Chapter 2, Problem 1E

Answer the following descriptive questions.

a. Explain the difference between the essential and natural boundary conditions.

b. The beam-bending problem is governed by a fourth-order differential equation. Explain what are the essential and natural boundary conditions for the beam-bending problem.

c. What is the basic requirement of the trial functions in the Galerkin method?

d. What is the advantage of the Galerkin method compared to other weighted residual methods?

e. A tip load is applied to a cantilevered beam. When $w\left(x\right)=c_1{x}^2+c_2{x}^3$ is used as a trial function, will the Galerkin method yield the accurate solution or an approximate one?

f. A uniformly distributed load is applied to a cantilevered beam. When $w\left(x\right)=c_1{x}^2+c_2{x}^3$ is used as a trial function, will the Galerkin method yield the accurate solution or an approximate one?

g. For a simply supported beam, can $w\left(x\right)=c_0+c_{1}x+c_2{x}^2+c_3{x}^3$ be a trial function or not? Explain why.

h. List at least three advantages of the finite element method compared to the weighted residual method.

i. Explain why the displacement is continuous across element boundary, but stress is not.

j. If a higher-order element, such as a quadratic element, is used, will stress be continuous across element boundary?

k. A cantilevered beam problem is solved using the Rayleigh-Ritz method with assumed deflection function $w\left(x\right)=c_0+c_{1}x+c_2{x}^2+c_3{x}^3$. What are the values of $c_0$ and $c_1$? Explain why.

l. For the Rayleigh-Ritz method, you can assume the form of solution, such as a combination of polynomials. What is the condition that this form needs to satisfy?

To determine

To find:Difference between the essential and natural condition.

Explanation of Solution

Essential boundary conditions:

The essential boundary conditions are the boundary conditions that are sufficient for solving the differential equation completely.Essential or geometric boundary conditions are imposed on the primary variable like displacement.The geometry boundary condition is displacement, slope.

Natural Boundary Conditions:

The natural boundary conditions having boundary conditions linking higher-order derivative terms and are not enough for differential equation solving completely, needful tominimum one essential boundary condition.Natural or force boundary conditions having compulsory on the secondary variables e.g.forces traction bending moment and shear force.

To determine

To find:fourth-order bending problems and explain the essential and natural condition for this problem.

Explanation of Solution

Given information:

Here, disscuss the fourth-order bending problems and explain the essential and natural condition By considering the 4^th Order ODE and the Elastic Beamsare the linear ODE that has significant applications in materials sciences is that for the deflection of a beam. The beam deflection y(x) is a linear fourth-order ODE can be shown like below DE equation.

  $\frac{d^2}{d{x}^2}\left(EI\frac{d^{2}y(x)}{d{x}^2}\right)=w(x)$

Here IfM.I. (moment of inertia) and the E (Young’s modulus) do not be contingent on the beam (homogenous material), then the beam equation shown below:

  $\left(EI\frac{d^{4}y(x)}{d{x}^4}\right)=w(x)$

The homogenous solution can be obtained by inspection—it is a general cubic equation $y_{\mathrm{hom}og}(x)=C_0+C_{1}x+C_2{x}^2+C_3{x}^3$ which has the four constants that are expected from-order ODE. The P.I solution is found by integrate with w(x) four time and constants of integration then the P.I solution naturally the homogeneous solution.

Free − Considering mainly No applied moments or applied shearing force:

Point Loaded - Considering mainly Local applied moment, displacement specified.

Clamped- Considering mainly Displacement specified; slope specified

To determine

To find:Basic requirements of the trail function in the Galerkin method.

Explanation of Solution

The space comprising the trail function is identified as the trial space. The function v ingoing the orthogonality condition in the Galerkin method and the technique of weighted residuals is named test function.The Ψ_i or w_ifunctions used as weighted in internal products with the residual.

To determine

To find:advantage of the galerkin method compared to other weighted residual methods.

Explanation of Solution

Galekin’s Method considering the Weighted residual methods by taking the Idea of the Project residual of D.E.

Galekin’s Method considering the Weighted residual methods by taking the Idea of theProject residual of D.E.

The least square and Galerkin methods are more widely used than collection and subdomain, that actually global methods.For least square method the equation solve for always symmetric but tend to be ill-conditioned and Estimated solution wishes very smooth.ForGalerkin’s method the equations solve for typically symmetric but additional robust.Integration by parts produces less smooth version of estimated solution are more useful for FEA method .

To determine

To find:The solution of the cantilever beam DE with using Galerkin method the accurate solution or using approximation solution.

Explanation of Solution

Galekin’s Method considering the Weighted residual methods by taking the Idea of the Project residual of D.E. onto original resembling functions shown below.

  $W_k(x)=\frac{\partial u_{approx}(x)}{\partial a_k}=N_k(x),k=1,2,\mathrm{...},n.$

Here, to become W’, necessity integrate derivatives in volume by partsintegral done shown below.

Let $\begin{array}{l}{R}_E(X,a)=R_{E,deriv}(X,a)+R_{E,noderiv}(X,a);\\ R_E(X,a)={\displaystyle \int }{R}_{E,deriv}(X,a)dx\end{array}$ (Assume derivative involves x.)

To determine

To find:the solution of the uniformly distribution load over cantilever beam DE with using Galerkin method the accurate solution or using approximation solution.

Explanation of Solution

Galekin’s Method considering the Weighted residual methods by taking the Idea of the Project residual of D.E. Galekin’s Method considering the Weighted residual methods by taking the Idea of the Project residual of D.E. onto original resembling functions and , to become W’, necessity integrate derivatives in volume by parts shown below.

  $W_k(x)=\frac{\partial u_{approx}(x)}{\partial a_k}=N_k(x),k=1,2,\mathrm{...},n.$

Here, to become W’, necessity integrate derivatives in volume by parts integral done shown below.

Let $\begin{array}{l}{R}_E(X,a)=R_{E,deriv}(X,a)+R_{E,noderiv}(X,a);\\ R_E(X,a)={\displaystyle \int }{R}_{E,deriv}(X,a)dx\end{array}$ (Assume derivative involves x.)

To determine

ToFind :The solution of the simply supported beam DE with using Galerkin method the accurate solution or using approximation solution.

Explanation of Solution

  $\begin{array}{l}\text{Two term approximation}\\ \stackrel{-}{u}(x)=(1-x)+c_1{\varphi }_1(x)+c_2{\varphi }_2(x)\\ =(1-x)+c_1{x}_1(1-x)+c_2{x}_2{}^2(1-x)\end{array}$

The approximate solution is split into two parts. In the Galerkin’s method the trial function is considered as the weighting function. The first satisfied the given essential boundary condition exactly.

Here, the trial function is,

  $\begin{array}{l}\text{using boundary value condition to the problem}\\ c_1{x}_1+c_2{x}_2{}^2=0\\ c_1{x}_1(0)+c_2{x}_2{}^2(0)=0\end{array}$

To determine

To find:The solution of the simply supported beam DE with using Galerkin method the accurate solution or using approximation solution.

Explanation of Solution

Given information:

Galekin’s Method considering the Weighted residual methods by taking the Idea of the Project residual of D.E.

Explantion:

Galekin’s Method considering the Weighted residual methods by taking the Idea of the Project residual of D.E.The least square and Galerkin methods are more widely used than collection and subdomain, that actually global methods.For least square method the equation solve for always symmetric but tend to be ill-conditioned and Estimated solution wishes very smooth.ForGalerkin’s method the equations solve for typically symmetric but additional robust.Integration by parts produces less smooth version of estimated solution are more useful for FEA method .

To determine

To find:Advantage of the galerkin method compared to other weighted residual methods.

Explanation of Solution

Given information:

Galekin’s Method considering the Weighted residual methods by taking the Idea of the Project residual of D.E.

Explantion:

The least square and Galerkin methods are more widely used than collection and subdomain, that actually global methods.For least square method the equation solve for always symmetric but tend to be ill-conditioned and Estimated solution wishes very smooth.ForGalerkin’s method the equations solve for typically symmetric but additional robust.Integration by parts produces less smooth version of estimated solution are more useful for FEA method .

To determine

To find: the solution of the uniformly distribution load over cantilever beam DE with using Galerkin method the accurate solution or using approximation solution.

Explanation of Solution

Given information:

Galekin’s Method considering the Weighted residual methods by taking the Idea of the Project residual of D.E.

The least square and Galerkin methods are more widely used than collection and subdomain, that actually global methods.For least square method the equation solve for always symmetric but tend to be ill-conditioned and Estimated solution wishes very smooth.ForGalerkin’s method the equations solve for typically symmetric but additional robust.Integration by parts produces less smooth version of estimated solution are more useful for FEA method .

To determine

To find:The solution of the uniformly distribution load over cantilever beam DE with using Galerkin method the accurate solution or using approximation solution.

Explanation of Solution

Given information:

The condition of the Rayleigh Ritz method for the given solution and try to satisfied the condition of Rayleigh Ritz method with assumed deflection function.

In the Galerkin’s method the trial function is considered as the weighting function and can be done approximate solution is split into two parts. The first satisfied the given essential boundary condition exactly. Here, the trial function is, Rayleigh Ritz method with assumed deflection function $\begin{array}{l}\text{using boundary value condition to the problem}\\ c_1{x}_1+c_2{x}_2{}^2=0\\ c_1{x}_1(0)+c_2{x}_2{}^2(0)=0\end{array}$

To determine

To find:The condition of the Rayleigh Ritz method for the given solution and try to satisfied the condition.

Explanation of Solution

The condition of the Rayleigh Ritz method for the given solution and try to satisfied the condition of Rayleigh Ritz method with assumed deflection function.

Rayleigh Ritz method with assumed deflection function It is integral approach method which is useful for solving complex structural problem, encountered in finite element analysis. This method is possible only if a suitable function is available.