Let us assume a cantilever beam with one end fixed and the other end has 1 unit load acting vertically downward. The length and height of the beam are 5 and 1 unit respectively. The Young's modulus is 10,000 unit. At first, we will solve this problem analytically using the famous Euler-Bernoulli beam equation to find the deflection at the end of the beam. Next, we will use Abaqus with 4 node quadrilateral reduced integration element. Then, we will solve the same problem using 8 node quadrilateral reduced integration element. Our goal is to see which element performs better in terms of convergence and accuracy for the finite element analysis (FEA) compared to the analytical solution.

The following Mathematica program solves the Euler-Bernoulli equation for the deflection at the tip.

**(*Exact Solution of Euler-Bernoulli Cantilever Beam*)**

**Clear[y, P, x, X2, EI, L];**

**y[x];**

**(*Given Property Values*)**

**L = 5;**

**h = 1;**

**P = 1;**

**Ie = 1/12;**

**Ee = 10000;**

**EI = Ee*Ie;**

**theta = D[y[x], x];**

**V = EI*D[y[x], {x, 3}];**

**M = EI*D[y[x], {x, 2}];**

**y1 = y[x] /. x -> 0; y2 = y[x] /. x -> L; theta1 =**

**theta /. x -> 0; theta2 = theta /. x -> L; M1 = M /. x -> 0; M2 =**

**M /. x -> L;**

**V1 = V /. x -> 0; V2 = V /. x -> L;**

**(*Solving the Equation*)**

**s = DSolve[{EI*y''''[x] == 0, y1 == 0, theta1 == 0, V2 == P, M2 == 0},**

**y, x];**

**Print["Analytical Solution of the Exact Displacement"]**

**y = Simplify[y[x] /. s[[1]]]**

**Plot[y, {x, 0, 5}, PlotStyle -> {Blue, Thick}]**

**Print["Displacement Variation at Point A in both X and Y Direction"]**

**DA = y /. x -> 5;**

**theta = D[y, x] /. x -> L;**

**a1 = N[L - (-h/2)*Sin[theta]];**

**a2 = N[DA + (-h/2)*Cos[theta]];**

**A1 = a1 - 5**

**A2 = a2 + h/2**

**Print["The Stress at Point B, S11"]**

**S11 = -Ee*X2*D[y, {x, 2}];**

**StressatB = S11 /. {x -> 0, X2 -> 0.5}**

Now, we begin to simulate the same problem with two types of elements: 4 and 8 node quadrilateral reduced integration. As in any FEA simulation, we are required to define the meshing parameters and elements prior to the analysis, and these two elements are frequently used in Abaqus.

The above results highlight that there is a trend of increasing convergence of the results starting from the fourth elements. We see that the stresses have minute changes in its absolute values, which are also visible for the displacements. With the increase of the element numbers, the faster the different plots to converge. It is evident that the elements are not converging to the same solutions always. Moreover, for this problem, the rate of convergence and accuracy is better with the 8 node quadrilateral reduced integration element; however, with the increase in number of elements, both 4 and 8 node quadrilateral reduced integration elements have similar performance.

If you would like to gather more knowledge on FEA, I would encourage you to visit this site as this tutorial is based on the discussions and examples of that.

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