151 
 

 American Scientific Research Journal for Engineering, Technology,  and Sciences  (ASRJETS) 
ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 

© Global Society of Scientific Research and Researchers  
http://asrjetsjournal.org/  

 

The Orthotropic Spline Finite Strip Technique in the 

Linear Analysis of Ribbed Bridge Decks 

Laith Khalid Al-hadithya*, Imad Abdulwahid alib 

aAsst. Prof. Dr. Civil Engrng. Dpt. /Al-Nahrain University, Baghdad, Iraq 
bState Company for Grain Board, Baghdad, Iraq 

aEmail: lthadithy@yahoo.com 
bEmail: Emad_aldlyme@yahoo.com 

 

 

 

Abstract 

This study presents an efficient simplification on the utility of the spline finite strip method SFSM in the 

analysis of orthotropic and ribbed bridge decks by conducting two stages of analogy. The actual ribbed bridge 

deck (practically subjected to combined plate-bending and plane stress actions) is, first, converted into and 

equivalent orthotropic plate, which is subsequently divided into orthotropic finite strips submitted to plate-

bending action only with the use of B3-spline function to express the displacement function in the longitudinal 

direction of strips. A programming code using MATLAB computer package is constructed for the analysis of 

orthotropic plates by the spline finite strip method, where the fewest admissible number of longitudinal sections 

and local B3-spline functions has been employed to check its efficiency. When applied to a ribbed RC bridge 

deck of four transverse diaphragms, the present orthotropic spline finite strip technique has proved its reliability 

in the analysis of such decking system where very high degrees of coincidence of its results with those of the 

finely meshed sophisticated finite element method attaining 98.4, 86.9 and 95.6 for midspan deflection and 

longitudinal and transvers bending moments, respectively, have been obtained. 

Keywords: Spline Finite Strip Method (SFSM);Geometrical Orthotropy; Ribbed Bridge Decks; Plate-Bending 

action; B3-Spline Function. 

 

 

------------------------------------------------------------------------ 

* Corresponding author.  

http://asrjetsjournal.org/


American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No  4, pp 151-168 

 

152 
 

1. Introduction 

The analysis of bridge decks can be carried out by  several methods. The  finite strip method (FSM) is one of 

those methods. The first use of the finite strip method was in 1968 by Cheung[1] for analysis of rectangular 

plates  with two opposite simply supported ends. In this method the structure is divided into a number of strips 

by longitudinal lines named nodal lines. It is a special form of the finite element method (FEM) where , both 

methods  use the displacement approach, while the (FEM) uses polynomial displacement functions in all 

directions, the (FSM) uses a simple polynomial in the transverse  direction and continuously differential smooth 

series in the longitudinal (major) direction where their a product together gives the displacement function of the 

strip. 

2. Basic concept and development the spline finite strip method (SFSM) 

 

Figure 1: Development and derivation of the basic concept of the spline finite strip method [4] 

The conventional finite strip method,  introduced by Cheung, depends on using free vibration function series of 

a beam as a basic function in the longitudinal direction, and osculated polynomials as the interpolation function 

in the transvers  direction [2] The beam vibration function is an infinite series which cannot be obtained by 

calculation so that  numerous  numerical experiments  must be done to find a suitable and acceptable truncation 

of the series for all loading cases. Moreover, the infinite continuity of the free vibration functions, hinders its 

further development, hence the introduction of the highly continuous vibration functions will result in a poor 

approximation if the series is truncated too early or yielding an oscillated value, when more terms are taken, 

Ritz- Galerkin Approach 
Solution represented by trial function 

satisfying all boundary condition. 

Variational Method 

Finite Element Approach 
Solution represented by piecewise osculated 

polynomials.  

Finite Strip Method 
Solution represented by a product of beam vibration function, satisfying 

the boundary condition in that direction, and the piecewise osculated 
polynomials in another direction. 

Spline Finite Strip Method 
Solution represented by a product of B3- spline function, satisfying both 

boundary and interior conditions in that direction and the piecewise 
osculated polynomials in another direction.  

Spline 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No  4, pp 151-168 

 

153 
 

Gibb's phenomenon will be valid. On the other hand “the SFSM produces monotonic convergence of stress 

instead of oscillatory convergence of stress (encountered when using FSM as identified by Gibb's phenomenon) 

" [3]. This is attributed to the development and structure of the spline finite strip method (SFSM)  shown in 

Figure 1. 

3.  Displacement function 

The longitudinal B3-spline representation and the transverse interpolation polynomial  are  multiplied  to 

introduce the displacement function. They are discussed separately 

3.1 B3-Spline representation for longitudinal direction  

Several types of splines, were developed in the elapsing century. The most efficient and versatile one is  the 

basic cubic B3- spline, defined in equation(1) and adopted in this paper.  

𝜑𝜑𝑖𝑖     1
66ℎ3

 .

⎩
⎪⎪
⎨

⎪⎪
⎧

0                                                                                                             , 𝑥𝑥 < 𝑥𝑥𝑖𝑖−2                        
    (𝑥𝑥 − 𝑥𝑥𝑖𝑖−2)3   ,𝑥𝑥𝑖𝑖−2  ≤ 𝑥𝑥 ≤ 𝑥𝑥𝑖𝑖−1

ℎ3 + 3ℎ2(𝑥𝑥 − 𝑥𝑥𝑖𝑖−1) + 3ℎ(𝑥𝑥 − 𝑥𝑥𝑖𝑖−1)2 − 3(𝑥𝑥 − 𝑥𝑥𝑖𝑖−1)3                  , 𝑥𝑥𝑖𝑖−1  ≤ 𝑥𝑥 ≤ 𝑥𝑥𝑖𝑖                   
ℎ3 + 3ℎ2(𝑥𝑥𝑖𝑖+1 − 𝑥𝑥) + 3ℎ(𝑥𝑥𝑖𝑖+1 − 𝑥𝑥)2 − 3(𝑥𝑥𝑖𝑖+1 − 𝑥𝑥)3 ,                    , 𝑥𝑥𝑖𝑖  ≤ 𝑥𝑥 ≤ 𝑥𝑥𝑖𝑖+1

 (𝑥𝑥𝑖𝑖+2 − 𝑥𝑥)3                 , 𝑥𝑥𝑖𝑖−1  ≤ 𝑥𝑥 ≤ 𝑥𝑥𝑖𝑖
  0                                                                                              ,   𝑥𝑥𝑖𝑖+2 < 𝑥𝑥                                    

 

 ……(1) 

Derivation of the B3- spline and the synthesis processes are shown in Figure 2 

 

Figure 2:  Derivation of B3-spline expression from natural spline through discretization and synthesis processes 

3.1.2 Amendments of boundary conditions 

In order to adapt  various boundary conditions at end knots, only three boundary local spline centered at each 

end  have to be amended, while the standard B3-spline,  defined by equation (1) is used for other knots. 

= 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No  4, pp 151-168 

 

154 
 

i.e. 

Y=[𝜑𝜑�−1, 𝜑𝜑�0, 𝜑𝜑�1, 𝜑𝜑2,𝜑𝜑3,……, 𝜑𝜑𝑚𝑚−3,𝜑𝜑𝑚𝑚−2, 𝜑𝜑�𝑚𝑚−1, 𝜑𝜑�𝑚𝑚, 𝜑𝜑�𝑚𝑚+1]

⎩
⎪
⎪
⎪
⎨

⎪
⎪
⎪
⎧
𝛼𝛼−1
𝛼𝛼0
𝛼𝛼1
𝛼𝛼2
𝛼𝛼3
⋮

𝛼𝛼𝑚𝑚−3
𝛼𝛼𝑚𝑚−2
𝛼𝛼𝑚𝑚−1
𝛼𝛼𝑚𝑚
𝛼𝛼𝑚𝑚+1⎭

⎪
⎪
⎪
⎬

⎪
⎪
⎪
⎫

               ..…(2) 

In short form,       Y =𝜑𝜑 . {𝛼𝛼} . 

Shapes of amended local splines 𝜑𝜑�0 and  𝜑𝜑�1 are shown in the Figure 3 

 

Figure 3:  Shapes of amended local splines 𝜑𝜑�0 and  𝜑𝜑�1. [4] 

Table 1: Amendment scheme for boundary local splines satisfying both rigid and natural conditions 

 
 

𝜑𝜑�−1 𝜑𝜑�0 𝜑𝜑�1 

              y(x0)  ≠ 0 
Free End           y'(x0)   ≠ 0 

              y"(x0) = 0 
Eliminated 𝜑𝜑0+2𝜑𝜑−1 𝜑𝜑1+𝜑𝜑0+𝜑𝜑−1 

 
  
 Eliminated Eliminated 𝜑𝜑1- 𝜑𝜑−1 

  

 

Eliminated Eliminated 𝜑𝜑1- 1
2
𝜑𝜑0+𝜑𝜑−1 

 

Eliminated 𝜑𝜑0 𝜑𝜑1+𝜑𝜑−1 

Amended                                   
Local spline 
Boundary 

condition 

Simply 
Supported                 

End                       

y(x0)  ≠ 0 
y'(x0) ≠ 0 

y"(x0) = 0 

  
Clamped 
 End             

y(x0)  = 0 
y'(x0) = 0 
y"(x0) ≠ 0 

 

Sliding  
Clamped 

End 

y(x0)  ≠ 0   
y'(x0) = 0 
y(x0) ≠ 0 

 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No  4, pp 151-168 

 

155 
 

3.2 Transverse interpolation polynomials   

The transverse interpolation polynomial of two nodal lines of finite strip is used to describe its interior behavior, 

and it is transformed to the form of shape function [5]. 

There are many  shape functions can obtained by either direct inspection or matrix transformation which are 

suitable for transverse  representation as they satisfy some  conditions. The cubic interpolation   defining two 

degrees (the displacement and its derivative) of freedom  for  each nodal line  of a strip is adopted. The  four 

appropriate shape functions are: 

                 N1 = 1- 3�̅�𝑥2 + 2 �̅�𝑥3           ,     N2 = 𝑥𝑥 ( 1-2 �̅�𝑥 +�̅�𝑥2) 

                 N3 =  3�̅�𝑥2 - 2 �̅�𝑥3                ,     N4 = 𝑥𝑥 (  �̅�𝑥2- �̅�𝑥 ) 

where   �̅�𝑥 = x
b
 

These shape functions, are derived from  a straight line  connecting two nodes, with displacement and transverse 

rotation as shown in Figure 4, they are  derived with their derivatives  by direct inspection. 

 

Figure 4: Transverse shape function in connection with the degrees of freedom of two edge nodal lines. 

4. Formulation of the strip characteristics 

After choosing the displacement function for describing the behavior of the strip element, the analysis can then 

proceed simply by following the standard procedures for discrete systems using displacement approach. The 

strip characteristics including stiffness matrix, load matrix and mass matrix can then obtained by variational 

method or its equivalence- the principle of minimum total potential energy. The basic procedures are as 

following:  

• Choose a suitable displacement function [f] for all defined degrees of freedom. 

• Construct the strain matrix [B] according to the kinematic relationship between strains {𝜀𝜀}and displacement 

parameters {𝜹𝜹} 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No  4, pp 151-168 

 

156 
 

i.e.  {𝜺𝜺} = [B]  {𝜹𝜹}     …..(3) 

• Connect the stress {𝝈𝝈} to the displacement parameters {𝜹𝜹} through constitutive equation  

{𝝈𝝈} = [D]  {𝜺𝜺}                 …..(4) 

In which [D] is the property matrix. Hence 

{𝝈𝝈} = [D]  [B]  {𝜹𝜹}     …..(5) 

• Drive the stiffness matrix and load matrix through the principle of minimum total potential energy. The 

potential energy consist of two parts: 

(i) Strain energy stored in the strips of volume V 

i.e. 𝝅𝝅𝟏𝟏 = 𝟏𝟏
𝟐𝟐
 ∫ {𝜺𝜺}𝑻𝑻𝑽𝑽 .  {𝝈𝝈} dv 

=  𝟏𝟏
𝟐𝟐
 ∫ {𝜹𝜹}𝑻𝑻𝑽𝑽 . [B]T [D] [B] {𝜹𝜹}dv                  …..(6) 

(ii) Potential energy of the applied loads 𝑞𝑞𝑖𝑖 over area 𝐴𝐴𝑖𝑖 respectively. 

i.e. 𝝅𝝅𝟐𝟐 = - ∑ ∫  {𝐟𝐟}𝑨𝑨𝑨𝑨𝑨𝑨
T. 𝒒𝒒𝑨𝑨 d𝑨𝑨𝑨𝑨 

= - ∑ ∫  {𝛅𝛅}𝑨𝑨𝑨𝑨𝑨𝑨
T [𝚽𝚽]T [N]T . 𝒒𝒒𝑨𝑨 d𝑨𝑨𝑨𝑨                  …..(7) 

  Hence, the total potential energy of the system is 

  𝝅𝝅 = 𝝅𝝅𝟏𝟏 + 𝝅𝝅𝟐𝟐    

=  𝟏𝟏
𝟐𝟐
 ∫ {𝜹𝜹}𝑻𝑻𝑽𝑽 . [B]T [D] [B] {𝜹𝜹}dv - ∑ ∫  {𝛅𝛅}𝑨𝑨𝑨𝑨𝑨𝑨

T [𝚽𝚽]T [N]T . 𝒒𝒒𝑨𝑨 d𝑨𝑨𝑨𝑨                 …..(8) 

It depends on the variables {𝜹𝜹} only. 

By taking first variation, 𝝏𝝏𝝅𝝅 = 0 , it gives 

[∫  [𝐁𝐁]𝑽𝑽
T  [D] [B] dv] {𝜹𝜹} = ∑ ∫  𝑨𝑨𝑨𝑨𝑨𝑨 [𝚽𝚽]T [N]T  𝒒𝒒𝑨𝑨 d𝑨𝑨𝑨𝑨                 …..(9) 

In short form, 

[K] . {𝜹𝜹} = {f}                               …..(10) 

• Carry out the integrations over the whole volume of each strip or the loaded area and use the following 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No  4, pp 151-168 

 

157 
 

expressions to obtain the stiffness matrix [K]e, mass matrix [M]e and the load matrix {f}e respectively. 

i.e.  [K]e = ∫  [𝐁𝐁]𝑽𝑽
T  [D] [B] dv                   …..(11) 

5. Analysis of thin rectangular plates in bending by  spline finite strip method (SFSM) 

The behavior of a strip in the spline finite strip analysis is described by its two boundary nodal lines. So, the 

prescribed conditions have been incorporated into the B3-Spline expression for the nodal lines, hence, there  will 

be no difference in solution procedures for different boundary or interior conditions. The solution presented here 

is based on Kirchhoff's theory, which enables the problem of thin plate bending to be treated as a two- 

dimensional one. 

5.1 Degrees of freedom   

Two degree of freedom for each nodal line (according to the convergence criteria) are necessary to acquire the 

minimum compatibility conditions. i.e. the lateral displacement, w, and the first derivative, 𝛉𝛉𝒙𝒙𝒙𝒙  = 𝝏𝝏𝒘𝒘
𝝏𝝏𝒙𝒙

   with 

respect to the transverse x-axis, as shown in Figure 5. 

 

Figure 5: Rectangular plate strip analysis. (a) Plate as an assembly of strips; (b) Typical strip 

5.2 Stiffness matrices 

The stiffness matrix [k] can be worked manually and expressed in explicit form, after deriving the displacement 

function and the expressions of relationships between the strain– displacement and the stress–strain. Finally 

Table 2 is got.  

where: 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No  4, pp 151-168 

 

158 
 

                𝐃𝐃𝒙𝒙 = 𝐄𝐄𝐱𝐱 .  𝐭𝐭𝟑𝟑

𝟏𝟏𝟐𝟐(𝟏𝟏−𝐯𝐯𝐱𝐱𝐯𝐯𝐲𝐲)
 

                𝐃𝐃𝒚𝒚 = 𝐄𝐄𝐲𝐲 .  𝐭𝐭𝟑𝟑

𝟏𝟏𝟐𝟐(𝟏𝟏−𝐯𝐯𝐱𝐱𝐯𝐯𝐲𝐲)
                                                                                                    ……(12) 

                𝐃𝐃𝟏𝟏= 𝒗𝒗𝒙𝒙. 𝐃𝐃𝒚𝒚 = 𝒗𝒗𝒚𝒚 . 𝐃𝐃𝒙𝒙 

                 𝐃𝐃𝒙𝒙𝒚𝒚 = 𝑮𝑮𝒙𝒙𝒚𝒚 .  𝒕𝒕𝟑𝟑

𝟏𝟏𝟐𝟐
 

  and, 𝐄𝐄𝐱𝐱 , 𝐄𝐄𝐲𝐲 are Young's moduli, 𝑣𝑣𝑥𝑥 , 𝑣𝑣𝑦𝑦 are Poisson's ratios, 

 and  𝑮𝑮𝒙𝒙𝒚𝒚 is shear modulus = 
�𝑬𝑬𝒙𝒙𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆 𝑬𝑬𝒚𝒚𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆

𝟐𝟐(𝟏𝟏+�𝒗𝒗𝒙𝒙𝒗𝒗𝒚𝒚 )
            [6] 

Poisson's ratio in the y direction 𝑣𝑣𝑦𝑦 May be considered as:  

              (𝒗𝒗𝒚𝒚=𝒗𝒗𝒙𝒙
𝑬𝑬𝒚𝒚
𝑬𝑬𝒙𝒙

 )                                             [7] 

Table 2: Bending stiffness matrix of a rectangular strip [4] 

5040  Dx I111 
-504 b2 D1 I211  
-504 b2 D1 I311  
+156 b4 Dy I411  
+2016 b2 Dxy I511  

 

2520 b D𝑥𝑥 I112  
-462 b3 D1 I212  
-42 b3 D1 I312  
+22 b5 D𝑦𝑦 I412  
+168 b3 D𝑥𝑥𝑦𝑦 I512  

 

-5040  D𝑥𝑥 I113 
+504 b2 D1 I213  
+504 b2 D1 I313  
+54 b4 D𝑦𝑦 I413  
-2016 b2 D𝑥𝑥𝑦𝑦 I513  

 

2520 b D𝑥𝑥 I114 
-42 b3 D1 I214  
-42 b3 D1 I314  
-13 b5 D𝑦𝑦 I414  
+168 b3 D𝑥𝑥𝑦𝑦 I514  

 

 1680 b2 D𝑥𝑥 I122 
-56 b4 D1 I222 
-56 b4 D1 I322  
+4 b6 D𝑦𝑦 I422  

+224 b4 D𝑥𝑥𝑦𝑦 I522  
 

-2520 b D𝑥𝑥 I123 
+42 b3 D1 I223 
+42 b3 D1 I323  
+13 b5 D𝑦𝑦 I423  
-168 b3 D𝑥𝑥𝑦𝑦 I523  

 

840 b2 D𝑥𝑥 I124 
+14 b4 D1 I224 
+14 b4 D1 I324  
-3 b6 D𝑦𝑦 I424  
-56 b4 D𝑥𝑥𝑦𝑦 I524  

 

Where: 
            I1𝑖𝑖𝑖𝑖 =  ∫ 𝜑𝜑T

𝑖𝑖
𝑎𝑎
0 𝜑𝜑𝑗𝑗d𝑦𝑦 ; 

          I2𝑖𝑖𝑖𝑖  =  ∫ 𝜑𝜑"T
𝑖𝑖

𝑎𝑎
0 𝜑𝜑𝑗𝑗d𝑦𝑦 ;             

 
           I3𝑖𝑖𝑖𝑖  =  ∫ 𝜑𝜑T

𝑖𝑖
𝑎𝑎
0 𝜑𝜑"

𝑗𝑗d𝑦𝑦 ; 
          
          I4𝑖𝑖𝑖𝑖  =  ∫ 𝜑𝜑"T

𝑖𝑖
𝑎𝑎
0 𝜑𝜑"

𝑗𝑗d𝑦𝑦 ; 
           
         I5𝑖𝑖𝑖𝑖  =  ∫ 𝜑𝜑′T

𝑖𝑖
𝑎𝑎
0 𝜑𝜑′

𝑗𝑗d𝑦𝑦  .   
    

5040  D𝑥𝑥 I133 
-504 b2 D1 I233  
-504 b2 D1 I333  
+156 b4 D𝑦𝑦 I433  
+2016 b2 D𝑥𝑥𝑦𝑦 I533  

 

-
2520 b D𝑥𝑥 I134  
+462 b3 D1 I234 
+42 b3 D1 I334  
-22 b5 D𝑦𝑦 I434  
-168 b3 D𝑥𝑥𝑦𝑦 I534  

 

 1680 b2 D𝑥𝑥 I144 
-56 b4 D1 I244 
-56 b4 D1 I344 

+4 b6 D𝑦𝑦 I444 

+224 b4 D𝑥𝑥𝑦𝑦 I544 
 

 

5.3 Load matrices 

 (i) For a patch load linearly distributed along y- direction from y1 to y2 (Figure 6): The load vector {F} can be 

[K]= 1
420b3

 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No  4, pp 151-168 

 

159 
 

expressed as: 

{F} = ∫ ∫ [𝝋𝝋]𝑻𝑻𝒚𝒚𝟐𝟐
𝒚𝒚𝟏𝟏

𝒃𝒃
𝟎𝟎  [𝐍𝐍]𝑻𝑻 { 𝒒𝒒𝟏𝟏  + ( 𝐪𝐪𝟐𝟐−𝐪𝐪𝟏𝟏

𝐲𝐲𝟐𝟐−𝐲𝐲𝟏𝟏
 ) (𝒚𝒚-𝒚𝒚𝟏𝟏 )} 𝐝𝐝𝒙𝒙𝐝𝐝𝒚𝒚                                     ……(13) 

{F}= 𝒒𝒒𝟏𝟏(∫ [𝝋𝝋]𝑻𝑻𝒚𝒚𝟐𝟐
𝒚𝒚𝟏𝟏 𝐝𝐝𝒚𝒚. 

⎩
⎨

⎧
𝐛𝐛/𝟐𝟐
𝐛𝐛𝟐𝟐/𝟏𝟏𝟐𝟐
𝐛𝐛/𝟐𝟐

 −𝐛𝐛𝟐𝟐 /𝟏𝟏𝟐𝟐⎭
⎬

⎫
 + (𝐪𝐪𝟐𝟐−𝐪𝐪𝟏𝟏

𝐲𝐲𝟐𝟐−𝐲𝐲𝟏𝟏
)( ∫ [𝝋𝝋]𝑻𝑻𝒚𝒚𝟐𝟐

𝒚𝒚𝟏𝟏 (𝒚𝒚-𝒚𝒚𝟏𝟏) 𝐝𝐝𝒚𝒚) 

⎩
⎨

⎧
𝐛𝐛/𝟐𝟐
𝐛𝐛𝟐𝟐/𝟏𝟏𝟐𝟐
𝐛𝐛/𝟐𝟐

 −𝐛𝐛𝟐𝟐 /𝟏𝟏𝟐𝟐⎭
⎬

⎫
              ……(14) 

 
Figure 6: Linearly distributed patch load on a rectangular plate strip 

(ii) For a concentrated load on a nodal line: 

To get  more accuracy it is recommended to locate a nodal line passing through the concentrated load, otherwise 

its effect can only be transferred to its two boundary nodal lines through the cubic interpolation polynomial of 

the assumed displacement function. With reference to Figure 7 the comprehensive load vector  for the 

combination of  a concentrated load P, moment 𝐦𝐦𝒙𝒙 and moment 𝐦𝐦𝒚𝒚 at locations 𝐲𝐲𝟏𝟏, 𝐲𝐲𝟐𝟐, 𝐲𝐲𝟑𝟑  on nodal line  i, 

respectively, is expressed in matrix form as follows: 

{F}= [ 𝝋𝝋(𝒚𝒚𝟏𝟏)]𝐓𝐓 �
𝐏𝐏
𝟎𝟎
𝟎𝟎
 𝟎𝟎
� + [ 𝝋𝝋(𝒚𝒚𝟐𝟐)]𝐓𝐓 �

𝟎𝟎
𝐌𝐌𝒙𝒙
𝟎𝟎
 𝟎𝟎

� + [ 𝝋𝝋′(𝒚𝒚𝟑𝟑)]𝐓𝐓 �

𝐌𝐌𝒚𝒚
𝟎𝟎
𝟎𝟎
 𝟎𝟎

�                                                    ……(15) 

 

Figure 7:  Concentrated loads on nodal line of rectangular plate strip. 



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160 
 

In Eq. 15 each of the three components  forms a column vector with four non-zero entries only, since there are 

only four local splines contributing non-zero values to it at the specified location y𝑖𝑖 , as shown in Figure 8. 

 

Figure 8: Local splines associated with concentrated load p and their contributions to load matrix. 

 (iii) For uniform distribution load q  [8] 

        {F} = ∫ ∫ 𝐪𝐪 [𝝋𝝋]𝑻𝑻𝒃𝒃
𝟎𝟎

𝒂𝒂
𝟎𝟎  [𝐍𝐍]𝑻𝑻 dx dy 

= q∫ [𝝋𝝋]𝑻𝑻𝒂𝒂
𝟎𝟎  dy .∫ [𝐍𝐍]𝑻𝑻𝒃𝒃

𝟎𝟎 dx  = q∫ [𝝋𝝋]𝑻𝑻𝒂𝒂
𝟎𝟎  dy . 

⎩
⎨

⎧
𝐛𝐛/𝟐𝟐
𝐛𝐛𝟐𝟐/𝟏𝟏𝟐𝟐
𝐛𝐛/𝟐𝟐

 −𝐛𝐛𝟐𝟐 /𝟏𝟏𝟐𝟐⎭
⎬

⎫
                       ……(16) 

6. Orthotropic plate analogy 

The term " Orthotropic Plate" explicitly refers to plates of either   material  or geometrical orthotropy. "A 

materially orthotropic" plate is composed of a homogeneous material which has different elastic properties in 

two orthogonal directions but the same geometric properties",[Eugene,2005]𝟐𝟐 ,i.e. the plate has the same  

second moment of area in  both orthogonal principal  directions but with different moduli of elasticity. It  should 

be noted that such a slab(materially orthotropic plate) is not found in bridge decks, it is an accurate 

approximation of some practical condition like timber. Most bridge decks slabs have different values of the 

second moment of area in  the two orthogonal principal  directions, such as voided slabs, and reinforced 

concrete ones( since  different amounts  of reinforcement are used in those two directions), such types of slabs 

are referred to as geometrically orthotropic  [6,10] 

7. Procedure of the present analysis of ribbed bridge decks 

The present analysis consist of two discrete stages; a preparatory stage (previously treated by researchers in the 

field of deck simulation), and the major stage concerning orthotropic SFSM analysis  solely treated by the 

present work.  



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161 
 

7.1 First (preparatory) stage: Modeling the ribbed bridge deck by equivalent materially  orthotropic plate  

This stage is summarized in the following steps:- 

1- Calculating the second moments of area per unit width (or length) of the deck cross-sections in  both 

orthogonal directions of the deck (Ix  and  Iy, respectively ).  

2- Finding the new equivalent  thickness by using  Equation 17, which means getting the equivalent 

orthotropic plate (solid plate) that has a constant thickness ( Ix = Iy ). 

d = √12𝐼𝐼3                                                      …….(17) 

3- Calculation of the new modulus of elasticity Ey and Poisson's ratio 𝒗𝒗y in the transverse (y) direction,  

using  Eqs. 18 and 19, respectively.  

  𝑬𝑬𝒚𝒚𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆 = 𝑬𝑬𝒔𝒔𝒆𝒆𝒂𝒂𝒃𝒃 𝑰𝑰𝒚𝒚
𝒔𝒔𝒆𝒆𝒂𝒂𝒃𝒃

𝑰𝑰𝒙𝒙𝒔𝒔𝒆𝒆𝒂𝒂𝒃𝒃
                   ……(18) 

  𝒗𝒗𝒚𝒚 = 𝒗𝒗𝒙𝒙
𝑬𝑬𝒙𝒙
𝑬𝑬𝒚𝒚

                                                                    …….(19) 

4- Finding the equivalent rigidities of the equivalent materially orthotropic plate, which is found by using 

Eq.12. This technique is schematically shown  in Figure 9 . 

 

 

Figure 9:  Materially orthotropic plate technique. 



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162 
 

7.2 The second (main) stage 

The analysis of equivalent orthotropic plates by using the B3-spline finite strip method under bending action in 

this study is formulated by MATLAB computer package where a programming code named RAOBDB3SFSM 

has been constructed. Formulation of the program consists of two stages :- 

A- Stage I   

This stage involves formulation of the bending stiffness matrix of a rectangular strip according to the following 

steps: 

1. Calculating  values of the coupling integrations  represented by  I1𝑖𝑖𝑖𝑖, I2𝑖𝑖𝑖𝑖,  I3𝑖𝑖𝑖𝑖, I4𝑖𝑖𝑖𝑖 and  I5𝑖𝑖𝑖𝑖 .  

2. Formulation of the submatrices: 

The dimensions of each (4x4) bending stiffness submatrix,  previously  determined and given in Table 3,will be 

according to the number of sections into which  the nodal line has been discretized. In the present work a fixed 

number of equally spaced sections (five equally spaced sections; m=5) has been adopted. Hence, the dimensions 

of each submatrix will be (m+3) × (m+3) = [ 8x8 ], therefore the dimensions of each rectangular strip  will be  

32x32 . 

3. Formulating the global stiffness matrix:  

The global stiffness matrix dimensions will be according to the number of strips to which the plate was divided.  

Primary four strips will be taken, so  dimensions of the global stiffness matrix will become 80 x 80 . 

B- Stage II  

This stage involves formulation of the load matrices by the following steps: 

1- According to the cases of load which are explained previously,  the integration of each local B3-spline 

function (hill) along the nodal line will be carried out, where each nodal line contains 8 hills;  𝜑𝜑−1,  𝜑𝜑0, 

. . . . . . , 𝜑𝜑6 .This function integration is also performed utilizing the facilities of  MATLAB program.  

2- The B3- spline is considered  to be amended with the  aid of table 2. 

3- Combination of all the integration values  common to each knot. 

4- Constructing the [10x80] load vector  {F}, for each of the four  strips. 

5- The unknown displacement parameters are carried out by solving the equation:  

                                      {F} = [K ]  . {𝛿𝛿} 

    using the Gauss elimination procedure. 

 



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163 
 

8. Application: Full scale R.C. slab-beam ridge Deck (seven ribs +four diaphragms) 

8.1 Description 

This case is represented by a simply supported reinforced concrete slab-beam bridge deck, with seven ribs and 

four diaphragms  as shown in Figure 10. The bridge is under a central 150 kN vertical concentrated load.  

Geometrical features and material properties are as follows: 

Span × Width                          =  25000×15000          mm 

Slab thickness           =  200                          mm 

28- day compressive strength; fc'            =  30                 N/mm2 

Modulus of elasticity           =  28                            kN/mm2 

Poisson's ratio                          =  0.15 

These values are in compliance with BS 5400 

 

 
 

Figure 10: Reinforced concrete slab-beam bridge deck with seven ribs and four diaphragm. 

 

8.2 Analysis methods 

Rigidities of the substitute  orthotropic plate have been computed as previously instructed in this paper, they are 

given in Table 3 below. 

 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No  4, pp 151-168 

 

164 
 

Table 3: Rigidities of the slab-beam bridge deck of seven ribs and four diaphragms. 

Flexural rigidities 
Dx  1079239551           kN.mm 

Dy   201570929            kN.mm 

Torsional rigidity Dxy   299573891            kN.mm 

Coupling rigidity 

 
D1    11287972             kN.mm 

 

According to the new technique explained  in the elapsing section,  the equivalent elastic   properties of the 

substitute orthotropic plate are: teq =478mm , 𝐸𝐸𝑥𝑥 = 74.96 kN.mm, 𝑣𝑣𝑥𝑥 = 0.4 

                    and considering that  D1= 𝑣𝑣 × ( the smallest value of 𝐷𝐷𝑥𝑥 or 𝐷𝐷𝑦𝑦  ). 

This ribbed bridge deck was previously  analyzed by Mehdi [11] using the grillage analogy. 

 

8.3 Analysis results 

Values of deflection computed by three  methods of analysis are given in Tables 4 and 5, then shown in figure11 

below. 

Table 4: Deflection values (in mm) of R.C ribbed bridge deck  

x-coordinate(mm) -12500 -7500 -2500 0 2500 7500 12500 

SFSM(present study) 

No. of strip=4 

No. of section=5 

0 1.85 2.96 3.10 2.95 1.85 0 

FEM (SAP2000) 

(15x25 meshes) 
0 1.73 2.95 3.15 2.95 1.73 

0 

 

Grillage analysis 

Mehdi [11] 
   3.21    

 

 



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165 
 

Table 5: Comparison of maximum moments Mx and My (in kN.mm/mm) of the R.C bridge deck.   

Method of analysis 
Mx 

(kN.mm/mm) 

My 

(kN.mm/mm) 

Present SFSM 41.17 21.25 

FEM (SAP2000) 

(15x25 meshes) 
47.37 20.36 

 

 

Figure 11: Variation of deflection along x-centerline of the R.C. ribbed bridge deck. 

8.4 Discussion of analysis results  

a) Agreement of deflection results 

i) Inspection of  Table 4 which gives deflection values by the three methods of analysis, high agreements 

between values obtained by the present SFSM (in its  least number of strips and sections) and those 

obtained by the other methods -especially the finite element method FEM implemented by SAP2000 

rather fine mesh- . 

ii)  The percentage differences of the maximum  deflection value, as shown in the bar chart of Figure 12  

is 1.6 % . 

iii) Observing Figure 11  which concerns spanwise variations of deflection along centerlines of the  bridge 

deck, quite close variation curves are produced which definitely indicate the high accuracy and 

efficiency of the present SFSM treatment of ribbed bridge decks by the equivalent orthotropic plate 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No  4, pp 151-168 

 

166 
 

replacement.  

b) Agreement of bending moment results 

i) Referring to Table 5 specialized in comparative bending moment values computed by the two method 

analysis (with utilizing the finite difference technique, obviously  good agreements between values 

produced  by the present SFSM and FEM  (SAP2000) (in spite of incorporation of the approximate 

finite difference technique) . 

 

Figure 12: Comparative chart showing values of : (a) The max. deflection, (b) The max. moment at centerline 

of bridge deck. 

9. Conclusions 

Based on the results of the present SFSM analysis the following conclusions have been drawn so far: 

1. The use of spline finite strip method (SFSM) in the analysis of  ribbed bridge decks by initially treating 

them as equivalent orthotropic plates  has just been  proved to be simpler, faster, more feasible and 

more economical than   the finite element method. Meanwhile, it is more accurate and  more versatile 

than the grid framework analogy. 

2. Proposing  a new technique for analysis orthotropic plates based on  converting the section of 

geometrical orthotropic plate to equivalent section of materially orthotropic plate using  simplified 

expressions for  elastic rigidities in that purpose has proved to be efficient and reliable 

3. In the analysis  of ribbed bridge decks more accuracy can be obtained when the equivalent thicknesses 

in the two orthogonal directions(i.e. tx and ty) of such plates are calculated twice, first by taking the 

41
.1

7 
47

.3
7 

21
.2

5 
20

.3
6 

0

10

20

30

40

50

60

kN
.m

/m
 

% of Variation = -13.1 

Present SFSM

FEM (SAP2000)

 1عمود

Present
SFSM(Mx)

Mx 

My 
% of Variation = + 4.4 

(b) (a) 

3.
1 

3.
15

 

3.
21

 

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

m
m

 

Present SFSM

FEM (SAP2000)

GFM (Mehdi)

    % of Variation = - 1.6 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No  4, pp 151-168 

 

167 
 

thickness  obtained from the equation (tx = �12𝐼𝐼𝑥𝑥
3   ), and second by taking the thickness obtained   

from the equation (ty = �12𝐼𝐼𝑦𝑦3  ) . Consequently, deflection values are obtained by averaging its  two 

different values calculated from two different plate thicknesses.  

4. Based on its results of  analysis of the full scale reinforced concrete slab-beam bridge deck has four 

transvers diaphragms, the present SFSM has also revealed extremely high degrees of accuracy for 

deflection, where the gained levels of coincidence of the present SFSM  with the finely meshed 

SAP2000 model is  98.4%. 

5. High levels of coincidence for the results of bending moment Mx and My outputs from the present 

SFSM and finely meshed SAP2000 model for the full scale reinforced concrete slab-beam bridge deck 

has four transvers diaphragms full scale RC slab-beam bridge deck, where 86.9% and 95.6%, 

respectively. 

10. Recommendation 

• The proposed "Orthotropic Spline Finite Strip" technique used in the present study for the analysis of 

ribbed bridge deck can be extended to study other types of  bridge deck such as  waffle slab decks , 

voided slab decks, cellular deck slab and box girder bridge decks 

• More investigation is needed on the least number of finite strips and /or sections per nodal line giving           

satisfactory accuracy with a minimum effort  of input data preparation and  execution time for analysis 

of  ribbed bridge decks by the present orthotropic SFSM 

• Investigation on the efficiency of the present orthotropic SFSM in analyzing  ribbed bridge decks of 

varying ribs depth.    

References    

[1] Cheung, Y.K. "The Finite Strip Method in the Analysis of Elastic Plates with Two Opposite Simply 

Supported Ends". Proc. Inst. Civ. Engr, Vol. 40, No.7, pp.1-7,1968. 

[2] Cheung, M.S., Li, W. and Chidiac, S.E. Finite Strip Analysis of Bridges Bridges. First Edition. 

London: E&FN Spon,1996. 

[3] Razzaq, Raja. javed. "Nonlinear Static and Dynamic Analysis of Composite, Layered plates and shells 

using Finite Strip methods". Ph. D. Thesis, Granfield university,2003. 

[4] Cheong, F.S. "Spline Finite Strip in Structural Analysis". Ph. D. Thesis, University of Hong-

Kong,1982. 

[5] Cheung, Y. K.  Finite Strip Method in Structural Analysis. First edition.  : pergamon  press,1976. 

[6] Eugene J. O'Brien and Damien L. Keogh. Bridge Deck Analysis . first edition. Ireland: E&FN 

Spoon,2005. 

[7] Emran, Amir Habeeb "Experimental and Analytical Study on   Orthotropic Plate Theory in Bridge 

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[8] Cheung, Y.K., and tham, L.G. Finite Strip Method. fourth edition.UK: CRC Press,1997. 

[9] Logan, Daryl. L. A First Course in the Finite Element Method. 4th Ed. University of Wisconsin–



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Platteville: Thomson,2010. 

[10]  Al-Hadithy, Laith "Analysis of Indeterminate Bridges by the Orthotropic Plate  Theory with 

Experimental Study". M. Sc. Thesis, University of Baghdad,1985. 

[11]  Mehdi, Wafa Sadiq "Elastic Analysis of Reinforced Concrete  Multiple T- Section  Bridge Decks by 

the Grillage Methods". M. sc. Thesis, University of Technology- Baghdad,1996. 

[12]  AL-Safarjalani, Modar Tawfik ."Analysis of Skew Plates by the  Spline Finite Strip Method". M. Sc.  

Thesis, University of Baghdad,1988. 

[13]  Al-Dawar, Mohammed.A.AL-Khaliq ."A Study on the Use of  Orthotropic Plate Theory in Bridge 

Deck Analysis". Ph. D. Thesis, University of Baghdad,1998. 

[14]  Hassan, Rafea Flaih ."Bridge Deck Analysis using Orthotropic Plate  Theory". M. Sc. Thesis, 

University of Technology- Baghdad,2005. 

[15]  A.Kadir, Benan Naji. "Analysis of Simply Supported Concrete Highway Bridges using the Orthotropic 

Plate Theory and Iraqi Specifications". M. sc.   Thesis, University of Baghdad,1979. 

[16]  Timoshenko, S.P. and Kreiger, S.W. Theory of Plate  and Shells. 2edEd. New Yourk: McGraw-Hill, 

1959. 

[17]  Cusens, A. R. and Pama, R.P. Bridge Deck Analysis. London: John Wile and Sons, 1975.