Abstract
A structural beam which is subjected to shear forces acting perpendicularly to its longitudinal axis will experience longitudinal and transverse shear stresses. In beams where failure in the transverse direction is plausible, it is desirable to maintain a constant transverse shear stress over the beam cross-section to avoid stress concentrations and to use the least amount of material. A numerical approach to the inverse problem of solving for a beam cross-section with a constant transverse shear stress distribution was investigated in this study using Microsoft Excel’s Solver and Matlab. The efficiency and shape of the developed cross-section were dependent on the number of elements used to discretize the cross-section. As the number of elements approached infinity, the shape of the cross-section became infinitely thin at the top and infinitely wide at the neutral axis, while also approaching an efficiency of 100%. It is therefore determined that this is an ill-posed inverse problem and no such perfect cross-section exists.
Highlights
In the design of a structural beam, there are several geometric and material parameters that must be considered and optimized in order to yield the desired strength, lifespan, and reliability
The rectangular cross-section transverse shear stress distribution is a parabola that is derived from Equation (7) and has an efficiency of 66.7% using Equation (10) [8]
The reason that the rectangular cross-section is not efficient is that the maximum transverse shear stress is well above the average transverse shear stress
Summary
In the design of a structural beam, there are several geometric and material parameters that must be considered and optimized in order to yield the desired strength, lifespan, and reliability. Though this is not always the case, it can be derived that the maximum carry no longitudinal load [7] The determination the shape the cross-section beam achieve maximum overalloverall transverse shear stress can be completed using general of the to beam to achieve maximum transverse shearefficiency stress efficiency can be completed using general analytical or numerical methods. Both methods are discussed, butthe only the solution the analytical or numerical methods. Other approaches optimization of cylindrical bar cross-sections [10], and open beam using the boundary element method have shown how the transverse shear stress can vary within an arbitrary cross-section [11]
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