The following differential equation is available for a physical phenomenon \(\dfrac{d^{2}u}{dx^{2}}+u+x=0\)

,\(0â‰¤ x â‰¤1\), \(u(0)=u(1)=0\). Solve above equation,using subdomain method and Galerkin method.

Derive the cubic shape function of Largrange's family.what are the characteristic of the shape function? Plot the shape function along of the element. State the difference shape function and interpolation function?

Explain the following:

1.Global Local and Natural Co-ordinate system

2.Boundary condition and its types.

^{11}N/m

^{2}and \(\rho=7800kg/m^3\). Estimate the natural frequencies of axial vibration of the bar using both consistent and lumped mass matrices. Use a two element mesh. If the exact solition is given by the relation \(\omega_{i} =\frac{i\pi}{2L}\sqrt{\frac{E}{\rho}} \) where \( i=1,2,3,4,5...\)

Area of each member =1000mm

^{2}

E for each member=210 Gpa

P

_{1}=10 KN

P

_{2}=20kN.

Evaluate the following integral \(I=\int_{0}^{4} x^{3}dx\\\)

Using (a) trapezoidal rule

(b) Simpsons one third rule

Compare the solution with the exact solution.

The governing differential equation for a rod loaded axial force is

\(\frac{d}{dx}(AE\frac{du}{dx})+q =0\), Where E is Young's modulus of elasticity, A is the area of cross section. q is the load intensity and u is the axial displacement. Obtain the variational form for this equation

Assume that the boundary condition are

At \(x=0,u =0\)

at , \(x=l, EA \dfrac{du}{dx}=P\)

where \(l\) is the length of the rod.

==IMAGE===

All length and diameter are in mm.

Generalised Jacobi method