Introduction to Bending
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Introduction to Bending
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Stress Transformations
General Problem Approach
Beam shearing questions consist of static systems in which a beam is subjected to shearing forces. The general approach to these types of problems is as follows:
- Solve for system reaction forces
- Apply a force analysis on sections of the beam
- Plot shear force diagram and bending moment diagram
- Calculate maximum normal stress from bending
- Calculate maximum shear stress from shear force
Shear, Moment, and Deformation
The question remains, how are shear, moment, and overall beam deflection related? The answer lies in performing a static balance over infinitesimal sections of the beam. Shear, V, can be plotted on a shear diagram after reaction forces are solved for the system.
Moment throughout the beam is simply the product of reaction shear force and distance between the coupling forces. Therefore we can derive beam bending moment from integrating shear force with respect to moment and applying the correct boundary conditions.
\displaystyle M=\int Vdx
The shear force and bending moment diagrams are sufficient for basic beam stress analysis. However, in some instances the deformation of the beam may be of interest. In such a case the slope, \theta, of the beam is derived as follows, again note that appropriate boundary conditions must be applied.
\displaystyle\theta=\dfrac{1}{EI}\int Mdx
Finally, the beam deflection, U_y, can be derived by integrating the beam slope and applying the correct boundary conditions.
\displaystyle U_y=\int\theta dx
Beam Selection
The stress associated with bending moment can be writing in terms of the beam section modulus, S. The section modulus is simply the ratio of the cross section second moment to the maximum distance from the neutral axis.
S=\dfrac{I}{y_{max}}
With this definition of section modulus, the maximum bending stress can be written:
\sigma_{max}=\dfrac{y_{max}}{I}\left|M_{max}\right|=\dfrac{\left|M_{max}\right|}{S}
Considering that the maximum stress must be below the allowable stress, \sigma_{max}\le\sigma_{all}, we can determine an acceptable minimum section modulus for the beam.
S_{min}=\dfrac{\left|M_{max}\right|}{\sigma_{all}}
All that remains to finalize the design of a beam is to account for stress caused by shear force.
Shear in Beams
The application of a shear force necessarily causes the propagation of shear stress through the beam in the form of \tau_{xy} and \tau_{yx}. Average shear stress through a horizontal strip of a beam cross-section can be calculated as follows:
\tau_{avg}=\dfrac{VQ}{It}
With the following definitions:
- V: Shear force
- Q: First moment of the area from the top/bottom of the beam to the strip in question
- I: Second moment of inertia of the cross-section
- t: Thickness of cross-section strip in question
By applying the above equation to a beam a general shear stress distribution can be found, allowing analysis of the beam design relative to the maximum shear stress.
Typically beams will be made of thin members, the typical I-beam as an example. In the case of thin members where shear stress can not flow along the shear force line of action the beam requires more analysis. Examples of this case include I-beams and C-beams, where the shear can not flow vertically in the flanges of the beam ad must therefore flow horizontally along the flanges. The shear flow, q, can be calculated by the following:
q=\dfrac{VQ}{I}
When calculating the shear stress flowing through a section not aligned with the shear force (\tau_{xz}) the shear flow should be used in tandem with the thickness of the section which the shear flows through.
\tau=\dfrac{q}{t}
The shear stress through the web of a beam can be approximated by the following:
\tau_{web}\approx\dfrac{V}{A_{web}}
