Parametric Study of the Behavior of Longitudinally and Transversally Prestressed Concrete under Pure Torsion

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1 designs Artile Parametri Study of Behavior of Longitudinally and ransversally Prestressed Conrete under Pure orsion Luís Bernardo * ID, Jorge Andrade and Mafalda eixeira C-MADE-Centre of Materials and Building ehnologies, Department of Civil Engineering and Arhiteture, University of Beira Interior, Edifíio II das Engenharias, Calçada Fonte do Lameiro, Covilhã, Portugal; jandrade@ubi.pt (J.A.); mafalda.m.teixeira@ubi.pt (M..) * Correspondene: lfb@ubi.pt; Fax: Reeived: 28 Marh 2018; Aepted: 20 April 2018; Published: 24 April 2018 Abstrat: Among several analytial models to desribe behavior of reinfored onrete (RC) and longitudinally prestressed onrete (LPC) beams under torsion, Modified Variable Angle russ Model (MVAM) is partiularly effiient to apture behavioral states of member until failure and agree well with experimental results. his artile aims to extend MVAM to over transversally prestressed onrete (PC) beams under torsion. he hanges in formulation and alulation proedure of original VAM, in order to inlude influene of transversal prestress, are presented. he extended MAVM is n used to ompute global response of LPC and PC beams under torsion with similar total prestress reinforement ratios, namely torque twist urves. he obtained preditions are n ompared and disussed. It is shown that for ultimate loading, transversal prestress onstitutes also an effetive solution to improve behavior of beams under torsion. However, transversal prestress is less effetive to delay raked state. Finally, it is also shown that when prestress is distributed in both longitudinal and transversal diretion, global response of beams under torsion is furr improved, namely resistane torque and torsional stiffness in raked state. Keywords: Prestress Conrete; beams; torsion; Modified russ Model; parametri study 1. Introdution Several oretial models have been developed to predit behavior of Reinfored Conrete (RC) beams under torsion. Among those models, Spae russ Analogy (SA), originally proposed by Raush in 1929 [1], has a high historial value and is base model of European Model Code sine 1978 and also ACI Code sine Suessive developments of original SA have been proposed by several authors, suh as Andersen in 1935 [2], Cowan in 1950 [3], Lampert and hurlimann in 1969 [4], Elfgren in 1972 [5] and Collins and Mithell in 1980 [6], among ors. Among referred developments of SA, Variable Angle russ Model (VAM) proposed by Hsu and Mo in 1985 [7,8] has been widely used by researhers. VAM was first model whih aimed to unify treatment of RC and Longitudinally Prestressed Conrete (LPC) beams. Some simplified versions of VAM were proposed later by or authors, namely Rahal and Collins in 1996 [9], Bhatti and Almughrabi in 1996 [10] and Wang and Hsu in 1997 [11]. However, se models only allow omputing ultimate torsional strength of beams and usually do not inorporate influene of prestress. he original VAM is able to predit behavioral state of RC and LPC beams throughout entire loading history, although good results are observed only for high loading levels [7,8]. his is Designs 2018, 2, 12; doi: /designs

2 Designs 2018, 2, 12 2 of 22 beause VAM assumes that response of beam starts from raked state and also beause it neglets influene of onrete ore for plain beams. o solve this issue, Jeng and Hsu in 2009 [12], Jeng et al. in 2010 [13] and Jeng in 2015 [14] proposed Softened Membrane Model for orsion (SMM) to model RC and LPC beams. his model predits well entire torque () twist (θ) urve of beams. Sine SMM onstitutes an extension of Softened Membrane Model (SMM) for RC membrane elements under shear [15], mamatial treatment and solution proedure is somehow omplex. When ompared with VAM, this latter provides a simpler physial understanding of how a raked RC or LPC beam behaves under torsion. For this reason, Bernardo et al. in 2012 [16] and in 2015 [17] proposed new developments of original VAM, namely Modified VAM (MVAM) and Generalized Softened VAM (GSVAM). hese models are also able to predit well entire θ urves of RC and LPC beams under torsion [18 20]. Nowadays it is well known that prestress, if rationally applied, inreases resistane to θ raking and also stiffness of onrete beams, inluding for beams under shear or torsion. his is beause prestress generates a ompressive stress state whih, in ombination with shear stress due to torsional moment, results in a biaxial stress state (shear + ompression). his biaxial stress state delays onrete raking. Furrmore, sine tensile strength of onrete does not inrease in same proportion as ompressive strength inreases, it is not possible to reah full potential of onrete in strutures for whih resistane is governed by tensile stresses due often to shear or torsion. herefore, prestress an also inrease torsional strength and allow for a larger area of onrete ross setion to be effetive. For se reasons, longitudinal prestress is a urrent tehnique used in beams under high torsional loads, suh as in girders of urved bridges [21]. ransversal prestress is also sometimes used in box girders to solve problems related with high shear stress in vertial walls due to shear fores. In suh members, high torsional moments an simultaneously exist, whih inreases even more shear stress in walls. In this ase, transversal prestress an be used to solve problem due to ombined shear fores and torsional moments. None of previously referred oretial models for torsion over transversal prestress. In this study, MVAM is extended to over PC beams under torsion. he hanges in original VAM formulation are presented separately for eah behavioral state of beam (as also performed in original MVAM), namely: non-raked state, transition between non-raked and raked state and raked state until failure. Calulation algorithms and solution proedures are also presented. Sine no experimental results with PC beams under torsion were found in literature, and sine preditions from MVAM were already heked against experimental results for LPC beams under torsion [18,20], this study presents a oretial parametri study to ompare global behavior of LPC and PC beams under torsion. his study aims to hek if transversal prestress is also an effiient tehnique to improve torsional behavior of beams. 2. Modified Variable Angle russ Model (MVAM) In previous studies [16,18,20], MVAM was validated for RC and LPC beams under torsion. his setion summarizes MVAM formulation (ables 1 and 2) and solution proedure for LPC beams. he MVAM is extended to PC beams in Setion Non-Craked State Figure 1 shows a typial θ urve for a RC or LPC beam under torsion. he omputation of oretial θ urve from MVAM [16,18,20] for non-raked state (Zone 1 in Figure 1) requires 3 equilibrium equations to ompute torque,, effetive thikness of equivalent hollow setion, t d, and angle of inlined onrete struts from horizontal axis of beam, α, (see Equations (1) (3) in able 1). MVAM also requires 3 ompatibility equations to ompute

3 Designs 2018, 2, 12 3 of 22 strain in transversal reinforement, ε t, strain in longitudinal reinforement, ε l, and twist, θ (Equations (6) (8) in able 1). Designs 2018, 2, 12 3 of 22 able 1. MVAM equations for LPC beams: non-raked state [7,8,16,18,20]. able 1. MVAM equations for LPC beams: non-raked state [7,8,16,18,20]. = onrete ompressive strength (uniaxial); f = onrete ompressive strength (uniaxial); A o = Ao area limited = area limited by by enter enter line ofline of flow of flow shear of shear stresses; stresses; A = area limited A = area limited by outer by perimeter outer perimeter of of setion setion (inludes (inludes hollow hollow area): area): A = xy, with A xy, x with and x y and y width and height of ross setion; A width and height of ross setion; lh ; A th = homogenized steel areas in longitudinal and transversal diretion, respetively; A l,eq Alh; ; A t,eq Ath = equivalent homogenized areasteel of effetive areas in onrete longitudinal in tension and for transversal longitudinal diretion, and respetively; transversal diretion, p = perimeter Al,eq; At,eq of = equivalent area A ; area of effetive onrete in tension for longitudinal and transversal diretion, p o = perimeter of area A p = perimeter of o ; area A; t = thikness of wall; u = perimeter po = perimeter of enterlines of area Ao; of losed stirrups: u = 2(x 1 + y 1 ), with x 1 and y 1 width and height of t enterlines = thikness of legs of of wall; losed stirrups; s = spaing u = of perimeter transversal of enterlines reinforement; of losed stirrups: u 2( x1y 1), with x1 and y1 width and f d = stress in diagonal onrete strut; A height of enterlines of legs of losed stirrups; sl ; A pl = total area of longitudinal ordinary and prestress reinforement, respetively; A st = s area of = spaing one unit of of transversal reinforement; ε ds = fd maximum = stress ompressive in diagonal strain onrete in strut; outer fiber in strut diretion; f sl ; f pl = stress in longitudinal ordinary and prestress reinforement, respetively; Asl; Apl = total area of longitudinal ordinary and prestress reinforement, respetively; f st = stress in transversal reinforement; f p = Ast stress= inarea of onrete one unit due of to prestress; transversal reinforement; n = Eεds /E s, = with maximum E andompressive E s Young s strain Modulus in outer for onrete fiber in andstrut ordinary diretion; reinforement, respetively; n p = fsl; E p /E fpl s =, with stress E in p and longitudinal E s Young s ordinary Modulus and for prestress prestress reinforement, and ordinary respetively; reinforement, respetively; ϱ l ; ϱ t = longitudinal and transversal reinforement ratio, respetively (see Setion 3.1); fst K = stress in transversal reinforement; t,eq = equivalent seant torsional stiffness; K t,eq,tot fp = equivalent = stress in total onrete torsional due stiffness to prestress; of full setion (inluding onrete ore); K t, = n torsional = E/Es, stiffness with E of and onrete Es Young s ore; Modulus for onrete and ordinary reinforement, respetively; β = St. Venant s oeffiient. np = Ep/Es, with Ep and Es Young s Modulus for prestress and ordinary reinforement, respetively; f ρl; ρt = longitudinal and transversal reinforement ratio, respetively (see Setion 3.1); Kt,eq = equivalent seant torsional stiffness; Kt,eq,tot = equivalent total torsional stiffness of full setion (inluding onrete ore); Kt, β = torsional stiffness of onrete ore; = St. Venant s oeffiient.

4 Designs 2018, 2, 12 4 of 22 Designs 2018, 2, 12 Designs 2018, 2, 12 4 of 22 able 2. Equations of MVAM for LPC beams: raked state [7,8,16]. able 2. Equations of MVAM for LPC beams: raked state [7,8,16]. 4 of 22 able 2. Equations of MVAM for LPC beams: raked state [7,8,16]. x; y = width and height of ross setion ore, respetively; xx;;δθ y =max width and height of ross setion ore, respetively; = orretion of of twists. width and height ross setion ore, respetively; y = Δθ max = orretion of twists. θ = orretion of twists. max u u y y r = Craking torque torque wist orresponding to r (non-raked r r ==Craking to r (non-raked r = wist orresponding state) to r (raked state) r = wist orresponding to r (raked r = wist orresponding state) r r y = Yielding torque state) θ wist orresponding to y yy == Yielding torque u = Resistane torque θy = wist orresponding to y θu = wist orresponding to u I II r r I1 2.a II r r 1 2.a y 2.b y u u3 u = Resistane torque θu = wist orresponding to u 3 2.bFigure 1. ypial θ foraarc RCororLPC LPC beam under torsion. Figure 1. ypial θ urve urve for beam under torsion. Figure 1. ypial θ urve for a RC or LPC beam under torsion.

5 Designs 2018, 2, 12 5 of 22 Designs 2018, 2, 12 5 of 22 he raking torque (r) ) sets upper limit for Zone 1 (Figure 1). r is alulated from an equation based on Bredt s hin ube heory, whih inorporates a prestress a fator fator aount to aount for for prestress prestress (Equation (Equation (9) in (9) able in able 1). Bonded 1). Bonded reinforement reinforement delays delays onrete onrete raking, raking, so alulation so alulation of ofeffetive effetive raking raking torque torque (r,ef) r,ef aounts ) for for influene of of torsional reinforement ratio (both longitudinal, ρl, ϱ l, and transversal, ρt) ϱ t ) (Equation (10) in able 1). Prestress reinforement is is also aounted if if bonded to to onrete onrete and if and loated if loated in outer in area outer of area onrete of onrete ross setion. ross setion. For non-raked state, MVAM inorporates ontribution of of onrete onrete in tension in tension (whih (whih is negleted is negleted in VAM) in for VAM) both longitudinal for both longitudinal and transversal and transversal diretions. diretions. In addition, In addition, MVAM MVAM also inorporates also inorporates ontribution ontribution of onrete of ore onrete for plain ore beams for plain (whih beams is also (whih negleted is also in negleted VAM). in his VAM). is performed his is using performed homogenization using homogenization tehnique for tehnique both diretions for both and diretions also by and omputing also by an omputing equivalent an thikness equivalent h eq thikness for onrete heq for in tension onrete (Figure in tension 2), whih (Figure is 2), assumed whih to is assumed be effetive to be to arry effetive to internal arry fores internal alongfores with along torsional with reinforement torsional reinforement (Equations (Equations (4) and (5) (4) in able and (5) 1). in Figure able 2 1). illustrates Figure 2 illustrates equivalent hollow equivalent setion hollow with setion wall s thikness with wall s h eq thikness assumed by heq assumed MVAM and by MVAM also ross and also setion of ross setion onreteof strut (with onrete unitary strut width) (with unitary with width) orresponding with orresponding strain and stresstrain diagrams and stress alongdiagrams its effetive along thikness its effetive t d. thikness td. ds B A F t,tot A st f st t t d/2 k 2 t d d= ds t d C A sl f sl F l,tot A pl f pl heq f d 1.00 neutral axis Figure 2. Equivalent hollow setion assumed by MVAM (LPC beams). he ross setion illustrated in Figure 2 is used to alulate depth of enter line of he ross setion illustrated in Figure 2 is used to alulate depth of enter line of shear shear flow, area of effetive tensile onrete and dimensions of onrete ore (for plain flow, area of effetive tensile onrete and dimensions of onrete ore (for plain beams). beams). In this study, ACI ode [22] is used to ompute heq (Equation (13) in able 1). In this study, ACI ode [22] is used to ompute h eq (Equation (13) in able 1). o haraterize behavior of onrete in ompression in diagonal struts and o haraterize behavior of onrete in ompression in diagonal struts and behavior behavior of reinforement in tension, average stress (σ) strain (ε) relationships are assumed of reinforement in tension, average stress (σ) strain (ε) relationships are assumed aounting for aounting for softening (onrete) and stiffening (reinforement) effets. he σ ε relationship for softening (onrete) and stiffening (reinforement) effets. he σ ε relationship for onrete in ompression onrete in ompression used in this study is one proposed by Belarbi and Hsu in 1991 [23] used in this study is one proposed by Belarbi and Hsu in 1991 [23] (Equations (45) and (46) in (Equations (45) and (46) in able 3) with softening oeffiients for maximum stress (βσ) and for able 3) with softening oeffiients for maximum stress (β σ ) and for strain orresponding to strain orresponding to maximum stress (βε) proposed by Zhang and Hsu in 1998 [24] (Equations maximum stress (β ε ) proposed by Zhang and Hsu in 1998 [24] (Equations (47) (50) in able 3). In addition, (47) (50) in able 3). In addition, σ ε relationship for non-prestress reinforement in tension σ ε relationship for non-prestress reinforement in tension proposed by Belarbi and Hsu in 1995 [25] proposed by Belarbi and Hsu in 1995 [25] (Equations (58) (61) in able 4) and σ ε relationship (Equations (58) (61) in able 4) and σ ε relationship for prestress reinforement in tension proposed by for prestress reinforement in tension proposed by Ramberg Osgood, as presented in [8] (Equations Ramberg Osgood, as presented in [8] (Equations (65) and (66) in able 5), are also used. (65) and (66) in able 5), are also used. he ompressive stress in onrete diagonal struts, f d, is defined as average stress of he ompressive stress in onrete diagonal struts, fd, is defined as average stress of a a non-uniform stress diagram (Figure 2) (Equation (51) in able 3). In Figure 2, parameters A, B, non-uniform and C are stress maximum diagram stress, (Figure average 2) (Equation stress and(51) in stress able diagram 3). In Figure resultant 2, parameters (Equations A, (52) (54) B, and in C able are 3), maximum respetively. stress, Parameter average kstress and stress diagram resultant (Equations (52) (54) in able 1 in Equation (51) is omputed by integrating Equations (45) and (46) (k 3), respetively. Parameter k1 in Equation (51) is omputed by integrating Equations (45) and (46) (k1 1 = B/A aording to Figure 2). = B/A aording to Figure 2).

6 Designs 2018, 2, 12 6 of 22 Designs 2018, 2, 12 Designs 2018, 2, 12 6 of 22 6 of 22 able able σ ε σ ε relationship relationship for for foronrete in in inompression. εo o = strain = strain orresponding to to f ; ; εo ε strain orresponding to ; 1 = tensile strain in perpendiular diretion to strut; ε1 = tensile strain in perpendiular diretion to strut; ε1 ε tensile strain in perpendiular diretion to strut; εdsi dsi = initial ompressive strain in outer fiber of onrete strut due to prestress; = initial ompressive strain in outer fiber of onrete strut due to prestress; εdsi ε ds = effetive initial ompressive strain in outer outer fiber fiber of of onrete strut strut; due to prestress; effetive ompressive strain in outer fiber of onrete strut; ϱ l ; ϱ t = effetive longitudinal ompressive and transversal strain in reinforement outer fiber ratio, of respetively onrete strut; (see Setion 3.1); ρl; ρt = and transversal reinforement ratio, respetively (see Setion 3.1); ρl; ϱ p = longitudinal ρt longitudinal prestress and transversal reinforement reinforement ratio (see ratio, Setion respetively 3.1); (see Setion 3.1); fρp = longitudinal prestress reinforement ratio (see Setion 3.1); ρp sly ; f sty longitudinal = yielding stress prestress in reinforement longitudinal ratio and (see transversal Setion 3.1); reinforement, respetively; ffsly; fsty yielding stress in longitudinal and transversal reinforement, respetively; fsly; pl0.1% = onventional proportional stress of longitudinal prestress reinforement. fsty yielding stress in longitudinal and transversal reinforement, respetively; fpl0.1% = onventional proportional stress of longitudinal prestress reinforement. fpl0.1% onventional proportional stress of longitudinal prestress reinforement. ds ds f f For For non-raked non-raked state, state, strains strains longitudinal in longitudinal non-prestress non-prestress and prestress and reinforement, prestress For non-raked state, strains in longitudinal non-prestress and prestress ε l reinforement, and ε p, whih are εl and need εp, to whih alulate are need stresses, to alulate f l and f p stresses,, must be omputed fl and fp, must aounting be omputed for reinforement, εl and εp, whih are need to alulate stresses, fl and fp, must be omputed initial aounting ompressive for strain initial due ompressive to prestress, strain ε dsi, and due also to prestress, for state εdsi, before and also onrete s for deompression state before (Equations aounting for initial ompressive strain due to prestress, εdsi, and also for state before onrete s (62) deompression and (63) in (Equations able 4 and(62) Equations and (63)(67) in able and 4 (68) and inequations able 5). (67) heand effetive (68) in ompressive able 5). onrete s deompression (Equations (62) and (63) in able and Equations (67) and (68) in able 5). strain he ε effetive ds in ompressive outer fiber of strain onrete he effetive ompressive strain in strut outer must fiber alsoof be omputed onrete strut aounting must also for this be omputed initial strain ds in outer fiber of onrete strut must also be omputed and ds aounting state (Equations for this initial (55) and strain (56) and in state able (Equations 3). (55) and (56) in able 3). aounting he for this initial strain and state (Equations (55) and (56) in able 3). he influene influene of of onrete onrete ore ore in in torsional torsional stiffness stiffness of of plain plain beams beams is is inorporated inorporated by by orreting he influene of onrete ore in torsional stiffness of plain beams is inorporated by orreting twists θ previouslyalulated with with oretial oretial model model for for non-raked non-raked state. he state. orreting twists θ previously alulated with oretial model for non-raked state. he he orretion orretion onsists onsists to add to add torsional torsional stiffness stiffness of of onrete onrete ore ore to to torsional torsional stiffness stiffness of of orretion onsists to add torsional stiffness of onrete ore to torsional stiffness of equivalent hollow setion (Equations (18) (21) in able 1). equivalent hollow setion (Equations (18) (21) in able 1).

7 Designs 2018, 2, 12 7 of 22 Designs 2018, 2, 12 7 of 22 able 4. σ ε relationship for forordinary ordinary reinforement in tension. in tension. Es; E s ; E E = Young s modulusof of ordinary reinforement and and onrete, onrete, respetively; respetively; A A = area = area limited by by external perimeter of setion; Ah A h = hollow = hollow area area of of setion (for plain setions: Ah A h = 0); 0); fr f r = onrete = onrete raking raking stress; stress; ρ ϱ = reinforement = reinforement ratio ratio (see (see Setion Setion 3.1); 3.1); εsli ε sli = initial = initial ompressive strain in in longitudinal ordinary reinforement; ε sl = effetive = effetive strain strain in in longitudinal ordinary reinforement; f sy = reinforement yielding stress. fsy = reinforement yielding stress. sl o o ompute solution points of oretial θ θ urve in in non-raked state, state, an iterative an iterative solution proedure based on a trial-and-error tehnique is is used. used. Figure Figure 3 shows 3 shows flowhart flowhart of of alulation algorithm for hollow (Figure3a) 3a) and andplain plain (Figure (Figure 3b) 3b) setions. setions. Figure Figure 3b only 3b only presents presents needed needed additional part of flowhartwith withrespet to to Figure Figure 3a. 3a. he he alulation alulation proedure proedure ends ends when when effetive effetive raking raking torque torque is is reahed. reahed Craked Craked State When When effetive raking torque (r,ef) is reahed, influene of onrete in tension vanishes r,ef ) is reahed, influene of onrete in tension vanishes from from longitudinal longitudinal and and transversal transversal equilibrium equilibrium equations. equations. From From this this point, point, resultant resultant fores fores in in both diretions are equilibrated only by longitudinal and transversal reinforement, inluding both diretions are equilibrated only by longitudinal and transversal reinforement, inluding prestress reinforement (Equations (23) and (24) in able 2). his indues a sudden inrease of prestress reinforement (Equations (23) and (24) in able 2). his indues a sudden inrease of twist (Zone 2.a in Figure 1), as it is experimentally observed for plain beams. For hollow beams, twist (Zone 2.a in Figure 1), as it is experimentally observed for plain beams. For hollow beams, experimental tests show that Zone 2.a is not learly observed [26] (Figure 1). For suh beams, experimental tests show that Zone 2.a is not learly observed [26] (Figure 1). For suh beams, MVAM MVAM usually estimates a negligible length for Zone 2.a [16]. usually Despite estimates onrete a negligible reahes its length tensile for strength Zone 2.a in [16]. outer area of ross setion, for plain beams Despite influene onrete of onrete reahesore its tensile still remains strength after inraking outer [27]. area of ross setion, for plain beams influene In raked of onrete state, ore equivalent still remains hollow after setion (with[27]. thikness heq) assumed for nonraked In state raked is no state, longer used equivalent to ompute hollow area setion limited (with by thikness enter hline eq ) assumed of shear forflow, non-raked neir state its perimeter. is no longer After used onrete to ompute raking, area effetive limitedwall by thikness enter is line onsidered of shear to be equal flow, neir to its perimeter. effetive thikness After onrete of raking, onrete struts, effetive td, omputed wall thikness from isvam onsidered [7,8]. to he beenter equal to line of effetive thikness shear flow ofis now onrete assumed struts, to be t d loated, omputed at td/2 from (Equations VAM (25) [7,8]. and (26) hein enter able 2). line of shear flow is now assumed to be loated at t d /2 (Equations (25) and (26) in able 2).

8 Designs 2018, 2, 12 8 of 22 Designs 2018, 2, 12 8 of 22 able 5. σ ε relationship for prestress reinforement in tension. able 5. σ ε relationship for prestress reinforement in tension. Ep = Young s modulus of prestress reinforement; E p = Young s modulus of prestress reinforement; fpi = initial stress due to prestress in prestress reinforement; f pi = initial stress due to prestress in prestress reinforement; fpt f = ultimate stress of prestress reinforement; pt = ultimate stress of prestress reinforement; fp0.1% f = onventional proportional stress of longitudinal p0.1% = onventional proportional stress of longitudinal prestress reinforement; ε εpl = strain in prestress reinforement; pl = strain prestress reinforement; ε εde = strain in prestress reinforement at onrete s de = strain prestress reinforement at onrete s deompression; ε εpli pli = initial = initial tensile tensile strain strain in prestress prestress reinforement; ε εsli sli = initial = initial ompressive ompressive strain strain in longitudinal ordinary ordinary reinforement; ε = effetive strain in longitudinal ordinary l = effetive strain longitudinal ordinary reinforement; ε p0.1% εp0.1% = strain = strain orresponding to to f fp0.1%.. l Sine in raked state beam already reahed onrete s deompression, alulation Sine in raked state beam already reahed onrete s deompression, alulation of strains in longitudinal non-prestress and prestress reinforement, εsl and εp, to alulate of strains in longitudinal non-prestress and prestress reinforement, ε sl and ε p, to alulate stresses, fsl and fp, is performed with Equation (64) in able 4 and Equation (69) in able 5, stresses, f sl and f p, is performed with Equation (64) in able 4 and Equation (69) in able 5, respetively. respetively. In same way, effetive ompressive strain in outer fiber of onrete In same way, effetive ompressive strain ε ds ds in outer fiber of onrete strut is omputed fromstrut Equation is omputed (57) in from ableequation 3. (57) in able 3. o estimate o estimate Zone Zone 2.a in 2.a Figure in Figure 1, alulation 1, alulation proedure proedure presented presented in able 2 in is able only performed 2 is only performed until r,ef is reahed ( alulation proedure for Zones 2.b and 3 is desribed in Setion until r,ef is reahed ( alulation proedure for Zones 2.b and 3 is desribed in Setion 2.3). 2.3). From obtained results, only point of θ urve orresponding to = r,ef is used sine From obtained results, only point of θ urve orresponding to = r,ef is used sine objetive of this setion is only to estimate Zone 2.a. objetive of this setion is only to estimate Zone 2.a. he target of this setion is illustrated in Figure 4b for plain setions. Zone 2a orresponds to he target of this setion is illustratedin Figure4b for plain setions. Zone 2a orresponds to horizontal landing, with = r,ef and r. he value of r orresponds to absissa of r horizontal landing, with = r,ef and θ I r θ θ II r. he value of θ I r orresponds to absissa of intersetion point between horizontal level for = r,ef and oretial θ urve omputed intersetion point between horizontal level for = r,ef and oretial θ urve omputed with MVAM for non-raked state (Setion 2.1). he value of orresponds to absissa r with MVAM for non-raked state (Setion 2.1). he value of θ II r orresponds to absissa of intersetion of intersetion point between point between same horizontal same horizontal level and level and oretial oretial θ urve θ urve omputed omputed with with MVAM for raked state. MVAM for raked state. Figure 4 illustrates how orretions are implemented to draw final oretial θ for RC Figure 4 illustrates how orretions are implemented to draw final oretial θ for hollow and plain beams (Figure 4a,b), respetively), starting from point orresponding to RC hollow and plain beams (Figure 4a,b), respetively), starting from point orresponding to raking torque (r,ef). raking torque ( Based on r,ef ). aforementioned, iterative alulation proedure to ompute θ urve Based on aforementioned, iterative alulation proedure to ompute θ urve presented in Setion 2.1 needs to be orreted for raked state (able 2). he flowhart for presented alulation in Setion algorithm 2.1 needs is illustrated to be orreted in Figure 5a for (hollow raked setions) state and (able Figure 2). 5b he (plain flowhart setions), for with alulation algorithm is illustrated in Figure 5a (hollow setions) and Figure 5b (plain setions), with α α and td alulated from Equations (2) and (3) (see able 1), respetively. and t d alulated from Equations (2) and (3) (see able 1), respetively.

9 Designs 2018, 2, 12 9 of 22 Designs 2018, 2, 12 9 of 22 Designs 2018, 2, 12 9 of 22 Figure 3. Flowhart to ompute θ urve from MVAM for non-raked state: (a) hollow Figure 3. Flowhart to ompute θ urve from MVAM for non-raked state: (a) hollow setions, Figure setions, 3. Flowhart (b) plain setions. to ompute θ urve from MVAM for non-raked state: (a) hollow (b) plain setions, setions. (b) plain setions. n n y y r,ef r,ef MVAM (non-raked state) MVAM MVAM orreted (Final urve) MVAM orreted (Final urve) MVAM (raked state ) MVAM (raked state ) max n max n (Final urve) (Final urve) (non-raked state) (non-raked state) (Final urve) (Final urve) (Zone 2.a) (Zone 2.a) I II I II r r y,or y n,or n r r y y,or n n,or I II I II r r y,or y (a) n,or n r r y y,or (b) n n,or (a) (b) Figure 4. Corretions of θ urves from MVAM after raking: (a) hollow setions, (b) plain Figure setions. 4. Corretions of θ urves from MVAM after raking: (a) hollow setions, (b) plain Figure 4. Corretions of θ urves from MVAM after raking: (a) hollow setions, (b) plain setions. setions. y y r,ef r,ef MVAM (non-raked state) MVAM MVAM (raked state) MVAM (raked state) MVAM (raked state) MVAM (raked state) MVAM orreted (Final urve) MVAM orreted (Final urve)

10 Designs 2018, 2, of 22 Designs 2018, 2, of 22 Figure 5. Flowhart to ompute θ urve from MVAM (raked state): (a) (a) hollow hollow setions, (b) (b) plain plain setions Zones 2.b and 3 (Craked and Ultimate States) he oretial θ urve from from MVAM for for raked raked state, state, omputed omputed as presented as presented in Setion in Setion 2.2, is 2.2, not is fully not fully valid valid for Zone for 2.b Zone and2.b 3 (Figure and 3 (Figure 1), sine1), sine model inorporates model inorporates influene influene of onrete of ore onrete for plain ore setions. for plain setions. Earlier studies Earlier [27,28] studies shown [27,28] that shown that influene influene of onrete of onrete ore forore raked for raked torsional torsional stiffness stiffness of beam of dereases beam dereases torsional as moment torsional inreases. moment inreases. his influene his is influene residualis inresidual ultimate in state. ultimate hen, state. forhen, plain for beams plain a riterion beams a riterion is adopted is adopted in this study in this tostudy orret to orret θ urve θ alulated urve alulated from MVAM from MVAM for raked for raked state (Setion state (Setion 2.2), starting 2.2), starting from r,ef from to r,ef maximum to maximum torque torque u. Sine u. Sine onrete onrete ore mainly ore influenes mainly influenes torsional torsional stiffness stiffness in raked in state raked [27,28], state only [27,28], only twists θ are twists orreted θ are orreted in θ in urve. θ he urve. orretion he orretion method method for plainfor beams plain onsists beams onsists as follows: as follows: from r,ef from to u r,ef to influene u influene of onrete of ore onrete is gradually ore is redued gradually byredued dereasing by linearly dereasing external linearly dimensions external ofdimensions onreteof ore. onrete Full dimensions ore. Full anddimensions null dimensions and null are dimensions are onsidered for r,ef and u, respetively (Equations (35) (41) in able 2). For θ θu orretions are made by assuming that influene of onrete ore no more exists (Equation

11 Designs 2018, 2, of 22 onsidered for r,ef and u, respetively (Equations (35) (41) in able 2). For θ θ u orretions are made by assuming that influene of onrete ore no more exists (Equation (42) in able 2). he riterion to orret θ urve of plain beams, for raked and ultimate state, is illustrated in Figure 4b. A linear variation for θ is assumed from r,ef to u. For hollow beams, θ urve is firstly alulated with MVAM for raked state (Setion 2.2). Sine no onrete ore exists and sine MVAM always predits a narrow Zone 2.a ( θ in Figure 1), a small orretion of twists is performed beause, as previously referred, no experimental Zone 2.a is usually observed for hollow beams. For this, all twists of θ urve, starting from point (θ II r; r ), are translated to left by θ (Equations (43) and (44) in able 2). he riterion to orret θ urve for hollow beams, for raked and ultimate state, is illustrated in Figure 4a. he flowharts for alulation algorithm for MVAM in raked state, inluding orretion riteria for twists, are illustrated in Figure 5a (hollow setions) and Figure 5b (plain setions). Only new/different part of flowhart is presented in Figure 5b. he oretial failure point of beam under torsion is defined as follows: eir maximum ompressive strain in outer fiber of onrete struts, ε ds (Figure 1), or tensile strain for torsional reinforement, ε s, reahes its ultimate onventional value (ε u and ε su, respetively). 3. Extension of MVAM for PC Beams In this setion, MVAM is extended to over both longitudinally and transversally prestressed onrete beams. he aim is to propose a generalization of MVAM for PC beams for whih RC, LPC and PC beams onstitutes partiular ases. hus, MVAM is generalized to beams with ombined longitudinal and transversal prestress (LPC beams) Non-Craked State As previously referred, upper limit of Zone 1 (Figure 1) is set up with effetive raking torque r,ef, whih is omputed from Equation (9) (able 1) and remains valid both for RC and LPC beams. For PC or LPC beams, failure riterion for onrete under biaxial state, from whih Hsu in 1984 [29] proposed prestress fator inorporated in Equation (9) ( ( f p / f )) for LPC beams, must be reviewed. Let us first onsider a hollow beam under a torsional moment,, ombined with a longitudinal prestress stress state, σ, as illustrated in Figure 6a. he biaxial stress state in element A of beam (Figure 6b) an be illustrated by Mohr s irle in a σ (normal stress) τ (shear stress) oordinate system, as illustrated in Figure 6. Failure of element A will our when biaxial stress state reahes a ritial value. For LPC beams, Hsu in 1984 [29] used Cowen s failure riterion from 1952 [3], whih onstitutes a simplifiation of Mohr s failure riterion. his riterion is defined by following equations whih allow to ompute maximum shear stress τ: τ f = σ ( ) σ 2 f (71) τ f f = ( 1 ( ) f ) f 1 + σ f, (72) f t Equation (71) is applied if failure is governed by ompression, while Equation (72) is applied if failure is governed by tension. f t is onrete strength in tension and σ is onrete ompression stress due to longitudinal prestress. f t

12 Designs 2018, 2, of 22 Designs 2018, 2, 12 Designs 2018, 2, of of 22 A A y y (a) (a) y 1 y 1 (b) (b) y y (-) (-) (-) (-) (+) (+) (-) (-) (-) (-) (+) (+) m xm x P(-,) P(-,) R(,0) min R(,0) min () () C C Q(,0) max Q(,0) max P (0,-) P (0,-) Figure 6. Stress state for LPC beams under torsion: (a) beam, (b) element A, () Mohr s irle. Figure 6. Stress state for LPC beams under torsion: (a) beam, (b) element A, () Mohr s irle. For usual prestress levels, failure is generally governed by tension [29]. hen, prestress fator is For omputed usual prestress from levels, Equation failure (72). Assuming is generally governed by t 10, Equation tension [29]. (72) an hen, be written prestress fator as follows: is omputed from Equation (72). Assuming f / f f / f t t= 10,, Equation (72) (72) an an be be written as as follows: τ 10 f = 1 1 ( ) f 1 + σ 110 f (72a) f f f f t }{{} (72a) t γ ft he prestress fator is ratio between strength of beam with and without prestress. he prestress fator γ is ratio between strength of a beam with and without prestress. Suh prestress fator is inorporated in Equation (9) (able 1) to ompute raking torque. Suh prestress fator γ is inorporated in Equation (9) (able 1) to ompute raking torque. In this study, for for PC PC or LPC or LPC beams beams prestress prestress fator is fator obtained is obtained a similar in way similar as previously way as In this study, for PC or LPC beams prestress fator is obtained in a similar way as previously desribed. Let us now onsider hollow PC beam under torsion (Figure 7a). he stress previously desribed. Let desribed. us nowlet onsider us now a hollow onsider PC a hollow beam under PC beam torsion under (Figure torsion 7a). (Figure he stress 7a). state he of stress state element of A (Figure element 7b) (Figure is illustrated 7b) is by illustrated Mohr s by irle Mohr s in Figure irle 7. in Figure 7. state of element A (Figure 7b) is illustrated by Mohr s irle in Figure 7. A A (a) (a) (b) (b) y y (-) (-) (-) (-) (+) (+) (+) (-) (+) (-) (-) (-) x x () () R(,0) min R(,0) min P(-,-) P(-,-) C C P (0,) P (0,) Q(,0) max Q(,0) max Figure 7. Stress state for PC beams under torsion: (a) beam, (b) element A, () Mohr s irle. Figure 7. Stress state for PC beams under torsion: (a) beam, (b) element A, () Mohr s irle. Figure 7. Stress state for PC beams under torsion: (a) beam, (b) element A, () Mohr s irle. Comparing Figures 7 and 6, it an be stated that unique differene between Mohr s irles Comparing Figures 7 and 6, it an be stated that unique differene between Mohr s irles is that Points and P are symmetrial with respet to axis. hen, Cowen s failure riterion is that Comparing Points P and Figures P are 7 symmetrial and 6, it an with be stated respet that to unique σ axis. differene hen, Cowen s between Mohr s failure riterion irles is gives same prestress fator as one given from Equation (72). Equation (9) from able is still gives that Points same P and prestress P are symmetrial fator as one with given respet from to Equation σ axis. (72). hen, Equation Cowen s (9) from failure able riterion 1 is still valid to ompute raking torque for PC beams, being onrete ompression stress due to valid gives to ompute same prestress raking fator torque as for one PC given beams, from being Equation σ (72). onrete Equation ompression (9) from able stress 1 due is still to transversal prestress. transversal valid to ompute prestress. raking torque for PC beams, being σ onrete ompression stress due to transversal For LPC hollow beam under torsion (Figure 8a), stress state of element (Figure 8b) For a LPC prestress. hollow beam under torsion (Figure 8a), stress state of element A (Figure 8b) is illustrated by Mohr s irle in Figure 8, where σl and σt are onrete stresses due to is illustrated by Mohr s irle in Figure 8, where σl and σt are onrete stresses due to longitudinal and transversal prestress, respetively. Comparing Figures 8 and 6 it an be stated that longitudinal and transversal prestress, respetively. Comparing Figures 8 and 6 it an be stated that element must rotate at an angle (Figure 8) to meet same stress onditions (normal stress element A must rotate at an angle Ø (Figure 8) to meet same stress onditions (normal stress

13 Designs 2018, 2, of 22 For a LPC hollow beam under torsion (Figure 8a), stress state of element A (Figure 8b) is illustrated by Mohr s irle in Figure 8, where σ l and σ t are onrete stresses due to longitudinal and transversal prestress, respetively. Comparing Figures 8 and 6 it an be stated that element ADesigns must 2018, rotate 2, 12 at an angle Ø (Figure 8) to meet same stress onditions (normal stress only13 inof one 22 diretion) as for element A for LPC beam in Figure 6. After element A is rotated, normal stress only in one diretion) as for element A for LPC beam in Figure 6. After element A is rotated, in vertial fae is σ l + σ t (Figure 8). hen, for LPC beams new prestress fator is: normal stress in vertial fae is σl + σt (Figure 8). hen, for LPC beams new prestress fator is: γ = 1 + σl + l σ t t 1 (73) (73) f One an onlude that for LPC beams (general ase), raking torque should be omputed from Equation (74) instead of Equation (9) in able t ( ) r = 166.1A t f (MPa) + σ l + σ 2.5 (MPa) 1 10 l t t ( 0.85 if f r At f if f 50 MPa) ) (74) f f t l l A y (a) (-) l (-) (-) t (-) l (-, ) P(,-) l t (b) (+) (-) t (+) (-) x R(,0) min () C P (-,-) t O Q(,0) max (0,- ) = 2OC = l t Figure 8. Stress state for LPC beams under torsion: (a) beam, (b) element A, () Mohr s irle. Figure 8. Stress state for LPC beams under torsion: (a) beam, (b) element A, () Mohr s irle. If transversal prestress reinforement is bonded to onrete and also loated in outer area If of transversal ross setion, prestress it is onsidered reinforement effetive is bonded to delay to onrete onrete raks. and also hen loated in transversal outer area of ross setion, it is onsidered effetive to delay onrete raks. hen transversal reinforement ratio, ρt, must also inorporate this reinforement as follows: reinforement ratio, ϱ t, must also inorporate this reinforement as follows: Au n st p Aptu p t A s ρ A s (75) t = A stu p A s + n pa pt u p (75) A s p where Apt is area of one unit of transversal prestress reinforement, up is perimeter and sp where A longitudinal pt is area of one unit of transversal prestress reinforement, u spaing of reinforement. Equation (75) should substitute p is perimeter and s Equation (12). p longitudinal spaing of reinforement. Equation (75) should substitute Equation (12). For PC beams, homogenization of ross setion to inlude influene of onrete For PC beams, homogenization of ross setion to inlude influene of onrete in in tension, for both longitudinal and transversal diretions, must also onsider transversal tension, for both longitudinal and transversal diretions, must also onsider transversal prestress prestress reinforement. In same way as presented in Setion 2.1 for LPC beams, now total reinforement. In same way as presented in Setion 2.1 for LPC beams, now total distributed distributed transversal fore, Ft,tot, onsiders ontribution of transversal prestress transversal fore, F t,tot, onsiders ontribution of transversal prestress reinforement as follows reinforement as follows (see Figure 9): (see Figure 9): F t,tot F t, tot = A A th th f t f/s t / = s (A A t /s t / s+ na t,eq, eq /s / s+ n p A pt /s / s p p ) f t f t (76) Equation (76) substitutes Equation (5) (5) in in able 1 to ompute effetive depth of struts ttd d (Equation (3) in able 1). 1). From VAM [7,8] for RC and LPC beams, alulation of angle of onrete struts, α, an be performed with Equation (77) whih is related with fore in transversal diretion, instead of Equation (2) in able 1. 2 Ft, tot sen (77) ft d d

14 Designs 2018, 2, of 22 From VAM [7,8] for RC and LPC beams, alulation of angle of onrete struts, α, an be performed with Equation (77) whih is related with fore in transversal diretion, instead of Equation (2) in able 1. sen 2 α = F t,tot (77) f d t d Designs 2018, 2, of 22 For PC beams, F t,tot must be omputed from Equation (76) in order to inorporate ontribution For PC beams, Ft,tot must be omputed from Equation (76) in order to inorporate of ontribution transversal of prestress transversal reinforement. prestress reinforement. A st f st Apt f pt F t,tot A sl f sl F l,tot Apl f pl f heq d Figure 9. Equivalent hollow setion assumed by MVAM (LPC beams). Figure 9. Equivalent hollow setion assumed by MVAM (LPC beams). In order to use one simple equation to alulate α and also to generalize suh equation for any In type order of beam to use(rc, onelpc, simple PC equation or LPC), to alulate an alternative α and and alsogeneral to generalize equation suh is equation proposed. for For any this, type of beam Equation (RC, LPC, (77) PC is divided or LPC), by Equation an alternative (2) from and able general 1. he equation new Equation is proposed. (78) substitutes For this, Equations (77) is divided (77) and by(2) Equation from able (2) from 1. able 1. he new Equation (78) substitutes Equations (77) and (2) from able sen Ft, tot p tg 2 (78) os Fl, tot tg 2 α = sen2 α All or equations from able 1 remain os 2 unhanged α = F t,tot p o (78) F l,tot and valid for PC and LPC beams. he redution fators βσ = βε depends on parameter η (Equation (48) in able 3), whih represents All or equations from able 1 remain unhanged and valid for PC and LPC beams. ratio between resisting fores in longitudinal and transversal reinforement. For PC he beams, redution parameter fators η must β σ = βalso ε depends aount onfor parameter resistane η (Equation fore (48) in in able transversal 3), whih prestress represents ratio reinforement. between hen resisting η alulated fores inas follows: longitudinal and transversal reinforement. For PC beams, parameter η must also aount for resistane fore in transversal prestress reinforement. sl fsly pl fpl 0.1% hen η is alulated as follows: (79) st fsty pt pt η = ρ sl f sly + ρ pl f pl0.1% (79) ρ st f sty + ρ pt f pt0.1% Au pt p pt (80) ρ As pt = A ptu p p (80) A s p ρpt = transversal prestress reinforement ratio; ϱ pt = fpt0.1% transversal = onventional prestress proportional reinforement stress ratio; of transversal prestress reinforement. f pt0.1% = onventional proportional stress of transversal prestress reinforement. Equations (79) and (80) substitute Equations (48) and (48b), respetively, in able 3. Equations For PC (79) beams, and (80) substitute alulation Equations proedure to (48) ompute and (48b), strain respetively, in transversal in able 3. prestress For reinforement, PC beams, εpt, whih alulation is need to ompute proedure stress, to ompute fpt, should follows strainidential transversal steps as ones prestress presented in Setion 2.1 for LPC beams. he equations are following ones: reinforement, ε pt, whih is need to ompute stress, f pt, should follows idential steps as ones presented in Setion 2.1 for LPC beams. he equations, are following ones: (81) pt pi t t f pi, t ε pt = ε pi,t + ε t (81) pi, t (82) E pt ε pi,t st = pi,t sti st E pt (83) (82) ε st = ε sti + ε st (83)

15 Designs Designs 2018, 2018, 2, 12 2, of of 22 Apt pt 2 s p f pi,t f, t ε sti = ( s 2 A p st s (E s E ) + A A h 2 A ) (84) sti pt s p E (84) A A st 2 E pt s E A A h 2 E For LPC beams, Equations (81) and (82) s must be used with s Equations p (67) and (68) in able 5, while Equations For LPC (83) beams, and (84) Equations must (81) be used and (82) withmust Equations be used (62) with and Equations (63) in(67) able and 4. (68) in able 5, In while PCEquations beams, (83) effetive and (84) must strain be in used with transversal Equations ordinary (62) and (63) reinforement able 4. must be alulated aounting In that PC itbeams, undergoes effetive an initial strain ompressive in transversal strain εordinary sti due toreinforement transversal must prestress. be alulated he strain in aounting transversal that reinforement it undergoes an ε st initial due toompressive external strain torque εsti due isto alulated transversal with Equation prestress. he (6) from ablestrain 1. in transversal reinforement εst due to external torque is alulated with Equation (6) he from outer able fiber 1. of onrete struts also undergoes an initial ompressive strain ε dsi due to transversal he prestress. outer fiber If of a 45 degrees onrete angle struts isalso assumed undergoes for an initial diretion ompressive of struts strain for εdsi due pre-raked to state, transversal n ε dsi an prestress. be simply If a 45 omputed degrees angle as follows is assumed (see for Figure diretion 10b): of struts for pre-raked state, n εdsi an be simply omputed as follows (see Figure 10b): ε sti ε dsi (85) sti os 45 dsi (85) os 45 dsi dsi ti 45º 45º li (a) (b) Figure 10. Initial ompressive strain in onrete struts due to prestress: (a) LPC beams, (b) PC Figure 10. Initial ompressive strain in onrete struts due to prestress: (a) LPC beams, (b) PC beams. beams. For LPC For LPC beams, beams, outer outer fiber fiber of of onrete struts undergoes an initial ompressive strain due to both due longitudinal to both longitudinal and transversal and transversal prestress. prestress. Again, Again, by assuming by assuming a 45 degrees a 45 degrees angle angle for for diretion of diretion struts, εof dsi anstruts, be omputed εdsi an be as omputed follows as (see follows Figure (see 10): Figure 10): sli sti dsi (86) ε dsi ε sli + ε sti os 45 (86) os 45 All or equations presented in Setion 2.1 remains valid for PC and LPC beams, All or equations presented in Setion 2.1 remains valid for PC and LPC beams, inluding inluding proedure to orret twists in order to inorporate influene of onrete ore proedure for plain beams to orret (able 1). twists in order to inorporate influene of onrete ore for plain beams (able For LPC 1). beams, flowhart for alulation algorithm is same as one illustrated in For Figure LPC 3a (hollow beams, setions) flowhart and Figure for 3b (plain alulation setions). algorithm he differenes is same are only as related one illustrated with in Figure orretion 3a (hollow of some setions) steps in and order Figure to substitute 3b (plain some setions). equations he and differenes add new ones: are only related with orretion - 1st of step: some Selet steps εds. in Calulate order to substitute εdsi (Equation some (86)) equations and (Equation and add (55)); new ones: ds - - 1st step: Selet ε ds. Calulate ε dsi (Equation (86)) and ε ds Equation (55)); - 4th step: Calulate (Equation (1)), εst (Equation (6)), (Equation (63)), sl (Equation (83)), -... st εpl (Equation (67), εpt (Equation (81)), fst and fsl (Equations (58)), fpl and fpt - 4th step: Calulate (Equation (1)), ε (Equations (65) and st (Equation (6)), ε sl (66)); (Equation (63)), ε st (Equation (83)), ε - pl (Equation (67), ε pt (Equation (81)), f st and f sl (Equations (58)), f pl and f pt (Equations (65) and (66)); th step: Calulate α (Equation (78)); - 7th step: Calulate α (Equation (78)); th step: > r,ef? (Equation (74)); -...

16 Designs 2018, 2, of Zone 2a (Craked State) In same way as desribed for LPC beams (Setion 2.2), after raking torque is reahed, influene of tensile onrete vanishes. hus, equilibrium equations inorporate only fores in reinforements, inluding prestress reinforement (A sl f sl + A pl f pl and A st f st /s + A pt f pt /s p for longitudinal and transversal reinforements, respetively). From VAM and for RC an LPC beams [7,8], alulation of angle of onrete struts, α, an be performed with Equation (87), whih is related with fore in transversal diretion, instead of Equation (23) from able 2. sen 2 α = A st f st s f d t d (87) For PC beams, Equation (87) must be rewritten in order to replae transversal fore (per unit length) in ordinary reinforement (A st σ st/ s) by total transversal fore (also per unit length) inluding both ordinary and prestress reinforement (A st f st /s + A pt f pt /s p ): sen 2 α = A st f st s f d t d + A pt f pt s p f d t d (88) where f pt is stress in transversal prestress reinforement. In order to use one simple equation to alulate α and to generalize this equation for any type of beams (RC, LPC, PC or LPC beams), an alternative equation is proposed by taking ratio between Equations (88) and (23) from able 2. his new equation substitutes Equations (88) and (23) from able 2. tg 2 α = sen2 α os 2 α = A st f st s A sl f sl p o + A pt f pt s p + A pl f pl p o (89) In same way, sine longitudinal equilibrium equation is also used to ompute effetive depth of onrete struts (t d ), Equation (24) must also inorporate both total longitudinal and transversal fores (general ase): t d = A sl f sl p o f d + A st f st s f d + A pl f pl p o f d + A pt f pt s p f d (90) Equation (90) substitutes Equation (24) from able 2. All or equations presented in Setion 2.2 remains valid for PC and LPC beams. he flowhart for alulation algorithm to ompute new θ urve for raked state is same as ones illustrated in Figure 5a (hollow setions) and Figure 5b (plain setions), with t d and α alulated with Equations (90) and (89), respetively Zones 2.b and 3 (Craked and Ultimate States) he alulation proedure presented in Setion 2.3 remains valid, inluding proedure to orret twists in raked state to inorporate influene of onrete ore for plain setions. he flowhart for alulation algorithm to ompute new θ urve for raked state is same as ones illustrated in Figure 5a (hollow setions) and Figure 5b (plain setions), with orretions presented in Setion 3.2.

17 Designs 2018, 2, of heoretial Parametri Study 4.1. Referene Beams and Study Variables In this setion, some oretial results obtained with MVAM extended to PC beams are used to ompare global behavior under torsion of PC, LPC and LPC beams. For this purpose, series of squared RC hollow beams tested by Bernardo and Lopes in 2009 [26] are used as referene beams. Among sixteen beams tested by authors, a set of nine beams were hosen in order to allow grouping beams with similar onrete strength and different torsional reinforement ratio (inluding beams with dutile and brittle failure) and also beams with similar torsional reinforement ratio and different onrete strength (inluding beams with normal-strength onrete NSC and high-strength onrete HSC). able 6 summarizes geometrial and mehanial properties of nine hosen squared RC hollow beams, namely: external width (x) and height (y) of ross setion, thikness of walls (t), distanes between enterlines of legs of losed stirrups (x 1 and y 1 ), area of longitudinal reinforement (A sl ), distributed area of one branh of transversal reinforement (A st/ s, where s is spaing of transversal reinforement), longitudinal reinforement ratio (ρ sl = A sl /A, with A = xy) and transversal reinforement ratio (ρ st = A st u/(a s), with u = 2(x 1 + y 1 )), average onrete ompressive and tensile strength ( f m f and f tm f r ), average yielding stress of longitudinal and transversal reinforement (f lym and f tym ), onrete Young s Modulus (E ), ompressive strains for onrete (peak stress value, ε o, and maximum value, ε u ). For reinforement, usual values were adopted for onventional maximum tensile strain (ε lu = ε tu = 10) and Young s Modulus (E s = 200 GPa). It should be noted that, for all nine beams, longitudinal and transversal reinforement ratios are equilibrated. able 6. Properties of referene RC hollow beams [26]. Beam x; y m m x 1 m y 1 m A sl A st /s m 2 m 2 /m ϱ sl % A A A B B C C C C ϱ st % f m MPa f tm MPa f lym MPa f tym MPa E GPa ε o % ε u % For objetives of this setion, referene beams will be oretially prestressed in longitudinal diretion (LPC), in transversal diretion (PC) and in both diretions (LPC). A moderate prestress level is hosen with respet to onrete ompressive strength (f p ) for LPC and PC beams: f p = f, with f p ompressive stress in onrete due to prestress. his value was hosen from range of allowed values imposed by ACI Code ( fp,max ACI = 0.45 f ). o evaluate effiieny of transversal prestress, when ompared with longitudinal prestress, hosen riteria was to impose same stress state in onrete (f p ) for LPC and PC beams. he areas of prestress reinforement were hosen so that prestress reinforement ratios for eah diretion are similar. For LPC beams, hosen riterion was to assume half of previously referred prestress level for eah diretion. With this riterion, total prestress reinforement ratio is same for all studied beams (LPC, PC and LPC), sine this quantity is related with total prestress fore to be applied. For parametri analysis, following study variables were onsidered: onrete ompressive strength (f ) and total reinforement ratios (ρ l,tot = ρ sl + ρ pl, ρ t,tot = ρ st + ρ pt, with ϱ sl, ϱ st, ϱ pl and ϱ pt

18 Designs 2018, 2, of 22 omputed with Equation (48a) able 3, Equation (12) able 1, Equation (48b) able 3 and Equation (80), respetively). o alulate ϱ pt, it is assumed that u p is equal to perimeter of enter line of walls. able 7 presents information about prestress for studied referene beams, namely: total area of longitudinal prestress reinforement (A pl ), distributed area of one branh of transversal prestress reinforement (A pt /s p, where s p is spaing of transversal prestress reinforement), ratio of longitudinal and transversal prestress reinforement (ρ pl and ρ pt, respetively), average stress in onrete due to prestress for eah diretion (f pl and f pt, respetively). For prestress reinforement, usual values were adopted for proportional onventional limit stress to 0.1% (f p0.1% = 1670 MPa) and Young s Modulus (E p = 195 GPa). he initial stress in prestress reinforement for eah diretion (f pli and f pti ) was onsidered onstant and equal to 1350 MPa. his value orresponds to a urrent hosen design riteria to ompute area of prestress reinforement, it orresponds to 0.75f pu with f pu tensile strength of prestress reinforement, whih is assumed to be equal to 1800 MPa. able 7. Prestress values for f p = f (LPC and PC) and f p = f (LPC). ype LPC PC LPC Beam A pl ϱ pl f pli f pl A pt /s p ϱ pt f pti f pt A pl A pt /s p ϱ pl ϱ pt f pli f pl f pti f pt m 2 % MPa MPa m 2 /m % MPa MPa m 2 m 2 /m % % MPa MPa MPa MPa A A A B B C C C C Comparative Analysis Based on extended MVAM presented in this study, oretial θ urves for all referene beams presented in ables 6 and 7 were omputed. he results are presented in Figures he θ urves are grouped for beams with similar onrete strength (Figure 11 for NSC beams and Figure 12 for HSC beams), as well as for beams with similar total reinforement ratio (Figure 13 for high torsional reinforement ratio torsional brittle failure, and Figure 14 for low torsional reinforement ratio torsional dutile failure). From Figures 11 14, it an be stated that raking torque is higher for LPC beams, when ompared with PC beams. his seems to show that longitudinal prestress is more effetive for low loading levels, namely to delay raked state. Sine initial average ompressive stress in onrete due to prestress is same (in magnitude) for both LPC and PC beams, this result show that raking torque strongly depends on prestressing diretion. For LPC beams, raking torque is also higher when ompared with PC beams, and similar when ompared with LPC beams. Despite LPC beams inorporate only half of prestress level for eah diretion, when ompared with LPC and PC beams, biaxial ompressive stress state in LPC beams shows to be also effetive to delay raking state. For high loading levels, Figures show that torsional behavior of LPC and PC beams with same prestress level is very similar, namely for resistane torque. In fat, it should be remembered that for high loading levels, PC beams behaves as urrent RC beams. he total reinforement ratio in prestressing diretion is very similar for both LPC and PC beams in Figures hen, from previous observations it an be stated that transversal prestress seems to be equally effetive for resistane torque when ompared with longitudinal prestress.

19 Designs 2018, 2, 12 Designs Designs2018, 2018,2,2, of of of [kn.m/m] [kn.m/m] [kn.m/m] A2-LPC A2-LPC A3-LPC A3-LPC A2-PC A2-PC A3-PC A3-PC A2-LPC A2-LPC A3-LPC A3-LPC A5-LPC A5-LPC A5-PC A5-PC A5-LPC A5-LPC θθ[rad/m] [rad/m] Figure Figure θ θurves urvesfor fornsc NSCbeams. beams [kn.m/m] [kn.m/m] [kn.m/m] C2-LPC C2-LPC C3-LPC C3-LPC C2-PC C2-PC C3-PC C3-PC C2-LPC C2-LPC C3-LPC C3-LPC C4-LPC C4-LPC C6-LPC C6-LPC C4-PC C4-PC C6-PC C6-PC C4-LPC C4-LPC C6-LPC C6-LPC θθ[rad/m] [rad/m] Figure Figure θ θurves urvesfor forhsc HSCbeams. beams. Figure θ urves urves for high high torsional reinforement reinforement ratio. Figure Figure 13. θ θ urvesfor for hightorsional torsional reinforementratio. ratio.

20 Designs 2018, 2, of 22 Designs 2018, 2, of [kn.m/m] Figure 14. θ urves for low torsional reinforement ratio. For LPC beams, resistane torque and torsional stiffness in raked state are higher when ompared with those of LPC and PC beams. Despite LPC beams inorporate same total amount of prestress reinforement, as for LPC and PC beams, results show that distribution of prestress reinforement in in both both diretions (longitudinal and and transversal) transversal) is better is better to enhane to enhane torsional torsional behavior behavior of of beams beams for for ultimate ultimate state. state. 5. Conlusions C2-LPC C2-PC C2-LPC θ [rad/m] In this study, MVAM was extended to over PC beams under torsion. For this, hanges in VAM formulation were were presented presented in order in order to inorporate to inorporate influene influene of transversal of transversal prestress, in prestress, additionin toaddition longitudinal to longitudinal prestress. prestress. Based on extended MVAM, a global alulation proedure was defined and implemented with an appropriate programming language (DELPHI). he extended MVAM allows to predit entire θ urve for beams under torsion, inluding RC, LPC, PC and LPC beams. he results from a oretial parametri study performed with extended MVAM where presented in order to ompare global behavior of LPC, PC and LPC beams with similar total prestress reinforement ratios. From oretial θ urves and for beams with prestress in only one diretion (LPC and and PC PC beams), beams), it was it observed was observed that, for that, for ultimate ultimate loading and loading whenand ompared when with ompared longitudinal with longitudinal prestress, transversal prestress, prestress transversal onstitutes prestress an onstitutes effetive solution an effetive for beams solution under for torsion. beams under However, torsion. forhowever, low loading for low levels, loading transversal levels, prestress transversal seems prestress to beseems less effetive, to be less namely effetive, to delay namely to raking delay torque. raking torque. For beams with prestress in both diretions (LPC beams), results shown that biaxial ompressive stress state due to prestress is also effetive to delay raking torque. Moreover, LPC beams shownto to have higher resistane torque torque and and torsional torsional stiffness stiffness in inraked raked state, when state, when ompared ompared with LPC withand LPCPC andbeams. PC beams. his observation his observation shows that, shows even that, with even with same total same amount total amount of prestress of prestress reinforement, reinforement, it is better it is to better distribute to distribute prestress prestress in both in both diretions diretions (longitudinal and and transversal). he extended MVAM proposed in in this study represents an an important advane in in attempt to generalize to Spae-russ Analogy for for PC PC beams under torsion. However, it it should be noted that experimental results with PC and LPC beams under torsion are of great need, namely to help to onfirm main findings of this study. Author Contributions: Luís Bernardo and Jorge Andrade worked oretial model, developed algorithms and programs. Mafalda eixeira alulated referene beams and analyzed data. Luís Bernardo wrote paper with reviews by Jorge Andrade and Mafalda eixeira. Conflits of Interest: he authors delare no onflit of interest. A2-LPC A2-PC A2-LPC B2-LPC B2-PC B2-LPC

21 Designs 2018, 2, of 22 Author Contributions: Luís Bernardo and Jorge Andrade worked oretial model, developed algorithms and programs. Mafalda eixeira alulated referene beams and analyzed data. Luís Bernardo wrote paper with reviews by Jorge Andrade and Mafalda eixeira. Conflits of Interest: he authors delare no onflit of interest. Referenes 1. Raush, E. Berehnung des Eisenbetons gegen Verdrehung (Design of Reinfored Conrete in orsion). Ph.D. hesis, ehnishe Hohshule, Berlin, German, Andersen, P. Experiments with Conrete in orsion. rans. ASCE 1935, 100, Cowan, H.J. Elasti heory for orsional Strength of Retangular Reinfored Conrete Beams. Mag. Conr. Res. 1950, 2, 3 8. [CrossRef] 4. Lampert, P.; hurlimann, B. orsionsversuhe an Stahlbetonbalken (orsion ests of Reinfored Conrete Beams) Beriht, Nr Inst. Fur Baustatik EH Zurih. 1969, 101. (In Germany) 5. Elfgren, L. Reinfored Conrete Beams Loaded in Combined orsion, Bending and Shear; Division of Conrete Strutures, Chalmers University of ehnology: Goteborg, Sweden, Collins, M.P.; Mithell, D. Shear and orsion Design of Prestressed and Non-Prestressed Conrete Beams. J. Prestress. Conr. Inst. 1980, 25, [CrossRef] 7. Hsu,..C.; Mo, Y.L. Softening of Conrete in orsional Members heory and ests. J. Am. Conr. Inst. 1985, 82, Hsu,..C.; Mo, Y.L. Softening of Conrete in orsional Members Prestressed Conrete. J. Am. Conr. Inst. 1985, 82, Rahal, K.N.; Collins, M.P. Simple Model for Prediting orsional Strength of Reinfored and Prestressed Conrete Setions. J. Am. Conr. Inst. 1996, 93, Bhatti, M.A.; Almughrabi, A. Refined Model to Estimate orsional Strength of Reinfored Conrete Beams. J. Am. Conr. Inst. 1996, 93, Wang, W.; Hsu,..C. Limit Analysis of Reinfored Conrete Beams Subjeted to Pure orsion. J. Strut. Eng. 1997, 123, [CrossRef] 12. Jeng, C.H.; Hsu,..C. A Softened Membrane Model for orsion in Reinfored Conrete Members. Eng. Strut. 2009, 31, [CrossRef] 13. Jeng, C.H.; Chiu, H.J.; Chen, C.S. Modelling Initial Stresses in Prestressed Conrete Members under orsion. In Strutures Congress; ASCE: Orlando, FL, USA, 2010; pp Jeng, C.H. Unified Softened Membrane Model for orsion in Hollow and Solid Reinfored Conrete Members: Modeling Preraking and Postraking Behavior. J. Strut. Eng. 2015, 141, [CrossRef] 15. Hsu,..C.; Zhu, R.R.H. Softened Membrane Model for Reinfored Conrete Elements in Shear. Strut. J. Am. Conr. Inst. 2002, 99, Bernardo, L.F.A.; Andrade, J.M.A.; Lopes, S.M.R. Modified variable angle truss-model for torsion in reinfored onrete beams. Mater. Strut. 2012, 45, [CrossRef] 17. Bernardo, L.F.A.; Andrade, J.M.A.; Nunes, N.C.G. Generalized softened variable angle truss-model for reinforement onrete beams under torsion. Mater. Strut. 2015, 48, [CrossRef] 18. Andrade, J.M.A. Modelling of Global Behavior of Beams under orsion Generalization of Spae russ Analogy with Variable Angle. Ph.D. hesis, Department of Civil Engineering and Arhiteture, Faulty of Engineering of University of Beira Interior, Covilhã, Portugal, (In Portuguese) 19. aborda, C.S.B. Generalization of VAM and GSVAM to Model Strutural Conrete Beams under orsion Combined with Uniforme Axial Stress State. Ph.D. hesis, Department of Civil Engineering and Arhiteture, Faulty of Engineering of University of Beira Interior, Covilhã, Portugal, (In Portuguese) 20. Bernardo, L.F.A.; eixeira, M.M. Modified Softened russ-model for PC Beams under orsion. J. Build. Eng. 2018, in review. 21. Laffranhi, M. Zur Konzeption Gekrümmter Brüken; IBK Report; Institut für Baustatik und Konstruktion: Zürih, Switzerland, (In Germany) 22. ACI Committee 318. Building Code Requirements for Reinfored Conrete, (ACI ); Amerian Conrete Institute: Farmington Hills, MI, USA, 2014.

22 Designs 2018, 2, of Belarbi, A.; Hsu,..C. Constitutive Laws of Softened Conrete in Biaxial ension-compression; Researh Report UHCEE 91-2; University of Houston: Houston, X, USA, Zhang, L.X.; Hsu,..C. Behavior and Analysis of 100 MPa Conrete Membrane Elements. J. Strut. Eng. 1998, 124, [CrossRef] 25. Belarbi, A.; Hsu,..C. Constitutive Laws of Softened Conrete in Biaxial ension-compression. Strut. J. Am. Conr. Inst. 1995, 92, Bernardo, L.F.A.; Lopes, S.M.R. orsion in HSC Hollow Beams: Strength and Dutility Analysis. ACI Strut. J. 2009, 106, Bernardo, L.F.A.; Lopes, S.M.R. heoretial Behavior of HSC Setions under orsion. Eng. Strut. 2011, 33, [CrossRef] 28. Bernardo, L.F.A.; Lopes, S.M.R. Behavior of Conrete Beams under orsion NSC Plain and Hollow Beams. Mater. Strut. 2008, 41, [CrossRef] 29. Hsu,..C. orsion of Reinfored Conrete; Van Nostrand Reinhold Company: New York, NY, USA, by authors. Liensee MDPI, Basel, Switzerland. his artile is an open aess artile distributed under terms and onditions of Creative Commons Attribution (CC BY) liense (