03-03-2011, 09:59 AM
PRESENTED BY
BRIJESH KUMAR SINGH
Galerkin’s method for structure analysis.pptx (Size: 162.37 KB / Downloads: 51)
STRUCTURE ANALYSIS
Structural analysis comprises the set of physical laws and mathematics required to study and predict the behavior of structures .
In struture analysis , we find deformation , internal forces and stresses in any structure.
Analysis Methods
Analytical
Numerical
GALERKIN’S METHOD
Galerkin’s method is a method for converting a continious operator problem (diffrential equation ) to a discrete problem . It is used for numerical analysis of various structure in different industries.
• Galerkin’s method is also known as Finite element method.
GALERKIN’S METHOD STEPS
STEP 1 : Select a basis function .
Ū= ∑QiGi ( where U = a0+a1X+a2X2 )
STEP 2: Get weight field .
W= ∑φiGi
(W should satisfy boundary condition of U)
GALERKIN’S METHOD STEPS
STEP 3: Find Qis such that
∫v Wedv = 0
(where ‘e’ is error term e = L(Ū) – P )
∫v W(L(Ū) – P )dv = 0
GALERKIN’S METHOD
Let U be the deformation in the
rod due to point load .
Є=∂U/∂X
σ =EЄ
Governing equation
σ=E ∂U/∂X
For one dimension case
∂σx = 0
d(EdU/dX)/dx =0
GALERKIN’S METHOD
Weight field
∫WedV =0
∫W(d(EdU/dX)/dX )dV=0
By using divergence Theorem ,we integrate above
equation .
2d1L2 (P – EAC1L/3)=0
Finding the value of C1
We find deformation
[U=3P(X(2L-X))/4EAL ]
GALERKIN’S METHOD
Analysis of vibrating shaft
A shaft with constant cross section is clamped at one end and free on other side with density ρ .It has angle of dislocation φ(x,t). Shaft is exited by hamonic motion M(t)
The equation of motion of shaft
Cφ ∂2φ/∂X2 = J∂2φ/∂t2
Cφ = GI/L (Torsional rigidity)
Analysis of vibrating shaft
Boundry condition for shaft
φ = 0 when x=0
Cφ∂φ/∂x=M0 cosωt when x=L
Now applying the Galerkin’s method we obtain a final solution
GALERKIN’S METHOD
Galerkin’s method is also used in meshless method in which continous change in structure takes place like dynamic fracture .
In this method discretization is based on set of nodes . This method is known as ELEMENT-FREE GALERKIN’S METHOD .
• This method is used for solving a problem of thin plate bending .
• EFG method is also used for analysis of shell like cylidrical shape , hemispherical shape .
APPLICATION OF GALERKIN’S METHOD
This method is used in aeronautical engineering in lift-line equation for obtaining approximate solution.
This method is used in automobile industry for analysis of various part of vehicle (like piston ring) and it is also used for crash simulation.
This method is also used in manufacturing industry for performance enhancement of a high speed chuck.