Name: principal and shear stresses in a bent tube
Uploaded: 2024-05-23T12:59:58.000Z
Duration: 1 min 23 s
Channel: JoVE

A bent tube with a circular cross-section subjected to several forces will produce stresses at certain points.
Calculating stresses involves passing a section through specific points and assessing the force-couple system at the centroid, representing internal forces and maintaining the member&#39;s equilibrium.
The centric axial force produces normal stresses in the section, while couple vectors cause the member to bend, producing normal stresses. The sum of the normal stresses produced is calculated using Saint-Venant&#39;s principles.
The twisting couple and shearing forces generate shearing stresses. The summation of these components is calculated, combined, and then displayed at specific points on the member&#39;s surface.
The normal and shearing stresses determine the principal stresses and orientation of the principal planes at these points. The maximum shearing stress at each of these points is then determined.
The superposition principle and Saint-Venant&#39;s principle guide stress determination. However, they are valid only if stresses are within the material&#39;s proportional limit, deformations from one loading do not affect others, and the analysis section is far from the points of force

<ol>
	<li>Ferdinand P. Beer, Russell E. Johnston, John T. Dewolf, and David F. Mazurek. (2012) Mechanics of Materials. New York, McGraw-Hill. PP [527-528].</li>
</ol>

stresses under combined loadings

Stresses under Combined Loadings

When analyzing a bent tube with a circular cross-section subjected to multiple forces, it is crucial to determine the stress distribution in order to maintain structural integrity under varied load conditions.
The process begins by slicing the tube at critical points and analyzing the internal forces and stress components at these sections, focusing on the centroid. Normal stresses, generated by axial forces and bending moments, are either compressive or tensile and vary across the section from the neutral axis to the outer edges.
Shearing stresses, which act tangentially to the section, arise from shear forces and torsional or twisting moments. These stresses provide insights into the overall stress state within the tube. In addition, the calculated stresses are combined to identify the principal stresses at specific points, which are the maximum and minimum normal stresses without shear components.
<img alt="Static equilibrium equation for stress analysis; formula includes &#963; and &#964;, relevant for mechanics." src="/files/ftp_upload/15665/15665_Equation_1.svg" style="text-align: center; height: 40px;" data-altupdatedat="1762173717000" data-alt="Equation 1" />
From these, the maximum shearing stress is determined, a crucial factor in assessing the potential for failure.
<img alt="Shear stress equation, formula for &#964;max, static equilibrium, mathematical concept, educational diagram." src="/files/ftp_upload/15665/15665_Equation_2.svg" style="text-align: center; height: 40px;" data-altupdatedat="1762173717000" data-alt="Equation 2" />
Saint-Venant's principle allows for the assumption that stresses become uniformly distributed in sections far from the points of load application. This principle allows the use of superposition to combine effects from different loads, provided the stresses remain within the material's proportional limits. This method ensures accurate predictions of how structural components behave under real-world conditions.

A bent tube with a circular cross-section subjected to several forces will produce stresses at certain points.
Calculating stresses involves passing a ...

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principal and shear stresses in a bent tube

A bent tube with a circular cross-section can experience normal stress, shear stress, principal stress, and maximum shearing stress when several loads act on it. To analyze these stresses, the tube is sliced at key points and the internal forces at each section are examined at the centroid. This helps show how the material responds under combined loading.

Normal stress comes from axial forces and bending moments. It can be tensile or compressive, and it changes across the cross-section from the neutral axis to the outer edges. Shearing stress acts tangentially along the section, and it can result from shear forces as well as torsional or twisting moments.

The calculated stresses are then combined to find the principal stresses at selected points. Principal stresses are the maximum and minimum normal stresses, and they have no shear component. From these values, the maximum shearing stress can be determined, which is important when checking whether the tube may fail.

Saint-Venant's principle helps simplify the analysis by assuming that stresses become more evenly distributed in sections far from where the load is applied. This makes it possible to use superposition, or the addition of stress effects from separate loads. That approach is valid as long as the stresses stay within the material's proportional limits, which supports accurate predictions of how the tube will behave in real conditions.

Principal and Shear Stresses in a Bent Tube

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A bent tube with a circular cross-section can experience normal stress, shear stress, principal stress, and maximum shearing stress when several loads act on it. To analyze these stresses, the tube is sliced at key points and the internal forces at each section are examined at the centroid. This helps show how the material responds under combined loading. Normal stress comes from axial forces and bending moments. It can be tensile or compressive, and it changes across the cross-section from...

key terms and concepts

Analyze a bent circular tube under combined loads by finding normal, shear, and principal stresses, then using superposition and Saint-Venant's principle.

Principal and Shear Stresses in a Bent Tube | JoVE K12

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Principal and Shear Stresses in a Bent Tube

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