Flexural-Distortional Performance of Thin- Walled Mono Symmetric Box Girder Structures

Flexural-Distortional Performance of Thin- Walled Mono Symmetric Box Girder Structures

Flexural-Distortional Performance of Thin- Walled Mono Symmetric Box Girder Structures

Osadebe N.N.

Dept of Civil Engineering, University of Nigeria, Nsukka.

Chidolue C.A.

Dept of Civil Engineering, Nnamdi Azikiwe University, Awka, Nigeria

Abstract

Thin-walled mono symmetric box girder structures are commonly found in the form of trapezoidal cross sections of either concrete or steel. Such structures resist eccentric vertical loads in bending action and torsion. The torsional component of eccentric loads on such structures give rise to pure torsion (Saint Venant torsion), distortion and flexure about the non symmetric axis of the box girder section. In order to provide an improved understanding of the complex interactions between these strain fields, this paper examined the interaction between the distortional strain mode and flexural strain mode and derived a general differential equations of equilibrium for flexural- distortional analysis of mono symmetric box girder structures. In addition the derived equations were used to analyze a double cell mono symmetric box girder section to obtain flexural and distortional deformations.

1. Introduction

   Thin walled structures are structures in which the ratio of the thickness t, to the two other linear dimensions (length l, and width w) ranges within the limits t/l or t/w = 1/50 to 1/10, Rekach [1]. Thus a thin walled structure has two dimensions of the structural element much larger than the third one, i.e., the thickness. When two or more plates are joined together to form an open or closed structure strength and rigidity are increased. For example, tanks, boilers, etc, are cylindrical shell structures with increased strength and rigidity. Conical shell structures are also common features in construction, mechanical engineering and aeronautical design. Thin-walled structures are used extensively in steel and concrete bridges, ships, air crafts, mining head frames and gantry frames. These are seen in the form of box girders, plate girders, box columns and purlins (z and channel sections). Because

   of their thin wall thicknesses, the shearing resistances are constant across the thickness of the plate. On the other hand thin walled box structures may be subjected to bending, torsional and distortional stresses. Distortion alters the geometry of the cross section and generates some additional stresses thereby reducing the bearing capability of the box structural component.

   Research [2], has shown that a mono symmetric thin walled box girder has three strain modes interactions: torsion interacts with distortion and each of these interacts with flexure about the non axis of symmetry. Thus we have torsional-distortional interaction, flexural-torsional interaction, and flexural- distortional interaction. In this work, the interaction of flexural strain mode about the non axis of symmetry with distortional strain mode of a mono symmetric box girder structure is examined.

2. Literature review

   Recent literatures, Hsu et al [3], Fan and Helwig [4], Sennah and Kennedy [5], on straight and curved box girder bridges deal with analytical formulations to better understand the behaviour of these complex structural systems. Few authors, Okil and El-tawil [6], Sennah and Kennedy [5], have undertaken experimental studies to investigate the accuracy of existing methods. Before the advent of Vlasovs theory of thin-walled beams, [7], the conventional method of predicting warping and distortional stresses is by beam on elastic foundation (BEF) analogy. This analogy ignores the effect of shear deformations and takes no account of the cross sectional deformations which are likely to occur in a thin walled box girder structure

   Several investigators; Bazant and El-Nimeiri [8], Zhang and Lyons [9], Boswell and Zhang [10], Usuki [11], Waldron [12], Paavola [13], Razaqpur and Lui [14], Fu and Hsu [15], Tesar [16], have combined thin-

   walled beam theory of Vlasov and the finite element technique to develop a thin walled box element for

   From the theory of elasticity the strains in the longitudinal and transverse directions are given by;

   elastic analysis of straight and curved cellular bridges. Osadebe and Chidolue [17], [18], obtained fourth order differential equations of torsional-distortional

   u(x, s)

   x

   m

   U

   i1

   i '(x)i (s)

   (3)

   k k

   equilibrium and flexural-torsional equilibrium for the

   v(x, s) n V

   '(x)

   (s)

   analysis of mono symmetric box girder structures using Vlasovs theory with modifications by Varbanov [19].

   Various theories were therefore postulated by different authors examining methods of analysis, both classical and numerical. A few others however carried

   x k 1

   The expression for shear strain is

   (x, s) u v

   s x

   out tests on prototype models to verify the authenticity of the theories. The authors are of the view that Vlasovs theory captures all peculiarities of cross

   sectional deformation such as warping, torsion, distortion etc, and is therefore adopted in this work.

   (x, s) i '(s)Ui (x) k (s)Vk

   m n

   i1 k 1

   '(x)

   (4)

   The objective of this study is to derive a set of differential equations governing the flexural- distortional behaviour of thin- walled mono symmetric box girder structures on the basis of Vlasovs theory

   Using the above displacement fields i and i ,

   and basic stress-strain relationships of the theory of elasticity, the expressions for normal and shear stresses become:

   and to apply the obtained equations in the analysis of

   double cell mono symmetric box girder structure to obtain flexural and distortional deformations.

   (x, s) E

   u(x, s)

   x

   m

   Ei (s)Ui '(x)

   i1

   m

   (5)

3. Vlasovs stress strain relations

   The longitudinal warping and transverse (distortional) displacements given by Vlasov [7], are

   (x, s) G (x, s) Gi '(s)Ui (x)

   i1

   n

   u(x, s) U (x)(s)

   v(x, s) V (x) (s)

   +

   (1)

   G k (s)Vk '(x)

   k 1

   (6)

   The displacements may be represented in series form as;

   m

   Transverse bending moment generated in the box structure due to distortion is given by;

   n

   u(x, s) Ui (x)i (s)

   i1

   (2)

   M x, s M k (s)Vk (x)

   k 1

   (7)

   n

   v(x, s) Vk (x) k (s)

   k 1

   where, Ui(x) and Vk(x) are unknown functions which express the laws governing the variation of the displacements along the length of the box girder frame.

   i (s) and k (s) are elementary displacements of the

   strip frame, respectively out of the plane (m

   where Mk(s) = bending moment generated in the cross sectional frame of unit with due to a unit distortion V(x) = 1.

4. Energy formulation of the equilibrium equations

   The potential energy of a box structure under the action of a distortional load of intensity q is given by:

   displacements) and in the plane (n displacements). These displacements are chosen among all displacements possible, and are called the generalized strain coordinates of a strip frame.

   U WE

   where,

   = the total potential energy of the box structure, U = Strain energy,

   (8)

   VE = External potential or work done by the external

   loads.

   Fromstrength of materials, the strain energy U, of a structure is given by:

   E a U '(x)U

   2 ij i j

   '(x)dx

   2 2

   G b U (x)U (x) c U (x)V '(x) dx

   (x, s) (x, s) t(s)

   2 ij i j kj k j

   1 E G

   U dxds

   (9) G

   2 LS

   M 2 (x, s)

   + cihUi (x)Vh '(x) rkhVk '(x)Vh '(x)dx +

   2

   EI( s )

   E s V (x)V (x)dx – q V dx

   (12)

   Work done by external load is given by:

   2 hk k h h h

   WE qv(x, s)dxds

   where the (Vlasovs) coefficients are defined as follows.

   ij ji i j

   = qV (x)

   (s)dsdx

   aij aji i (s)j (s)dA

   (a)

   h h s x

   b b ' (s)' (s)dA

   (b)

   = q V dx

   (10)

   c c

   ' (s)

   (s)dA

   (c)

   h h

   kj jk

   k j

   x c c

   ' (s)

   (s)dA

   (d) (13)

   Substituting eqns (9) and (10) into eqn. (8) we obtain that:

   ih hi

   r r

   i k

   (s)

   (s)dA;

   (e)

   kh hk

   k h

   2 (x, s)

   2 (x, s)

   s s

   1 M k (s)M h (s) ds

   (f)

   2E

   L S

   t(s)dxds

   2G

   kh hk E

   EI( s )

   2

   (11)

   qh q hds

   (g)

   1 M (x, s) – qv(x, s)dxds

   2 L S

   where,

   EI (s)

   The governing equations of flexural-distortional equilibrium are obtained by minimizing the energy

   functional eqn. (12), with respect to its functional

   (x, s) = normal stress

   (x, s) = shear stress

   M (x, s) = transverse distortional bending moment

   variables u(x) and v(x) using Euler Lagrange technique [20]. Minimizing with respect to u(x) we obtain;

   m m n

   k aijUi ''(x) – bijUi (x) – ckjVk '(x) 0 (14)

   i1

   i1

   k 1

   q = line load per unit area applied in the plane of the plate

   3

   Minimizing with respect to v(x) we have;

   c U '(x) s V (x)

   I t (s)

   1

   ih i hk k

   ( s )

   12(1- 2 ) = moment of inertia

   rkhVk ''(x)

   qh 0

   (15)

   E = modulus of elasticity G = shear modulus

   = poisson ratio

   t = thickness of plate

   where

   G

   E 2(1 )

   G

   Substituting eqns (1), (5), (6), and (7) into eqn.(11) and simplifying, noting that t(s)ds dA we obtain

   Equations (14) and (15) are Vlasovs generalized differential equations of distortional equilibrium for a box girder structure.

   the potential energy of the box structure as follows.

5. Generation of Strain Modes Diagrams

   Consider a simply supported girder loaded as shown in Fig 1(a). If we assume the normal beam theory, i.e., neutral axis remaining neutral before and after bending, then the distortion of the cross section will be as shown in Fig. 1(b) where, is the distortion angle (rotation of the vertical axis). The displacement 1 at any

   distance R, from the centroid is given by 1 R . If

   we assume a unit rotation of the vertical (z) axis then

   1 R , at any point on the cross section. Note that

   The generalized strain modes for the double cell mono-symmetric frame are shown in Fig 2

   x

   y

   z

   1. Simply supported girder section

      1 can be positive or negative depending on the value

      of R, in the tension or compression zone of the girder. –

      Thus, 1 is a property of the cross section obtained by y

      plotting the displacement of the members of the cross +

      section when the vertical (z-z) axis is rotated through a unit radian.

      Similarly, if the load is acting in horizontal (y- y) direction, normal to the x-z plane in Fig.1(a), then the bending is in x-z plane and y axis is rotated through

   2. Cross section distortion

Fig. 1 Simply Supported Girder and Cross Section Distortion

angle 2 giving rise to 2 displacement out of plane.

915 3660 3660

915

The values of 2 are obtained for the members of the cross section by plotting the displacement of the cross section when y-axis is rotated through a unit radian.

The warping function 3 of the beam cross section

is obtained as detailed in [1] and [2]. It has been explained that the warping function is the out of plane

203 203

203

203

2745 1830 1830

2745

3050

displacement of the cross section when the beam is twisted about its axis through the pole, one radian per

1. Double cell box girder section

   unit length without bending in either x or y direction and without longitudinal extension.

   1 and 2 are in-plane displacements of the cross section in x-z and x-y planes respectively while 3 is the distortion of the cross section.

   1.16

   –

   6 5 – –

   1.16

   1

   – 1 2

   y

   The authors have shown that these in-plane displacement quantities 1, 2 and 3 are the same as the derivatives of their corresponding out of plane

   + 4

   1.889

   + 3 +

   + 1.889

   z

   displacements. Consequently, 1, 2 and 3 are obtained by numerical differentiation of 1 , 2 and 3

   diagrams respectively.

2. Longitudinal strain mode diagram (Bending about y-y Axis)

   4 is the displacement diagram of the beam cross section when the section is rotated one radian in say, a clockwise direction, about its centroidal axis. Thus, 4 is directly proportional to the perpendicular distance ( radius of rotation) from the centroidal axis to the members of the cross section. It is assumed to be

   positive if the member moves in the positive directions

   0.857

   +

   1.00

   ' +

   1 1

   0.857

   z

   +

   0.857

   y

   of the coordinate axis and negative otherwise.

3. Transverse stain mode in y-y direction

   4.575 + 3.66

   + 2

   3.66

   –

   –

   4.575

   y

   1.83 +

   z

   – 1.83

4. Longitudinal strain mode (Bending about z-z axis)

   0.514 0.514

   –

   1.00

   – ' +

   0.514

   2 2 0.514

   – 1.00

   Computation of Vlasovs coefficients

5. Transverse strain mode in z-direction

   The coefficients

   aij , bij , ckj , cih and

   rkh , of the

   0.966 – 5 0.582

   6

   0.582

   – –

   1 +2

   differential equations of equilibrium are computed with the aid of Morhs integral chart. Thus:

   0.582

   – 0.966

   aij a ji i (s) j (s)dA

   3 a (s).

   (s)dA 25.073

   0.901 22 s 2 2

   — –

   0.901 4

   3 +

   + 0.901

   a23 a32 2 (s)3(s)dA 0.425

   a33 3(s).3(s)dA 0.750

6. Warping function diagram

   s

   b b ' (s). ' (s)dA

   ij ji i j

   0.417

   0.417

   b ' s '

   s

   _ 0.159

   6 5 1 2

   22

   2 (s).2 (s)dA 2.982

   ' '

   '

   b23 b32 2 (s).3(s)dA 0.449

   3 3 +

   ' s '

   b33 3(s).3(s)dA 1.533

   4 3 s

   0.492 '

   + ckj c jk k (s). j (s)dA

7. Distortion diagram s '

   c22 2 (s).2 (s)dA 2.982

   2.570 s '

   + 1.105

   c23 c32 2 (s).3(s)dA 0.449

   s

   s

   s

   s '

   _

   1.945

   4 +

   _

   y

   2.570

   c33 3(s).3(s)dA 1.533 rkh rhk k (s). h (s)dA r22 2 (s). 2 (s)dA 2.982

8. Pure rotation diagram

Fig. 2 Generalized strain modes for double cell mono-symmetric box girder frame

r23 r32 2 (s).3(s)dA 0.449

s

s

r33 3(s).3(s)dA 1.533

6.1 Evaluation of distortional bending moment oefficients, shk

1. Flexural-distortional equilibrium equations

   The distortional bending moment coefficients

   shk ,

   The relevant coefficients for flexural-distortional

   given by eqn. (13f) depend on the bending deformation of the strip frame characterized by the distortional

   bending moment, M (for k = 1, 2, 3, 4). To compute

   equilibrium are those involving strain modes 2 and 3 as shown in the computation of Vlasovs coefficients. These are:

   k

   the coefficients we need to construct the diagram of the

   bending moments due to strain modes , ,

   a22 , a23 , a33 , b22 , b23 , b33 , c23 , c33 ,

   and s33 . All other coefficients are zero.

   r22 ,

   r23 ,

   r33

   and

   4 . Incidentally, 1 , 2 and 4

   1 2 3

   strain modes

   Substituting these into eqns. [14] and [15] and adopting matrix notation of the equations we obtain:

   do not generate distortional bending moment on the

   0 0 0

   U1 ''

   0 0 0

   U1

   box girder structure as they involve pure bending and

   0 a a

   b U –

   pure rotation. Only

   strain mode generates

   22 23 U2 '' – 0

   b22 23 2

   3

   distortional bending moment which can be evaluated using the distortion diagram for the relevant cross

   0

   a32

   a33 U3 ''

   0

   b32

   b33 U3

   section. Consequently the relevant expression for the coefficient becomes:

   V1 '

   0 0 0 0 V2 '

   1 M3(s)M3(s)

   0 c22 c23

   0 0

   (16)

   shk skh s33 (19)

   0 c c

   0 V3 '

   E s EIs

   32 33

   V '

   where M3 (s) is the distortional bending moment of

   4

   the relevant cross section due to strain mode 3.

   0 0 0

   U '

   The procedure for evaluation of distortional bending

   0 c22

   c23 1

   moments is given in literatures [1], [17]. Fig. 3 shows

   0 c c

   U2 ' –

   the distortional bending moment for evaluation of Shk for the double cell mono symmetric frame of Fig. 2(a).

   32 33 U '

   3

   0 0 0

   The computed value of

   shk s33

   for the single cell

   0 0 0 0 V

   mono symmetric frame example was :

   0 0 0 0 1

   S33 0.723* IS .

   0 0 s

   0 V2 +

   33

   V

   0.243

   -0.458

   0.229

   -0.243

   0 0 0 0 3

   V4

   0 0 0 0

   + +

   +

   0 r r

   0 V1 ''

   q1

   –

   – 1 2-

   0.22

   + – 3

   22 23

   V

   ''

   1 q

   0

   r32

   r33

   0 2

   2 0

   (17)

   4

   M3( S)

   0 0 0 0 V3 ''

   G q3

   – –

   +

   0.280 + 4 –

   6 + 5 +

   V

   ''

   q

   4

   0.324

   -0.560

   0.280

   0.324

   Multiplying out we obtain:

   ka22U2 '' ka23U3 ''- b22U2 – b23U3 –

   Fig. 3 Distortional bending moment for double cell mono-symmetric frame

   – c22V2 '- c23V3 ' 0

   (18)

   ka32U2 '' ka33U3 ''- b32U2 – b33U3 –

   V iv V iv – V '' K

   (a)

   1 2 2 3 1 3 3

   (22)

   – c32V2 '- c33V3 ' 0

   (19)

   V iv V iv – V ''- V -K

   (b)

   22 2 23 3 22 2 23 3

   c U ' c U ' r V '' r V '' – q2

   (20)

   3 2 4 3 2 3 1 3 4

   The relevant coefficients are as follows:

   G a22 25.05;

   a23 a32 -0.270,

   a33 0.757

   c U ' c U '- ks V r V '' r V

   '' – q3

   b c r 2.982 b c r 1.407

   32 2 33 3 33 3 32 2 33 3

   G

   (21)

   22 22 22 33 33 33

   b23 b32 c23 c32 r23 r32 -0.153

   Simplifying further we obtain the coupled differential equations of flexural-distortional equilibrium for mono symmetric sections as follows:

   r44 14.616; s33 0.261* 6.9712 *10-4 1.8195 *10-4

   V iv V iv – V '' K (a)

   E 24 *109 N / m2 ; G 9.6 *109 N / m2 , k 2.5

   1 2 2 3 1 3 3

   (22)

   The coefficients of the governing equations are as

   V iv V iv – V ''- V -K

   (b)

   3 2 4 3 2 3 1 3 4

   follow:

   r

   where, 44 ,

   c ks

   2

   r c

   c43

   • c r ;

     1 43 33

     2

     b33r44 – c34c43 ,

     1 ka22 62.6825;

     3 Ka32 1.0625;

     2 ka23 1.0625

     4 Ka33 1.875

     1 34 43 33 44

     ka33c43

     a23c22 – a22c23 k 2s33

     c22 q3

     1

     c c – c2

     0.0283

     K1 –

     c c

     • c2 G

       33 22 32

       2

       33 22 32

       k s33 a33c22 – a32c23 -3

       c23

       q3

       2

       c c – c2

       1.750 *10

       K2 –

       c c

     • c2 G

   33 22 32

   33 22 32

   a32c23 – b33c22 ks33 -3

   K3 b23K1 – b22K2 ,

   K4 b32K2 b33K1

   1

   c c – c2

   -1.260 *10

   33 22 32

2. Flexural-distortional analysis of double cell mono symmetric section

   K1 –

   c22

   q

   3

   2 -1.115 *10-5

   c c – c G

   In this section the solutions of the differential

   33 22 32

   equations of equilibrium eqns. (22) are obtained for the double cell mono symmetric box girder structure

   K2 –

   c23

   q

   3 -6

   c c – c G

   2 -1.684 *10

   whose cross section is shown in Fig. 1(a). Live loads are considered according to AASHTO-LRFD [21], following the HL-93 loading: uniform lane load of 9.3N/mm distributed over a 3m width plus tandem load of two 110 KN axles. The loads are positioned at the outermost possible location to generate the maximum torsional effects. A 50m span simply supported bridge deck structure is considered. The obtained torsional

   33 22 32

   K3 b23K1 – b22K2 1.003 *10-5

   K4 b32K2 b33K1 -1.634 *10-5

   Substituting the coefficients into eqns (22 ) we obtain

   loads are ; q2 0.00KN , q3 196.46KN .

   The governing equations of equilibrium are:

   2 3 3

   62.683V iv 1.0625V iv – 0.0283V

   '' -1.003 *10-5

   Integrating by method of trigonometric series with

   1.0625V iv 1.875V iv -1.750 *10-3V ''1.26 *10-3V

   accelerated convergence we hav:

   2 3 3 3

   -5

   V2 (x) 8.626 *10-3 Sin x / 50

   (24)

   1.634 *10

   (23)

   V3 (x) 1.250 *10-2 Sin x 50

   30

   Distortional displacement

   25 Flexural Displacement

   Displacement (mm)

   20

   15

   10

   5

   0

   0 5 10 15 20 25 30 35 40 45 50

   Distance Along the Length of the Girder(m)

   Fig.4: Variation of flexural and distortional displacements along the length of the girder

3. Discussion of Results

   The derived governing differential equations of flexural-distortional equilibrium eqn. (22), is applicable to all mono symmetric box girder structures, both single cell and multi cell profiles. Along the axis of symmetry of the box girder structure, bending strain mode 1 does not interact with distortional strain mode 3 hence, there was no relationship between V1 and V3 as could be seen from the derived eqn. 22.

   Fig.4 shows the variation of flexural and distortional displacements along the length of the girder as described by eqn. (24). It should be recalled that flexural strain mode has interaction with

   distortional strain mode only on the non symmetric axis of the box girder structure. The results show that on ths non symmetric axis, the maximum (mid span) distortional deformation (12.5mm) was one and half times that of flexural deformation (8.5mm), for a simply supported box girder structure of 50m span.

4. Conclusions

In a mono symmetric box girder section flexural strain mode does not interact with distortional strain mode along the axis of symmetry. However, along the non symmetric axis, flexure interacts with distortion giving rise to coupled differential equations of flexural-

distortional equilibrium, eqn. (22) which when solved for a particular cross sectional profile yields the flexural and distortional deformations. For the double cell mono symmetric example frame we established that distortional deformation at mid span of the girder was about one and half times that of flexural deformation.

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21. American Association of State Highway and Transportation Officials (AASHTO), Load and Resistance Factor Design, LRFD, Bridge Design Specifications Washington , D.C, 1998.