Week 10 Challenge- Frame analysis and MATLAB Module for Gauss Elimination Method

1) Assume a structure frame. Write down a displacement matrix with variables corresponding to pre-selected degrees of freedom and degrees of freedom to be eliminated. Mathematically, derive the process that takes the original set of equations and condense it to the form shown below

Ans :- As original equation is [k][delta]=[P]

now using condensation process we gonna used specified degree of freedom [delta(b)] and degree of freedom to be eliminated [delta(c)].

so delta = [delta(b) delta(c)] ,

[K] = [Kbb Kbc; Kcb Kcc] and

[P] = [Pb Pc]

therefore original equation becomes

[Kbb Kbc; Kcb Kcc] [delta(b) delta(c)] = [Pb Pc]

solving above we get equatoins,

Kbb*delta(b) + Kbc*delta(c) = Pb ....(1)

Kcb*delta(b) + Kcc*delta(c) = Pc ....(2)

reducing equation (1) into value of delta(b) we get,

delta(b) = [(Kbb^-1)*Pb] - [(Kbb^-1)*Kbc*delta(c)]

putting delta(b) value in equation (2),

K(condensed) = [Kcc - Kcb*Kb^-1*Kbc]

so we get condesnsed equation as

2) Eliminate the rotational degree of freedom from the system of structure shown below and express stiffness matrix in terms of the vertical displacement variable

Ans :-

Ke = [ AE/L 0 -AE/L 0; 0 3EI/L^2 0 -3EI/L^2; -AE/L 0 AE/L 0; 0 -3EI/L^2 0 3EI/L^2 ]

3) Explain the concept of sub structuring with the example of a portal frame given below

Ans :- The substructuring methods are efficient in dealing with large-scale structures, as it takes the local area as an independent structure . First the global structure is divided rationally into several smaller substructures to make it much easier and faster to analyze the small substructures independently. Second the FE model of a substructure has much fewer uncertain parameters than the global structure, which helps to accelerate the convergence of optimization process to identify these parameters and alleviates the problems. Third the substructuring method is required to measure the local area of the practical structure and save the experimental instruments. Finally, the substructuring method can be more promising if combined with parallel computation.

We divide the structure into two parts with internal nodes and interface objects (common to both the structure). By this we can reduce degree of freedoms and make structure easy to understand by dividing big structures into individual structure.

4) Explain the type of flexible connections applied in the structural framing systems

Ans :-
● Unrestrained

● Partially Restrained

● Fully Restrained

5) Read online about Gauss Elimination Method ,Develop a matlab module to solve linear
equations by the method of Gauss Elimination

Ans :- The Gaussian elimination method is known as the row reduction algorithm for solving linear equations systems. It consists of a sequence of operations performed on the corresponding matrix of coefficients.

Lets take example for solving linear equations-

C = [1 2 -1; 2 1 -2; -3 1 1]
b= [3 3 -6]'
A = [C b];                                                  %Augmented Matrix
n= size(A,1);                                               %number of eqns/variables
x = zeros(n,1);                                             %variable matrix [x1 x2 ... xn] coulmn
for i=1:n-1
    for j=i+1:n
        m = A(j,i)/A(i,i)
        A(j,:) = A(j,:) - m*A(i,:)
    end
end
x(n) = A(n,n+1)/A(n,n)
for i=n-1:-1:1
    summ = 0
    for j=i+1:n
        summ = summ + A(i,j)*x(j,:)
        x(i,:) = (A(i,n+1) - summ)/A(i,i)
    end
end