Von-Mises Stresses: Physical Significance and Application in Steel Bridges

July 28, 2021
BLOG BRIDGE INSIGHT

 

Von-Mises Stresses:

Physical Significance and Application in Steel Bridges

 

 

Table of Contents

 

1. Static Failure Theory - Ductile Materials

2. Recollecting Terminologies

  • Hydrostatic Loading
  • Strain Energy
  • Von-Mises Stress with Example
  • Safety Factor

3. Physical Significance of Von-Mises Stress & Application in Steel Bridges


 

1. STATIC FAILURE THEORY – DUCTILE MATERIALS

 

The main reason for observing stress is to determine if a part or structure fails. This article sees how parts fail under static or near static loading and how to predict it. Part failure is affected by the part's geometry, its loads, and its material.

 

In general, shear strength limits ductile isotropic materials (Ductile Materials εmax >= 5%) in static tensile loading, and the tensile strength limits brittle materials.

 

There are several failure theories, but the two most reliable methods are:

  • Von-Mises Stress ( Maximum Distortion )
  • Maximum Shear Stress

Let's look at the Stress-Strain Curves in Fig.1

Stress-Strain Curve of Ductile Material

Figure 1. Stress-Strain curve of ductile materials
 
 

Frequently these curves are generated by placing a specimen in a machine and applying a tensile force until the material fails.

Fig.2 Tensile Test & Mohrs CircleFigure 2. Tensile test & Mohr's Circle

 

The degree scale for Mohr's circle is twice what it is in the material. This is because the maximum Shear occurs 90o away from the Maximum Normal Stress but 45o away from the material.


 

2. Recollecting Terminologies

 

1. Hydrostatic Loading

 

Materials, though stressed beyond their ultimate strength, do not fail under hydrostatic loading.

 

Fig.3 Hydrostatic LoadingFigure 3. Hydrostatic loading
 
 

With hydrostatic loading, there is compressive stress (normal) but no shear stress. This leads us to suspect that when the parts fail, shear plays a strange role. In fact, it is related to the distortion n of the objects. It allows the molecules to slide next to each other in the structural lattice.

 

2. Strain Energy

 

Fig. 4 gives the relationship between stress-strain and energy.

Fig.4 Strain EnergyFigure 4. Strain Energy
 
 

The energy (U) at any point is:

Equation 1.1

 

Using the relationship between stress and strain & substituting these into the equation for strain energy

 

equation 2.2

 

We saw earlier that hydrostatic loading alone did not cause failure. Hence the strain energy can be divided and developed above into two parts, Hydrostatic & Distortion Energy.

 

equation 3.3

 

The stresses causing hydrostatic energy are the same in all directions so that we can divide the principal stresses into:

equation 4.4

Adding these together yields:

equation 5.1

 

For volumetric change with no distortion, the terms in parenthesis must equal to zero so

 

equation 6.1

Now the Hydrostatic Energy (Uh) can be found. We know that hydrostatic stress is the same in all directions. Substituting in the total energy, one may obtain the Distortion Energy (Ud) as,

 

equation 7.1

Using this, we set,

equation 8.1

 

We call this the Von-Mises stress. It is based upon Distortion Energy, and it seems to be better at predicting failure than the other stresses.

 

3. Von-Mises Stress

 

Von-Mises effective stress is defined as the "uniaxial tensile stress that would create the same distortion energy as is created by the actual combined applied stresses".

Figure 5. Von Mises StressesFigure 5. Von-Mises Stresses
 

 

Example

We have the following stress in a material,         Fig.6 Example for computing Von Mises Stresses

Figure 6. Example for computing Von-Mises Stresses
 

 

4. Safety Factor

 

If we define the safety factor as:

N = Failure Stress / Analysed Stress

 

This N value should always be larger than 1. In fact, it is usually significantly larger than 1.

 

1.3 <= N <= 5 or higher

 

1.3 when everything is known and carefully analyzed & 5 when there are many things unknown. For Von-Mises Stresses, the safety factor is computed with

N = σy / σ= Yield stress / Von mises stress


 

3. Physical Significance of Von Mises Stress & Application in Steel Bridges

 

When we check the failure using the Von-Mises Stress, we are applying the Von-Mises yield criterion to determine yielding. Von Mises Stress Theory ( N = σy / σ ') is good for computing the Safety Factor against failure. Hence the material has to be checked if it fulfills the Von-Mises Criteria.

 

Application in Steel Bridges

 

Von Mises criterion is widely applied in the case of steel bridges and is the most preferred failure theory. Von Mises stresses are checked and ensured they are well within the elastic yield limit.

 

Consider a Bow-String Girder Bridge model in midas Civil as in Fig. 7. The Yield Strength of the Girder of Material E250 is 250 N/mm2. Two cases are examined for Train Load, Double Track. The Stringer beam & the Cross Bracing at the mid-span as these elements are expected to experience maximum stress.

 

Fig. 7  midas Civil model of Bow-String Girder Bridge with applied Train LoadFigure 7. midas Civil model of Bow-String girder bridge with applied train loads

 

The Von-Mises stresses can be obtained from midas Civil, under Results > Stresses > Beam Stresses (Equivalent) > Von- Mises.

Fig. 8  Von-Mises Stress option in midas Civil modelFigure 8. Von-Mises Stress option in midas Civil model

 

The Von-Mises stress for the Stringer Beam near Mid Span (Dark Yellow Contour ) is 159 N/mm2, less than the Yield Strength value. Hence the design is SAFE. However, one may revise the section to consider a higher Factor of Safety.

 

Fig.9 Von-Mises Stress for Stringer Beam near mid-spanFigure 9. Von-Mises Stress for Stringer Beam near mid-span

 

The Von-Mises stress for the Cross Bracing near the mid-span (Pale Green Contour ) is 152 N/mm2, lesser than the Yield Strength value. Hence the design is SAFE. However, it is hard not to notice that the Von Mises for the Arch element near the support is greater than the Yield Strength, and one needs to revise the section.

 
Fig. 10  Von-Mises Stress for Cross Bracing near mid-spanFigure 10. Von-Mises Stress for Cross Bracing near mid-span
 

In short, an engineer's duty is to keep the maximum value of Von-Mises stress induced in the material less than its yield strength.

 

Before Von-Mises researched looked at the tensile test of ductile materials, it was apparent that the failure was due to shear, not torsion. Both the Maximum Shear Stress Theory ( N = σys / τmax ) and Von Mises Stress Theory ( N = σy / σ ') are good for computing the safety factor against failure. However, the max shear stress theory is a little more conservative than Von-Mises stress.

 

Note: The Maximum Normal Stress Theory ( N = σy / σ1 ) is not valid. Materials can fail even though N is greater than 1.

Subscribe S.O.S Newsletter

About the Author
Sindhu Bharathi |Senior Structural Engineer |MIDAS IT India

Sindhu has 7+ years of experience as a Civil Engineer (Structures) with extensive technical reviewing and consulting experience with bridge & building projects. Equally conversant with RCC, prestress, steel, and composite design. Predominantly interested in software workflow optimization.

Comments
DOWNLOAD Ebook Download

Please fill out the form below
to download the Ebook

Read all