This report presents the static analysis of a beam that is going to be used for lifting equipment in a building. A commercial software package, SolidWorks, is used to perform a set of FEA simulations. Numerical results are compared with theoretical values of maximum stress and deflection for a cantilever beam.
1. Part A
The first part of this report describes a cantilever beam with one end fully constrained and a point force of 5 kN applied at the opposite end. Three cross sections are considered, namely, a wide flange section (I-shape), a T section and a channel cross sections. The beam is made of Mild Steel with a yield strength of 248 MPa. The modulus of elasticity (E) is 200 GPa and the density of steel is 7800 kg/m3. A safety factor of 2.0 is considered, and the maximum allowable deflection is 40 mm. Based on the safety factor, the maximum allowable stress in the beam is,
In order to compute the stress analytically, we need to determine the section properties. The moment of inertia of each cross section is computed about the neutral axis. The parallel axis theorem is used,
I = ΣI+Ad2
Values of I for each sectional area, distance from the neutral axis and the weight of the beam are calculated and summarized in the following table,
The beam's free body diagram, shear force and bending moment diagram are shown in Figure 1.
Figure 1. Free body, shear force and bending moment diagrams. W, F and R represent the weight, point force and resultant force, respectively. The total moment is also shown.
The flexural formula is used to calculate the maximum bending stress (Patnaik and Hopkins, 2004)
σmax=McI (2)
where M is the (maximum) bending moment at the fixed end.
The method of superposition is used to compute the cantilever's beam deflection, i.e. the deflection due to the point force at the tip of the beam is added to the deflection caused by the beam's weight acting through the centre of mass (at x = 2 m).
ymax = y5kN+yW=FL33EI+5WL348EI (3)
The aim of the FEA simulation is to estimate the maximum stress and deflection of the cantilever beam and evaluate three different cross sections. A three-dimensional model is created in SolidWorks 2012 (“SolidWorks Tutorials – Videos, guides, lessons and project files I SolidWorks,” n.d.). The Simulation suite is used to generate a mesh, apply constrains and loads, and calculate the results. Contours of normal stress (x-direction) are shown in Figures 2 to 4. A deflected beam is also illustrated. Theoretical values of the stress and deflection are compared with numerical results in Table 2.
Figure 2. Normal stress distribution in the case of an I-section
Figure 3. Normal stress distribution in the case of a T-section
Figure 4. Normal stress distribution in the case of a channel-section
The maximum stress calculated by SolidWorks is somewhat higher. However, the predicted deflection for each cross section is in excellent agreement with the theoretical value.
2. Part B
Equation (3) is modified so as to calculate the theoretical deflection of the beam when a uniformly distributed load of 10 kN/m is applied,
ymax = y10kN+yW=Fa26EI3L-a+5WL348EI (4)
where a = 3.5 m is the position of an equivalent point force of 10kN. The free body diagram, shear force and bending moment diagram are shown in Figure 5.
The FEA model was modified in order to account for the uniformly distributed load. Contours of normal stress on a deflected beam are shown in Figures 6 to 8. Theoretical and numerical values of maximum stress and deflection are summarized in Table 3.
Figure 5. Free body, shear force and bending moment diagrams. The uniformly distributed load is w = 10 kN/m.
Figure 6. Normal stress distribution in the case of an I-section
Figure 7. Normal stress distribution in the case of a T-section
Figure 8. Normal stress distribution in the case of a channel-section
The discrepancies may be explained by the stretching an compression of the material which changes the cross-sectional area of the beam. This slight variation is not taken into account by the theoretical formulas but it is correctly captured by the FEA simulations
Based on the normal stress and deflection results shown in Table 2 and 3. The I-shape (wide-flange section) is the only geometry that is able to withstand the point and distributed loads. Although the maximum normal stress (numerical result, part B) is greater than the allowable stress computed in (1), it is significantly lower than the yield stress. A maximum deflection of 40 mm is slightly exceeded when a uniformly distributed load is applied. This condition is marginally fulfilled by the I-shape beam. The I-shape should be selected as it holds acceptable levels of stress and deflections.
Reference List
Patnaik, S., Hopkins, D., 2004. Strength of Materials: A New Unified Theory for the 21st Century. Elsevier Science.
SolidWorks Tutorials – Videos, guides, lessons and project files I SolidWorks [WWW Document], n.d. URL http://www.solidworks.com/sw/resources/solidworks-tutorials.htm (accessed 3.7.13).
