Bending Strength of Metals – Bending Stress, Bending Moment and Bending Test

Flexural strength describes the behavior of a material or sample under the influence of a load that causes it to bend. During bending, tensile and compressive stresses simultaneously occur in the cross-section of the element, with a region associated with the neutral axis between them. For ductile metals, the concept of flexural strength requires careful interpretation. A sample can reach its yield point and undergo significant permanent deformation without fracture. Therefore, depending on the test objective, stress at a specific deflection, the onset of yielding, the maximum nominal stress, or the sample's ability to bend without fracture are analyzed, among other things.

What is bending?

Bending is a loading mode in which a bending moment causes a curved member. Typically, one side of the cross-section is in tension and the opposite side is in compression. Between them lies a neutral axis , where, under classical beam theory assumptions, the longitudinal normal stress due to bending is zero. Bending affects, among other things:

beams,
shafts and axles,
sheet metal,
profiles,
springs made of, among others, carbon spring steel i alloy spring steels,
steel structure elements,
machine parts,
tools.

Bending moment M

The basic quantity describing the bending load is the bending moment M . The moment is created by the action of a force at a specific distance from the cross-section under consideration. In the simple case: M = F × l , where:

M – bending moment,
F - force,
l – the arm of the force.

The unit of torque can be, for example, N m or N mm.

Bending stress

In the linear-elastic range, the classical theory of beam bending allows us to determine the normal stress in the cross-section from the formula: σ = M × y / I , where:

σ – normal stress from bending,
M – bending moment,
y – distance of the analyzed point from the neutral axis,
I – geometric moment of inertia of the cross-section relative to the bending axis.

The highest absolute stress value occurs in the cross-sectional fibers that are furthest from the neutral axis.

Section modulus of strength W

For the extreme fiber, the relationship can be written in a more convenient form: σ _max = M / W , where: W = I / y _max . W is the flexural strength of the cross-section . This is a geometrical property of the cross-section, not a material property. Two elements made of the same material can therefore resist very different bending moments if they have different cross-sectional shapes and dimensions.

Example of bending stress

If the bending moment is M = 1,000,000 N·mm and the section modulus W = 10,000 mm³ , then the maximum nominal bending stress is: σ = M/W = 1,000,000/10,000 = 100 MPa . The example shows that the bending stress depends on both the load and the cross-section geometry.

Stretching and compressing during bending

In pure bending, one part of the cross-section is in tension and the other is in compression. For the classical linear-elastic beam model, the stress varies linearly with the cross-section height:

on one surface it reaches the maximum stretching,
on the neutral axis is zero,
on the opposite surface it reaches its compressive maximum.

It is the simultaneous occurrence of tension and compression that distinguishes bending from simple uniaxial tension or compression.

What is bending strength?

Flexural strength is the stress determined by applying a load to a specimen in a bending test according to an established procedure. For brittle materials, it may correspond to the nominal stress at fracture of the specimen. For ductile materials, the situation is more complex, as the material can reach a plastic state and undergo significant deformation without fracture. Therefore, the term "flexural strength" should always be interpreted in conjunction with the test method and criterion used to determine the result.

Why is the bending strength of ductile metals problematic?

When ductile steel is bent, it can:

first deform elastically,
reach the beginning of plasticization in the extreme fibers,
undergo increasingly greater plastic deformation,
create a plastic joint,
not break even at a very large bending angle.

In such cases, there is not always a clear point corresponding to the classic ultimate strength. Therefore, in the case of metals, the bending test , the yield point, or the plastic moment analysis are often more important than the "flexural strength" value itself.

Bending yield strength

As the moment increases, the highest stresses initially occur in the extreme fibers of the cross-section. When the local stress reaches the material's yield strength , permanent deformation begins in these areas. With further increases in moment, the plasticized region can gradually expand toward the neutral axis. This means that the onset of plasticization of the entire element does not occur simultaneously across the entire cross-section.

Elastic moment and plastic moment

For the cross-section, one can distinguish, among others:

the moment corresponding to the start of plasticization extreme fibers,
plastic moment related to the plasticization of the entire cross-section in an idealized model of an elastic-ideally plastic material.

The plastic moment depends on the material's yield strength and the cross-section's geometry. The cross-section's ability to resist moments after local yielding has begun is used, among other things, in the plastic analysis of structures.

Young's modulus and bending

Young's modulus E describes the stiffness of a material in the linear-elastic range. The larger E, the smaller the material's elastic deformation at a given stress level. However, this does not mean that a material with a larger E automatically has greater bending strength. E describes stiffness, not strength.

Bending stiffness EI

In the case of a beam, the product EI is of particular importance , where:

E Young's modulus of the material,
I – geometric moment of inertia of the cross-section.

The EI product determines the flexural stiffness of the element. This means that the deflection depends on both the material and the cross-section geometry.

Why is cross-section geometry so important?

The moment of inertia I strongly depends on the material's location relative to the neutral axis. Material positioned far from the neutral axis very effectively increases the stiffness and bending resistance of the cross-section. Therefore, structures use, among other things:

I-beams,
channel sections,
closed profiles,
tubing,
box profiles.

Appropriate material arrangement allows for high bending stiffness without proportionally increasing the mass of the element.

Moment of inertia of a rectangular cross-section

For a rectangular cross-section of width b and height h, bent about an axis passing through the center of the cross-section parallel to the width: I = b × h³ / 12 . The strength modulus is: W = b × h² / 6 . This shows that increasing the height of the cross-section very strongly increases its resistance to bending.

Moment of inertia of a circular cross-section

For a full circular cross-section with diameter d: I = πd⁴ / 64 , and the strength modulus: W = πd³ / 32 . These relationships are important, among others, in the analysis of shafts and axles subjected to bending loads.

Metal bending test

A bending test can be used both to determine mechanical characteristics and to technologically assess a material's ability to deform without cracking. Depending on the method, the sample is loaded against a specific support and force system. Tests include:

three-point bending,
four-point bending,
technological tests of bending to a specific angle.

Three-point bending

In the three-point bending test, the sample rests on two supports, and the force is applied between them, usually at mid-span. The greatest bending moment occurs in the area of the central force. This setup is simple and widely used in testing various materials.

Four-point bending

In the four-point bending test, the specimen rests on two supports and the load is applied at two points. Between the force application points, there may be an area with a constant bending moment and—in an idealized case—no shear force. This increases the length of the area subjected to high moments compared to classical three-point bending.

Three-point vs. four-point bending

Characteristic	Three-point bending	Four-point bending
Number of support points	2	2
Number of load application points	1	2
Maximum torque	In a given cross-section under force	It can occur in the section between forces
High torque area	More local	Larger sample area

Technological bending test

For metals, the bending process is crucial . Its purpose isn't necessarily to determine a single strength value. It can be used to verify whether a sample can be bent to a specific angle or around a mandrel of a specific diameter without unacceptable cracking. Such a test can be used to assess:

blach,
rods,
metallurgical products,
welded joints,
materials after technological processing.

Bending radius and material deformation

The smaller the bend radius, the greater the deformation in the outer layers of the material. Too small a radius can lead to:

local narrowing,
crack formation,
surface damage,
cracking of the material on the outside of the bend.

The minimum permissible bending radius depends, among other things, on the type of material, its condition, thickness, rolling direction and bending technology.

Rolling direction and sheet bending

Rolled sheets can exhibit anisotropy . The bending ability of a material may therefore depend on the orientation of the bend line relative to the rolling direction. Requirements regarding the sampling and bending direction should be derived from the relevant material standard or product specification.

Springback after bending

When the force is removed, some of the elastic deformation disappears, causing the element to partially return to its original shape. This phenomenon is called springback . It is of great importance during the technological bending of sheet metal and profiles. The magnitude of springback is influenced by, among other factors:

Young's modulus E,
yield point,
material thickness,
bending radius,
tool geometry,
degree of hardening of the material.

Influence of strain hardening

During plastic bending, a material can undergo strain hardening . In the most deformed areas, the dislocation density increases and local mechanical properties change . This phenomenon is particularly important during cold bending.

Bending and Rm

The tensile strength Rm is not the bending strength. Rm defines the maximum engineering stress achieved in a static tensile test . During bending, a stress gradient occurs – only a portion of the cross-section is at the maximum stress value. Therefore, bending test results should not be directly equated with Rm.

Bending and Re and Rp0,2

The yield strength Re or Rp0,2 is important when analyzing the onset of permanent deformation in a bending element. In a simple elastic model, the onset of yielding occurs when the stress in the most heavily loaded fiber reaches a level corresponding to the material's yield strength. Further increases in moment cause the development of plastic zones in the cross-section.

Bending and hardness

Hardness and flexural behavior are distinct material properties. There is no single, universal relationship that allows for the direct conversion of HBW , HV , or HRC to flexural strength. A material with high hardness may simultaneously exhibit limited plastic bending capacity and greater susceptibility to fracture, depending on the microstructure and loading conditions.

Bending and impact strength

The bend test and the Charpy impact test are not the same test. The Charpy test uses a dynamic impact on a notched specimen and measures the energy absorbed during fracture. The bend test follows a different procedure and can be used to determine the mechanical characteristics or technological ability of a material to bend without fracture.

Bending and material fatigue

If the bending moment changes repeatedly over time, flexural fatigue can occur . This is particularly important in:

rotating shafts,
axes,
springs,
suspension elements,
machine parts subjected to cyclic loads.

Fatigue cracks often initiate at the surface where bending stresses are highest.

Rotational bending

A classic case of fatigue loading is rotational bending . If a shaft is subjected to a constant bending moment and simultaneously rotates, a given point on its surface cycles through a region of tension and compression. This creates a cyclic stress that can lead to fatigue fracture even when the stress is much less than Rm.

Notches and bending

Holes, grooves, threads, sharp changes in cross-section, and other discontinuities cause stress concentrations . This is particularly important in bending because the maximum nominal stresses occur at the surface of the component. A notch located in such an area can significantly increase local stress and facilitate crack initiation.

Surface condition and fatigue bending

Surface quality is crucial in cyclic bending. Scratches, machining marks, corrosion , pitting , and other surface defects can act as local stress concentrators and reduce the fatigue life of a component. Therefore, surface condition is particularly important in shafts, springs, and other components subjected to repeated bending.

Effect of temperature on bending

Temperature affects the material properties responsible for its bending behavior. Temperature changes can affect, among other things:

yield strength,
Young's modulus,
ductility,
resistance to cracking.

Therefore, the ability of a material to bend safely at room temperature may not be the same at very low or high temperatures.

Effect of heat treatment

Heat treatment can significantly affect a material's bending behavior. Hardening can increase hardness and strength, but also limit the ability to undergo large plastic deformations. Tempering , annealing , or other processes can alter the trade-off between strength, hardness, and ductility. The effect depends on the material grade and processing parameters.

Bending of brittle materials

For brittle materials, bend testing is particularly important because the sample can fracture without significant plastic deformation. In such materials, the bending strength can be determined based on the load that causes the sample to fracture. For this reason, bend testing is also widely used for materials for which classical tensile testing is difficult to perform.

Bending of ductile materials

Ductile materials can achieve significant bending angles before fracture. For such materials, the test can primarily be used to assess:

ability to undergo plastic deformation,
material quality,
presence of defects,
quality of the welded joint,
suitability for further technological processing.

Bending test of welded joints

Bending is also used to evaluate the properties of welded joints. A sample containing a weld can be bent so that a specific portion of the joint is subjected to tension. This test reveals certain imperfections and assesses the joint's ability to deform without unacceptable cracking. The exact sample preparation method, bending direction, and evaluation criteria are derived from the relevant standard or specification.

Standards for metal bending tests

One of the basic standards for technological bend testing of metallic materials is ISO 7438 – Metallic materials – Bend test . For welded joints, separate standards are used for bend testing of welds and joint materials. The relevant standard specifies, among other things:

sample geometry,
the way it is supported,
mandrel diameter or bending conditions,
bending angle,
the method of conducting the study,
principles of sample evaluation.

A distinction should be made between technological bending tests and tests used to quantitatively determine mechanical properties during bending.

What determines the behavior of metal during bending?

The behavior of the material and element is influenced by, among other things:

yield strength,
material strength,
Young's modulus,
ductility,
microstructure,
heat treatment,
surface condition,
temperature,
rolling direction,
bending radius,
cross-section geometry,
notches and other stress concentrators,
type and method of applying the load.

Material strength and bending capacity of the element

The strength of a material and the load-bearing capacity of the entire element are not the same. Both material properties and cross-sectional geometry influence the load-bearing capacity of a element. Therefore, two elements made of the same steel can have completely different bending resistances if they differ in height, width, or cross-sectional shape. Similarly, two elements with identical geometry can have different load-bearing capacities if they are made of materials with different yield strengths.

Does higher flexural strength always mean a better material?

No. The right material should be selected for the specific application. In addition to bending load resistance, the following may be important:

stiffness,
ductility,
fatigue strength,
resistance to cracking,
impact strength,
hardness,
weldability,
corrosion resistance,
mass of the element.

A material with a very high yield strength may be advantageous in one application but unsuitable in another where high deformation capacity or high fracture toughness is required.

Bending strength – the most important information

During bending, one part of the cross-section is stretched and the other is compressed.
Between them there is a neutral axis.
In the linear-elastic range, the stress can be determined from the formula σ = My/I.
For the extreme fiber the following relationship applies σ_Max = M/W.
W is a geometric indicator of the cross-sectional strength, not a material property.
EI determines the bending stiffness of the element.
Young's modulus describes stiffness, not bending strength.
In ductile metals, once yielding begins, the sample may continue to bear load and deform significantly.
The technological bending test is often used to assess the ability of a material to deform without unacceptable cracking.
The three-point and four-point tests produce different bending moment distributions.
The cross-section geometry has a significant influence on the load-bearing capacity and stiffness of the element.
A small bending radius increases the deformation of the outer layers of the material.
Fatigue failure may occur with cyclic bending.
Notches and surface defects are particularly important because the maximum bending stresses occur at the surface.
Bending strength, Rm, Rp0,2 and hardness are not interchangeable parameters.