Chapter 20: Bending

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20.11:

Plastic Deformations of Members with a Single Plane of Symmetry

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Plastic Deformations of Members with a Single Plane of Symmetry

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20.10: Members Made of Elastoplastic Material

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20.12: Residual Stresses in Bending

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Consider a member experiencing plastic deformation due to bending and having one plane of symmetry. The stresses are uniformly distributed above and below the neutral axis, with stress being -σy above the neutral axis and +σy below the neutral axis. Here, the neutral axis of the member does not coincide with the centroid of the member. To locate the neutral axis, consider the resultant of the compressive forces above the neutral axis R1 and the resultant of the tensile forces R2 below the neutral axis. The compressive and tensile resultant forces form a couple, which is equivalent to the couple applied to the member. This shows that the neutral axis divides the member into cross-sections of equal areas. The line of action of the resultant compressive and tensile forces passes through the centroids of the two equal areas. If the distance between these two centroids is defined as d, then the plastic moment of the member is expressed as half of the product of the area of the cross-section, the magnitude of the stress, and d.

20.11:

Plastic Deformations of Members with a Single Plane of Symmetry

When a structural member undergoes plastic deformation due to bending, it is crucial to understand the position of the neutral axis and the stress distribution. This member, characterized by a single plane of symmetry, exhibits a uniform stress distribution, with negative stress above the neutral axis and positive stress below. Notably, the neutral axis does not align with the centroid of the cross-section. This misalignment is typical in cases where the cross-section is not rectangular or experiences severe deformation.

The resultant forces generated by the bending of the member are analyzed to locate the neutral axis. The compressive forces acting above the neutral axis and the tensile forces below it generate resultants with equal magnitude but opposite directions, forming a couple. This configuration indicates that the neutral axis divides the cross-section into two equal areas, each contributing equally to force equilibrium.

The plastic moment of the member M_p, a critical structural parameter, is derived from the relationship between the cross-sectional area, the material's yield stress, and the distance between the centroids of the areas d created by the neutral axis.

Calculating the plastic moment is essential for predicting the maximum moment the section can handle before significant plastic deformations occur. This consideration is particularly crucial for structural components in dynamically loaded areas, such as seismic zones, emphasizing how material properties and cross-sectional geometry influence a structure's capacity to withstand bending loads.