This document discusses the finite element analysis of trusses. It explains that truss elements can be determinate or indeterminate, and that joint displacements are unknown variables. It presents the formulation of the stiffness matrix for a truss element, including the nodal displacement vector and elemental stiffness matrix. It also discusses the transformation matrix used to relate local and global coordinate systems for truss elements, and how to assemble the global stiffness matrix for a truss from the elemental stiffness matrices.

1. FINITE ELEMENT ANALYSIS
OF TRUSS
BY-
Ms. JAPE ANUJA S.
ASSISTANT PROFESSOR,
CIVIL ENGINEERING DEPARTMENT,
SRES, SANJIVANI COLLEGE OF ENGINEERING,
KOPARGAON-423603.
MAID ID: anujajape@gmail.com
japeanujacivil@sanjivani.org.in

2. FINITE ELEMENT ANALYSIS OF TRUSS
• Truss may be determinate or indeterminate truss
• Joint displacements are Unknown variables.
• Formulation of stiffness matrix of truss:
• The nodal displacement vector for the bar element is
• The stiffness matrix of a bar element is
1 2
AE, L
 
'
' 1
'
2
e
u
x
u
 
  
 
'
' 1
'
2
1 1
1 1
uAE
K
L u
 
       
' '
1 2u u

3. Transformation matrix for the truss:
• x' y'= Local coordinate systems
• x, y = global coordinate system
u1' , u2' = Displacements in local
coordinate system
• u1, u2, u3, u4= Displacements in
global coordinate system
• Ɵ=Angle measured in
anticlockwise sense w.r.t. positive
x-axis
Since axial directions of all members of truss are not same, hence in global coordinate
system (x-y) there are two displacement components at every node. Hence the nodal
displacement vector for typical truss element is
 
1
1
2
2
e
u
v
x
u
v
 
 
 
  
 
  

4. At Node1 At Node 2
Therefore, in matrix form above relation are
Where =vector of local unknowns
=vector of global unknowns
=Transformation matrix
where
'
1 1 1cos sinu u v   '
2 2 2cos sinu u v  
    
1
'
11
'
22
2
'
cos sin 0 0
0 0 cos sin
ee
u
vu
uu
v
x L x
 
 
 
      
    
    
  

 '
e
x
 e
x
 L
 
0 0
0 0
l m
L
l m
 
  
 
2 1
2 1
cos
sin
x x
l
L
y y
m
L



 

 

5. Stiffness matrix of truss element in global coordinate system:
     
 
 
'
0
0 1 1 0 0
0 1 1 0 0
0
0
0
0
0
T
K L K L
l
m l mAE
K
l l mL
m
l
m l m l mAE
K
l l m l mL
m
   
 
               
 
 
 
           
 
 
 
2 2
2 2
2 2
2 2
l lm l lm
lm m lm mAE
K
L l lm l lm
lm m lm m
  
 
  
  
 
  

14. THANK YOU