, Portugal

e-mail: [email protected]

S. M. R. Lopes

CEMUC - University of Coimbra, Coimbra, Portugal

Materials and Structures (2012) 45:1877–1902

DOI 10.1617/s11527-012-9876-4

strength of the beam while VATM is able to predict the

general state of the beam throughout the entire loading

history. However, very good results are only observed

for high loading levels [3, 21, 22].

Recently, Jeng and Hsu [23] extended their Soft-

ened Membrane Model, developed for RC membrane

elements under shear, to RC members under torsion.

This new analytical method, called Softened Mem-

brane Model for Torsion can predict the entire Torque

(T)–Twist (h) curve. However, according to these

authors such model is not able to simulate adequately

the global behaviour of hollow beams under torsion.

Moreover, the predicted behaviour immediately after

cracking is still different when compared with exper-

imental observations and the mathematical formula-

tion of the model is somehow complex and does not

explicit a simple concept on how a RC beam behaves

under torsion after cracking. VATM is recognized as a

model which provides a simple physical understand-

ing of the torsion phenomenon for any RC beam.

In this study, the VATM is modified in order to turn

it able to predict the global behaviour of RC beams

under torsion. The changes in the original VATM

formulation are made by studying separately each

particular behaviour state of the RC beam under

torsion, namely:

– non-cracked state;

– transition between non-cracked and cracked state;

– cracked state until failure.

Calculation algorithms that incorporate the modifi-

cations to VATM are developed. Such algorithms are

computationally implemented by means of DELPHI

program language. At the end of each study, the

predictions obtained from the theoretical model are

compared with experimental results of reference

beams under torsion, whose results are available in

the literature.

Finally, it should be pointed out that this study deals

exclusively with pure torsion. In actual structures,

torsion normally occurs associated with other internal

forces, such as, bending, shear, and axial forces.

However, in some cases, such as in curved bridges,

torsion might be an important action. On the other

hand, the behaviour of current RC members under

pure torsion needs to be well known before theoretical

studies with special members under torsion or under

interaction between torsion and other internal forces

are carried out [10, 35].

2 Previous studies and research significance

In previous studies, some authors predicted the

behaviour o Normal-Strength Concrete (NSC) beams

under torsion by using the VATM. For example, Hsu

and Mo [22] used the VATM to predict the theoretical

T–h curve in order to compare it with available

experimental results. The authors showed that the

ultimate values of the T–h curves, both experimental

and theoretical, were quite similar. The same obser-

vation was made for some other similar beams.

For NSC beams under torsion, Bernardo and Lopes

[7] showed that the computing procedure based on

VATM was shown to be quite appropriate to predict

the ultimate behaviour. However, for High-Strength

Concrete (HSC) beams these authors showed that the

original computing procedure no longer could be

considered adequate, since resistances were overesti-

mated for HSC beams with high torsional reinforce-

ment ratio [11]. The computing procedure was

reviewed by Bernardo and Lopes [11] in order to

incorporate specific stress (r)–strain (e) relationships

for HSC. In fact, for HSC the shape of the r–erelationships are different when compared with those

for NSC. This observation was firstly observed in

beams under flexure [8, 26]. In their study, Bernardo

et al. tested several r–e relationships to characterize

the mechanical behaviour of the concrete in com-

pression (struts) and the steel in tension (reinforce-

ment). For concrete in compression, the r–erelationship were derived from the experimental

results with plates under shear and took into account

the softening effect. Bernardo and Lopes [11] found

appropriate r–e relationships for HSC beams (both

plain and hollow section) to incorporate in the

prediction model.

The success of VATM to predict the points of the

T–h curve for the ultimate behaviour is understand-

able, since, for high level of loading, the concrete is

extensively cracked. In this state, the theoretical model

approaches the real model. For low level of loading,

the beam is not extensively cracked (or not cracked at

all). Furthermore, before and after cracking, the

concrete core of plain sections (neglected in VATM)

also influences the torsional stiffness of the beams.

That explains the deviations between theoretical and

experimental T–h curves.

Based on the previous studies, Bernardo and Lopes

[7, 11] developed a calculation procedure in order to

1878 Materials and Structures (2012) 45:1877–1902

predict the overall theoretical behaviour (not just the

ultimate behaviour) of RC beams under torsion. The

theoretical approach was firstly performed by studying

different behavioural states, each of one identified

with the states that can be observed experimentally.

These states were characterized individually by using

different theories:

– Linear elastic analysis in non cracked state:

Theory of Elasticity, Skew-Bending Theory and

Bredt’s Thin-Tube Theory;

– Linear elastic analysis in cracked state: Space

Truss Analogy with an angle of 45� for the

concrete struts and considering linear behaviour

for the materials;

– Non linear analysis: VATM, considering a non

linear behaviour for the materials and the softening

effect.

To make the transition between the different

theoretical states, Bernardo and Lopes adopted semi-

empirical criteria. From the comparative analysis

between the theoretical predictions from the model

and the experimental results, the authors showed that

the procedure was adequate to predict the global

behaviour of RC beams under torsion [7, 11].

Despite the good results provided by the global

model of Bernardo and Lopes, the same authors

recognized that the model was not fully theoretically

consistent because different torsional theories were

used to characterize the behaviour of each state. This

option led the authors to adopt special criteria to make

the transition between the behavioural states, in order

to obtain the full T–h curve.

This study presents an alternative theoretical and

global torsional model based on a modification of the

VATM. This new approach is theoretically more

acceptable since this model is mainly based in one

torsional theory.

Furthermore, a theoretical and reliable global

model to predict the behaviour of RC beams under

torsion for low loading level does not exist yet. This

aspect makes this study particularly important.

3 Variable Angle Truss-Model for RC beams

The computation of the theoretical T–h curve from the

VATM [21, 22] (Fig. 1) requires three equilibrium

equations to compute the torque, T, the effective

thickness, td, of the equivalent hollow section and the

angle of the inclined concrete struts, a, from the

horizontal axis of the beam [Eqs. (1)–(3) in Table 1].

The VATM [21, 22] also needs three compatibility

equations to compute the strain of the transversal

reinforcement, et, the strain of the longitudinal rein-

forcement, el, and the twist, h [Eqs. (4)–(6) in Table 1].

To characterize the compression concrete diagonal

struts and the tension reinforcement, r–e relationships

Al fl T

At ft

T

t

1.00

t /2d

td

k2td

εds

ε =ε /2dsd

A B

Cσ

σdα

neutral axis

σ

Fig. 1 VATM: strains and stresses in the cross-section concrete struts

Materials and Structures (2012) 45:1877–1902 1879

must be adopted taking into account the unfavorable

effect of the softening effect for concrete and the effect

of stiffening effect for reinforcement. The r–e rela-

tionships for concrete are derived from experimental

tests of panels under shear.

In a previous study Andrade [3] tested several r–erelationships for the materials (concrete in compres-

sion and reinforcement in tension). This study was

based on numerical simulations with VATM formu-

lation in order to calculate the ultimate behaviour of

RC beams under torsion. Softening effect was con-

sidered for concrete in compression in the struts and

stiffening effect was considered for reinforcement in

tension. From comparative analysis with experimental

results, some r–e relationships were found to provide

good theoretical predictions of the ultimate behaviour

of RC beams under torsion. One of the most suitable

theoretical model is the one that incorporates the r–erelationship for compressed concrete in struts pro-

posed by Belarbi and Hsu [5] [Eqs. (7) and (8) in

Table 2] with softening factors for maximum stress in

concrete (br) and for strain corresponding to maxi-

mum stress (be) proposed by Zhang and Hsu [38] [Eq.

(9)–(12) in Table 2] and the r–e relationship rein-

forcement in tension proposed by Belarbi and Hsu [6]

(Eqs. (17)–(20) in Table 3).

Table 1 Equations of VATM

VATM with softening effect

Equilibrium equations [21, 22]:

T ¼ 2Aotdrd sin a cos a ð1Þcos2 a ¼ Alfl

pordtdð2Þ td ¼

Alflpord

þ Atftsrd

ð3Þ

Compatibility equations [21, 22]:

et ¼A2

ord

poT tg a� 1

2

� �eds ð4Þ el ¼

A2ord

poT cotg a� 1

2

� �eds ð5Þ h ¼ eds

2td sin a cos að6Þ

Ao: area limited by the centre line of the flow of shear stresses, which coincides with the centre line of the strut’s thickness td:

Ao = (x - td)(y - td) (x and y are the outer dimensions of the cross section), po = perimeter of area Ao: po = 2(x - td) ? (y - td), rd:

stress in the diagonal concrete strut, Al: total area of the longitudinal reinforcement, At: area of one unit of the transversal

reinforcement, s = spacing of the transversal reinforcement, ed: compressive strain in the strut direction, eds: maximum compressive

strain in the external surface in the strut direction, fl: longitudinal reinforcement stress, ft: transversal reinforcement stress

Table 2 r–e relationship for concrete in compression


Based on VATM, the stress of concrete diagonal

struts, rd, is defined as the average stress of a non-

uniform compression stress diagram on concrete strut

(Fig. 1) and is calculated from r–e relationship for

concrete in compression [Eq. (13) in Table 2]. In

Fig. 1, parameters A, B, and C, corresponds respec-

tively to maximum stress, average stress and the stress

diagram resultant [Eqs. (14)–(16) in Table 2]. The k1

parameter in Eq. (13), which corresponds to the

quotient between average stress and maximum stress

for the stress diagram of the concrete strut (k1 = B/

A according to Fig. 1), is obtained by integrating Eqs.

(7) and (8). In this study, this integration is numeri-

cally performed by the computational model.

Some unknown and interdependent variables exist.

Therefore, to start the computing procedure, it is

necessary to run an iterative calculation algorithm to

compute the points of the T–h curve. This procedure

can be formulated by a calculation algorithm whose

flowchart is shown in Fig. 2.

The theoretical failure point of the beam under

torsion is usually defined by either the case of the

maximum compressive strain on the surface of

concrete struts, eds (Fig. 1), reaching its ultimate value

(ecu) or the case of the tensile strain for the torsion

reinforcement, es, reaching its ultimate value (esu).

4 Key points and properties of T–h curve

In general, the T–h curves obtained from laboratorial

tests on RC beams (for normal reinforcement ratios)

under pure torsion up to failure can lead to a typical

T–h curve, as presented in Fig. 3. This curve shows 3

different zones (zone 1, 2 and 3 of Fig. 3).

The key points to limit the 3 zones of a T–h curve

are fully defined by their (h; T) coordinates (Fig. 3).

After cracking, the beams suffer a sudden increase

of the twist. This zone, identified as Zone 2.a in Fig. 3,

starts at (hcrI ; Tcr) and ends at a certain level of twist

(hcrII).This point correspond to the transition from the

non-cracked state to the cracked state. Experimental

tests show that this behavioural zone is not observed in

RC hollow beams [9].

All the key points and properties of T–h curve

presented in Fig. 3 will be used for comparative

analysis in this study (Sect. 6).

5 Reference beams for comparative analysis

The theoretical results obtained from the modified

VATM will be compared with the results of test beams

under pure torsion found in literature.

Table 3 r–e relationship for reinforcement in tension


The same beams used by Bernardo et al. [7, 11] will

be used for the comparative analysis. Not all the

experimental results available in our bibliography can

be used due to various reasons. For instance, some

older studies present a range of dates that is insuffi-

cient for this kind of comparative study or even the test

beams do not meet basic design rules fount in current

codes of practice. In this last case, such beams show

behaviours under torsion very different from normal.

In other experimental studies, including some recent

ones, the authors presented medium twists for the

whole length of the beams, and not the local twists of

the critical section. Theoretical twists based on a cross

section analysis, cannot be compared with such

experimental values of twists (average oven a long

length). This aspect is particularly relevant in slender

beams.

Table 4 summarizes the geometrical and mechan-

ical properties of 28 beams found in our bibliography,

including the external width (x) and height (y) of the

rectangular cross section, the thickness of the walls of

the cross hollow sections (t), the distances between

centerlines of legs of the closed stirrups (x1 and y1), the

total area of longitudinal reinforcement (Asl), the

distributed area of one branch of the transversal

reinforcement (Ast/s, where s is the spacing of

transversal reinforcement), the longitudinal reinforce-

ment ratio (ql = Asl/Ac, with Ac = xy) and the trans-

versal reinforcement ratio [qt = Astu/(Acs), with

u = 2(x1 ? y1)], the average concrete compressive

and tensile strength (fcm � f 0c and fctm), the average

yielding stress of longitudinal and transversal rein-

forcement (flym and ftym), the concrete Young Modu-

lus’s (Ec), the compressive strains for concrete (peak

stress value, e0, and maximum value, ecu). For the

reinforcement, usual values were adopted for maxi-

mum tensile strain (elu = etu = 10 %) and Young’s

Modulus (Es = 200 GPa).

Parameters fctm, Ec, eo, and ecu were computed from

EC2 [30] [Eqs. (21a, 21b)–(24)]. For NSC

(fck = fcm - 8 (MPa) \ 50 MPa), the compressive

maximum strain for concrete, ecu, is constant (3.5 %).

fctm ¼ 0:3ðfckðMPaÞÞ2=3ðMPaÞ if fck� 50 MPa

ð21aÞ

fctm ¼ 2:12 ln 1þ fcmðMPaÞ10

� �ðMPa)

if fck [ 50 MPa

ð21bÞ

Ecm ¼ 22fcmðMPaÞ

10

� �0:3

ðGPaÞ ð22Þ

eo � ec1 ¼ 0:7ðfcmðMPaÞÞ0:31\2:8 ð0=00Þ ð23Þ

Select εds

Estimate td, α, β

Calculate k1

and σd (Eq. (13))

Calculate T (Eq. (1)), εt (Eq. (4)), εl (Eq. (5)), σt and σl (Tab. 3)

Calculate td´ (Eq. (3))

td = td´ ?No

Yes

Calculate α´ (Eq. (2))

α = α´ ?No

Yes

Calculate β´ (Eq. (9))

β = β´ ? No

Yes

Calculate θ (Eq. (6))

εds > εcu ? εl or εt > εsu ?

No

Yes

END

Fig. 2 Flowchart to compute T–h for RC beams (VATM)


ecu � ecu1 ¼ 2:8þ 2798� fcmðMPaÞ

100

� �4

ð0=00Þ if

fck ¼ fcm � 8 ðMPaÞ[ 50 MPa;

ð24Þ

where fck is the characteristic value of the compressive

concrete strength.

Table 4 does not include HSC plain beams. Two

experimental studies were found in our bibliography

with HSC plain beams under torsion: Rasmussen and

Baker [32] and Fang and Shiau [18]. Those beams

were not included in Table 4 by the same reasons

presented above.

6 Modified Variable Angle Truss-Model

(MVATM)

6.1 Zone 1 (non-cracked state)

To correct the formulation of the VATM for Zone 1

(Fig. 3) it is necessary to know the cracking torque

(Tcr) to set the upper limit of the zone for which the

beam is non-cracked.

Several torsional theories to calculate Tcr exist

namely; the Theory of Elasticity, The Skew-Bending

Theory and the Bredt’s Thin Tube Theory. To know

which of these theories provide the best predictions for

Tcr, Bernardo and Lopes [7, 11] performed a compar-

ative analysis based on the experimental results of

several beams found in the literature. These authors

found that the Bredt’s Thin Tube Theory [13] was

appropriate regardless of the section type of the beam

(plain or hollow). The equation to compute Tcr of RC

rectangular hollow sections, based on the Bredt’s Thin

Tube Theory, was proposed by Hsu and Mo [22]:

Tcr ¼ 2Act 2:5ffiffiffiffiffiffiffiffiffiffiffiffiffiffif 0c ðpsiÞ

q� �

¼ 2Act 0:2076

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 0c ðMPaÞ

q� �ð25Þ

In Eq. (25), Ac is the area limited by the outer

perimeter of the section (includes hollow area) and t is

the thickness of the wall.

Hsu [21] showed that Eq. (25) could also be applied

for RC rectangular plain sections by taking t = 1.2Ac/

pc, where pc is the outer perimeter of the cross section.

Bernardo and Lopes [11] showed that Eq. (25) must

be multiplied by 0.85 when f 0c [ 50 MPa:

Experimental studies also show that the torsional

reinforcement (both longitudinal ql and transversal qt)

slightly increases the cracking torque. In 1968, Hsu

[19] proposed an empirical equation to compute the

effective cracking torque (Tcr,ef):

Tcr;ef ¼ 1þ 4ðql þ qtÞ½ �Tcr ð26Þ

Equation (26) will be used in this study to compute

the effective cracking torque of RC beams under

torsion.

The twist h and the torsional stiffness for non-

cracked state will be calculated by using the modified

VATM. For this, the theoretical model should incor-

porate the contribution of the concrete in tension and

the contribution of the concrete core for plain sections

(both neglected in VATM). These options will allow

the theoretical model to simulate the transition to non-

T

θ

Tcr

θcr θly

Tly

(GC) I

(GC)II

1

1

θmax

Zone 1

Zone 3

Zone 2.a

Zone 2.b

I θcrII θn

Tn

Tty ;

θty ;

To

Theoretical curve(VATM)

Typical experimental curve

Tcr = Cracking torque Ιθcr = Twist corresponding to Tcr (non-cracked state) ΙΙθcr = Twist corresponding to Tcr (cracked state)

Tly = Torque corresp. to yielding of long. reinf. θly = Twist corresponding to Tly

Tty = Torque corresp. to yielding of transv.reinf. θty = Twist corresponding to Tty

Tn = Resistance torque θn = Twist corresponding to Tn

θmax = Maximum twist at beam´s failure (GC)I = Torsional stiffness in non-cracked state (GC)II = Torsional stiffness in cracked state

Fig. 3 T–h curve for a RC beam under pure torsion


Ta

ble

4P

rop

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eso

fex

per

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tal

RC

bea

ms

Bea

mS

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on

typ

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x (cm

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y (cm

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t (cm

)

x 1 (cm

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y 1 (cm

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Asl

(cm

2)

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/s(c

m2/m

)

q l(%

)q t

(%)

f cm

(MP

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f ctm

(MP

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f lym

(MP

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Ec

(GP

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(%)

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.70

.31

B4

[9]

11

.25

2.3

53

.63

2.2

15

.10

.90

.97

9.8

4.4

72

46

72

41

.00

.31

B5

[9]

11

.75

1.8

51

.84

0.2

18

.91

.11

.17

6.4

4.3

72

46

72

40

.50

.31

C2

[9]

10

.05

3.2

53

.31

4.0

6.3

0.4

0.4

94

.84

.96

72

69

64

3.2

0.2

20

.28

C3

[9]

10

.35

4.5

54

.02

3.8

10

.50

.70

.69

1.6

4.8

72

47

15

42

.80

.28

C4

[9]

10

.35

4.6

54

.53

0.7

14

.10

.90

.99

1.4

4.8

72

46

72

42

.70

.28

C5

[9]

10

.45

4.0

54

.33

6.7

17

.41

.01

.19

6.7

4.9

67

26

72

43

.50

.27

C6

[9]

10

.45

3.3

52

.94

8.3

22

.61

.31

.38

7.5

4.7

72

47

24

42

.20

.29


cracked state. Such transition is due to the instanta-

neous loss of the contribution of the concrete in

tension.

Firstly, to maintain some consistency with the basic

assumptions of VATM, an equivalent hollow section

will be assumed for the non-cracked state. In this way,

the influence of the concrete core will be introduced

later. For plain sections the recommendations by ACI

318R-05 [1] will be assumed. Such code also assumes

an equivalent hollow section for the non-cracked state,

by defining an equivalent thickness for the wall (heq)

equal to 0.75Acp/pcp, where Acp is the area limited by

the outer perimeter of the section and pcp is the

external perimeter of the section. For hollow sections,

if the real thickness is lesser that heq then the real

thickness will be adopted. The transformation of a

plain section into an equivalent hollow section for the

non-cracked state is illustrated in Fig. 4a.

The equivalent thickness heq will be used to

calculate some properties of the section. For the

computation of the points of the T–h curve, the

variable td (thickness of the struts) will be used in

accordance with the general formulation of VATM to

maintain consistency with the theoretical model. The

same option is assumed for the angle of the concrete

struts (a) although some theories assume a constant

value of 45� for non-cracked state.

In this phase, the non-linear r–e relationships for

the materials are incorporated in the computing

procedure, despite r–e relationships being almost

linear and softening and stiffening effects being

negligible for this low level of loading.

In the non-cracked state, it is also assumed that the

torsional reinforcement has no influence on the

position of the center line of the shear flow. Thus,

the area limited by the center line of the equivalent

thickness of the wall is assumed to be Ao (Fig. 4b):

Ao ¼ ðx� heqÞðy� heqÞ ð27Þ

The new perimeter of the center line of the shear

flow po is calculated as follow (Fig. 4b):

po ¼ 2ðx� heqÞ þ 2ðy� heqÞ ð28Þ

In the non-cracked state, the participation of the

concrete in tension must be introduced in the longi-

tudinal and transversal equilibrium equations of

VATM. For this purpose, the RC section is homog-

enized for both longitudinal and transversal directions

by considering the equivalent thickness heq of the

concrete in tension as effective to carry the forces

along with the torsional reinforcement. Thus, it is

assumed that only concrete around bars will partici-

pate for the longitudinal and transversal equilibrium.

Since the original equilibrium equations of the VATM

are written with respect to the forces in the reinforce-

ment, Al fl for longitudinal reinforcement (Eqs. (2) and

(3) in Table 1) and At ft/s for transversal reinforcement

[Eq. (3) in Table 1)], the area of concrete in tension is

‘‘transformed’’ in an equivalent steel area. Figure 5a

and b illustrates the effective area of concrete in

tension in the longitudinal and transversal direction,

respectively. Thus, the total longitudinal force (Fl,tot)

and the total distributed transversal force (Ft,tot) are

computed as follow:

Fl;tot ¼ Alhfl ¼ Al þ nAcl;eq

� �fl ð29Þ

Ft;tot ¼ Athft=s ¼ At þ nAct;eq

� �ft=s ð30Þ

Acl;eq ¼ xy� x� heq

� �y� heq

� �ð31Þ

Act;eq ¼ s heq ð32Þ

T heq Aº

pº heq

x

y

(a) (b) (c)

Fig. 4 Section for non-cracked state: a equivalent hollow section; b definition of Ao and po; c concrete core


In the above equations, n = Ec/Es is the ratio

between Young’s modulus for concrete and steel, Alh

and Ath are the homogenized steel areas in the

longitudinal and transversal direction, respectively,

Acl,eq and Act,eq are the equivalent area of effective

concrete in tension for the longitudinal and transversal

direction, respectively.

From the original formulation of VATM pre-

sented in Sect. 3 and Table 1, the new theoretical

model for non cracked state incorporates the follow-

ing changes:

• The geometric parameters Ao and po are calculated

using new Eqs. (27) and (28), respectively;

• The equilibrium Eqs. (2) and (3) from Table 1

must be rewritten to incorporate the total

longitudinal force Fl,tot [Eq. (29)] and the total

distributed transversal force Ft,tot [Eq. (30)]. These

forces substitutes the forces in the longitudinal

reinforcement Al fl and in the transversal rein-

forcement At ft/s, respectively. The new equilib-

rium equations are:

cos2 a ¼ Fl;tot

pordtdð33Þ

td ¼Fl;tot

pord

þ Ft;tot

rd

ð34Þ

• The new computing procedure must end when the

torsional moment, calculated with very small

increments, is higher than the effective cracking

torque Tcr,ef from Eq. (26).

Let us now consider the influence of the concrete

core.

For beams with hollow section, a hollow concrete

core must be considered only if the actual thickness of

the walls of the section is much higher than the

equivalent thickness (heq). For example, among

the test beams of Table 4, for the majority of them

the actual thickness of the wall is less than heq. Only

three beams do not pass this condition (Beams D4

[19], VH1 [25] and B5 [9]), although by a very small

margin. In view of this, it was assumed that, for hollow

sections, the influence of any eventual hollow concrete

core would be negligible.

It was assumed that the influence of the concrete

core in plain sections only affected the torsional

stiffness of the beam. It should be remembered that the

Bredt’s Thin Tube Theory provides good results for

Tcr,ef. Since the VATM formulation neglects the

influence of the concrete core, it was decided to add

to the torsional stiffness of the equivalent hollow

section (calculated by the theoretical model) the

torsional stiffness of the concrete core (calculated

separately) in order to correct the final twists hcalculated with the theoretical model for the non-

cracked state. The correction method consists, for each

increment of eds, to perform the following steps:

1. From the computing procedure and for each

increment of eds obtain the value of the twist hand the correspondent torque T;

2. Calculate an equivalent secant torsional stiffness

based on the previous values;

Kt;eq ¼ T=h ð35Þ

3. Using Theory of Elasticity, calculate the torsional

stiffness of the concrete core: Kt,c;

Al fl

T

T

heq

Fl,tot

T

s

T

heq

s

Ft,tot

At ft

s

(a) (b)

Fig. 5 Effective areas of concrete in tension (non-cracked state): a longitudinal direction; b transversal direction


4. Calculate the equivalent total torsional stiffness of

the entire section, Kt,eq,tot, by adding the torsional

stiffness of the concrete core, Kt,c:

Kt;eq;tot ¼ Kt;eq þ Kt;c ð36Þ

5. Calculate the corrected twist hcor, based on

Kt,eq,tot, [Eq. (37)]. This leads to the final

point(hcor; T) for the theoretical T–h curve;

hcor ¼ T=Kt;eq;tot ð37Þ

6. Repeat Steps 1–5 for each point (h; T) in order to

draw the entire T–h curve for the non-cracked state.

This correction method is incorporated in the

computing procedure for plain sections (Fig. 6b).

To calculate the torsional stiffness of the concrete

core it is necessary to define its dimensions. After

some simulations and comparative analysis with

experimental results, it was found that it is necessary

to consider some overlap between the equivalent

hollow section and the section of the concrete core in

order to take into account the real connection between

them. The chosen criterion consisted on considering

an overlap area until the core area reached the center

line of the equivalent thickness of the wall (located at

heq/2 from the outer surface of the real section), as

illustrated in Fig. 4c. Thus, the ‘‘external’’ dimensions

of the rectangular concrete core were x - heq/2 and

y - heq/2.

(b)(a)

Calculate td´ (Eq. (34))

td = td´ ?No

Yes

Calculate α´ (Eq. (33))

α = α´ ?No

Yes

Calculate β´ (Eq. (9))

β = β´ ? No

Yes


T > Tcr,ef ? (Eq (26))

No

Yes

END

T > Tcr,ef ? (Eq (26))

Yes

END

β = β´ ? No

Yes


Calculate Kt,eq (Eq. (35)) and Kt,eq,tot (Eq. (36))

Calculate θcor (Eq. (37))

No

Fig. 6 Flowchart to compute T–h curve (zone 1—non-cracked state)—MVATM: a hollow sections; b plain sections


To calculate the torsional stiffness of the concrete

core (Kt,c = K(GC)), Theory of Elasticity was used.

According to St. Venant’s Theory [34], for a

rectangular section the torsional factor C can be

computed as Xx3y, where x and y are the smallest

and the largest dimension of the rectangular section

(concrete core) and X is a St. Venant’s coefficient.

The shear modulus G is equal to Ec/[2(1 ? m)],

where Ec is the Young’s modulus of concrete and mthe Poisson’s coefficient. Parameter K is a reduction

factor (K & 0.7) to account for a lightly drop of the

torsional stiffness due to micro cracking as observed

in experimental tests before effective cracking is

reached [7]. Assuming m = 0.2 for non cracked

state, Kt,c = 0.292EcC.

A new iterative computing procedure was imple-

mented to compute the T–h curve for non-cracked

state (Zone 1 of Fig. 3). The new theoretical model is a

modification of the VATM, and it was named

MVATM. The new flowchart for the calculation

algorithm is illustrated in Fig. 6a (hollow sections)

and Fig. 6b (plain sections). Only the new part of the

flowchart is presented in Fig. 6. The computing

procedure of Fig. 6 was implemented computationally

with the programming language Delphi.

To check the validity of MVATM for non-cracked

state, a comparative analysis between theoretical and

experimental results of test beams was performed. The

secant torsional stiffness was analyzed. To calculate

this parameter from the theoretical T–h curve, Eq. (38)

was used, (hcr is the twist correspondent to Tcr,ef).

Kt ¼ KðGCÞ ¼ Tcr;ef

hcr

ð38Þ

Tables 5 and 6 present, for each test beam (plain

and hollow section, respectively), the results obtained

for the torsional stiffness in non-cracked state. The

results for plain and hollow sections were treated

separately due to the concrete core influence. Each

table presents, for each test beam, the experimental

value of torsional stiffness (Kt,expI ) and the correspon-

dent theoretical value (Kt,thI ). The ratios Kt,exp

I /Kt,thI are

also presented, as well as the average value xmean, the

sample standard deviation s and the coefficient of

variation cv. A bar graph is also presented for a visual

analysis of the dispersion of the results.

The analysis of Table 5 for plain sections shows that

the criterion to introduce the influence of the concrete

core seems to be appropriate. The predictions of the

torsional stiffness are very good (xmean & 1) and the

dispersion is acceptable (cv = 9 %, see graph).

The results in Table 6 for hollow sections show that

the theoretical torsional stiffness is close to the

experimental one (xmean = 1.17), although the disper-

sion is somehow relevant (cv = 16 %, see graph).

Since the values of the twist are very small in the

non-cracked state, it is expected that the observed

dispersion of the results is less important.

Table 5 Comparative analysis for the torsional stiffness in non-cracked stage (plain sections)


6.2 Zone 2.a (cracked state)

The simulation of the Zone 2.a of the T–h curve

(Fig. 3) by using the MVATM computing procedure

of Sect. 6.1 will be based on the cracking of the

concrete. When the effective cracking torque (Tcr,ef) is

reached, the effective concrete in tension is no longer

considered for the equilibrium in the longitudinal and

transversal direction. This will cause an instantaneous

increase of the twist, as observed experimentally.

For hollow beams, experimental observations show

that Zone 2.a is not perceptible [9]. Therefore, for

these beams, it is expected that the length of the Zone

2.a is very small, as observed by Bernardo and Lopes

[7, 11] by using Hsu’s model for torsion in cracked

state [20].

For plain beams the influence of the concrete core

persists beyond the cracking point, at least for some

levels of loading not much higher than the cracking

load [7].

It should be noted that the equivalent hollow

section (with heq) for non-cracked state is no longer

valid. After cracking, the effective wall thickness will

be attributed to the td parameter (effective thickness of

the concrete struts). Then, the center line of the shear

flow will be assumed as located at td/2, as assumed by

VATM (Table 1).

In order to simulate Zone 2.a, MVATM for non-

cracked state (Sect. 6.1) will incorporate two changes:

• The geometric parameters Ao and po are calculated

as for VATM (see Table 1);

• Equations (33) and (34) are replaced by Eqs. (2)

and (3) (the force in the longitudinal and transver-

sal direction are totally carried by the longitudinal

and the transversal reinforcement);

• The T–h curve for cracked state is calculated with

MVATM that incorporates the above changes, but

only the results for Tcr,ef are retained, since the only

objective of this section is to calculate Zone 2.a.

The iterative computing procedure of Sect. 6.1 is

modified to compute the T–h curve for the cracked

state and for Tcr,ef. The flowchart for the calculation

algorithm to compute the new T–h curve for cracked

state is the same as illustrated in Fig. 6a (hollow

sections) and Fig. 6b (plain sections), with t0d and a0

calculated through Eqs. (3) and (2), respectively.

The key points of this phase are illustrated in Fig. 7.

Zone 2.a corresponds to a horizontal line, with

T = Tcr,ef, and defined by the range of twist

hcrI B h B hcr

II . The value of hcrI corresponds to the

abscissa of the intersection point between the hori-

zontal level for T = Tcr,ef and the theoretical T–hcurve computed with MVATM for non-cracked state

Table 6 Comparative analysis for the torsional stiffness in non-cracked stage (hollow sections)


(Sect. 6.1). The value of hcrII corresponds to the abscissa

of the intersection point of the horizontal line with the

theoretical T–h curve computed with MVATM for

cracked state.

The new computing procedure was implemented

with programming language Delphi in order to

calculate the theoretical points of the T–h curve for

cracked state (Zone 2.a).

Table 7 is similar to Tables 5 and 6 and summarizes

the results obtained for the length of Zone 2.a,

Dh = hcrII - hcr

I , only for the plain beams for which it

was possible to calculate Dh from experimental results.

Table 7 presents the experimental values of Dh (Dhexp)

and the corresponding theoretical values (Dhth).

For hollow beams, since Dh is not observed in

experimental tests, a comparative analysis with

theoretical previsions is not possible. However, it

was found that for the hollow beams that were studied,

the Dhth values were very small, as would be expected.

Despite the limited number of beams, the results

from Table 7 show that the length of Dh is not

reasonably predicted. The dispersion of the results is

high. This shows that the physical phenomena

involved in Zone 2.a is complex to simulate. However,

the values of twists are very small in this state, and

consequently, this dispersion may not be considered

very important.

6.3 Zones 2.b and 3 (cracked state and ultimate

state)

As discussed in Sect. 2, some previous studies [3, 22]

show that the theoretical T–h curve computed with

VATM does not fit accurately to the correspondent

experimental T–h curve for Zone 2.b. The differences

are particularly pronounced in plain beams. For these

beams, the deviations can be explained by the

influence of the concrete core in the torsional stiffness

after cracking [7] (such influence is neglected in

theoretical VATM model). These deviations can be

seen in Fig. 3.

By using MVATM, the theoretical T–h curve for

cracked state presented in Sect. 6.2 cannot be used to

draw the entire T–h curve for Zone 2.b and 3, since the

referred model incorporates the fully influence of the

concrete core for plain sections. Bernardo and Lopes

[7, 11], based on theoretical simulations of Zone 2.b

by using the torsional Hsu’s model for cracked state

[20] found that the influence of the concrete core into

the cracked torsional stiffness of the beam decreases as

Table 7 Comparative analysis for the length of Subzone 2.a in cracked stage (plain sections)

T

θ

Tcr

θcr

2.a

I θcrII

MVATM for non-craked state

MVATM for craked state

Fig. 7 Theoretical modelation of Zone 2.a (plain sections)


the torsional moment (from Tcr,ef) increases. This

influence is residual in the ultimate behaviour. For

plain beams, it is therefore necessary to adopt a

criterion to correct the torsional stiffness, calculated

with MVATM for cracked state (Sect. 6.2), from Tcr,ef

to maximum toque Tn in order to take into account the

decrease of the concrete core influence. The correction

method for plain beams consists as follows:

• Calculate the entire T–h curve from Tcr,ef with

MVATM for cracked state (Sect. 6.2). The

torsional moments of this T–h curve can be

considered as good values, because it is assumed

that the concrete core only influences the torsional

stiffness. Then, only the twists h must be affected

by the correction method;

• From Tcr,ef to Tn the influence of the concrete core

is gradually reduced by reducing linearly the

‘‘external’’ dimensions of the concrete core (con-

sidering the full area for Tcr,ef and a null area for

Tn). Then, from point (hcrII ; Tcr,ef) all the points of

the T–h curve will be corrected in the h axis.

The criterion to correct the T–h curve of plain

beams is illustrated in Fig. 8b. This figure shows that

the correction method consists on adding increasing

twists from Tcr,ef to Tn, assuming a linear variation.

This technique simulates the decreasing of stiffness

due to the loss of influence of the concrete core.

For hollow beams, all the T–h curve is firstly

calculated with MVATM for cracked state (Sect. 6.2).

Since no concrete core exists and since a little

theoretical Zone 2.a (Dh) exists, a correction needs

to be implemented because Zone 2.a is not experi-

mentally observed for such beams. All the points of

the T–h curve, starting from point (hcrII ; Tcr), will be

translated to the left by Dh. The criterion to correct T–hcurve of hollow beams is illustrated in Fig. 8a.

It should be pointed that this correction is not

relevant since it was observed that the theoretical Dhvalues for hollow beams are very small.

The correction technique described above was

implemented into the computing procedure of

MVATM for cracked state in order to calculate the

final points of T–h curve above Tcr,ef.

The theoretical failure of the section (last point of

the T–h curve) was defined from the maximum strains

of the materials (concrete and steel). The strain of the

concrete struts, eds (Fig. 1), reaches its maximum

value ecu (Table 4) and the steel strain, es, reaches the

value of esu = 0.01.

Tables 8, 9, 10 and 11 summarize the values of the

torsional stiffness in cracked state in Zone 2.b (KII)

and of the ordinate at the origin (ToII) of the straight line

representing the equivalent torsional stiffness in

cracked state in Zone 2.b. The results for plain and

hollow sections are separated due to the concrete core

influence. KII and ToII were obtained by calculating a

straight line by linear regression of the theoretical and

experimental T–h curve points corresponding to Zone

2.b. Only the points located along the T–h curve with

T

θ

Tcr,ef

θcrI θcr

II θn

TnTmax ≡

θ y

Ty

θy,cor

VATM non-cracked state

θn,cor

VATMcracked stateVATM corrected

Δθcor

T

θ

Tcr,ef

θcrI θcr

II θn

TnTmax ≡

θ y

Ty

θy,cor

VATM non-cracked state

θn,cor

VATMcracked state

(Final curve)

(final curve)

Δθ Δθ Δθ

Δθ

Δθ(a) (b)

Fig. 8 Correction of T–h curves (Zone 2.b): a hollow sections; b plain sections


an approximately straight development were consid-

ered. While parameter KII measures the slope of the

T–h curve in the Zone 2.b, parameter ToII (see Fig. 3)

measures the ‘‘position’’ of the T–h curve in the graph.

Tables 8, 9, 10, and 11 present the experimental value

of the torsional rigidity in cracked state (KexpII ) and

the ordinate at the origin (To,expII ) as well as their

correspondent theoretical values (KthII and To,th

II ).

The analysis of Table 8 (plain sections) shows that

the prediction of torsional rigidity in cracked state

(KII) is quite good (xmean close to unity), although the

dispersion of the results is not negligible (cv � 10 %)

mainly because of Beam G6 [19] (see graph). The

analysis of Table 10 (plain sections) shows that the

ordinate at the origin (To) is slightly underestimated

(xmean = 0.87). The levels of dispersion of the results

are acceptable (cv = 13 %, see graph). The analysis

of Table 9 (hollow sections) shows that the torsional

stiffness in cracked state (KII) is slightly underesti-

mated (xmean = 0.85). The dispersion of results are

much more acceptable when compared with those

observed for plain sections. Finally, the analysis of

Table 8 Comparative analysis for the torsional stiffness in cracked stage (plain sections)

Table 9 Comparative analysis for the torsional stiffness in cracked stage (hollow sections)


Table 11 (hollow sections) shows that the predictions

of the ordinate at the origin (To) are very good (xmean

close to unity) and the dispersion levels of the results is

acceptable.

Tables 12 and 13 present the experimental values

of the yielding point (only for beams with ductile

behaviour) and ultimate values (Tly,exp, hly,exp, Tty,exp,

hty,exp, Tn,exp, hn,exp) and their correspondent theoret-

ical values (Tly,th, hly,th, Tty,th, hty,th, Tn,th, hn,th). The

results for plain and hollow sections are now presented

together since it is assumed that in the ultimate state

(Zone 3 of Fig. 3) the influence of concrete core is

negligible.

The results show that the predictions of torsional

moments (Tly, Tty and Tn) are very good, mainly for Tn,

being the dispersion of the results satisfactorily low

except for Tly. This is not a surprise since VATM gives

good predictions for the ultimate behaviour, as

Table 10 Comparative analysis for the ordinate at the origin (plain sections)

Table 11 Comparative analysis for the ordinate at the origin (hollow sections)


discussed in Sect. 2. For the correspondent twists (hly,

hty and hn) the results show that these parameters are

acceptable but dispersion of the results is high

(cv � 10 %).

7 Comparative analysis with T–h curves

Based on the previous Sect. 6, a final computing

procedure was implemented with the programming

language DELPHI. The computer application [4] gives

the full theoretical T–h curve of RC beam under torsion.

Figures 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,

21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35

and 36 present the T–h curves concerning the test

beams analyzed in Sect. 6. These figures include the

experimental curve, the theoretical curve from

MVATM and the theoretical curve from the global

model by Bernardo and Lopes [7, 11] (Sect. 2) for

comparison.

Figures 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,

21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35

and 36 include in each T–h curve some key points,

Table 12 Comparative analysis for the yielding parameters


Table 13 Comparative analysis for the ultimate parameters


θ

0

5

10

15

20

25

30

35

0,0 1,0 2,0 3,0 4,0 5,0

T [k

N.m

]

[º/m]

experimentalMVATMBern. & LopesTcr,efTtyTlyTn

25,40

38,10

ρl = 0,83 %

tρ = 0,82 %fcm = 28,6 MPa

(cm)

Fig. 9 T–h curves—Beam B2 [19]

θ

0

5

10

15

20

25

30

35

40

45

T [k

N.m

]

[º/m]


25,40

38,10

ρ l = 1,17 %

tρ = 1,17 %fcm = 28,1 MPa

(cm)

0,0 1,0 2,0 3,0 4,0


θ

0

10

20

30

40

50

60

T [k

N.m

]

[º/m]

experimentalMVATMBern. & LopesTcr,efTtyTlyTn25,40

38,10

ρ l = 1,60 %

tρ = 1,62 %fcm = 29,2 MPa

(cm)

0,0 1,0 2,0 3,0 4,0 5,0


θ

0

10

20

30

40

50

60

0,0 2,0 4,0 6,0

T [k

N.m

]

[º/m]


25,40

38,10

ρ l = 2,11 %

tρ = 2,04 %fcm = 30,6 MPa

(cm)


θ

0

5

10

15

20

25

30

35

40

45

T [k

N.m

]

[º/m]


25,40

50,80

ρ l = 0,60 %

tρ = 0,59 %fcm = 29,9 MPa

(cm)

0,0 2,0 4,0 6,0

Fig. 13 T–h curves—Beam G6 [19]

θ

0

10

20

30

40

50

60

70

80

T [k

N.m

]

[º/m]


25,40

50,80

ρ l = 1,32 %

tρ = 1,31 %fcm = 28,4 MPa

(cm)

0,0 1,0 2,0 3,0 4,0 5,0

Fig. 14 T–h curves—Beam G8 [19]


θ

0

5

10

15

20

25

30

35

40

45

0,0 1,0 2,0 3,0 4,0

T [k

N.m

]

[º/m]


25,40

38,10

ρ l = 1,17 %

tρ = 0,78 %fcm = 30,6 MPa

(cm)

Fig. 15 T–h curves—Beam M2 [19]

θ

0

20

40

60

80

100

120

140

160

T [k

N.m

]

[º/m]


50,00

50,00

ρ l = 0,72 %

tρ = 0,75 %fcm = 35,3 MPa

(cm)

0,0 1,0 2,0 3,0 4,0

Fig. 16 T–h curves—Beam T4 [24]

θ

0

5

10

15

20

25

30

35

T [k

N.m

]

[º/m]


25,40

25,40

ρ l = 1,24 %

tρ = 1,22 %fcm = 39,4 MPa

(cm)

0,0 2,0 4,0 6,0 8,0

Fig. 17 T–h curves—Beam A3 [27]

θ

0

5

10

15

20

25

0,0 2,0 4,0 6,0 8,0

T [k

N.m

]

[º/m]


35,56

ρ l = 0,82 %

tρ = 0,98 %fcm = 39,7 MPa

(cm)


θ

0

5

10

15

20

25

30

T [k

N.m

]

[º/m]


35,56

ρ l = 1,27 %

tρ = 1,26 %fcm = 38,6 MPa

(cm)

0,0 2,0 4,0 6,0 8,0


θ

0

5

10

15

20

25

30

35

T [k

N.m

]

[º/m]


35,56

ρ l = 1,80 %

tρ = 1,73 %fcm = 38,5 MPa

(cm)

0,0 2,0 4,0 6,0 8,0



θ

0

10

20

30

40

50

60

0,0 1,0 2,0 3,0 4,0

T [k

N.m

]

[º/m]


38,10

l = 1,60 %

tρ = 1,62 %fcm = 30,6 MPa

(cm)

6,35

Fig. 21 T–h curves—Beam D4 [19]

θ

0

20

40

60

80

100

120

140

160

T [k

N.m

]

[º/m]


50,00

l

tρ = 0,75 %fcm = 35,4 MPa

(cm)

8,00

0,0 1,0 2,0 3,0 4,0

Fig. 22 T–h curves—Beam T1 [24]

θ

0

5

10

15

20

25

T [k

N.m

]

[º/m]


32,40

32,40

l = 0,33 %

tρ = 0,30 %fcm = 17,2 MPa

(cm)

6,50

0,0 1,0 2,0 3,0 4,0

Fig. 23 T–h curves—Beam VH1 [25]

θ

0

50

100

150

200

250

300

0,0 1,0 2,0 3,0 4,0 5,0

T [

kN.m

]

[º/m]


60,00(cm)

60,0010,00

= 0,37 %= 47,3 MPa

= 0,39 %

tρf

l

cm


θ

0

50

100

150

200

250

300

350

T [

kN.m

]

[º/m]

experimentalMVATMBern. & LopesTcr,efTtyTlyTn60,00(cm)

60,0010,00

= 0,49 %= 46,2 MPa

tρf

l

cm

0,0 1,0 2,0 3,0


θ

0

50

100

150

200

250

300

350

400

450

T [

kN.m

]

[º/m]


60,0010,00

= 0,65 %= 54,8 MPa

tρf

l

cm

0,0 1,0 2,0 3,0



θ

0

50

100

150

200

250

300

350

400

450

0,0 0,5 1,0 1,5 2,0 2,5

T [k

N.m

]

[º/m]


60,0010,00

= 0,83 %

= 53,1 MPa

= 0,85 %

tρf

l

cm


θ

0

50

100

150

200

250

300

T [k

N.m

]

[º/m]


60,00

= 0,40 %= 69,8 MPa

10,00

tρfcm

l

0,0 1,0 2,0 3,0 4,0


θ

0

50

100

150

200

250

300

350

400

450

T [k

N.m

]

[º/m]


60,00

= 0,67 %= 77,8 MPa

10,00

tρfcm

l

0,0 1,0 2,0 3,0


θ

0

100

200

300

400

500

0,0 0,5 1,0 1,5 2,0

T [

kN.m

]

[º/m]


60,00

= 0,89 %= 79,8 MPa

10,00

tρfcm

l


θ

0

100

200

300

400

500

600T

[kN

.m]

[º/m]


60,00(cm)

60,00

= 1,09 %= 76,4 MPa

10,00

tρfcm

l

0,0 0,5 1,0 1,5 2,0


θ

0

50

100

150

200

250

300

T [

kN.m

]

[º/m]


= 0,37 %= 94,8 MPa

10,00

tρfcm

l

(cm) 60,00

60,00

0,0 1,0 2,0 3,0 4,0

Fig. 32 T–h curves—Beam C2 [9]


namely the points correspondent to cracking torque,

yielding torque and maximum torque.

The analysis of Figures 9, 10, 11, 12, 13, 14, 15, 16,

17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,

32, 33, 34, 35 and 36 shows that the theoretical T–hcurves obtained from MVATM are generally very

close to the correspondent experimental curves. This

confirms the global conclusions presented in Sect. 6

and validates the MVATM. Beam T4 [24] (Fig. 16) is

an exception. For this beam, high deviations between

the theoretical and experimental curves were

observed. These deviations result from the large

difference between the theoretical and experimental

cracking torque.

8 Conclusions

The principal objective of this study was to generalize

the Space-Truss Analogy, by using the VATM

formulation, in order to make it able to predict the

entire T–h curve of RC beams under torsion, and not

only the points corresponding to the ultimate behav-

iour. This attempt consisted on implementing some

modifications on the VATM formulation in order to

make it capable of covering low levels of loading.

Based on this work, a global computing procedure

was established and implemented with programming

language DELPHI. The new model allows the calcu-

lation of the full theoretical T–h curve for plain or

hollow RC beams under torsion (both NSC and HSC

beams are included).

θ

0

50

100

150

200

250

300

350

400

450

0 ,0 0 ,5 1 ,0 1 ,5 2 ,0 2 ,5

T [

kN.m

]

[º/m]


= 0,37 %= 94,8 MPa

10,00

tρfcm

l

(cm) 60,00

60,00


θ

0

50

100

150

200

250

300

350

400

450

500

T [k

N.m

]

[º/m]


= 0,86 %= 91,4 MPa

10,00

tρfcm

l

(cm) 60,00

60,00

0,0 0,5 1,0 1,5 2,0 2,5


θ

0

100

200

300

400

500

600

T [k

N.m

]

[º/m]


= 1,05 %= 96,7 MPa

10,00

tρfcm

l

(cm) 60,00

60,00

0 ,0 0 ,5 1 ,0 1 ,5 2 ,0 2 ,5


θ

0

100

200

300

400

500

600

700

0,0 0,5 1,0 1,5 2,0

T [

kN.m

]

[º/m]


= 1,34 %= 87,5 MPa

10,00

tρfcm

l

(cm) 60,00

60,00



It was shown that MVATM provides very good

results when its predictions are compare with exper-

imental results of RC beams under torsion and with

another previous model from Bernardo and Lopes

[7, 11].

Compared with the original VATM from Hsu and

Mo [22], the MVATM, by incorporating the influence

of the tensile concrete for the non-cracked state and also

the influence of the concrete core (plain sections) for

low loading levels, is able to predicts the full theoretical

T–h curve. As referred in Sect. 2, the VATM is only able

to predict the ultimate zone of the T–h curves (as

illustrated in Fig. 3), because only for high level loading

the concrete is extensively cracked and the concrete

core of plain sections no longer influences the torsional

stiffness of the beams. The VATM approaches the real

model only for the ultimate state, because VATM

neglects both the influence of the tensile concrete and

the influence of the concrete core. From this point of

view, the MVATM constitutes a more global model for

RC beams under torsion when compared with the

VATM.

When compared with the previous model from

Bernardo and Lopes [7, 11], the full theoretical T–hcurves obtained from MVATM are also generally very

close to the correspondent experimental curves. The

results from both theoretical models are generally very

similar (see Sect. 7). However, as referred in the Sect.

2, the theoretical approach from Bernardo and Lopes

[7, 11] was firstly performed by characterizing the

different behavioural states by using different tor-

sional theories. Then, to make the transition between

the different theoretical states, Bernardo and Lopes

adopted semi-empirical criteria in order to draw the

full T–h curve.

Despite the good predictions from the model of

Bernardo and Lopes [7, 11], it can be state that

MVATM is theoretically more acceptable since this

model is mainly based in one torsional theory (based

on a modification of the VATM).

The MVATM represents an important advance in

the attempt to generalize the Space-Truss Analogy in

order to make it able to predict the overall behaviour of

RC beams under torsion.

Such work is important before proceeding to a more

advanced of the model for other important studies

cases, such as special beams under torsion and beams

under interaction of forces.

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