Finite element formulation of a composite double T-beam subjected to torsion

Engineering Structures 29 (2007) 2935–2945 www.elsevier.com/locate/engstruct

Finite element formulation of a composite double T-beam subjected to torsion Yong-Hak Lee a , Won-Jin Sung b , Tae-Hyung Lee a,∗ , Kee-Won Seong a a Civil Engineering Department, Konkuk University, 1-Hwayang-dong, Kwangjin-gu, Seoul 143-701, Republic of Korea b GS Engineering & Construction Co., Seoul, Republic of Korea

Received 19 July 2006; received in revised form 2 February 2007; accepted 5 February 2007 Available online 26 March 2007

Abstract When a composite double T-beam is subjected to torsion, a pair of prestressing tendons resist torsional twisting due to the coupled restoring forces provided by the restoring action of the upward and downward displaced prestressing tendons. In addition, the composite action of the composite double T-beams provides an additional pure and warping torsional resistance. A three dimensional finite beam element for the composite double T-beam is formulated to account for the torsional stiffness due to the restoring action of a pair of prestressing tendons and the composite action. The finite element formulation is based on Vlasov’s hypothesis that considers the warping displacement in open sections. Strain energies stored in concrete, encasing steel, reinforcing bars, and a pair of prestressing tendons are included in the total potential energy of the composite double T-beam. Two-noded beam elements with seven-degrees of freedom per node approximate the axial, flexural, and torsional displacements. The torsional resistance due to the restoring action of the pair of prestressing tendons is discussed by comparing the two warping stiffness terms calculated with and without the consideration of their action. As numerical examples, two-span and three-span steel composite double T-beams are analyzed, and their bimoments and angles of twist are compared with those calculated from the conventional three dimensional finite element analysis and the analytical method of solving the governing differential equations. c 2007 Elsevier Ltd. All rights reserved.

Keywords: Composite double T-beam; Warping torsion; A pair of prestressing tendons; Finite element analysis; Bimoment

1. Introduction Due to its aesthetic design, the use of the double T-beam, in particular, in designing bridges is gradually increasing. However, it is recognized that the double T-beam is an expensive structural system because it is usually constructed by the Moving Scaffolding System that is complex and slow in construction. To relieve the disadvantages of constructing the double T-beam, a composite double T-beam has been proposed by Park [13], where steel plates are used to encase concrete webs of the double T-beams. The new type of double T-beam takes advantage of web steels for replacing the form to support the concrete during curing and contributing to the structural stiffness after curing. Moreover, this structural system provides a longer span for the same cross-section, or slimmer webs for ∗ Corresponding author. Tel.: +82 2 450 4093; fax: +82 2 2201 0783.

E-mail address: [email protected] (T.-H. Lee). c 2007 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.02.002

the same span length. However, the torsional stiffness of the double T-beam is relatively small when compared to the closedsection of box-girders. Therefore, a refined method is required to analyze torsional behaviors of composite double T-beams considering the contribution of the prestressing forces and the composite action. A number of researches have been performed to investigate the flexural behaviors of prestressed concrete beams including the recent works by El-Ariss [4] and Lou and Xiang [7]. El-Ariss [4] proposed an analytical model for predicting the flexural behavior of externally prestressed concrete beams under service loads. Lou and Xiang [7] presented a nonlinear numerical model for externally prestressed concrete beams based on the finite element method. The early studies on the torsional behavior of structural members, in particular, thin-webbed open-section girders, subjected to combined axial, flexural, and torsional loads were conducted by Vlasov [17] and Timoshenko and Goodier [16].

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Notation Ac , Ab , As , A p cross-sectional areas of concrete, encasing steel, re-bar, and PS tendon Bo , Bv , Bw , Bθ strain–displacement matrices corresponding to do , dv , dw , dθ do , dv , dw , dθ vectors of axial, flexural about y and z axes, torsional degrees of freedom E c , E b , E s , E p modulus of elasticity of concrete, encasing steel, re-bar, and PS tendon (E A)x transformed area (E S y )x , (E Sz )x first moments of the transformed area with respect to y and z axes (E I y )x , (E Iz )x second moments of the transformed area with respect to y and z axes (E Iω )g warping stiffness of concrete and encasing steel sections (Eω)g warping stiffness of re-bar and PS tendon G c , G b shear modulus of concrete and encasing steel (G K )g torsional stiffness of concrete, encasing steel, and a pair of PS tendons Ic , Ib second moments of cross-sectional areas of concrete and encasing steel Icω , Ibω warping constants of concrete and encasing steel K c , K b pure torsional constants of concrete and encasing steel L se , L pe lengths of re-bar and PS tendon Mx , M y , Mz moments along x, y, z axes Mω Bimoment No , Nv , Nw , Nθ shape function vectors corresponding to do , dv , dw , dθ Peh , Plh horizontal components of effective prestressing force and prestressing force increment S location of shear center Sc , Sb first moments of cross-sectional areas of concrete and encasing steel ysc , y pc directional components of distances from centroid to re-bar and PS tendon along y axis yss , y ps directional components of distances from shear center to re-bar and PS tendon along y axis z sc , z pc directional components of distances from centroid to re-bar and PS tendon along z axis z ss , z ps directional components of distances from shear center to re-bar and PS tendon along z axis εo uniform strain along the depth of cross-section φx warping displacement due to torsional warping φ y , φz curvatures along y and z axes θ y , θz rotational angles corresponding to M y , Mz σxb , σxω flexural and warping normal stresses ω(y, ¯ z) warping function

behavior of open-section prestressed concrete girders are rarely found. Luccioni et al. [10] proposed the use of transformed cross-sectional areas of prestressing tendons and reinforcing bars to compute the torsional stiffness of prestressed concrete girders. This “transformed section” approach is widely used to evaluate the torsional behavior of open-section prestressed concrete girders. A composite double T-beam has a pair of prestressing tendons that are symmetric about the weak axis of the cross-section. When such a member is subjected to torsion, a pair of prestressing tendons induces coupled restoring forces that improve the torsional stiffness in addition to that considered in the “transformed section” approach. In this regard, the transformed section approach may underestimate the torsional stiffness of composite double T-beams. Comparing the resistant mechanism of torsion to that of flexure, the prestressing forces in the torsional case improve the torsional stiffness, not the load-carrying capacity, while those in the flexural case improve the load-carrying capacity, not the flexural stiffness. In this study, a three-dimensional (3D) finite beam element is formulated to simulate the behavior of the composite double Tbeam with bonded prestressing tendons subjected to combined flexural and torsional loads. In the formulation, the coupled restoring forces in a pair of prestressing tendons and their contribution of the encasing steel are considered in an explicit manner. A large deformation assumption, commonly used in analyzing cables of suspension bridges [1], is applied to evaluate the coupled restoring forces. For this, the strain energies of the concrete, encasing steel, reinforcing bars, and the pair of prestressing tendons are obtained from the deformed geometry of a composite double T-beam to form the total potential energy. The stiffness matrix of the element is derived based on the principle of minimum total potential energy. The beam element has two nodes where each of them has seven degrees of freedom (DOFs), including the warping DOF as well as the conventional 6 DOFs of the beam element. Sectional properties defining the stiffness matrix are computed at two reference axes, one through the centroid and the other one through the shear center, based on the assumption that the torsional and the flexural modes of deformation are uncoupled. The presented finite element formulation evaluates the behavior of two-span and three-span continuous composite double T-beams subjected to combined flexural and torsional loads. The results are compared to those of 3D finite element analyses using conventional continuum elements and analytical solutions of the governing differential equation to verify the accuracy and the efficiency of the proposed formulation. To evaluate the contributions of prestressing tendons and composite action, the results are also compared to those of the transformed section approach. 2. Basics in deformation geometry

Since then, numerous researches [2,3,5,6,8,14,15,18,19] have been conducted on the torsional behavior of thin-webbed opensection steel girders where the torsion of the member is considered as the superposition of the pure torsion and the warping torsion. Meanwhile, the researches on the torsional

Let’s consider a prismatic beam with an arbitrary crosssection. Fig. 1 shows the displacement of an arbitrary point A to A0 of a cross-section due to the torsional moment Mx where S and θx indicate the shear center and the rotation angle of the

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3. Strain energies stored in composite double T-beams 3.1. Strain energy of the concrete section Considering the axial deformation and shear distortions, one can express the strain energy stored in a concrete section Uco as Z 1 Uco = {σx x εx x + τx y γx y + τx z γx z }dVc . (4) Vc 2

Fig. 1. Deformed geometry of the cross-section subjected to torsional moment.

cross-section, respectively. When the cross-section is subjected to combined axial load and bending moments about y and z axes in addition to the torsional moment, the displacements u, v, and w in x, y, and z directions, respectively, can be expressed as u = u 0 + θ y z − θz y + φx ω(y, ¯ z) v = v0 + vθ

(1)

w = w0 + wθ where, vθ = −(z − z s )θx , wθ = (y − ys )θx , u 0 , v0 , and w0 are the displacement components of the centroid along x, y, and z axes, respectively. θ y and θz are the rotational angles corresponding to M y and Mz , respectively, φx is the warping displacement, and ω(y, ¯ z) is the warping function. The axial strain εx x and shear distortions γx y and γx z can be derived from (1) as εx x = ε0 + φ y z − φz y + f ω¯   ∂ ω¯ − (z − z s ) φx γx y = ∂y   ∂ ω¯ γx z = − (y − ys ) φx ∂z

(2)

+ E c Scy φ y ε0 + E c Icy φ y2 − E c Scz φz ε0 + E c Icz φz2 + E c Icw f 2 + G c K c φx2 ]dx

(5)

where Ac is the cross-sectional area of the concrete, Scy and Scz are the first moments of the cross-sectional area of the concrete about the y and z axes, respectively. Icy and Icz are the second moments of the cross-sectional area of concrete about the y and z axes, respectively, and Icω and K c are the warping and pure torsional constants of the concrete section, respectively. 3.2. Strain energy of the steel section Similarly to the strain energy of the concrete section, the strain energy of the steel section Ub can be written as Z 1 {σx x εx x + τx y γx y + τx z γx z }dVb . (6) Ub = Vb 2 Substituting (2) into (6) leads to Z l 1 [E b Ab ε02 + E b Sby ε0 φ y − E b Sbz ε0 φz Ub = 0 2 + E b Sby φ y ε0 + E b Iby φ y2

where φx = dθx /dx, φ y = −d2 w/dx 2 , φz = −d2 v/dx 2 , f = d2 θx /dx 2 , and ys and z s are the components of the distance between the centroid and the shear center in the y and z directions, respectively. Stress resultants at the cross-section can be derived by integrating the corresponding stresses over the cross-sectional area, as given by Z Z Fx = σx x dA, Fy = τx y dA, ZA Z A Fz = τx z dA, B= σx x ωdA ¯ A ZA (3) Mx = {τx z (y − ys ) − τx y (z − z s )}dA, ZA Z My = σx x zdA, Mz = − σx x ydA A

By substituting (2) into (4), expressing stresses in terms of strains using Hooke’s law, and transforming the volumetric integration into line integration, one can rewrite (4) as Z l 1 Uco = [E c Ac ε02 + E c Scy ε0 φ y − E c Scz ε0 φz 2 0

A

where the normal stress σx x and shear stresses τx y and τx z are obtained from (2) using Hooke’s law.

− E b Sbz φz ε0 + E b Ibz φz2 + E b Ibw f 2 + G b K b φx2 ]dx (7) where E b is the modulus of elasticity of the steel, Ab is the cross-sectional area of the steel, Sby and Sbz are the first moments about the y and z axes, respectively. Iby and Ibz are the second moments about the y and z axes, respectively, and Ibω and K b are the warping and pure torsional constants of the steel section, respectively. 3.3. Strain energy of prestressing tendons When a composite double T-beam is subjected to torsion, the cross-section rotates about the shear center by the twist angle θx , and prestressing tendons on the left and right hand sides of y axis are displaced by −v p (x) and +v p (x), respectively, as shown in Fig. 2(a). In this case, the tendon that is depressed by −v p (x) will be elongated, while the other tendon, which is elevated, will be shortened. Consequently, the strain energy of prestressing tendons is the sum of those of the elongated and shortened tendons.

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Fig. 2. Deformed geometry of the double T-beam subjected to torsion: (a) torsional rotation of cross-section; (b) depressed prestressing tendon; (c) elevated prestressing tendon.

Denoting the change in the twist angle between two cross-sections with the distance dx as dθx , the change of an infinitesimal element length of the depressed PS tendon [Fig. 2(b)] can be expressed as (ds p + ∆ds dp )2 = dx 2 + (dy psi − dv p )2 + dw 2p

(8)

where y psi is the y component of the distance between the ith prestressing tendon and the shear center. The infinitesimal d of the infinitesimal element of the deformation energy dU ps depressed prestressing tendon can be derived from the product of the prestressing forces in the tendon and the elongation of the infinitesimal element ∆ds dp as    ds 1 d ∆ds dp dU ps = Peh + Plh 2 dx   1 = Peh + Plh 2 (  )     dv p dy psi 1 dv p 2 × − dx + dx (9) dx dx 2 dx where Peh is the horizontal component of the effective prestressing force in the tendon, and Plh is the horizontal component of the increase in the prestressing force due to the depression of the tendon. It is noted that 1/2 is multiplied to Plh in (9) because Plh is the resultant force due to the applied load.

The deformation energy of the depressed prestressing tendon can be obtained by integrating (9) as   1 d U ps = Peh + Plh 2 ( Z  )   Z  l dv   dy 1 l dv p 2 p psi × − dx + dx . (10) dx dx 2 0 dx 0 The terms in the second parenthesis in (10) represent the elongation of the tendon that can be expressed in terms of the axial stiffness of the tendon as    Z l Z  Plh L pe dv p dy psi 1 l dv p 2 dx + dx(11) =− E p A pi dx dx 2 0 dx 0 where L pe , A pi , and E p are the length, area, and modulus of elasticity of the prestressing tendon, respectively. The strain energy of the elevated prestressing tendon [Fig. 2(c)] can be obtained by substituting −Plh and −v p to Plh and v p in (10), respectively, as   1 e U ps = Peh − Plh 2 (Z  )   Z  l dv   dy 1 l dv p 2 p psi × dx + dx . (12) dx dx 2 0 dx 0

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The total strain energy U ps of the prestressing tendons at the cross-section of the composite double T-beam subjected d of the to torsion is the summation of the strain energies U ps e depressed and U ps of the elevated prestressing tendons. That is,  Z l dv p 2 U ps = Peh dx dx 0  Z l    dv p dy psi + Plh − dx . (13) dx dx 0 Substituting Plh of (11) and v p = −z psi θx from the geometry in Fig. 2(a) and integrating the second term of the right hand side by parts, (13) can be rewritten as  Z l E p A pi 2 dθx 2 2 dx + U ps = Peh z psi z dx L pe psi 0 Z l  2   Z l  2   d θx d θx × y psi dx y psi dx (14) dx 2 dx 2 0 0 where the first and second terms represent strain energies of prestressing tendons due to pure torsion and warping torsion, respectively. It should be noted that the restoring force of prestressing tendons subjected to torsion is proportional to the initial prestressing force. It is also noted that the strain energy is proportional to the axial stiffness of the prestressing tendon as written in the second term of (14). In addition to the strain energy of the prestressing tendons due to the twist of the beam about x axis, the axial and the flexural strain energies due to rotations of the cross-section about the y and z axes obtained as Z l " X n n X 1 Ep A pi εo2 + E p A pi z pci εo φ y U ps = 0 2 i=1 i=1 − Ep + Ep − Ep

n X i=1 n X i=1 n X

A pi y pci εo φz A pi z pci φ y εo + E p A pi y pci φz εo + E p

i=1

+

n X i=1

+

n X i=1

Peh z 2pci

n X i=1 n X

A pi z 2pci φ y2

flexural and torsional loads. The beam element has two nodes where each of them has seven DOFs, including the warping DOF. Based on the uncoupled assumption between the torsional and the flexural modes of deformation, the formulation adopts two reference axes through the centroid and the shear center to simplify the complicated calculations of the sectional properties regarding the pure torsion and warping torsion. 4.1. Approximation of strains The displacement field values of u(x), v(x), and w(x) and the angle of twist of the cross-section θ (x) are approximated using interpolation functions and nodal displacement vectors [Fig. 3(a)] as u(x) = No do , w(x) = Nw dw ,

v(x) = Nv dv

(16)

θ (x) = Nθ dθ

where the axial DOFs vector do = {u 1 , u 2 }, flexural DOFs vector about y axis dv = {v1 , θz1 , v2 , θz2 }, flexural DOFs vector about z axis dw = {w1 , θ y1 , w2 , θ y2 }, torsional DOFs vector about x axis dθ = {θx1 , φx1 , θx2 , φx2 } and the corresponding shape function vectors No , Nv , Nw , and Nθ . It is noted that Nv = Nθ , Nw1 = Nv1 , Nw2 = −Nv2 , Nw3 = Nv3 , and Nw4 = −Nv4 , where the second subscript i indicates the ith entry of vectors Nv or Nw . Decomposing the axial strain into the uniform strain along the depth of cross-section εo and the curvature strain φ y y and φz z in (16), and substituting into (2) give the axial strain εx x and shear distortions γx y and γx z in terms of nodal displacements as   ¯ φ do εx x = Bo zBw yBv ωB   ∂ ω¯ − (z − z s ) Bθ dTθ γx y = ∂y   ∂ ω¯ γx z = + (y − ys ) Bθ dTθ ∂z

dw

dv

dθ

T (17)

where Bo = dNo /dx, Bw = −d2 Nw /dx 2 , Bv = −d2 Nv /dx 2 , and Bθ = dNθ /dx, Bφ = d2 Nθ /dx.

# A pi y 2pci φx2 dx

i=1

Z

l

0

E p A pi 2 z L pe psi

4.2. Equilibrium equations of the finite element approach

φx2 dx l

Z

l

 Z f y psi dx

0

 f y psi dx

(15)

0

where y psi and z psi are the directional components of the distance of the ith prestressing tendon to the shear center along the y and z axes, respectively, and y pci and z pci are the directional components of the distance of the ith prestressing tendon to the centroid along the y and z axes, respectively. 4. Finite element formulation A 3D finite beam element is formulated to represent the behavior of a composite double T-beam under combined

The external work of the beam element due to distributed loads qv (x) and qw (x), and equivalent nodal forces f x j , f y j , and f z j representing the prestressing force in prestressing tendons [Fig. 3(b)] can be written as l

Z W = 0

(qv v + qw w)dx +

2 X

( fx j u j + f y j v j + fz j w j

j=1

+ m x j θx j + m y j θ y j + m z j θz j + B j φx j ).

(18)

The total strain energy U is the summation of strain energies of the concrete, steel web, prestressing tendons, and reinforcing bars. The total potential energy Π can be expressed as Π = U − W . Using the principle of the minimum total potential

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energy δΠ (d) = (∂Π /∂d)δd = 0, one can derive Kd = F where K is given as Z l {BTo (E A)x Bo K= 0

+ BTo (E S y )x Bw + BTo (E S Z )x Bv + BTw (E S y )x Bo + BTw (E I y )x Bw + BTv (E Sz )x Bo + BTv (E Iz )x Bv + BTφ (E Iω )g Bφ  Z l Z l T T Bφ dx . Bφ dx + Bθ (G K )g Bθ }dx + 2(Eω)g 0

0

(19) In (19), (E A)x , (E S y )x , (E Sz )x , (E I y )x , and (E Iz )x are transformed sectional properties of the composite double Tbeam and can be expressed as (E A)x = E c Ac + E b Ab + E s (E S y )x = E c Scy + E b Sby E s

n X

Asi + E p

i=1 n X

n X

Asi z sci + E p

i=1

(E Sz )x = E c Scz + E b Sbz + E s

A pi

i=1 n X

A pi z pci

i=1 n X

Asi ysci

i=1

+ Ep

n X

Fig. 3. Three-dimensional beam element: (a) nodal degrees of freedom; (b) external forces acting on the beam element.

A pi y pci

i=1

(E I y )x = E c Icy + E b Iby + E s

n X

(20) 2 Asi z sci

i=1

+ Ep

n X

A pi z 2pci

i=1

(E Iz )x = E c Icz + E b Ibz + E s

n X

2 Asi ysci

i=1

+ Ep

n X

A pi y 2pci

reinforcing bar. As shown in (21), the warping stiffness of the beam element consists of the warping stiffness of the concrete (E Iω )g and the warping stiffness of the prestressing tendon (Eω)g that consists of contributions of the reinforcing bars and prestressing tendons. It should be noted that the torsional stiffness (G K )g of the developed beam element includes the contribution from the restoring action of the elevated and depressed prestressing tendons as well as the torsional stiffnesses of the concrete G c K c and composite effect G b K b .

i=1

where E s is the modulus of elasticity of the reinforcing steel, Asi the area of the reinforcing bars, ysci and z sci the directional component of the distance of the ith reinforcing bar to the centroid along y and z axes, respectively. The warping stiffness (Eω)g and (E Iω )g , and torsional stiffness (G K )g in (19) can be expressed as (E Iω )g = E c Icω + E b Ibω n n X X E p A pi 2 2 E s Asi 2 2 (Eω)g = z ssi yssi + z y L L pe psi psi se i=1 i=1 (G K )g = G c K c + G b K b + 2

n X

(21)

Peh z 2psi

i=1

where G c and K c are the shear modulus and the torsional constant of concrete, respectively, yssi and z ssi are y and z components of the distance of the ith reinforcing bar to the shear center, respectively, and L se is the length of the

5. Numerical applications and discussions 5.1. Properties of the case study of composite double T-beams Two composite double T-beams with bonded prestressing tendons were analyzed using the developed 3D beam element to investigate the effects of coupled restoring forces in a pair of prestressing tendons and the encasing steel on the torsional resistance. The first numerical example of a 12 m long two-span continuous composite double T-beam shown in Figs. 4(a) and 5 was tested by Park [11] to develop design, construction, and evaluation procedures for composite double T-beams. Fig. 5 shows the design details of the T-beam with eight reinforcing bars of D13 (nominal diameter = 12.7 mm) placed in the top flange of the concrete, three prestressing tendons of 7-wirestrand (138.7 mm2 /strand), and a 4.5 mm-thick web steel plate. The shear studs of a 50 mm long D13 reinforcing bar welded on the top flange and the inside of the web were designed to

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Fig. 4. Longitudinal view of the composite double T-beam: (a) two-span double T-beam; (b) three-span double T-beam (unit: m). Table 1 Mechanical properties of the composite double T-beam Parameter

Value

Concrete

Modulus of elasticity E c Poisson’s ratio νc

2.41 × 104 MPa 0.2

Steel

Modulus of elasticity E b Poisson’s ratio νb

2.06 × 105 MPa 0.3

Prestressing tendon

Modulus of elasticity E p Prestressing force/web Pe

2.06 × 105 MPa 823 kN

Rebar

Modulus of elasticity E s

2.06 × 105 MPa

Fig. 5. Cross-sectional dimension of the composite double T-beam (unit: m).

provide a full composite action between concrete and the web steel. The geometry of the cross-section with a 1.6 m width of the top flange concrete slab and 0.6 m depth, is symmetric about the y axis. Material properties of the concrete, steel, prestressing tendons, and reinforcing steel are listed in Table 1. A line load of 1.47 kN/m along the left web in Fig. 5 on the two spans was applied to induce the torsional effect in the composite double T-beam as well as their flexure. The level of the applied load was selected so that the stress level of the analyzed structure stayed within the elastic range. The second numerical example is a 20 m long, three-span continuous composite double T-beam as shown in Fig. 4(b). The cross-sectional design of the selected structure is same as that of the first example. The span length was designed so that the maximum positive and negative moments are same as those in two-span composite double T-beams. A line load of 1.47 kN/m along the left web of the center span was applied to investigate the effects of the torsional resistance of side spans due to the torsional load.

Fig. 6. Decomposition of normal stresses.

All the numerical results from the developed beam element were compared to the analytical solutions of the governing differential equations of the beam subjected to warping torsion [12], and also to finite element analyses results by conventional 3D continuum elements. The warping normal stresses in the flange and the webs by conventional 3D elements were calculated by decomposing the total stresses into the flexural part of σxb and the warping part of σxω through the relations of σxb = (σx−z + σx+z )/2 and |σxω | = |σx−z − σx+z |/2 as shown in Fig. 6. The warping level was estimated based on the bimoment, Mω = σxω Iω /ω. ¯ The values of the warping function ω¯ i are listed in Table 2 where the location of the warping function is shownRin Fig. 7. The warping constant Iω was calculated by Iω = A ω¯ 2 tds and, in the present study, Iω = 32.47 m4 .

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Table 2 Values of the warping functions of the composite double T-beam Warping function

Value (mm2 )

ω¯ 1 ω¯ 2 ω¯ 3 ω¯ 4 ω¯ 5 ω¯ 6 ω¯ 7

0.0 52.050 104.100 −169.800 −52.050 −104.100 169.800

Fig. 7. Warping function diagram (unit: m).

5.2. Flexural behavior Fig. 8(a) and (b) compare the deflections calculated by two approaches of the developed beam element and the conventional 3D finite element analyses of the commercial program MIDAS [9] for the two-span and three-span composite double T-beams shown in Fig. 4(a) and (b). In the 3D finite element analysis, 8-node brick elements for concrete, 4-node shell elements for web steel, and 2-node bar elements for prestressing tendons and reinforcing bars were used to simulate the behavior of the composite double-T beam. The deflection at the top of the concrete flange is monitored where the node closest to the shear center of the cross-section is located. As shown in Fig. 8(a) and (b), deflections by the two methods agree well except at the center of each span, which is mainly due to the inherent difference of the two finite element approaches and the different monitoring points within the cross-section of the beam. 5.3. Torsional behavior The torsional behaviors of the two composite double T-beams were analyzed by three approaches, namely the developed finite element analysis, the conventional 3D finite element analysis, and the direct solution of governing differential equations regarding the torsional behavior of the double T-beam. In the conventional 3D finite element analysis, the twist angle was calculated by dividing the relative deflections calculated at the two points where the flange and web meet by the distance between the two points. Fig. 9(a)

Fig. 8. Comparison of deflections: (a) two-span double T-beam; (b) three-span double T-beam.

and (b) compare the torsional stiffness in terms of the angle of twist. In the case of the two-span double T-beam, the developed approach shows the stiffest result among the three approaches, while in case of the three-span double T-beam, the conventional 3D finite element analysis gives the stiffest result. The results of the developed approach match well with that of the analytical solution, while they show a relatively large difference from those of the 3D finite element analysis. The difference is mainly due to the boundary conditions: where the conventional 3D finite element analysis considers boundary conditions across the bottom of webs, it is not allowed in beam-type finite element analysis. The in-plane deformation in the conventional 3D continuum elements is another reason for the difference, because in this case, the twist angle was calculated in terms of the relative deflections between the two points. It is interesting to note that for both the double T-beams, the developed approach is always stiffer than the analytical approach due to the torsional stiffness provided by the restoring action of the pair of prestressing tendons and the encasing steel.

Y.-H. Lee et al. / Engineering Structures 29 (2007) 2935–2945

Fig. 9. Comparison of rotational angles: (a) two-span double T-beam; (b) threespan double T-beam.

To evaluate the effects of the coupled restoring forces of the pair of prestressing tendons on torsional stiffness, the twist angles were calculated in two cases of formulation, with and without consideration of the coupled restoring forces. In the latter case, the formulation was based on the transformedsection approach which neglects the torsional stiffness provided by the coupled restoring forces of a pair of prestressing tendons. Fig. 10(a) and (b) show that the twist angle in the case without the coupled restoring forces is 7%–9% larger than that in the other case. This observation illustrates the limitations of the conventional approach, which transforms the prestressing tendon into the equivalent concrete area. Since the torsional resistance is proportional to the vertical component of the prestressing force as shown in (19), this effect becomes crucial when the effective prestressing force is relatively large, or when the depth of cross-section is high enough to give the distance between the shear center and the location of the prestressing tendons.

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Fig. 10. Effects of the restoring action of the prestressing tendon on torsional stiffness: (a) two-span double T-beam; (b) three-span double T-beam.

The composite effect was investigated by evaluating the contribution of the web steel section to the torsional stiffness. Fig. 11(a) and (b) compare the twist angles of the composite double T-beam with and without the web steel for both the double T-beams. The angle of twist is 20% less in the case of having one web steel section than the other. The comparison shows that the web steel section considerably contributes to the torsional stiffness as well as to the flexural stiffness. 5.4. Warping behavior The effects of the web steel section on torsional warping stiffness were investigated by the results of the three approaches. Fig. 12(a) and (b) show bimoments of the three approaches. The three results agree well except for the regions of interior support for the two-span double T-beam and the center of the middle span for the three-span double T-beam. The reason for the discrepancy is attributed to the method of calculating the bimoment in conventional 3D finite element analysis, as described in Section 5.1 for Fig. 6.

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Fig. 11. Efects of the web steel composite on the torsional stiffness: (a) twospan double T-beam; (b) three-span double T-beam.

When a composite double T-beam is subjected to torsion due to the eccentric load away from the shear center, the prestressing force in the tendon increases or decreases depending on the direction of twist. Fig. 13(a) and (b) show the variations of the prestressing tendon stresses along the length of the beam for both the double T-beams. The variation of the prestressing tendon stress is caused by the flexure and torsional warping under the given load conditions of combining flexure and torsion. The counter-clockwise rotation of the T-beam due to the left-sided eccentric load results in the distribution of the warping functions of Fig. 7 and, therefore, the tensile stress in the bottom of the left-side of the web and the compressive stress in the other. The resulting stress increments of the prestressing tendons are obtained by combining the warping-induced tensile and compressive stresses to the flexure-induced stresses of the prestressing tendons. In Fig. 13(a) and (b), the stress increments of the prestressing tendon in the left-side of the web is positive, i.e. in tension, and that in the right-side of the web is negative, i.e. in compression. It indicates that the torsional warping due

Fig. 12. Comparison of bimoments: (a) two-span double T-beam; (b) threespan double T-beam.

to the applied load is dominant in this particular case, so that the stress increments in the two tendons take different signs. 6. Conclusions A 3D finite beam element was formulated to represent the behavior of a composite double T-beam subjected to combined flexural and torsional loads considering the coupled restoring forces of a pair of prestressing tendons and their encasing steel. The large deformation assumption was applied to the prestressing tendons to evaluate the coupled restoring forces. For this, the strain energies of the concrete, the encasing steel, the reinforcing bars, and the prestressing tendons were obtained from the deformed geometry of the cross-section of a composite double T-beam subjected to torsion. The element stiffness matrix was derived using the principle of the minimum total potential energy. The developed beam element was applied to analyze the flexural and torsional behaviors of two-span and three-span

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estimates give 7%–9% stiffer torsional resistance than that of the transformed approach. 5. The steel section, which is designed to increase the flexural stiffness, also increases the torsional stiffness of the composite double T-beam, e.g. by the maximum of 20%. 6. The proposed beam element can be useful in evaluating a prestressing beam subjected to a combined flexure and torsion; in particular, when torsion is dominant, the initial prestressing force is large, so the location of the prestressing tendon is far from the shear center. Acknowledgement The first author wishes to acknowledge the partial financial support of the Konkuk University Research Grant Committee (Seoul, Korea) in 2000 that made it possible to perform this study. References

Fig. 13. Changes of the prestressing tendon stress: (a) two-span double Tbeam; (b) three-span double T-beam.

composite double T-beams. Several conclusions were drawn, as listed below. 1. The coupled restoring forces of the pair of prestressing tendons provides a torsional stiffness which is proportional to the initial prestressing forces and the square of the distance from the tendon to the shear center of the crosssection. 2. The stiffness contributions of the pair of prestressing tendons and reinforcing bars with respect to warping torsion are derived using the characteristic equations of the cables. These stiffness terms are proportional to the axial stiffness and the square of the distance from the tendon or the reinforcing bar to the shear center of the cross-section. 3. The encasing steel provides additional warping and torsional stiffness terms. These stiffness terms are superimposed onto those of the double T-beam without the encasing steel. 4. The result of the proposed beam element is stiffer than that of the transformed approach, due to the coupled restoring forces of the pair of prestressing tendons. In the particular numerical examples considered, the proposed beam element

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