Plastic bending
Encyclopedia
Plastic bending is a nonlinear behaviour peculiar to members made of ductile materials that frequently achieve
much greater ultimate bending strength than indicated by a linear elastic bending analysis. In both the plastic and
elastic bending analyses of a straight beam, it is assumed that the strain distribution is linear about the neutral
axis (plane sections remain plane). In an elastic analysis this assumption leads to a linear stress distribution but
in a plastic analysis the resulting stress distribution is nonlinear and is dependent on the beam’s material.
The limiting plastic bending strength
can generally be thought of as an upper limit to a beam’s load–carrying capability as it only represents the strength at a particular cross–section and not the load–carrying capability of the overall beam. A beam may fail due to global or local instability before 
is reached at any point on its length. Therefore, beams should also be checked for local buckling, local crippling, and global lateral–torsional buckling modes of failure.
Note that the deflections necessary to develop the stresses indicated in a plastic analysis are generally excessive, frequently to the point of incompatibility with the function of the structure. Therefore, separate analysis may be required to ensure design deflection limits are not exceeded. Also, since working materials into the plastic range can lead to permanent deformation of the structure, additional analyses may be required at limit load to ensure no detrimental permanent deformations occur. The large deflections and stiffness changes usually associated with plastic bending can significantly change the internal load distribution, particularly in statically indeterminate beams. The internal load distribution associated with the deformed shape and stiffness should be used for calculations.
Plastic bending occurs when an applied moment causes the outside fibers of a cross-section to exceed the material's yield strength. Loaded with only a moment, the peak bending stress
es occurs at the outside fibers of a cross-section. The cross-section will not yield simultaneously through the section. Rather, outside regions will yield first, redistributing stress and delaying failure beyond what would be predicted by elastic analytical methods. The stress distribution from the neutral axis
is the same as the shape of the stress-strain curve of the material (this assumes a non-composite cross-section). After a structural member reaches a sufficiently high condition of plastic bending, it acts as a Plastic hinge
.
Elementary Elastic Bending theory requires that bending stress varies linearly with distance from the neutral axis
, but plastic bending shows a more accurate and complex stress distribution. The yielded areas of the cross-section will vary somewhere between the yield and ultimate strength of the material. In the elastic region of the cross-section, the stress distribution varies linearly from the neutral axis to the beginning of the yielded area. Predicted failure occurs when the stress distribution approximates the material's stress-strain curve. The largest value being that of the ultimate strength. Not every area of the cross-section will have exceeded the yield strength.
As in the basic Elastic Bending theory, the moment
at any section is equal to an area integral of bending stress across the cross-section. From this and the above additional assumptions, predictions of deflections and failure strength
are made.
Plastic theory was validated at the beginning of the last century by C. v. Bach.
much greater ultimate bending strength than indicated by a linear elastic bending analysis. In both the plastic and
elastic bending analyses of a straight beam, it is assumed that the strain distribution is linear about the neutral
axis (plane sections remain plane). In an elastic analysis this assumption leads to a linear stress distribution but
in a plastic analysis the resulting stress distribution is nonlinear and is dependent on the beam’s material.
The limiting plastic bending strength
can generally be thought of as an upper limit to a beam’s load–carrying capability as it only represents the strength at a particular cross–section and not the load–carrying capability of the overall beam. A beam may fail due to global or local instability before is reached at any point on its length. Therefore, beams should also be checked for local buckling, local crippling, and global lateral–torsional buckling modes of failure.Note that the deflections necessary to develop the stresses indicated in a plastic analysis are generally excessive, frequently to the point of incompatibility with the function of the structure. Therefore, separate analysis may be required to ensure design deflection limits are not exceeded. Also, since working materials into the plastic range can lead to permanent deformation of the structure, additional analyses may be required at limit load to ensure no detrimental permanent deformations occur. The large deflections and stiffness changes usually associated with plastic bending can significantly change the internal load distribution, particularly in statically indeterminate beams. The internal load distribution associated with the deformed shape and stiffness should be used for calculations.
Plastic bending occurs when an applied moment causes the outside fibers of a cross-section to exceed the material's yield strength. Loaded with only a moment, the peak bending stress
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...
es occurs at the outside fibers of a cross-section. The cross-section will not yield simultaneously through the section. Rather, outside regions will yield first, redistributing stress and delaying failure beyond what would be predicted by elastic analytical methods. The stress distribution from the neutral axis
Neutral axis
The neutral axis is an axis in the cross section of a beam or shaft along which there are no longitudinal stresses or strains. If the section is symmetric, isotropic and is not curved before a bend occurs, then the neutral axis is at the geometric centroid...
is the same as the shape of the stress-strain curve of the material (this assumes a non-composite cross-section). After a structural member reaches a sufficiently high condition of plastic bending, it acts as a Plastic hinge
Plastic hinge
In structural engineering beam theory the term, plastic hinge, is used to describe the deformation of a section of a beam where plastic bending occurs...
.
Elementary Elastic Bending theory requires that bending stress varies linearly with distance from the neutral axis
Neutral axis
The neutral axis is an axis in the cross section of a beam or shaft along which there are no longitudinal stresses or strains. If the section is symmetric, isotropic and is not curved before a bend occurs, then the neutral axis is at the geometric centroid...
, but plastic bending shows a more accurate and complex stress distribution. The yielded areas of the cross-section will vary somewhere between the yield and ultimate strength of the material. In the elastic region of the cross-section, the stress distribution varies linearly from the neutral axis to the beginning of the yielded area. Predicted failure occurs when the stress distribution approximates the material's stress-strain curve. The largest value being that of the ultimate strength. Not every area of the cross-section will have exceeded the yield strength.
As in the basic Elastic Bending theory, the moment
Moment (physics)
In physics, the term moment can refer to many different concepts:*Moment of force is the tendency of a force to twist or rotate an object; see the article torque for details. This is an important, basic concept in engineering and physics. A moment is valued mathematically as the product of the...
at any section is equal to an area integral of bending stress across the cross-section. From this and the above additional assumptions, predictions of deflections and failure strength
Strength of materials
In materials science, the strength of a material is its ability to withstand an applied stress without failure. The applied stress may be tensile, compressive, or shear. Strength of materials is a subject which deals with loads, deformations and the forces acting on a material. A load applied to a...
are made.
Plastic theory was validated at the beginning of the last century by C. v. Bach.