ed

Available online 27 March 2014

Keywords:T-beamsCombined loading

itte

on reinforced (RC) beams under exure, shear or torsion. The behavior of inverted T-shaped beams withboth web and ange closed stirrups are not fully explored. In this research paper, an innovative test setup

d T-sh

was studied by others [2,3]. Moreover, inverted T-shaped beamsare subjected to signicant torsional moments. Thus, these beamsmust be designed to resist signicant torsional moment combinedwith shear forces. In 1998, the ASCE-ACI Committee 445 on shearand torsion identied integrating and designating a physical signif-icance for the torsion design provisions, as well as reviewing com-bined shear and torsion, as an upcoming challenge [1]. Modeling of

and shear force,vely insignicant.load appl

Although diagonal tension cracks occur in RC beamstorsion or shear, the behavior of RC beams due to torsion isent from that under shear. In the case of shear forces, thepropagate in the same direction on both sides of the beam parallelto the applied shear plane. In case of torsion, the cracks follow aspiral pattern, propagating in opposite directions on the oppositesides of the beam. In addition, the assumptions used in modelingRC beams under shear are different from that used for modelingthose under torsion. In the case of shear forces, stresses are as-sumed to be in the plane of the applied shear and uniform across

Corresponding author.E-mail addresses: diffalaf@mcmaster.ca, ahmed.deifalla@bue.edu.eg (A. Deifalla).

Engineering Structures 68 (2014) 5770

Contents lists availab

g

lseshear, torsion, or a combination of these failure modes. In additionto the conventional modes of failure, inverted T-shaped beamscould fail due to other local causes such as hanger failure in theweb, cantilever action, or punching shear in the ange, which

This segment is subjected to signicant torsionwhile the values of bending moments are relatiIn addition, it is far from the local effect of themechanism.http://dx.doi.org/10.1016/j.engstruct.2014.02.0110141-0296/ 2014 Elsevier Ltd. All rights reserved.ication

due todiffer-cracksused as the main girders that support the lateral secondary precastbeams or slabs which is one of the popular structural systems formany existing bridges and parking garages as shown in Fig. 1.The behavior of inverted T-shaped beams is more complicated thanthat of conventional either rectangular or T-shaped RC beams.Conventional rectangular and T-shaped RC beams fail in exure,

differently, especially for cases that involve signicant torsion,which was indicated by many researchers [46]. Thus, a uniedpractical solution is required for the analysis of these invertedT-shaped beams. Fig. 3 shows a typical inverted T-shaped beamloading and forces. The test setup simulates the behavior at theinection point with zero bending moment highlighted in Fig. 3.TorsionShearGlobal behaviorFlange stirrup

1. Introduction

Reinforced concrete (RC) invertecapable of simulating the behavior of inverted T-shaped beams under combined shear and torsion wasdeveloped and implemented. The behavior of three inverted T-shaped beams tested under differentvalues for the ratios of the applied torque to the applied shear force is discussed. The value of the torqueto shear ratio signicantly affects the behavior of the inverted T-shaped beams in terms of cracking pat-tern; failure mode; strut angle of inclination; cracking and ultimate torque; post-cracking torsional rigid-ity; cracking and ultimate shear; ange and web stirrup strain. The ange stirrup is more efcient inresisting torsion moment over shear forces. A model capable of predicting the behavior of anged beamsunder combined actions was developed and implemented. The model showed good agreement with theexperimental results from three different experimental studies.

2014 Elsevier Ltd. All rights reserved.

aped beams are being

un-cracked anged beams is more complex than that of rectangu-lar beams as shown in Fig. 2. Conventional Design Codes approachthe design of RC beams subjected to combined shear and torsionRevised 19 February 2014Accepted 20 February 2014

torsion as well as giving physical signicance for torsion design as an upcoming challenge (ASCE-ACICommittee 445 on shear and torsion, 1998). Most of the previous experimental studies were focusedBehavior and analysis of inverted T-shapand torsion

A. Deifalla a,, A. Ghobarah baBUE, EL-Shourouk City, Postal No.11837, P.O. Box 43, EgyptbMcMaster University, Hamilton, Ontario, Canada L8S 4L7

a r t i c l e i n f o

Article history:Received 23 July 2013

a b s t r a c t

The 1998 ASCE-ACI Comm

Engineerin

journal homepage: www.eRC beams under shear

e 445 on shear and torsion identied researching combined shear and

le at ScienceDirect

Structures

vier .com/ locate /engstruct

Nomenclature

Ac the gross concrete cross section areaAo the area enclosed inside the center of the shear ow

loopAs area of each bar (j)Ec Youngs modulus of the concretef 0c the compressive strength of concretefy the yield stress of the steeli panel numberk number of concrete strips for moment calculationsl the length of the panel (i) parallel to the shear planem number of steel barsMx moment around the x-axisN the applied axial force on the cross sectionNvk the shear contribution from each panel (i)Pc the perimeter of the concrete cross sectionPo the perimeter of the centerline of the equivalent thin

tubeq the average shear ow of the panel (i)qs the shear ow due to the shear force (V)qt the uniform shear ow on the panel due to the torsionT the applied torsion moment on the whole cross sectionV the applied shear force on the whole cross sectionTi the applied torsion moment on the rectangular sub-

divisiont the effective thickness of each element resisting both

shear and torsionts the thickness of the element resisting the shear force (Vi)tt the thickness of the element resisting the torsional mo-

ment (Ti)Vi the applied shear force of each rectangular sub-division

(i)yci distance between the elastic centroid and the centroid

of each concrete panel (i)

ysj distance between the elastic centroid and the center ofeach bar (j)

ysk distance between the elastic centroid and the panel (i)centroid

t the shear stress/d the curvature in the direction of angle h/L the longitudinal curvature/t the transversal curvatureb1 softening coefcient of the concrete stressb2 strain softening coefcientc the shear strain of each panel (i)DAci the area of the stripe0c concrete strain at the peak stresse1 the principal average tension concrete straine2 the average principal compression straine2s the maximum compression principal strain at the sur-

face of the concreteex the average longitudinal strainey the average transverse strainh the inclination angle of the principal strainsqh the ratio of the transverse steel per unit length of the

span to the gross area of the concrete cross sectionr0ci the concrete stress at the centroid of the stripr0sj stress in the steel longitudinal reinforcement for each

bar (j)r1 the principal average tension stressesr2 the principal average compression stressesrst the steel reinforcement stressrx the average longitudinal stressry the average transverse stressesui the curvature for each panelW the twist rate

(a)

(b) (c)Fig. 1. Examples of inverted T-shaped beams under signicant torsion.

58 A. Deifalla, A. Ghobarah / Engineering Structures 68 (2014) 5770

eerin(b)

A. Deifalla, A. Ghobarah / Enginthe perpendicular plane to it. In the case of torsion, the diagonalconcrete compression strain is assumed to vary linearly acrossthe assumed effective thickness of the walls of the cross-sectiondue to lateral curvature that eventually causes the variation ofthe stress across the section, both vertically and horizontally[35]. In addition, according to the theory of hollow-tube space-truss analogy, the effective thickness of the tube varies based onthe applied torque, similar to the variation of the effective depthof the beam with the bending moment [35]. In theory, the concreteweb and the steel web stirrup carry most of the shear. However,the torsional moment must be distributed between the web andthe ange, which can vary based on the dimensions and reinforce-ments of the section.

In this research study, an experimental programwas conducted.An innovative test set-up that allows the beams to fail due to com-bined shear and torsion accompanied by relatively low levels ofbending moments, was developed and constructed. In addition,

were the effect of the torque to shear ratio on the behavior of the

(c)Fig. 2. R and T-shaped beams under torsion (a)

Fig. 3. Typical inverted T-beam loading and internal forces.RC inverted T-shaped beams subjected to shear, torsion, and anunavoidably small bending moment. In addition, a previous analyt-ical model developed by the authors was extended to predict thefull shear and torsional behavior of the inverted T-shaped beams.

2. Research signicance and previous work

The 1998 report by the ASCE-ACI Committee 445 on shear andtorsion outlined the challenges of reviewing RC beams under com-bined shear and torsion and integrating and designating a physicalsignicance for current torsion design provisions [1].

Behavior of RC inverted T-shaped beams, despite its frequentuse since the 1950s, remained as one of the least investigated untilthe test setup is capable of applying different shear to torsion ra-tios by varying the ratio between the applied loads. Three invertedT-shaped beams were designed, constructed, and tested while sub-jected to various torque to shear ratios. The tested beams repre-sented a scaled concrete inverted T-shaped beam model. Theinverted T-shaped beams were tested under torque to shear ratiosof 0.5 m, 1.0 m and 0.1 m while being referred to as TB1, TB2, andTB3, respectively. The parameters investigated by the test program

(a)

isometric; (b) uncracked and (c) cracked.

g Structures 68 (2014) 5770 59mid-1980s [2,3]. Until that time, no guidance for handling designissues specically those associated with the inverted-T sectionwas available in design standards. Therefore, engineers havetended to rely on personal judgment and discretion for design ofthese beams.

A careful examination of existing literature has shown the fol-lowing: (1) very valuable contributions concerning the behaviorof RC beams under combined shear and torsion were made byseveral researchers [29,11,24,17]. However, these studies focusedon rectangular beams rather than T-shaped beams with angestirrups; (2) pioneering works on the behavior of T-shaped beamswere conducted by several researchers [3639,2,34,40,41,3,26,27];however, they all focused on T-shaped beams under pure shear,pure bending, pure torsion, combined shear and moment, or com-bined moment and torsion. In addition, many recent investigationswere concerned with spandrel L-shaped beams [12,28,4245].Kaminski and Pawlak indicated that, despite all the extensive re-search conducted in the area of beams under combined torsion,not all the questions were answered. In addition, it was pointedout that the behavior of RC beams with a cross section other thanrectangular or circular is yet to be explored [45].

Experimental testing remains the most reliable research ap-proach compared to the use of numerical models. The tests provide

A milestone point in the analysis of RC beams under combined

eling of RC beams under torsion [19,24,25]. Their work focused on

eerincomprehensively examining previous experimental and analyticalmodels to verify and improve existing analytical models. Ulti-mately, a modied version of the Variable Angle Truss-Modelby Hsu and co-workers [34,16,22] that is capable of predictingthe behavior of the beams for all loading stages was presented.Moreover, they indicated that the next step would be dealing withspecial beams under combined straining actions.

The behavior of RC inverted T-shaped beams is different fromRC rectangular beams. The cross-section shape can have a signi-cant effect on the behavior and design, as shown by severalresearchers [26,27,5,31,46]. In addition, the inverted T-shapedbeams with ange stirrups are an asymmetricaly-reinforced sec-tion. Moreover, there is no unied approach for the design of RC in-verted T-shaped beams under combined loading. The rst step inreaching a unied approach is to conduct an experimental programin order to identify the signicance of the contribution of variousparameters to the behavior.

3. Testing inverted T-shaped beams

3.1. Scale model for the inverted T-beam

The concrete dimensions of the tested beams were chosen ashalf-scaled model for a commonly used precast inverted T-shapedbeam [47] or a typical 700 mm girder monolithically cast with a200 mm slab. Since the study focused on the effect of the torqueto shear ratio on the behavior, beams were heavily reinforced inthe longitudinal direction to minimize the effect of exure on thebehavior of the tested beams. The stirrups were designed accord-ing to the CSA [48]. The concrete dimension and steel reinforce-ments were kept the same for all tested beams.

3.2. Specimen details

All of the test beams had a total depth of 350 mm, a angethickness of 100 mm, a ange width of 450 mm, and a web widthof 150 mm. Fig. 4b shows a typical cross-section of the beam with-in the test region. The concrete cover was 25 mm for the web and15 mm for the ange. Fig. 4d shows a typical longitudinal sectionshear and torsion was the work presented by both Hsu, and Rahaland Collins [34,17]. Hsu presented a unied theory for combinedshear and torsion Softened Truss Model that was based on: (1)equilibrium equations; (2) compatibility equations; (3) the soft-ened constitutive laws of concrete [34]. Rahal and Collins [17] up-dated the existing space truss model to include; (1) concretesoftening; (2) tension stiffening; (3) improved modeling for thecover spalling; and (4) an equivalent uniform stress distributionblock for the concrete strut. Another key point in the history ofRC beams under combined actions was the work by Greene andBelarbi [21]. They presented a Combined-Action Softened TrussModel, which was based on the Softened Truss Model by Hsuand Mo for pure torsion with improvements over existing models[17,16]. More recently, Bernardo and co-workers studied the mod-physical knowledge and information about the behavior of the sys-tem studied [53,54]. Moreover, test results are essential in verify-ing analytical models such as (1) the skew bending theorymodels based on an inclined plane failure [713]; (2) the space hol-low tube truss models [1418,6,1925]; (3) the nite element andthe nite difference numerical models [2628]; and (4) the empir-ical models developed by tting experimental data [2933].

60 A. Deifalla, A. Ghobarah / Enginof the beams and the reinforcements. All transversal and longitudi-nal reinforcements were ribbed steel bars. The longitudinal rein-forcement is 420 M (i.e. 4 bars 20 mm diameter) at the bottomof the web and 215 M + 410 M (i.e. 2 bars 15 mm and 4 bars10 mm diameter) in the ange. The transverse reinforcement wasdetermined to be 10 M @ 170 mm (i.e. 10 mm stirrup every170 mm). The clear length of the central region was 1400 mm, asshown in Fig. 4, to ensure that at least one complete spiral crackwould occur within the central region. At the two ends of the testregion, an end block was created with a rectangular section havinga total depth of 350 mm, a width of 450 mm, and a length of250 mm. These two end blocks were used to apply torsion at oneend (active frame) and to restrain the torsion at the other end(reactive frame). To apply the required load and the proper bound-ary condition far from the test region, the beam was extended atboth ends. The extensions were either for applying load (loadingarm) or for applying the end restraints (roller arm). The loadingarm was 900 mm long while the roller arm was 750 mm as shownin Fig. 6. To ensure that failure would occur within the test region,both arms had additional longitudinal and transverse reinforce-ment. The shear reinforcement was 10-M @ 70 mm, the bottomreinforcement was 620 M, and the top longitudinal reinforcementwas 410 M + 215 M.

The concrete mix was designed using Type 10 cement, sand, and10 mm aggregate. The results from the compression testing ofstandard concrete cylinders are shown in Table 1. The 28-day con-crete compressive strength was 25.6 MPa. Compression tests con-ducted on the same day of the beam testing showed acompressive strength of 35.9 MPa for beams TB3 and TB1, and33.6 MPa for beam TB2. The longitudinal and transversal steel barswere ribbed high strength steel. The tensile testing of couponsmade from the reinforcement bars showed that 10 M bars yieldedat 465 MPa, while the 15 M and 20 M bars yielded at 450 MPa.

Linear variable differential transformers (LVDTs) were used tomeasure displacements at different locations of the beam. TenLVDTs measured the vertical displacements at ve sections of thebeamtwo at each section. The two LVDTs at the tip of the angeof each section were used to calculate the rotation and the averagevertical displacement. Strain gauges were used to measure thestrain in the longitudinal and transversal reinforcement at differ-ent locations, as shown in Fig. 5. Strain in the longitudinal rein-forcement was measured at the maximum and at the zeromoment section. Strain in the transverse reinforcement was mea-sured at the beginning, middle, and the end of the test region area.Strain gauges were installed at the same location in all the testedbeams.

3.3. Test set-up

Recently, Talaeitaba and Mostonejad proposed a test setupusing a simple beam with a cantilever in the middle for applyingcombined shear and torsion [49]. In their test setup, the combinedshear and torsion are accompanied by relatively large bending mo-ments. The test setup used in the present research was designed in2005 with the objective of minimizing the bending moments in thetorsion and shear interaction test region. The combined shear andtorsion is signicant at low bending moment values, including, butnot limited to, the following cases: (1) the case of inection pointfor a continuous beam; or (2) the case of a section at the supportof a simple beam. Fig. 6 shows a schematic of the structural systemfor the test set-up where three different actuators are used to applyloads to the beam (denoted as L1, L2, and L3), simulating a simplebeam with a cantilever at both ends. The middle section of the testregion is subjected to combined shear and torsion with zero ornear zero bending moment. The two hydraulic actuators L2 andL3 apply the load to the beam through 0.5 m long steel arms to ap-

g Structures 68 (2014) 5770ply the required torque. The hydraulic actuator L1 acts at the cen-ter of the cross-section of the beam. The top end condition foractuator L1 is a pin support. The middle region (test region) was

eerinA. Deifalla, A. Ghobarah / Enginsubjected to combined shear and torsion while the torque to shearratio was kept xed throughout the test by controlling the threedifferent applied loads. After installing the T-beams in the test set-up and attaching the instruments to the data acquisition system,the beam was loaded with low-level load combinations withinthe elastic range of concrete. Measurements from this test wereveried to ensure that all the instruments were correctly installedand functioning properly. The load values L1, L2, and L3 that givethe desired shear and torsion combination were calculated fromsimple structural analysis. The loads were applied in small stepsof 2 kN in order to exercise better control over the loading valuesand achieve the required torque to shear force ratio. After each

Fig. 4. Dimensions and reinforcem

Fig. 5. Strain gauge location on longitudg Structures 68 (2014) 5770 61load step, the beamwas inspected for cracks and any possible signsof failure. During the tests, it was possible to maintain good controlover the torque to shear ratio all the way to near failure of thebeam. Fig. 7 shows a photo for the test setup with a specimen inplace. Four load cells (L1, L2, L3, and L4) were used in the testset-up. Three of the load cells (L1, L2 and L3) were used to measurethe actual applied loads at Points A, D and E on the beam. Thefourth load cell (L4) was used at point F to measure the reactionat the support of the beam. Fig. 8 shows the boundary conditionsat points F, D, E and A. Due to the complexity of the test set-up,the assumptions made concerning the beam boundary conditionswere veried. This was done by comparing the measured values

ent details of tested beams.

inal and transversal reinforcement.

eerin62 A. Deifalla, A. Ghobarah / Enginof the reaction at point F (L4) to the theoretically predicted reac-tion at the same location (R1) using a linear structural analysis,assuming actual hinges at R2 and R3, and an actual roller at R1,as shown in Fig. 9.

3.4. Torque to shear ratio

Fig. 10 shows the applied torque versus the applied shear forthe tested inverted T-shaped beams. The beams (TB1, TB2, and

Fig. 6. Schematic structural system and interal forces for the tested beams.

Table 1Concrete strength at different dates.

Batch I Batch II

Date f 0c (Mpa) Date f0c (Mpa)

7 days 17.7 28 days 25.628 days 25.6 TB3 35.9TB1 35.9 TB2 33.6

AB

E

C

D

F H

G

Fig. 7. The test setup with a specimen in place.4.2. Torsional behavior

Fig. 14 shows the relationship between the applied torque andthe angle of twist for the tested beams. Before cracking, the behav-ior was similar for all of the tested inverted T-shaped beams, with apre-cracking torsional rigidity value of approximately 2110 kN m2.The value of the cracking strength was taken as the minimum of4. Experimental results

4.1. Cracking pattern and failure mode

The concrete cracking pattern for beams TB1, TB2, and TB3 areshown is Figs. 1113, respectively. In addition, the failure modesare listed in Table 3. For beam TB1 (T/V = 0.5 m), the onset of crack-ing was observed at the bottom of the web at a total load value of56 kN. Afterwards, more diagonal cracks were initiated within boththe web and the ange, which were spiral and uniformly distrib-uted, as shown in Fig. 11(ad). Before failure, signicant concretecover spalling from the ange (as shown in Fig. 11b and c) andadditional longitudinal cracks in the exure compression zone sidewere observed, as shown in Fig. 11a and b. These additional longi-tudinal cracks are due to the diagonal compression stress from theshear and torsion, and that from the exure. The major diagonalcracks were formed at an average angle of inclination with the lon-gitudinal axis of the beam (h) value of 51. Beam TB1 failed due tostirrup yielding before concrete compression at a load value of162 kN.

For beam TB2 (T/V = 1.0 m), the onset of cracking occurred at anapplied load of 33 kN. The cracks propagated in a helical formaround the beam in a similar manner to those of beam TB1 (asshown in Fig. 12ad), where concrete cover spalling from boththe ange and the web was observed. However, on average, themajor cracks formed at an average (h) value of 55, which is steeperthan beam TB1. Beam TB2 failed due to stirrup yielding before con-crete compression at a load value of 75 kN. In comparing Fig. 11band Fig. 12b, it is clear that beam TB2 exhibited signicant webspalling with respect to beam TB1.

For beam TB3 (T/V = 0.1 m), the onset of cracking was observedat a load value of 130 kN. Signicant diagonal cracks were ob-served in the web compared to that in the ange, as shown inFig. 13ad. The cracking pattern varied along the test region andbetween both sides of beam TB3. For the web side, where the shearstresses due to the torsion and shear were added together, theaverage (h) for the cracks was 30, which is lower than that ofthe other web side, where shear stresses due to the torsion andshear will subtract. Beam TB3 failed due to diagonal concrete com-pression before stirrup yielding at a load value of 342 kN. Compar-ing Fig. 13(ad) with Fig. 11(ad) and Fig. 12(ad), the angle ofinclination of the cracks of beam TB3 was lower than those ofeither beam TB1 or TB2. The spacing between the cracks of beamTB3 was smaller than that of either beam TB1 or TB2. The crackingpatterns of beams TB1, TB2, and TB3 were signicantly inuencedby the torque to shear ratio.TB3) were tested under torque to shear ratios of 0.5 m, 1.0 m and0.1 m, respectively. The torque to shear ratio was chosen to covera wide range of practical shear torsion interactions. In addition, Ta-ble 2 shows the ratio of the applied torque to shear ratio to the ulti-mate torque to shear ratio, which was chosen to vary from 1 to 10.Based on this range, the applied torque to shear ratios were chosento be either 0.1 m, 0.5 m, or 1.0 m.

g Structures 68 (2014) 5770either the strength at which the torsion behavior deviated fromthe initial linear behavior or the strength at which cracks wereobserved during the testing of the beam. The recorded values of

Fig. 8. Details of the test setup; (a) roller support at point F, (b) actuator used to apply load at points D and E and c) actuator used to apply load at point A.

0

20

40

60

80

100

120

140

160

0 20 40 60 80 100 120 140 160

Calc

ulat

ed re

actio

n fo

rce,

R1(k

N )

Measured reaction force, R1 (kN)

TB1 TB2 TB3

Fig. 9. Physical verication of the test setup.

0

20

40

60

80

100

120

0 5 10 15 20 25

Shea

r for

ce, Q

2 (kN

)

Torque, T (kN. m)

TB2 (1.0 m)

TB1 (0.5 m)

TB3 (0.1 m)

Fig. 10. The applied shear and torsion.

A. Deifalla, A. Ghobarah / Engineering Structures 68 (2014) 5770 63

5 0.510 1.0

(V - T)

(V - T)C B

(a)

(b)

(V + T)(V + T)

B

B

C

C

eering Structures 68 (2014) 5770(V - T) (V - T)

(V + T) (V + T)

C B

B C

(a)Table 2Selected torque to shear ratios.

(T/Tult)/(V/Vult) (T/V) (m)

1 0.1

64 A. Deifalla, A. Ghobarah / Enginthe cracking torque and corresponding twist for all the testedbeams are shown in Table 3. After cracking, the behavior of beamsTB1 and TB2 was similar because they were subjected to high tor-que to shear ratios. However, the behavior of beam TB3 was differ-ent compared to beams TB1 and TB2. In examining Fig. 15, it can beseen that the average post-cracking torsional rigidity of beam TB3was higher than that of either beam TB1 or TB2 that is due to widercracks associated with the high torsion to shear ratio for beamsTB1 and TB2. This is commonly observed after steel yielding, whichis the case for both TB1 and TB2. The value of the ultimate strengthwas taken as the maximum strength observed during the testing ofthe beam. Table 3 shows the ultimate torque and the correspond-ing twist for all the tested beams. As shown in Table 3, the sheartorsion interaction affected the value of the ultimate torque.

4.3. Shear behavior

The shear behavior of the tested beams was affected by thetorque to shear ratio. Fig. 13 shows the relationship between theapplied shear force and the maximum strain in the transverse steelreinforcement. The stirrup strain increased substantially with theincrease in the torque to shear ratio. The applied shear force at

B

B

C

C

(b)

(c)

(d)Fig. 11. Crack pattern for TB1 (0.5); (a) south, (b) north, (c) bottom and (d) top.

(c)

(d)

B C

Fig. 12. Cracking pattern for TB2 (1.0); (a) south, (b) north, (c) bottom and (d) top.

(c)

(d)

(a)

(V + T)

(V + T)

B

B

C

C

C

B

(b)

(V - T)(V - T)C B

Fig. 13. Crack pattern for TB3 (0.1); (a) south, (b) north, (c) bottom and (d) top.

the onset of cracking for all tested beams is shown in Table 3. Thevalue of the ultimate shear strength for all tested beams is alsoshown in Table 3. It is clear in the table that the sheartorsioninteraction affected the ultimate and cracking shear force.

Table 3Summary of the experimental results.

Beam T/V(m)

Cracking torque(kN m)

Twist at cracking(deg/m)

Cracking shearforce (kN)

Ultimate torque(kN m)

Twist at ultimate(deg/m)

Ultimateshear (kN)

Observed failure mode

TB1 0.5 11.6 0.25 17 23 2.82 46 Stirrup yield beforeconcrete crushing

TB2 1.00 11 0.33 11 22.7 3.16 21.4 Stirrup yield beforeconcrete crushing

TB3 0.1 4 0.13 42 10.8 0.5 105 Concrete diagonal crushing

0

5

10

15

20

25

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

Torq

ue, T

(kN.

m)

Angle of twist (deg/m)

TB2 (1.0 m)

TB1 (0.5 m)

TB3 (0.1 m)

Fig. 14. Torsional behavior.

0

20

40

60

80

100

120

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Shea

r for

ce, Q

2 (kN

)

Stirrup strain at the mid of the test zone (1000 microstrain)

Tran

sver

se st

eel y

ieldTB3 (0.1 m)

TB2 (1.0 m)

TB1 (0.5 m)

Fig. 15. Shear behavior.

(a)

Fig. 16. Ultimat and cracking experimentaly observed shea

A. Deifalla, A. Ghobarah / Engineering Structures 68 (2014) 5770 654.4. The sheartorsion interaction

Fig. 16a shows a plot for the observed absolute values of thecracking and the ultimate shear forces versus the torsion moment.We can see that the relationship is not linear and there is a clearcurvature in the interaction. Huang et al. proposed a circulardimensionless relationship for the torqueshear interaction basedon the theory of plasticity [30]. In an attempt to quantify thesheartorsion interaction, the shear forces and torsion momentswere normalized and compared with the interaction relationshipproposed by Huang et al. and are shown in Fig. 16b [30]. We cansee that the experimentally observed values agreed fairly well withthe relationship, with an error less than 10%.

4.5. Transverse steel strain

Fig. 17 shows the transverse steel strain for the ange and webstirrups versus the total load. In case of beam TB2 (high torque toshear ratio), the strain measured in the ange was similar to thestrain measured in the web. However, in the case of beam TB3(low torque to shear ratio), the strain in the web was larger thanthe strain in the ange. The ange was more effective in cases ofhigher torque to shear ratios.

Fig. 18 shows the relationship between the ange stirrup strainat both the top and bottom branch versus the total load. The straingauges were installed as shown in Fig. 5, with the exception ofbeam TB1, where the bottom strain gauge was installed in the mid-dle of the bottom branch within the overlapping zone of the angestirrup. The strain of beam TB1 (under low torque to shear ratio)was signicantly lower than that of TB3, which agrees well withthe assumption that the ange stirrup primarily carries forces fromtorsion.(b)rtorsion interaction (a) absolute and (b) normalized.

rectangular RC beams up to failure using a displacement controlsolution scheme rather than a force control solution scheme; (2) in-clude the FRP material modeling; (3) model external bonded rein-forcements with different arrangements; and (4) improved theconcrete constitutive modeling [17,23]. All of these models focusedon rectangular beams under combined torsion, although structuralmembers subjected to torsion may be of different congurations,such as rectangular beams, T-shaped beams, L-shaped beams, andbox beams.

In this study, the model by Deifalla and Ghobarah was adaptedand further extended to predict the behavior of cross-sections withdifferent shapes subjected to torsion, shear, and bending moments[23]. In the development of the proposed model, the followingassumptions were made:

(1) The longitudinal strain follows the BernoulliNavier hypoth-

0

50

100

150

200

250

300

350

400

450

0 0.5 1 1.5 2 2.5

Tota

l loa

d (L

1+L2

+L3

), kN

Strain (1000 microstrain)

66 A. Deifalla, A. Ghobarah / Engineering Structures 68 (2014) 5770Fig. 17. Stirrup strain versus total applied load (L1 + L2 + L3).

300

350

400

1+L2

+L3

), kN5. Analytical model

Several models were developed for predicting the behavior of RCbeams subjected to combined straining actions. Numerouscontributions bymany researchers attempting to improve the spacetruss model by Rausch were found in existing literature [50,1518,51,6,19,21,20,2225]. Deifalla and Ghobarah [23] adapted themodel by Rahal and Collins [17] to: (1) predict the behavior of

0

50

100

150

200

250

-1 -0.5 0 0.5 1 1.5 2 2.5 3

Tota

l loa

d (L

Strain (1000 microstrain)Fig. 18. Flange stirrup strain versus total applied load (L1 + L2 + L3).

(a)Fig. 19. Compression Stress distribution within the concrete strut (a) actual stress ddistribution.esis, which indicates that plane section before bending willremain plane after bending.

(2) Mohr Circle can be used to evaluate the strain, curvature,and stress status at any point in the plane.

(3) The direction of the principal stresses at any point in theplane is coincident with the direction of the principal strainevaluated at the same point in the plane [35].

(4) The torsional behavior is dominated by SaintVenants tor-sion, which indicates that the torsion will be resisted byshear ow in the perimeter of the cross section [34,17,23].

(5) The effective thickness of the diagonal concrete struts isfunction of the external loading [33,35] which is similar tobeams subjected to bending where the effective depth isfunction of the bending moment.

(6) The equivalent hollow tube is being divided into four panels;each panel is subjected to uniform bi-axial stresses[33,17,18,23,21].

(7) The diagonal compressive strain distribution within the con-crete diagonal struts is assumed to be linear and conse-quently the diagonal compressive stress is assumed to benon-uniform [33,35,20,23].

(8) The torsion stresses and the uniform shear stresses are beingreplaced by one equivalent uniform stress block as shown inFig. 19 [17].

5.1. Modeling T-shaped beams

This section describes the capability of predicting the behaviorof the anged beams. The anged cross-section is divided intoseveral rectangular sub-divisions. Each rectangular sub-division

(b) (c)

istribution, (b) equivalent stress distribution, and (c) equivalent uniform stress

b) (c)

eering Structures 68 (2014) 5770 67(a) (

A. Deifalla, A. Ghobarah / Enginis analyzed independently while subjected to the applied com-bined shear and torsion. For example, the T-shaped beam is di-vided into rectangular sub-divisions as shown in Fig. 20. Aftermodeling each rectangular section, the principle of superpositionis applied to obtain the strength and the deformations of the com-plete T-shaped beam, while assuming that the angle of twist forthe T-shaped beam and the sub-divisions are the same. The appliedtorque (T) on the whole cross section is calculated such that:

T Xni1

Ti 1

where Ti is the torsion carried by each rectangular sub-division (i) atthe same angle of twist and n is the total number of rectangularsubdivisions. The applied shear force (V) is calculated as follows:

V Xni1

Vi 2

where Vi is the shear carried by each rectangular sub-division (i) atthe same angle of twist. The stirrup strain e is calculated such that:

e Xni1ei 3

Fig. 20. Rectangular divisions (a) Solution I, (b) Solution II, (c) Solution III.

Input section details and applied internal actions.

Calculate using Eq. (18-20)

Arbitrary assume tt

Calculate average shear stress, Eqs. (4-7)

Set T and V as 0.1 of the applied internal actions.

Calculate Ao and po, Eqs. (19-20)

Calculate average stresses and strains for each panel, Eqs. (8-14 and 23-29)

Check tt , Eqs. (15-17)

End

yes

No

Increase the T and V by

0.1 applied actions.

Use the Panel subroutine shown in fig (20) to calculate longitudinal stresses and strain for the whole section, Eqs. (21-22).

Check T and V applied actions

yes

No

Fig. 21. Flow chart for the main program.where ei is the stirrup strain for each rectangular sub-division (i) atthe same angle of twist.

5.2. Modeling rectangular sub-divisions

For predicting the full behavior of each rectangular sub-divi-sion, the model proposed by Deifalla and Ghobarah is implemented[23]. The model is briey listed in Eqs. (4)(29); however, detailsregarding the development of the adapted model is to be foundin both Deifalla [6] and Deifalla and Ghobarah [6,23].

qt Ti2Ao

4

Input average shear stress and

longitudinal strain

Assume the diagonal strain

Assume the angle of inclination

Solve the wall element Check the angle

Check the diagonal stressNO

NO

Yes

Yes

Return to main program

Fig. 22. Flow chart for the Panel subroutine [23].

q t t s st

6

2 r2 rx

2Ao

r2 b1f 0c ife2b2e0c

P 1 24

b1 0:9

1:0 400e1p 25

b2 1:0

1:0 500e1p 26

r1 Ece1 27

r1 0:33f0c

1 500e1p 28rs Eses 6 fy 29

A ow chart for the solution technique is being shown in Figs. 21and 22. A force driven solution technique is being used limiting themodel predictions to the ultimate strength.

5.7. Model validation

Three rectangular RC beams (N1, N2, and N3) were found inthe literature. The beams were tested under combined signicant

5

10

15

Torq

ue, T

(kN.

m)

Experimental [52]

Anaylitical

eering Structures 68 (2014) 5770Xki1

r0ciDAciyci Xmj1

r0sjAsjysj Mx X4k1

Nvkysk 22

5.6. Material modeling

2" #Ao Ac X4i1

litti2

19

Po Pc X4i1

tti 20

Xki1r0ciDAci

Xmj1r0sjAsj N

X4k1

Nvk 21tan h r2 r1 12

r2 s tanh 1tanh

r1 13

ry qhrst r1 s tan h 14

ui w sin 2hi 15

tti e2sui16

/d /t sin2h /L cos2h w sin2h 17

5.5. Panel assemblage

w P4

i1lici 18m qt

7

5.3. Mohr circle for the average concrete strains of each panel

c 2e2 extanh 8

ey c2 tanh e2 9

e1 e2 ex ey 10

5.4. Equilibrium and compatibility conditions for each panel

rx r2 ry r1 11qs Vil

5

q t q t

68 A. Deifalla, A. Ghobarah / Enginr2 b1f 0c 2e2e0c e2

e0cif

e2b2e0c

6 1 23(a) (b) (c)Angle of twist (deg/m)

00 5 10 150 5 10 150 5 10 15

Fig. 23. Torque versus angle of twist for (a) N1; (b) N2 and (c) N3.

Angle of twist (deg/m)0.0 0.8 1.6 2.4 3.2 0.00 0.25 0.50

0

5

10

15

20

25

0 1 2 3

Torq

ue, T

(kN.

m)

Experimental

Solution I

Solution II

Solution III(a) (b) (c)Fig. 24. Torque versus angle of twist for (a) TB1; (b) TB2 and (c) TB3.

close agreement with the experimental results. However, onlyup to the ultimate strength as the model employs a force drivensolution technique. From the current study, three T-shapedbeams (TB1, TB2, and TB3) tested under torque to shear ratiovalues of 0.5, 1.0, and 0.1 m. Each T-shaped beam was dividedinto two rectangular sub-divisions using each of the three pro-posed solutions, as shown in Fig. 20(ac). The comparison be-tween the behavior (i.e., torque versus twist and shear forceversus stirrup strain) predicted by the model and the experimen-tally observed behavior is shown in Fig. 24(ac) and Fig. 25(ac).The gures show that the model prediction agrees well with theexperimental results. Two L-shaped beams were found in the lit-erature tested under combined torsion [45]. Each L-shaped beamwas divided into two rectangular sub-divisions using each of the

20

40

60

80

100

120

Shea

r for

ce, Q

2 (kN

)

Experimental

Solution ISolution IISolution III

A. Deifalla, A. Ghobarah / Engineering Structures 68 (2014) 5770 69Transversal steel strain (1000 micro-strain)

(a) (b) (c)

00.00 0.60 1.20 0.00 1.00 2.00 3.00 0.00 0.50 1.00

Fig. 25. Shear force versus transversal steel strain for (a) TB1; (b) TB2 and (c) TB3.torsion [52]. Beams had the same cross-section dimensions, butthe stirrups spacing were different. The model was used to pre-dict the torsional behavior of three rectangular RC beams up toultimate torsion. The comparison between the model predictionsand the experimental results for the tested RC beams are shownin Fig. 23(ac). The predicted behaviors were found to be in

T-shaped beams under combined shear and torsion wasdesigned and implemented.

Fig. 26. Torque versus angle of twist for L-shaped beams [45].

Table 4Strength and deformation predicted using the proposed model with solutions I, II, and III

Beam Experimentally observed ultimate Predicted by the model

Torque Angle of twist Torque A

(kN m) (/m) I II III I

TB1 23 2.82 25.4 23 20.3 2TB2 22.7 3.16 21.5 21.3 21.2 3TB3 10.8 0.5 9.3 11.8 8.6 0BK-Ta 16.8 1.9 16.2 16.8 17.3 1BK-TVM-1a 18.6 2.5 16.2 16.8 17.3 1

AverageCoefcient of variation95% Condence interval

a Ref. [45].2. The behavior of the tested inverted T-shaped beams wasaffected by the value of the torque to shear ratio. Decreasingthe applied torque to the applied shear force ratio resulted inthe following: (1) a signicant reduction for the spacingbetween diagonal cracks, the strut angle of inclination, crackingand ultimate torque, ange and web stirrup strain; (2) a signif-icant increase for the failure and cracking load, post-crackingtorsional rigidity, cracking and ultimate shear; and (3) the stir-rups efciency was reduced, thus, beams failed due to concretediagonal failure rather than stirrups yield.

3. The proposed analytical model showed remarkable agreementwith the experimental results for the behavior of anged beamsunder combined actions.

versus measured.

Experimental/predicted

ngle of twist Torque Angle of twist

II III I II III I II III

.8 2.6 2.82 0.91 1.00 1.14 1.00 1.07 1.00

.2 3.2 3.16 1.06 1.07 1.07 1.00 1.00 1.00

.49 0.49 0.49 1.17 0.98 1.26 1.01 1.01 1.01

.91 1.9 1.9 1.04 1.00 0.97 0.99 0.98 0.98

.91 1.9 1.9 1.15 1.10 1.08 1.31 1.29 1.29

1.07 1.03 1.10 1.06 1.07 1.0610% 5.0% 10% 13% 12% 12%three proposed solutions, as shown in Fig. 20(ac). The compar-ison between the torsional behavior predicted by the model andthe experimentally observed behavior is shown in Fig. 26. Thegure shows that the model predictions agree well with theexperimental results.

Table 4 shows the experimentally observed ultimate torqueand the corresponding angle of twist versus the analytically cal-culated ones using the three solutions shown in Fig. 20. Fromthe table, we can see that any of the three solutions showedgood compliance with the experimentally observed results foranged beams. However, solution II predictions were more con-sistent compared with those of solutions I and III for beams un-der combined torsion. This might be because solution II followsthe stirrups conguration.

6. Conclusions

1. An innovative test setup capable of simulating the behavior of0.1 0.05 0.1 0.14 0.13 0.13

References

[1] ASCE-ACI Committee 445 on shear and torsion. Recent approaches to sheardesign of structural concrete. J Struct Eng ASCE 1998;124(12):1375417.

[2] Mirza SA, Furlong RW. Serviceability behavior and failure mechanisms ofconcrete inverted T-shaped beam bridge bentcaps. ACI J 1983:294304 [JulyAugust].

[3] Mirza SA, Furlong RW, Ma JS. Flexural shear and ledge reinforcement inreinforced concrete inverted T-girders. ACI J 1988:50920 [SeptemberOctober].

[4] Rahal KN, Collins MP. Effect of the thickness of concrete cover on the sheartorsion interaction an experimental investigation. ACI Struct J1995;92(3):33442 [MayJune].

[29] Klus JP. Ultimate strength of reinforced concrete beams in combined torsionand shear. ACI Struct J 1968;65(3):2106.

[30] Deifalla A, Ghobarah A. Simplied analysis of RC beams torsionalystrengthened using FRP. In: Chen, Teng (Eds.), Proceedings of internationalsymposium on bond behavior of FRP in structures (BBFS 2005), Hong Kong,China; 79 December 2005. p. 3816.

[31] Deifalla A, Ghobarah A. Calculating the thickness of FRP jacket for shear andtorsion strengthening of RC T-girders. In: Third international conference onFRP composites in civil engineering (CICE 2006), December 1315, Miami,Florida, USA; 2006. Paper No. 203.

[32] Huang LH, Lu Y, Shi C. Unied calculations method for symmetricallyreinforced concrete section subjected to combined loading. ACI Struct J2013;2013:12736 [JanuaryFebruary].

[33] Hsu TTC. Unied theory of reinforced concrete. Boca Raton: CRC Press Inc., Fla;1993.

70 A. Deifalla, A. Ghobarah / Engineering Structures 68 (2014) 5770[5] Deifalla A, Ghobarah A. Assessing the North American bridge codes for thedesign of T-girders under torsion and shear. In: Proceeding of the 7thinternational conference on short & medium span bridges, Montreal, August2325, Paper No. 717; 2006a. 8pp.

[6] Deifalla AF. Behaviour and strengthening of RC T-girders in torsion and shear.PhD thesis, McMaster University, Canada; 2007. p. 280.

[7] Lessig NN. Theoretical and experimental investigation of reinforced concreteelements subjected to combined bending and torsion. In: Theory of design andconstruction of reinforced concrete structures, Moscow; 1958 [in Russian].

[8] Hsu TC. Torsion of structural concrete-interaction surface for combinedtorsion, shear, and bending in beams without stirrups. Portland CementAssociation, Research and Development Laboratories, Bulletin D319; 1968.

[9] Elfgren L, Karlsson I, Losberg A. Torsionbending-shear interaction for concretebeams. J Struct Div 1974;100(8):165776.

[10] Badawy HEI. Experimental investigation of the collapse of reinforced concretecurved beams. Magaz Concr Res 1977;29(99):5969.

[11] Ewida AA, McMullan AE. Torsionshear interaction in reinforced concretemembers. Magaz Concr Res 1981;23(115):11322.

[12] Zia P, Hsu T. Design for torsion and shear in prestressed concrete exuralmembers. PCI J 2004:3441 [MayJune].

[13] Klein G, Lucier G, Walter C, Rizkalla S, Zia P, Gleich H. Torsion simplied: afailure plane model for design of spandrel beams; 2013. 19pp.

[14] Rabat B, Collins MP. A variable space truss model for structural concretemembers subjected to complex loading. SP-55. Detroit: ACI; 1978. p. 54787.

[15] Collins MP, Mitchell D. Shear and torsion design of prestressed and non-prestressed concrete beams. J Prestress Concr Inst, Proc 1980;25(5):32100.

[16] Hsu TTC, Mo YL. Softening of concrete in torsional memberstheory and tests.ACI J 1985;82(3):290303.

[17] Rahal KN, Collins MP. Analysis of sections subjected to combined shear andtorsion a theoretical model. ACI Struct J 1995;92(4):45969.

[18] Cocchi GM, Volpi M. Inelastic analysis of reinforced concrete beams subjectedto combined torsion, exural and axial loads. Comput Struct1996;61(3):47994.

[19] Chalioris CE. Analytical model for the torsional behavior of reinforced concretebeams retrotted with FRP materials. Eng Struct 2007;29(12):326376.

[20] Bernardo LFA, Lopes SMR. Behavior of concrete beams under torsion NSCplain and hollow beams. Mater Struct 2008;41(6):114367.

[21] Greene Jr G, Belarbi A. Model for reinforced concrete members under torsion,bending, and shear I: theory. J Eng Mech ASCE 2009;135(9):9619 [September1].

[22] Jeng CH, Hsu TC. A softened membrane model for torsion in reinforcedconcrete members. Eng Struct 2009;31(9):194454.

[23] Deifalla A, Ghobarah A. Full torsional behavior of RC beams wrapped with FRP:analytical model. ACSE, Compos Construct 2010;14(3):289300.

[24] Bernardo LFA, Andrade JMA, Lopes SMR. Softened truss model for reinforcedNSC and HSC beams under torsion: a comparative study. Eng Struct2012;42:27896.

[25] Bernardo LFA, Andrade JMA, Lopes SMR. Modied variable angle truss-modelfor torsion in reinforced concrete beams. Mater Struct 2012;45:1877902.

[26] Karayannis CG. Torsional analysis of anged concrete elements with tensionsoftening. J Comput Struct 1995;54(1):97110.

[27] Karayannis CG, Chalioris CE. Experimental validation of smeared analysis forplain concrete in torsion. J Struct Eng ASCE 2000;126(6):64653.

[28] Hassan T, Lucier G, Rizkalla S, Zia P. Modeling of L-shaped, precast, prestressedconcrete spandrels. PCI J. 2007:7892 [MarchApril].[34] Hsu TC. Torsion of reinforced concrete. New York: Van Nostrand ReinholdCompany; 1984. 516p.

[35] Collins MP, Mitchell D. Prestressed concrete structures. Canada: ResponsePublication; 1997.

[36] Erosy U, Ferguson PM. Behavior and strength of concrete L-beams undercombined torsion and shear. ACI Struct J, Proc 1967;64(12):76776.

[37] Farmer LE, Ferguson PM. T-shaped beams under combined bending, shear andtorsion. ACI Struct J, Proc 1967;64(11):75766 [November].

[38] Lash SD, Kirk DW. Concrete tee-beams subjected to torsion and combinedbending and torsion. Report RR160. Department of Highways Ontario Canada;1970. 19p.

[39] Victor DJ, Aravindan PK. Prestressed and reinforced concrete T-shaped beamsunder combined bending and torsion. ACI J 1978:52632 [October].

[40] Razaqpur AG, Ghali A. Design of transverse reinforcement in ange of T-shaped beams. ACI J 1986;1986:6809 [JulyAugust].

[41] Zararis PD, Penelis gGr. Reinforced concrete T-shaped beams in torsion andbending. ACI J 1986:52632 [JanuaryFebruary].

[42] Logan Donald R. L-spandrels: can torsional distress be induced by eccentricvertical loading? PCI J 2007 [MarchApril].

[43] Lucier G, Walter C, Rizkalla S, Zia P, Klein G. Development of a rational designmethodology for precast concrete slender spandrel beams: part 1experimental program. PCI J 2011;56(2).

[44] Lucier G, Walter C, Rizkalla S, Zia P, Klein G. Development of a rational designmethodology for precast concrete slender spandrel beams: part 2 analysis anddesign guidelines. PCI J 2011;56(4):10633.

[45] Kaminski M, Pawlak W. Load capacity and stiffness of angular cross sectionreinforced concrete beams under torsion. Arch Civ Mech Eng2011;XI(4):885903.

[46] Chalioris CE, Karayannis CG. Effectiveness of the use of steel bres on thetorsional behavior of anged concrete beams. Cem Concr Compos2009;31:33141.

[47] PCI. PCI design handbook precast and pre-stressed concrete. Chicago, Illinois60606; 2006. 736pp.

[48] CSA-A23.3-04. Design of concrete structures for buildings. CSA-A23.3-04standard. Rexdale, Ontario, Canada: Canadian Standards Association; 2005.240pp.

[49] Talaeitaba SB, Mostonejad D. Fixed supports in assessment of RC beamsbehavior under combined shear and torsion. Int J Appl Sci Technol2011;1(5):11926.

[50] Rausch E. Berechnung des Eisenbetons gegen Verdrehung (design of reinforcedconcrete in torsion). Ph.D. thesis, Berlin; 1929. 53pp [in German].

[51] Pang X-BD, Hsu TTC. Fixed angle softened truss model for reinforced concrete.ACI Struct J 1996;93(2):197207 [MarchApril].

[52] Ghobarah A, Ghorbel M, Chidiac S. Upgrading torsional resistance of RC beamsusing FRP. J Compos Construct ASCE 2002;6(4):25763.

[53] Deifalla A, Hamed M, Saleh A, Ali T. Exploring GFRP bars as reinforcement forrectangular and L-shaped beams subjected to signicant torsion: anexperimental study. Eng Struct 2014;59:77686.

[54] Deifalla A, Awad A, El-Garhy M. Effectiveness of externally bonded CFRP stripsfor strengthening anged beams under torsion: an experimental study. EngStruct 2013;56:206575.

Behavior and analysis of inverted T-shaped RC beams under shear and torsion1 Introduction2 Research significance and previous work3 Testing inverted T-shaped beams3.1 Scale model for the inverted T-beam3.2 Specimen details3.3 Test set-up3.4 Torque to shear ratio

4 Experimental results4.1 Cracking pattern and failure mode4.2 Torsional behavior4.3 Shear behavior4.4 The sheartorsion interaction4.5 Transverse steel strain

5 Analytical model5.1 Modeling T-shaped beams5.2 Modeling rectangular sub-divisions5.3 Mohr circle for the average concrete strains of each panel5.4 Equilibrium and compatibility conditions for each panel5.5 Panel assemblage5.6 Material modeling5.7 Model validation

6 ConclusionsReferences