The document discusses the natural coordinates of a 1D bar element in finite element analysis. It defines the natural coordinate ξ as the ratio of the distance from any point P to the origin, over the maximum distance from the origin to node 2. This yields expressions for ξ at each node. At node 1, ξ = 0, and at node 2, ξ = 1. Therefore, the natural coordinates of a 1D bar element are ξ, which ranges from 0 at node 1 to 1 at node 2.

1. Dr. A. S. Sayyad
Professor & Head
Department of Structural Engineering
Sanjivani College of Engineering, Kopargaon 423603.
(An Autonomous Institute, Affiliated to Savitribai Phule Pune University, Pune)
Finite Element Method In Civil Engineering
Natural coordinates of 1D bar element
( coordinate system)
,
 
1
2
2
2
L x
x
L

  
 
  
 
 
 

2. Let us consider a two nodded bar/line element (1D). Node 1 and 2 have Cartesian
coordinates 1 x and 2 x respectively. Cartesian coordinate of any point ‘P’ is ‘ x ’.
Natural coordinate of any point ‘P’ are ( ).
Natural coordinates of 1D bar element ( -coordinate system)


 = It is ratio of distance of any point P from origin (OP) to its maximum
distance from origin (O2)

3. 1 2 2 1 1
1
/ 2
2
2 2
2 2
2
2
2
OP
L
x x x x x
x x
L L
L x
x
L

 

 
      
   
     
   
   
   
   
  
 
   
 
 
 
At Node 1, x = x1
1 1 1
1
2 2 2 2 2
2 2
L x x L x
x
L L
 
    
   
    
 
   
   
 
1

  

4. At Node 1, x = x2
1 2 1
2
2 2 2 2 2
2 2
2 2
2
L x x L x
x
L L
L L
L
 

    
   
     
 
   
   
 

 
   
 
1

 
Finally Natural Coordinates of 1D bar element in coordinate
system are:
,
 