Analytic Solutions of Shear Lag on Steel-Concrete Composite T-girder under Simple Bending

The composite T-girders include concrete flange plates and steel beams ,which are connected by shear connectors. The longitudinal stress about concrete plate is a non-uniform distribution on cross section because of the shear lag effect. A differential equation of longitudinal forces at transverse section flange and cantilever flange is separately established according to the strain compatibility and the force equilibrium conditions about a composite T-girder. The method of separation of variables is used to solve the differential equation about the simply supported composite T-girder. The shear lag coefficient is decided by the ratio between stress calculated by this method and stress decided by elementary beam theory. An example of such calculation is given to approve its applicability.


INTRODUCTION
The composite T-girders are commonly used in construction, due to taking advantage of the mechanical characteristic of steel and concrete, namely the use of concrete compressive and steel tensile capacity.It is well known that the uneven deformation of the wider T-girder flanges can produce an uneven distribution of the longitudinal stresses [1] under symmetrical bending.The shear lag effect can result in the obvious increase of longitudinal stress near the edge of the flange and cause stress concentration.Due to shear lag effects, it can cause stress concentration in structure, structural damage [2].
Those methods have different characteristics, but difficulty of more calculation and analysis exists with these methods to calculate composite section.The same method is used to calculate shear lag effect of composite box girders in reference [6].

DIFFERENTIAL EQUATIONS ABOUT FLANGE EQUILIBRIUM
As shown in Fig. (1), It is assumed that relative slip between the concrete plate and steel girder is not happened *Address correspondence to this author at the School of Civil Engineering, East China Jiaotong University, Nanchang, Jiangxi, 330013, P.R. China; Tel: 13879104029; E-mail: bridge407@126.com  2) ,bears a shear flow q e and a normal force n x,i (i=1 represents one cantilever flange, i=2 represents the other).It is assumed that the shear flow q e is resisted by the composite flange plate itself, without considering the help from the stiffeners.But both the composite flange plate and stiffeners bear the normal longitudinal forces.Without special provisions, the subscript i has the same meaning as the subscript i of n x,i in the followings.

Stress and Strain
In the elastic range，an equivalent thickness s t of steel flanges can be obtained according to the Match-plate method.Then, an equivalent thickness t for the composite flange can be obtained according to the equivalent conversion principle, where concrete slab thickness of flange is converted to steel equivalent thickness.
As in Fig. (1), t 1 represents the steel plate thickness of flange, a and a 1 are the spacing of longitudinal and transverse stiffener respectively, and A s is the transverse section area of each longitudinal stiffener, A sx is the transverse section area of each transverse stiffener, t c is the thickness of the concrete layer; E and E c represent the Young"s moduli of steel and concrete respectively.
As shown in Fig. (2), the direct strains in the longitudinal direction are then given by 2 , 1 , , , Also, it may be assumed that the transverse direct strain in the steel plate and concrete slab are the same so that Where i x,  represents the x-direction stress of different element,υs andυ c express the Poisson"s ratio of the steel and concrete respectively; n xs,i and n xc,i are the sections of the normal force (n x,i ) carried by the steel and concrete components respectively, such that According to equations (3),(4), equation ( 5 By substitution from equations (3),( 5), the i y, It can be simplified by using an "equivalent" Poisson"s ratio (r), equation( 6

Shear Stress and Strain
It is assumed that no relative slippage between the, the shear strains(γ i ) of elements about steel and concrete slab are the same, then Where, G s and G c express the shear moduli of the steel and concrete respectively.q s,i and q c,i are the sections of the shear flow(q i ) carried by the steel and concrete components respectively.The sum of q s,i and q c,i equals the shear flow(q i ).By substitution, equation ( 8 That may be written as Where the equivalent thickness of the shear action is given by

Compatibility and Equilibrium Equations
As shown in Fig. (2), the equilibrium of the flange element in the longitudinal direction is given by (10) As shown in Fig. (1), the composite flange can be regarded as a plane stress problem, by the Hooke"s Law [3], the equation controling the condition of compatibility may be written as [11] Substituting equations ( 2), ( 6) and ( 9) for the strains, equation (11) and the shear flow q i may be substituted by the normal force n x,i from equation (10), thus,

SOLUTION OF DIFFERENTIAL EQUATION
For the partial differential equation, the normal force n x,i (x,y) can be written as the Fourier series by adopting the method of separation of variables [8] under the simply supported border, such that Where, L represents the length of the simply supported span and N k,i (y) expresses an amplitude function which represents the variation in the cross-section direction of the normal force.Where, x represents the distance from the origin to the calculated section.k is the series term.
By substituting the k th term of the series into equation (13), the ordinary differential equation is written as , the common solution of the equation is written as where the coefficient C i,k1 and C i,k2 are constants that decided by the supporting and loading conditions at the longitudinal edges of the flange.As shown in Fig. (1), normal force n x,i (x,y) with shear lag has a symmetric distribution in the cross section under symmetrical bending.As shown in Fig. (2), the coefficient C i,k2 is zero due to symmetry.The constant C i,k1 can be determined according to the conditions of shear flow equivalent and normal force continuity.By substituting equation (14) into equation(13), the normal longitudinal force can be expressed as The normal longitudinal force of the cut-off point about two cantilever flange must meet continuous conditions.Then, the normal longitudinal force is written as By substituting equation(15) into equation (10), the equation of the first-order derivative of y on q i is given as integrating the y and noting the value of ξ k , the shear flow at any point may be written as At the central of T-girder flange(where y= c) as shown in Fig. (2), the sum of the shear flow on two cantilever flanges equals the shear flow (q e ) transfered from the web to the flange, which can be expressed as Where,the shear flow q e,1 (x) and q e,2 (x) expresses the shear flow transfered from the web to the flange, which can be expressed as The shear flow (q e ) can be approximately decided from elementary beam theory, it can be written as Where, V(x) is the total shear force imposed on the Tgirder cross section at the position x, and I is the inertial moment of the cross section, e is the distance from the neutral axial to the centroidal axis of the flange.For the simply supported span, the shear flow q e (x) can be expressed by the Fourier series, such that According to the equations( 16)，( 19) and (20), the constant C i,k1 can be gotten, such that

By substituting the constant C
i,,k1 into equation (13) and equation ( 14), the normal force n x,i (x,y) may be gotten.Thus, the normal stresses of the steel flange and the concrete plate can respectively be given, such that The shear stress of the steel flange and the concrete plate can be given as

CALCULATION AND CONCLUSIONS
According to the approximate calculation method for shear lag effect of composite T-girder, a simple manual or the program can get the shear lag coefficient of flange.In order to analyze the convergency, precision of the method and the parametric influence about the shear lag effect of composite T-girder, the transverse section of an example is written as in Fig. (3).The area of the longitudinal and horizontal stiffeners is 100Square millimeters.The depth of concrete flange is 60mm.The height of the composite Tgirder is 1200mm.The thickness of the steel slab is 12mm, and other parameters are illustrated in Table 1.According to the ratio of wide-span, the flange width c is respectively 600mm、800mm、1200mm and 1800mm.As shown in, Fig. (4), it shows the relationship between width-span ratio , number of term in Series and the stressratio of flange about mid-span section, by using the method to calculate the shear lag effect, where the vertical coordinate expresses the ratio of the longitudinal stress about concrete slab and about elementary beam theory.
As shown in Fig. (4), the convergency is quite fast for a uniformly distributed load.And the wide-span ratios of Tgirder flange has little influence on the convergency, where the convergency of the longitudinal stress about the concrete plate of mid-span is shown for different wide-span ratios and the shear lag coefficient(λ) is defined.Shear lag coefficients about concrete slab increase or reduce with respect to its wide-span ratio.
As shown in Fig. (5), it expresses the normal longitudinal distribution of concrete plate stress, where the finite elements method (FEM) refers to ANSYS program.The calculated values with FEM is larger than that with the proposed method because of stress concentration at the intersection with the flange of the Web.Relationship between width-span ratio ,number of term in series and stress-ratio about cross section of midspan (σ 1 -analytic solutions of this text, σ 2 -Solutions of primary beam theory).

CONCLUSIONS
The proposed method for solving the shear lag effect of the composite T-girders has fast convergency from simple calculations.The precision of result is enough for engineering .The has great effect on the shear lag effect of the composite T-girder.And the shear lag coefficient (λ) increases or decreases with the change of the wide-span ratios.Thus, it may be paid more attention to the wider Tgirders.The method can also be applied for multi-span girders which may be divided into the simply supported and cantilevered girders .
It is also applied to calculate shear lag effect of the composite T-girder.Parameters of a composite T-girder are shown in Fig. (1).