Ema405 - Finite Element Practicuum Project

EMA 405 Practicum in Finite Elements – Homework #2

Problem 1: Beam deflection due to Gravity

In this problem, the deflection of steel beam with a square cross-section due to its own weight is analyzed. An illustration of the problem is shown in Figure 1. The deflection and bending stresses can be calculated using basic mechanics of materials theory. The gravitational force is modeled as a distributed load acting on the beam is supported with a roller and a pin.

Figure 1. Simply supported beam exposed to a distributed load.

Length of the beam (L) Depth of the beam (b) Width of the beam (h)

Figure 2. Beam properties.

3 [m] 0.1 [m] 0.1 [m]

Young’s modulus (E) Density Moment of inertia (I)

200 [GPA] 7500 [kg/m3] 8.33 ∗ 107 [m4]

According to Roark’s Formulas for Stress and Strain, Page 189, Table 8.1, Ref. 2e, the maximum deflection ������������������������ at ������ =

������ 2

can be calculated by ������������������������ = −5 ������ ������4 384 ������������

with ������ = ������(������ ∙ ℎ) ∙ ������

yields to ������ = 735.75 ������/������ ������������������������ = 4.656 ∗ 10−4 ������

The maximum moment ������������������������ and bending stress, ������������������������ , at ������ = Ed., R.C. Hibbeler, Pg. 287, Equation 6-12) is calculated by ������������������������ = ������������������������ = with ������������������������ = 827.72 Nm ������ = ±0.05������ ������ ������2 8

������ 2

is (see Mechanics of Materials, 8th

−������������������������ ������ ������

������������������������ = ±4.966 ∗ 106 ������������ Modeling the problem in ANSYS, PLANE 182 elements were used. In order to achieve accurate results, an element count of more than 2000 is necessary. For the FE analysis, 200 elements along the beam and 10 elements across the thickness were defined yielding to an error of 0.5% when compared to the analytical solution. On the left hand side, the displacement in x and y were restricted to zero modeling a pin support....