Name: bending stress in a symmetric member
Uploaded: 2024-05-23T12:59:43.000Z
Duration: 1 min 7 s
Channel: JoVE

Consider a prismatic member subjected to two couples, equal in magnitude but in opposite directions, acting within the symmetric plane of the member.
A section is cut through an arbitrary point on the member. The equilibrium conditions for each part dictate that the internal forces at the section must be equivalent to the couple.
The internal forces acting on the section consist of the normal stress at a point of the cross-section and the shearing stress components.
Recall that a couple consists of two equal and opposite forces whose sum in any direction is zero. Furthermore, the couple moment is the same about any axis perpendicular to its plane but is zero about any axis in the plane.
By choosing the axes arbitrarily, the sums of the components and the moments of the forces equal the corresponding components and moments of the couple.
Here, the negative sign indicates that tensile stress contributes to a negative moment of the normal force about the z-axis.

symmetric member in bending

Symmetric Member in Bending

In the study of the mechanics of materials, analyzing the behavior of prismatic members under opposing couples is crucial for understanding internal stress distributions, which are essential for structural design. When subjected to couples, a prismatic member experiences internal forces that maintain equilibrium. A couple, characterized by two equal and opposite forces, creates a moment but no resultant force. The internal forces at any section cut of the member must balance these external couples and resolve them into normal and shear stress components.
Normal stresses, acting perpendicular to the cross-sectional area, result from axial forces due to the bending moment caused by the couple. The normal stresses are equal in magnitude but opposite in direction. Shear stress, tangential to the cross-sectional area, maintains translational equilibrium. By selecting appropriate axes, typically the principal axes of the cross-section, the moments due to internal stresses are equal to the moment of the external couples. The bending moment is countered by an equivalent moment from the normal stresses, where the distance from the neutral axis to the area of the cross-section is taken into account.
<img alt="Static equilibrium diagram; stress components &#964;xzdA, &#964;yzdA, &#963;xdA on 3D element with axes x, y, z." class="xfigimg" src="/files/ftp_upload/15625/15625_Figure_1.svg" data-altupdatedat="1762173717000" data-alt="Figure 1" />
<img alt="Static equilibrium equation &#8747;(-y&#963;xdA)=M, representing mechanical moment balance in mechanics." src="/files/ftp_upload/15625/15625_Equation_1.svg" style="text-align: center; height: 38px;" data-altupdatedat="1762173717000" data-alt="Equation 1" />
The sign convention indicates that positive normal stress, or tension, contributes negatively to the moment about the z-axis, where counter-clockwise moments are positive. Understanding these stress distributions is vital for predicting failure modes and optimizing material distribution, forming a cornerstone of structural engineering.

Consider a prismatic member subjected to two couples, equal in magnitude but in opposite directions, acting within the symmetric plane of the member.
A ...

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Bending

KEY TERMS AND CONCEPTS

bending stress in a symmetric member

Bending stress in a symmetric member comes from a prismatic member that is loaded by two couples acting in opposite directions. A couple is made of two equal and opposite forces. It creates a moment, but it does not create a resultant force.

To keep the member in equilibrium, internal forces appear at any section cut. These internal forces must balance the external couples. They are resolved into normal stress and shear stress components.

Normal stress acts perpendicular to the cross-sectional area. It comes from axial forces caused by the bending moment from the couple. The normal stresses are equal in size but opposite in direction.

Shear stress acts tangential to the cross-sectional area. It helps maintain translational equilibrium. In many cases, the axes are chosen along the principal axes of the cross-section. Then the moment from the internal stresses matches the moment from the external couples.

The bending moment is opposed by an equivalent moment from the normal stresses. The distance from the neutral axis to each area element is included in this balance. The sign convention also matters. Positive normal stress, or tension, contributes negatively to the moment about the z-axis, while counter-clockwise moments are taken as positive.

Understanding these stress distributions helps predict failure modes and improve material placement in structures. It is a basic idea in structural engineering and mechanics of materials.

Bending Stress in a Symmetric Member

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Pure bending describes how a beam bends under equal and opposite moments. It is a key idea in structural mechanics. Engineers use it to study how prismatic members, such as beams, deform under symmetrical loads without direct forces on the member. In pure bending, the beam experiences bending without axial force or shear force. This makes it possible to predict the stress pattern across the cross-section. The bending stress depends on the bending moment, the moment of inertia of the...

Bending stress in a symmetric member comes from a prismatic member that is loaded by two couples acting in opposite directions. A couple is made of two equal and opposite forces. It creates a moment, but it does not create a resultant force. To keep the member in equilibrium, internal forces appear at any section cut. These internal forces must balance the external couples. They are resolved into normal stress and shear stress components. Normal stress acts perpendicular to the...

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key terms and concepts

Learn how opposing couples create bending stress in a symmetric prismatic member, including normal and shear stress, internal moments, and equilibrium.

Bending Stress in a Symmetric Member | JoVE K12

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Bending Stress in a Symmetric Member

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