I(y)= …………………………….(1)
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L[y(x)]=
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S[z(x,y)] = dxdy
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S(q) = Where, q is the function to be found: q:[a,b] R t = q(t) such that q is differentiable, q(a)=
The Euler–Lagrange equation, then, is given by Here and denote the partial derivatives of L with respect to the second and third arguments, respectively.
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& which extremixes the functional. We assume that F is twice continuously differentiable A weaker assumption can be used but the proof becomes more difficult. Let i.e the result of such pertulation of f, where is small & n(x) is a differentiable function satisfying Then define, We now wish to calculate the total derivative of with respect to It follows from the total derivation that.
So, Where we have & has an extremum value so that Using the boundary conditions Applying the fundamental lemma of calculus of variations now yields the Euler- Lagrange eq.
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δWi= δWe Internal virtual work External virtual work
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Our ODF is sepretable , thus we, seprate Now, integrable both sides we get, Thus , So, Applying y© =1 we obtained Finally, |
We are seprating our differential eq. Applying y(x)=4 4=0-0+C C=4 Put this value in eq. 1
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We are seperating our differential equation Integrating both sides Apply boundary value conditions Y(0)=3 Take exponential function both sides C=3 Put in eq. 1
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Element | K(N/m) | Nodes | Displacements m | boundary conditions |
1 | 500 | 1-2 | ||
2 | 100 | 2-3 | ----- |
Step -2 Element stiffness matrices. Step -3 global stiffness matrix Assemble the element stiffness matrices to get global stiffness matrix Step -4 reduced stiffness matrix imposing boundary conditions i.e eliminate first raw and first column. Therefore, reduced stiffness matrix is. Step-5 determine unknown joint displacement Applying equation of equilibrium Step -6 calculation of spring force. Spring -1 Spring 2 –
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Element | K(N/m) | Nodes | Displacement | Boundary conditions | |
1 | 1000 | 1-2 | |||
2 | 2000 | 2-3 | - | ||
3 | 3000 | 3-4 | |||
Step -2 element stiffness matrices Step -3 global stiffness matrix Assemble the element stiffness matrices to get the global stiffness matrix Step -4 reduced stiffness matrix Imposing boundary conditions Eliminate first raw, first column & fourth raw & fourth column . Therefore reduced stiffness matrix is Step -5 determine unknown joint displacement Applying equation of equilibrium Step -6 calculate of spring force Spring 1 -
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