Our series of articles addresses possible sources of error when strain gauges are used in experimental stress analysis and shows how to successfully assess measurement uncertainty already in the design stage

# Series of articles: Measurement accuracy in experimental stress analysis – part 3

Strain gauge technology has been optimized over the course of decades with a wide range of options to compensate for errors. Yet there are still effects that have a negative impact on measurements. The objective of this paper is to point out the numerous (and often avoidable) sources of errors when using strain gauges in experimental stress analysis and to provide some help in estimating the measurement uncertainty early on in the planning phase.

## Estimating measurement uncertainty for non-zero-point related measurements

An important element of this measurement procedure is that the zero point for analyzing the measurement results is unnecessary. That is because only changes in the measured quantity are of interest and the zero point does not drift during the measurement (typical for relatively short measurement tests). Examples are crash tests, tensile tests and brief loading tests.

Material after-effects and strain gauge creep can be somewhat important in non-zero-point related measurements and are therefore covered in this section. On the other hand, phenomena such as thermal expansion, swelling of the adhesive, falling insulation resistance, temperature response of the strain gauge and strain gauge fatigue in non zero-point related measurements are almost completely irrelevant.

Of course, resistance will not drop so dramatically during a brief loading test of insulation resistance that failure of the measuring point would be possible.

## Tolerance of the modulus of elasticity

The modulus of elasticity (manufacturer specification) exhibits an uncertainty (tolerance of the modulus of elasticity) which may be several percent. Accurately determining the modulus of elasticity in a suitable laboratory is costly and often cannot be implemented.

In experimental stress measurements, or as we sometimes refer to it as experimental stress analysis (ESA), the relative uncertainty of the modulus of elasticity produces a relative uncertainty in the mechanical stress of the same amount.

This means that if the material has a modulus of elasticity with a value known within an uncertainty of 5%, that alone produces an uncertainty of 5% in the stated mechanical stress.

The modulus of elasticity also depends on temperature as an influence quantity and the temperature coefficient (TC) of the modulus of elasticity (for steel ≈ -2 • 10^-4/K). The relative change in the modulus of elasticity is derived from the product:

This is equivalent to the additional uncertainty of the mechanical stress.

Example: If the modulus of elasticity of steel is given for a temperature of 23 °C and the measurement is performed at 33 °C, the modulus of elasticity drops by 0.2%. If this effect is not compensated for by computations, there will be a deviation of 0.2% in addition to the tolerance specified for the modulus of elasticity. Note that the TC of the modulus of elasticity is itself temperature-dependent, which means that this effect can never be entirely compensated for.

### Index of formulas

## Radius for measurement objects subject to bending loads (increase in strain)

If the strain gauge is located on a component that bends longitudinally to the measuring grid, the strain of the measuring grid deviates from the surface strain of the component (Fig. 7). The measured values obtained are too large. The smaller the radius of curvature and the greater the distance of the measuring grid from the component surface, the greater the effect.

If the strain gauge is located in the concave area, the measured values would also be too large simply in terms of the amount. The factor describing the measurement error would be the same. This also results in a multiplicative deviation relative to the measured value. The equation for calculating is:

For a medium distance of 100 μm from the measuring grid to the component surface and a bending radius of 100 mm, the resulting increase in strain is 1/1000 relative to the current strain value. The actual strain of the component in this example is 0.1% lower than the measured strain. That means that the stress is measured 0.1% too large. This measurement error is clearly only relevant for small bending radii.

## Elastic after-effects

In many materials, the strain still increases somewhat further after spontaneous mechanical loading. This phenomenon is largely complete after about 30 minutes (steel at 23 °C) and also occurs when the load is removed. The quotient of the amount of this additional strain and the spontaneous strain depends heavily on the material. Material after-effects thus produce an additional (positive) measurement error. This only occurs when acquiring strain values. This deviation can therefore be almost completely avoided in many measurement tasks.

However, if the measured value is acquired long after the load is applied, and the strain of the material has increased by 1% (relative to spontaneous strain), the result will be that the measured value for the material strain is 1% too large.

## Misalignment of the strain gauge

If the strain gauge is not exactly aligned in the direction of the material stress (uniaxial stress state), a negative measurement error is produced. The measured strain will then be less than the material strain. The relative strain error is determined as follows:

An alignment error of 5 degrees and a Poisson's ratio of 0.3 (steel) results in a strain error of -1%. Thus, the actual strain and the material strain are 1% greater.

## Strain gauge creep

After material strain is induced spontaneously, the measuring grid of the strain gauge creeps back somewhat. The process, determined primarily by the properties of the adhesive and the geometry of the strain gauge (short measuring grids are critical, strain gauges with very long reversing lengths do not creep), is also temperature-dependent. After return creep the strain of the grid is somewhat less than the material strain. The strain gauge often used in ESA (HBM type LY11-6/120 with an active measuring grid length of 6 mm) when used with adhesive Z70 (HBM) at a temperature of 23 °C has a return creep of about 0.1% within one hour. This is equivalent to a negative measurement error of -0.1% relative to the measured stress. Of course the deviation will be less if the measured value is determined immediately after spontaneous loading. Due to the negative sign, the strain gauge creep compensates at least partially for the elastic after-effects and may therefore often be completely ignored in ESA. However, advise caution when using other adhesives at higher temperatures. For example, adhesive X60 (HBM) applied at 70 °C with a strain of 2000 μm/m, the resulting deviation after just one hour is -5%.

## Hysteresis of the strain gauge

The same applies to the hysteresis: short measuring grids tend to be critical and the adhesive has some effect. The hysteresis for strain gauge LY11-6/120 is only 0.1% with a strain of ±1000 μm/m if Z70 was used as the adhesive. It is therefore negligible.

If a very small strain gauge (LY11-0.6/120) with an active measuring grid length of 0.6 mm has to be used though, the hysteresis increases, and with it the uncertainty of the strain or stress measurement to 1%.

## The Gauge Factor

### Tolerance of the gauge factor

It is assumed that the measurement chain is exactly adjusted to the nominal value of the gauge factor (as specified by the manufacturer on the strain gauge package). This factor describes the correlation between the change in strain and the change in relative resistance. It has been determined experimentally by the manufacturer. The uncertainty of the gauge factor is generally 1%. The gauge factor is also specified on the package. It produces the same relative degree of uncertainty in both strain and stress measurements.

### Temperature coefficient (TC) of the gauge factor

The gauge factor is temperature-dependent. The sign and amount of the dependence are determined by the measuring grid alloy. The fact that the TC of the gauge factor is itself temperature-dependent can be ignored for purposes of ESA. The TC for a measuring grid made of Constantan is about 0.01% per Kelvin. Thus, the gauge factor decreases by 0.1% with a temperature increase of 10 K, which is generally negligible. If the measurements were performed at 33 °C, the strain or stress values would deviate upward by just 0.1%.

Although at 120 °C, it would be 1%, which is worth considering.

## Measuring Grid Length

As generally understood, a strain gauge integrates the strains under its active surface. If the stress field under that surface is non-homogeneous, the relative change in resistance will not correspond to the greatest local strain, but rather to the average strain under the active measuring grid. This is fatal, because it is especially the greatest stresses that are of interest. The measured values therefore deviate downward from the desired maximum values, leading to negative deviations.

Since this phenomenon is well known, as are suitable countermeasures (short measuring grid), major errors seldom occur in practical applications. Nevertheless, let’s take an example: The measurement is applied to bending stress at the beginning of the beam. The strain gauge acquires the average strain under its measuring grid (Fig. 8). The strains behave like stresses:

The maximum stress value that is actually wanted could easily be determined in this simple case with a correction calculation. If this is not done, a deviation of the measurement result from the maximum stress will be produced.

Its relative deviation is:

If a measuring grid with an active length of less than 2% of l2 is used in the example above, the deviation drops to less than 1% of the measured value.

Ultimately the ratio of the maximum strain and the measured strain always depends on the distribution of strain under the measuring grid. If this is known from a Finite Element Calculation, the desired maximum value can be calculated from the arithmetic mean of the stress.

*Of course, deviations will occur if the strain gauge is positioned incorrectly. This can also be largely avoided and it must be.*

## Linearity Deviations

### Linearity deviation of the strain gauge

Strain gauges with suitable measuring grid materials (Constantan, Karma, Nichrome V, Platinum-tungsten) exhibit excellent linearity. Although for large strains, appreciable deviations can be demonstrated in Constantan measuring grids. The actual static characteristic curve can be very adequately described (empirically) with a quadratic equation:

If the strains were determined with the relationship

there would be no linearity deviations at all. However, as the quadratic component is simply neglected in practical applications, the resulting error should be indicated here. The relative deviation of the determined strain value from the true value is as large as the strain itself:

For strains up to 1000 μm/m, the value of the relative strain deviation does not exceed 0.1%. This is equivalent to 1 μm/m, which is negligible.

Linearity deviation only becomes appreciable at greater strains:

10,000 μm/m results in 1%

100,000 μm/m results in 10%

To a large extent, this is fortunately compensated for by the linearity deviation of the quarter bridge circuit.

### Linearity deviation of quarter bridge circuit

Small relative changes in resistance are commonly analyzed with a Wheatstone bridge circuit. As noted above, only one strain gauge per measurement point is usually used in the ESA. Thus, the other bridge resistances are strain-independent. The correct relationship for the stress ratio in this case is:

Although the relationship is non-linear, linearity is assumed in practical measurement applications (whether or not this is known) and the approximation equation

is used. The relative deviation resulting from this simplification can be calculated with eq.

A strain of 1000 μm/m (with k = 2) results in a change of 0.2% in the relative resistance.

The relative measurement error as determined with eq. 17 is -0.1%. This is equivalent to an absolute deviation of -1 μm/m. The deviation from the true value is negligible.

Appreciable linearity deviations occur at greater strains however, as noted above:

10,000 μm/m results in a deviation of -1%,

100,000 μm/m results in a deviation of -9.1%.

When Constantan strain gauges are used (non-linearity similar in terms of magnitude, but with the opposite sign), the two deviations largely cancel each other out and therefore do not need to be considered any further.

*Note however that no compensation is ever completely successful, especially given that the gauge factor deviates somewhat from 2 and the actual static characteristic curve does not exactly match the empirical eq. 12.*

## Summary of partial uncertainties

The individual uncertainties are difficult to correlate with each other. However, to the extent they can be (material after-effects and strain gauge creep, linearity deviation of the strain gauge and quarter bridge circuit), their effects cancel each other out to some extent. Therefore, it is permissible to combine the individual uncertainties with root sum square. The values in bold type above are used to achieve a result for the example.

The uncertainty of the strain measurement is just under 3%. The stress measurement reaches almost 6% of the measured value.

That percentage multiplied by the measured value gives the deviation in μm/m or N/mm2. The uncertainty of the modulus of elasticity is generally responsible for the largest amount of error in non-zero-point related measurements in ESA. Additional uncertainties must be considered for zero-point related measurements.

#### Read on...

Read more about this topic in **Part 4 of our series of articles** on "Measurement accuracy in experimental stress analysis".