Uniaxial stress state

Pressure bar with uniaxial loading

Pull rod / push rod

The uniaxial stress state occurs for example in tension and compression members as in Fig. 1
In uniaxial bending there is also uniaxial stress state.

Druckstab, einachsiger Spannungszustand, Definition der Dehnung

The maximum tensile / compressive stresses are created at tension rod / compression rod in the direction of the force.
In all other directions tensions are smaller and follow the equation 1

Abhängigkeit der mechanischen Spannung vom Winkel bei der Messung des einachsigen Spannungszustands mit Dehnungsmessstreifen. (Equation 1)

φ:. Angle between the direction (here the line of action of force) and viewing direction (measuring direction)
σ 1: First main direction

Despite the uniaxial stress state can be found but a biaxial strain state, due to the lateral expansion of the material.

(equation 2)

ε1: strain in the 1. main direction
ε2: strain in the 2. main direction (perpendicular to 1. main direction)
ν: Poisson's ratio;
 
Note
  • The material supply may only be from the equation σ = E · ε are calculated when the strain was measured in the direction of force and the stress state is uniaxial.
  • An elongation in the transverse direction is measured, although there is no mechanical stress is present.

literature

Links in Forum: Links-stress analysis

The "E-MechLAB" of the Institute of Applied Mechanics of the Technical University of Berlin presents the basics in a clear way:
types of voltages: normal and shear stresses
principal stress, reference stress, Mohr's stress circle: principal stresses, equivalent stresses

in the TechNotes Vishay be shown the basics of strain gauge rosettes. The Grid "a", "b", "c" are here, however, with "1", "2", "3" indicated:
TechNote-515: TN515_DMS_Rosetten.pdf
TechNote-512-1: TN 512-1_ebene_Schubmessung.pdf

Stress analysis with strain gauges

The stress analysis with strain gauge is used to determine the stress in a single component.
In the stress analysis with strain gauges usually bridge circuits are used with only one active measurement grid.

In a uniaxial stress condition, it is sufficient to detect the strain with a single measuring grid. The direction of the mechanical stress is required in this case, as known. The strain gauge is used to determine the amount. To calculate the mechanical power from the measured strain of the elastic modulus of the material as well as the k-factor of the strain gauge must be known.

If it is a biaxial stress state in which the magnitude and direction of the principal stresses are unknown, the stretching must be measured in three directions become. For this strain rosettes are used with three active measuring grids. To calculate the mechanical power from the measured strain of the elastic modulus of the material, the Poisson's ratio of the material as well as the k-factor of the strain gauge must be known.

Biaxial stress state

When biaxial stress state the maximum stresses occur in two mutually perpendicular directions.
These directions are called principal stress directions, indicated with 1 and 2.

In general, the principal stress directions are unknown in the stress analysis.
In this case, a stress analysis is performed with rosettes.

The strain gage rosette is the expansion in three directions "a", "b" and "c" gemesen.

Gitter a, b und c der Dehnungsmessstreifen Rosette

The grids "b" and "c" are respectively oriented relative to the measuring grid "a" through 45 ° or 90 ° counterclockwise.
Alternatively, measuring grids 0, 60 ° and 120 ° are used. The equations listed here can then not be used.

Der Winkel phi wird gegen den Uhrzeigersinn relativ zu Messgitter a gemessen.

 

The angle φ is the angle between the measuring grid a and the first main direction.
For the 90 ° Rosette (0 °, 45 °, 90 °) in Figure 7 and Figure 8 following relationship applies for determining the principal stresses σ1 and σ2..:

To determine φ the angle, a case distinction must be carried out.
 

Because of the ambiguity of the tangent function must be determined on the basis of the case analysis,
in which the quadrants I to IV is the solution for this angle φ:

arctan (y/x)

Bestimmung des Arcustangens mit einer Fallunterscheidung

 

Zähler Y y ≥ 0 y > 0 y ≤ 0 y < 0
Nenner X x > 0 x ≤ 0 x < 0 x ≥ 0
Quadrant I II III IV
Hauptrichtung φ = 1/2 · (0° + |ψ|) φ = 1/2 · (180° - |ψ|) φ = 1/2 · (180° + |ψ|) φ = 1/2 · (360° - |ψ|)

Table 1: determining the angle φ from the auxiliary angle ψ using a case distinction.
Note: the sum of ψ is used.

Calculation of principal stresses
With this online calculator principal stresses are calculated from the measured strains of a rectangular rosette FAER (3x45°).