Deflection and Bending Stress in a Cantilever Beam

Introduction
This EngineeringPaper.xyz sheet shows how to calculated the maximum stress and displacement for a cantilever beam loaded by a vertical force at its end. The geometry of the beam is shown in the figure below. The beam has a rectangular cross section with height h and width b. The maximum stress in a cantilever with length much larger than its height is due to bending stress that occurs at the base of the beam. For a downward force, the maximum stress will be a tensile force at the top of the beam and a compressive stress at the bottom of the beam. For beam sections that are symmetric top to bottom, the maximum tensile and compressive stresses will have equal magnitude.

Select Beam Material
Steel

Aluminum
Select Beam Section
Filled Circle

Hollow Circle

Ellipse

Rectangle

Hollow Rectangle
Source for beam section equations and images: https://en.wikipedia.org/wiki/List_of_second_moments_of_area
Define the Other Beam Parameters
Define the Deflection and Stress Equations
The maximum displacement of a cantilever beam loaded at its end is defined by [1]:
where Ix is the area moment of inertia of the cross section about the x-axis as selected in the table above. The maximum displacement can now be queried:
=-0.165297794307557 mm
The maximum bending stress in a beam section is given by:
where M is the bending moment at the section of interest and c is distance from the neutral axis of the beam to the top or bottom of the beam. The maximum moment occurs at the base of the beam and is defined by:
where l is the length of the beam. Now that everything is defined, the maximum stress can be queried and converted to [MPa] units:
=5.33333333333333 MPa
These results can be compared to the results obtained from a finite element analysis using SolidWorks. The beam has the same dimensions, loading, and material as the rectangular cross-section and the aluminum table options above. The displacement plot is shown below. The maximum displacement magnitude obtained using finite element analysis is .175 [mm], which is close to the .165 [mm] obtained using the equations above.Similarly, the maximum stress calculation can be compared to the SolidWorks results. The maximum stress obtained from SolidWorks is 6.5 [MPa] as shown below. Note that this is higher than the 5.3 [MPa] obtained using the beam equation. This is the result of the stress concentration that occurs at the based of the cantilever where it meets the fixed boundary condition. A littler further from the fixed boundary condition, the finite element values are close to the calculated values.[1] Norton, R. L. "Machine design. A integrated approach, 5th Editi." (2013).

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Deflection and Bending Stress in a Cantilever Beam

Introduction

Select Beam Material

Select Beam Section

Define the Other Beam Parameters

Define the Deflection and Stress Equations