analysis of straight and skewed box girder bridge by finite strip method

7/18/2019 Analysis of Straight and Skewed Box Girder Bridge by Finite Strip Method

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 11, November 2012)

191

Analysis of Straight and Skewed box Girder Bridge by FiniteStrip Method

Prof. Dr. S.A.Halkude1, Prof. Akim C.Y.2

Principal, Walchand Institute of Technology, Sholapur

Asst. Professor in Civil Engg.Dept., Deogiri Institute of Engineering and Management Studies, Aurangabad

Abstract — Generally a box girder is a beam and hence

bending and shear actions exist in longitudinal direction. Due

to the thin walled and widespread cross-section, the shear is

not uniform even along the width of horizontal plates. When

subjected to eccentric loading, the section exhibits torsion. An

exact close form solution for such a continuum with a complex

behavior is almost impossible. Some approximate analytical

methods with simplifying assumptions for loading and

geometry have been used for analysis. The objective of the

research reported in this paper is to obtain analysis of box

Girder Bridge by Finite Strip Method considering various

values of skew angles considering their effect on each nodal

line at specific distance interval along the span of girder by

considering self weight and point load.

Keywords — Finite Strip Method, Box Girder, Skew angle,

Nodal lines, Strips, Load Vector, Stiffness matrix, Strain

energy, Finite Element Method.

I.

I NTRODUCTION

For a structure with constant cross-section and end

boundary conditions that do not change transversely, stress

analysis can be performed using finite strip instead of finite

elements. In each strip, the displacement component at any

point is expressed in terms of the displacement parameters

of nodal lines by means of simple polynomials in the

transverse direction and continuously differentiable smooth

series in the longitudinal direction. With the stipulation that

such series should satisfy the boundary conditions at the

ends of the strip. Using Strain-displacement relationships,

the strain energy of the structure and the potential energy of

external loads can be expressed in terms of the

displacement parameters should make the total potential

energy of the structure become minimum. This yields a set

of linear algebraic equations with the displacement parameters, the displacement and stress components at any

point in the structure can be obtained.

II.

FINITE STRIP MODELING OF BRIDGES

Selection of the best approach and the most suitable type

of strip is based mainly on the bridge geometry, namely the

shape and support conditions. The load conditions should

also be considered. For a right or skew slab of box Girder

Bridge with a constant cross-section and simply supported

ends, the finite strip method is the most efficient method. It

reduces to half bandwidth of the stiffness matrix

significantly. Therefore the time consumption in forming

this matrix is reduced

Generating a finite strip model

Depending upon the objective of analysis and the

loading conditions, the suitable finite strip model can be

chosen. In using the finite strip method for the overall

analysis of a single span skew box girder bridge, the

following modeling will provide adequate accuracy.

The flanges and webs are idealized as the orthotropic

plates with equivalent elastic properties.

Each web and flange is divided into minimum three

strips. If only the deflection and longitudinal stress are

required, few strips are sufficient. If more accurate

results are desired more strips should be used

considering load act on a nodal line.

In regions with a higher stress gradient, narrower strips

should be used. The width of strip should change

gradually from one strip to another.

If the cross section and loads are both symmetrical about

the centerline, only half of the bridge needs to be

analyzed.

If the bridge is subjected to uniform load only, five to

ten symmetrical harmonies are adequate for the analysis

Numbering nodal lines and strips

In order to minimize the half bandwidth of a stiffness

matrix in a finite strip model, the nodal lines should be

numbered so as to keep the difference between the numbers

of all the nodal lines within each strip at a minimum.





192

Modelling of box girder

The numbering of nodal lines and strips for the proposed

box girder model is done as shown below. The Fig.2.2

shows the Finite Strip Model of box girder whereas the

Fig.2.3 shows the Finite Element Model of box girder.

III.

THEORETICAL FORMULATION

Consider a simply supported beam carrying a central

point load at centre and having span L.

The boundary conditions at both ends y = 0 and y = l

are,

w(y) = 0

M(y) = - EI

= 0 3.1

The deflection of the beam by a half – sine wave is,

w(y) = δ sin

3.2

In order to obtain the best approximation the total

potential energy developed in the system become a

minimum.

П = U + W

П =

∫

3.3

The first derivative of П with respect to δ must be zero

П

δ = 0 3.4

П =

δ

∫

δ 3.5

П =

δ

δ 3.6

Then the value of is obtained by solving eq. 3.4

3.7

The deflection function proposed in Equation 1.2

becomes

w(y) = sin 3.8

From which the bending moment is obtained as

M(y) =

2

2

d w EI

dy = sin 3.9

To improve the accuracy using a series of sine function

w(y) =

1

sinr

m

m ym

l

3.10

Applying the energy approach again yields.

w(y) =4

1

1r

m m sin sin

M(y) =2

1

1r

m m sin sin 3.11

This will give resulting deflection and moment at mid

span





193

Proposed work for analysis of bridge is descritized into

no. of strips.

In each strip the displacement components at any pointis

w(x,y) =

1

( )sinr

m

m y fm x

l

=2 3

0 1 2 3

1

( ....) sinr

m

m ya a x a x a x

l

3.12

Where 0 1 2 3, , , ......a a a a are the unknown displacement

parameters.

Consider deflection amplitudes for a plate strip. Asshown above In this there are four displacement parameters

imw ,

jmw , im and jm i.e.

iw =

1

sinr

im

m

m yw

l

jw =

1

sinr

jm

m

m yw

l

3.13

i = ( )dw

idx

=

1

sinr

im

m

m y

l

j = ( )dw jdx

=

1

sinr

jm

m

m yl

3.14 3.14

To include the four displacement parameters

imw , jmw ,

im and

jm

a third order polynomial is

required in equation.

( ) fm x =2 3

0 1 2 3a a x a x a x

After solving the above equation for 0 1 2 3, , , ......a a a a

the displacement function 3.12 can be written as

Where

1( ) (1 N x 3 2 X +2

3 X )

2 ( ) (1 N x x 2 X + 2 X )

3( ) ( N x 3

2 X 2 3 X )

4 ( ) ( N x x 2

) X X

In a matrix form,

w(x,y) = 1 2 3 4

1

[ , , , ] sin

im

r im

jmm

jm

w

m y N N N N

w l

3.16

More concisely

w(x,y) = 1[ ] sin

r

mm

m y N l

………… 3.17

Energy Formulation

The strain energy of a plate strip is given by

The Equation 3.17 can be written in the following matrix

form.

U = 1

2

2

2

2

2

2

2

w

x

w

y

w

x y

dxdy

U = 1

2

dxdy





194

1

1

2

T r

mmm m

U k

1

T r

mm m

W p

1

3 2

4 5 1

5 6 3 2

ii ijb b

m

ij jjb b

symmetrical k

K K k k k

k k k K K

k k k k

4 2 21 1 3

13 12 6 670 5 5

m y m xy m xlb l l l k k D k D k D D

b b b

34 2 2

2 1

4 2 2

210 15 15m y m xy m x

lb lb lb l k k D k D k D D

b

24 2 2

3 1 2

11 3 3

420 5 5m y m xy m x

lb l l l k k D k D k D D

b

4 2 2

4 1 3

9 12 6 6

140 5 5m y m xy m x


b b b

24 2 2

5 1 2

13 3

840 5 10m y m xy m x


b

34 2 2

6 1280 15 30

m y m xy m x

lb lb lb l k k D k D k D D

b

Load vector m

p for point load0

P at (0

x , 0 y ) is,

1 0

2 0

0 0

3 0

4 0

( )

( )sin

( )

( )

mm

N x

N x p p k y

N x

N x

The strain energy and the potential energy of the entire

plate is,

1 1

1

2

T S r

t Im Im I m Im

U k

1 1

T S r

t Im I m Im

W p

Now the Total potential energy of the entire plate is now

expressed as

t t t U W

In order to obtain the best approximation the total

potential energy developed in the system become a

minimum.

0t

imw

, 0t

im

This yields a set of linear equation, after solving it gives

an unknown displacements and bending moments at any

point inside the structure

IV.

VALIDATION OF PROGRAM





195

Table No. 4.1

Node no. 01 to 08 and skew angle 15

0

displacement for distance alongbeam

Comments:

Displacement is obviously zero at both the supports and

nature of graph observed to be parabolic in nature.

The variation of displacement observed at 3.0 m, 6.0 m

and 9.0 m from the left to right support shown by the

graphs is plus or minus 5 % while comparing program to

staad-pro results.

Comments:

Bending Moment is obviously zero at both the supports

and nature of graph observed to be sinusoidal in nature.

Variation of Bending Moment (Mx, My and Mxy)

observed at 3.0 m, 6.0 m and 9.0 m from the left to right

support shown by the graphs is plus or minus 5 % while

comparing program to staad-pro results.

Results and discussion

Comparison of displacement and bending moment

(transverse, longitudinal and twisting) values as obtained

by the computer program and Staad-Pro at 3.0 m, 6.0 m

and 9.0 m from the left to right support shown by the

graphs. It is observed from these graphs that values of

downward displacement and bending moment obtained by

the computer program and that obtained by Staad-Pro is

comparable.

V.

PARAMETRIC WORK

The close agreement of the results of the computer

program based on the finite strip method with those given

by the Staad-Pro confirms that the computer program

developed is accurate and reliable.





196

On basis of this the parametric work is done by

considering the skew angles150

,300

&450

The resultsobtained are tabulated in the form of tables and graphs.

Table No. 5.1

Node no. 1 to 8 and skew angle 150, 300&450 displacement for distance

along beam

Comments:

At support their downward displacement is obviouslyzero.

However in the one third portion near the either support

as we go towards the centre displacement decreases with

increase in skew angle i.e. about 5 to 10%

In the central one third portion it is seen that

displacement is reducing

Comments :

Bending moment obviously zero at both the support.

For given skew angle Mx varies in a sinusoidal form

The maximum bending moment observe at centre.

With increase in skew angle Mx decreases up to skew

angle 300 then it again increases





197

Comments:

Bending moment obviously zero at both the support.

For given skew angle Mx varies in a sinusoidal form


With increase in skew angle bending moment increases

and its nature reverses.

Comments:

Bending moment obviously zero at both the support.For given skew angle Mxy varies in a sinusoidal form


With increase in skew angle bending moment increases

and its nature reverse

VI. CONCLUSION

The computer time and effort required for the analysis of

skew box girder using newly developed program is less

as compared to Finite Element Analysis using Staad-Pro.

Finite Strip Method require less input data because of

the smaller number of mesh lines involved due to the

reduction in dimensional analysis but in case of Finite

Element Method it is somewhat tedious and difficult.

Finite Strip Method involves less number of equations

and matrix with narrow bandwidth, consequently much

less computing time for solution, where as Finite

Element Method involves more no. of equations and

matrix so it is time consumable.

In Finite Strip Method it is easy to specify only those

locations at which displacement and bending momentsare required and then accordingly the computations

In an analysis of box Girder Bridge by Finite Strip

Method considering loadings i.e. point load at centre, the

variation of nodal displacement and nodal bending

moments as obtained by computer program is almost

same.

It is observed that the longitudinal bending moment My

decreases and the twisting moments M xy increases with

an increase in the skew angle.

VII.

NOTATIONS

P ……….. load

w ………. Displacement

M ………. Bending Moment

∏ ………. Total potential energy

W ………. Potential energy

U ………. Strain energy

r ……….. No. of series terms

[N] ……… Matrix of transverse shape function

Mx ……… Transverse bending moment

My ……… Longitudinal bending moment

Mxy ……. Twisting moment

{M} ……. Moment vector

Dx ……… Flexural rigidity

Dxy …….. Torsional rigidity

D1 ……… Coupling rigidity





198

{K} …….. Stiffness matrix

{P} ……… Load vector

{ᵟ} ………. Curvature vector

s ………… no. of strips

t ………… is subscript representing the whole structure.

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[2 ]

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analysis of straight and skewed box girder bridge by finite strip method

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