Karl Hoffmann

An Introduction

to Measurements

using Strain Gages

With 172 figures and tables

Publisher: Hottinger Baldwin Messtechnik GmbH, Darmstadt

Contents IX

Contents

1 1.0.1 1.0.2 1.0.3 1.0.4 1.0.5 1.0.6 1.0.7 1.0.8 1.1 1.1.1 1.1.2 1.2

2

2.1 2.1.1 2.1.2 2.1.3 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.4 2.4.1 2.4.2 2.4.3

3 3.1 3.1.1

Page

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1 Metal strain gages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2 Semiconductor strain gages 8 Vapor-deposited (thin-film) strain gages 9 Capacitive strain gages Piezoelectric strain gages Photoelastic strain gages Mechanical strain gages Other systems

10 11

" 11 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12

" 12 The operating principle of the strain gage 13 Metal strain gages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13 Semiconductor strain gages 15 The measurement system . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17

Terms and units of measurement used in strain-gage technology 19 Strain: Definition and unit of measurement " 19 Absolute change of length " 19 Relative change of length or strain 21 Unit of measurement for strain [2-1] 21 Mechanical stress: Definition and unit of measurement 23 Normal stress 24 Shear stress " 25 Residual stress, thermal stress 28 Stress states " 28 Material parameters 29 Modulus of elasticity, definition and units 29 Shear modulus 33 Poisson's ratio 33 Thermal expansion " 35 Strain-gage loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37 Static measurements (zero referenced) . . . . . . . . . . . . . . . . . .. 37 Quasi-static measurements " 38 Dynamic measurements (non-zero referenced) 38

Selection criteri.a for strain gages 39 Range of application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41 Stress analysis, model measurement techniques, biomechanics 41

ContentsXII Contents

7.1.2 Thermal output of a quarter bridge in a three-wire eonfiguration . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 176

8.2.2.4 0 al

7.1.3 Temperature eompensation of a quarter bridge 8.2.3 1\1

with eompensating strain gages. . . . . . . . . . . . . . . . . . . . . . . .. 178 8.3 D

7.1.4 Compensation of thermal output with the tt double quarter or diagonal bridge " 180 8.4 Sl

7.1.5 Compensation for thermal output using v. the half bridge circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 181 8.4.1 1\1

7.1.6 Compensation for thermal output with 8.4.2 1\1

the full bridge cireuit ... . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 182 8.4.3 S: 7.2 The influenee of lead resistanees 183 w

7.2.1 Simple quarter bridge cireuit . . . . . . . . . . . . . . . . . . . . . . . . .. 184 8.4.4 l\!

7.2.2 Quarter bridge in a three-wire cireuit 187 8.4.4.1 T

7.2.3 Quarter bridge with eompensating strain gage 188 8.4.5 1\1

7.2.4 7.2.5 7.2.6 7.2.7 7.3

7.3.1

Double quarter or diagonal bridge Half bridge cireuit Full bridge cireuit Error eorreetion using the gage faetor selector . . . . . . . . . . .. Eliminating eable effects with special cireuits in the measuring amplifier HBM bridge (citation from [5-2])*)

189 190 190 192

193 193

8.4.6 8.4.7 8.4.7.1

8.4.7.2 8.4.7.3

m

1\1

1\1

C o ~

S

7.3.2 7.4 7.4.1 7.4.2

The six-wire cireuit The influenee of eable eapacitanees . . . . . . . . . . . . . . . . . . .. Capacitive unsymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Phase rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

196 197 199 203

9 9.1 9.2

!I ( (

7.5 Correetion of the transverse sensitivity of astrain gage. . . .. 204 9.2.1 7.5.1 Correetions for linear strain gages. . . . . . . . . . . . . . . . . . . . .. 206 9.2.1.1 7.5.2 Correetion for strain gage rosettes . . . . . . . . . . . . . . . . . . . . .. 210 9.2.2 7.5.2.1 X rosettes 0°/90° 210 9.2.3 7.5.2.2 Rosettes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 212

9.2.4 8 Hooke's Law for the determination of

material stresses from strain measurements 215 9.2.5 8.1 The uniaxial stress state . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 215 9.2.6 8.2 The biaxial stress state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 219 8.2.1 The biaxial stress state with known principal direetions .... 219 10 8.2.2 The biaxial stress state with unknown principal direetions .. 222 8.2.2.1 Measurements with the 0°/45°/90° rosette 223 11

8.2.2.2 Measurements with the 0°/60°/120° rosette 224 8.2.2.3 The determination of the principal directions 224

Contents Contents XIII

176

178

180

181

182 183 184 187 188 189 190 190 192

193 193 196 197 199 203 204 206 210 210 212

215 215 219 219 222 223 224 224

8.2.2.4

8.2.3 8.3

8.4

8.4.1 8.4.2 8.4.3

8.4.4 8.4.4.1 8.4.5

8.4.6 8.4.7 8.4.7.1

8.4.7.2 8.4.7.3

9 9.1 9.2

9.2.1 9.2.1.1 9.2.2 9.2.3

9.2.4

9.2.5 9.2.6

10

11

Other ways of determining the principal normal stresses and their directions 227 Mohr's Stress Circle 227 Determination of residual stresses according to the drill-hole method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 231 Strain measurements and stress analysis for various loading cases 234 Measurement on a tension/compression bar 235 Measurements on a bending beam 238 Symmetrical and asymmetrical cross-sectional beams loaded with both an axial force and a bending moment . . . . . . . . .. 241 Measurements on a shaft under torsion (twisted shaft) 248 Transferring the measuring signal from rotating shafts 252 Measurement on a twisted shaft with superimposed axial force and bending moment. . . . . . . . . . . . . . . . . . . . . .. 256 Measurements on a shear beam 258 Measurement of thermal stresses 261 Comparison of measurements on a free and on a restrained object 263 Measurement with a compensating piece 264 Separate or later determination of the thermal output 264

Measurement accuracy 265 Causes of measurement errors 266 Calculating the degree of random deviation in a measurement series 267 Test requirements 267 The Gaussian distribution 268 Arithmetical mean 270 (Empirical) standard deviations sand coefficient of variance v 270 Confidence limits and confidence range for the expected value J.l •••••••••••••••••••••••••••••••• 271 Measurement uncertainty" 272 Measurement result 274

Literature , 275

Index 285

List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 291