Week - 2 - Explicit and Implicit Analysis

Most of the structural analysis are mainly depend upon the time and displacement relativity i..e,

When a load is applied we need to determine the displacement. To find out this variation we need to differentiate the time and force separately .

For every case there is two types of time and displacements are considered like in some cases like izod charpy test the load will fall on object suddenly like in milliseconds and the it comes into halt

And in some of the cases like metal forming and metal pressing the load will applied gradually

On the metal so these cases are to be considered as linear and non-linear case .  linear case where load is applied gradually and in nonlinear load is applied in milli seconds

To solve above two variations in FEM analysis is used .

In FEM analysis the above two variations are to be known as implicit and explicit analysis methods

Explicit analysis

As per the discussed in the class the explicit analysis or any other has to be solved in the form of differential equation as follows

                                               [K(x)]{x} = {f}

In the same way there is also a another case called dynamic equilibrium , some cases falls under these cases like automotive crash, blastic and meteor impact et..,

                                          Sum of all forces= force * acceleration

Explicit analysis is used to solve the problems where time is not independent to applied force and also used to solve dynamic solutions like in automotive crash stimulation .

Most of the structural element problems are the time independent problems in verey less cases we come across the linear problems where time is dependent . the time dependent problems are also known as implicit analysis.

IMPLICIT ANALYSIS:

 In implicit analysis there is a special case where we consider the force to be equilibrium. For a instance of time which is also called as quasi-static process . in the quais static-process we consider load and displacement as linear this is a iseal process so to prove this process we use some special itration methods like Raphson-iteration method and there will be some of the tolerance valve for every process where we use the tolerance valve to solve the equation by Raphson-iteration method .

Implicit analysis use the equation as below

                                                 [K]{x} = {f}

Above equation is linear equation used in implicit equation .

AIM :- Consider the case of a simple bar in tension .Suppose that the force in the bar is a nonlinear function of the displacement

Solution :-

EXPLICIT ANALYSIS

           Given equation = u³+9u²+4u  ( del = delta )

1. U₀=0.0 ;   k(u₀)=1  ; del F= 1 ;  de lU₁ = (U₀) = 1

• U₁= U₀+U₁ = 0+1 = 1

K(u₁) = 1+1²   delF= 1 ;   u₂ =  del F/K(U₁) = ½ = 0.5

       

• U₂ = U₁ + del U₂ = 1+ 0.5 = 1.5                                 

             K(u₂) = 1+(1.5)² =3.25   = 1 ;   u₃= (U₁) = 1/3.25

• U3 = u2 + del  U₃ = 1.5+1/1.35 =1.81

               Ku₃= 1+(1.81)² = 4.27  ;   = 1  u₃(U₃) = 1/4.27

• U₄= u₃ + del u₃ = 1.8 + 1\4.27 = 2.01

     F _ext = del F+ del F+ delF+ del F= 4

     F_int   = u³ + 9u² + 4u   = ( 2.01)²+ 9 (2.01) ² + 4(2.01)  = 52.5

Step (i)	del Fi	del ui	ui	Fext	F int	Fint-Fext
1	1.0	1	1	1.0	1.333	0.333
2	1	0.5	1.5	2.0	2.625	0.625
3	1	0.30	1.81	3.0	3.790	0.790
4	1.0	0.23	2.01	4.0	4.716	0.716

                                     Graph plot for explicit analysis 

Concusion : We cannot diffrentiate above the both analysis for the better one . for the time based analysis we use explict because it is simple and can be used as easy method to solve , but in case of implicit we have to do more itrations because every moment it will give result for the instant time and the T+1 time also . It is time indpendet method .

so from the both explicit is the good solving method for various dynamic problems .