Strain transformation equations and explanation; function, design and biomedical applications of strain gauges

1. Strain Transformations
BIOE 3200 - Fall 2015
Note: These slides were adapted from slides by Dr Abhay Kumar on
slideshare.com.
Image above from http://trilion.com/optical-strain-measurement-biomechanic/

2. Learning objectives
 Calculate strains in multiple planes
(strain transformations) within a rigid
body
 Describe the function, design, and
application of a strain gauge.
BIOE 3200 - Fall 2015

3. Plane Strain Loading
x
y
Consider the 2D case where all
elements of the body are
subjected to normal and shear
strains acting along a plane (x-y);
none perpendicular to the plane
(z-direction)
z = 0; xz = 0; zy = 0

4. x
y
x’
y’
A
State of Stress at A
 x
y
xy
xy
’x=?
’xy=?
Plane Strain Transformations
Similar to derivation of equations for stress
transformation, replacing:
 by , and
 by /2

5. Plane Strain Transformations
Sign Convention:
Shear strain ( ): decreasing angle is positive
Normal strains (x and y): extension is positive
x
y
before
x
y
after
x positive
y negative
 positive

6. Strain
Transformation
Equations
𝜖 𝑥𝑥
′
=
𝜖 𝑥𝑥 + 𝜖 𝑦𝑦
2
+
𝜖 𝑥𝑥 − 𝜖 𝑦𝑦
2
𝑐𝑜𝑠2𝜃 +
𝜸 𝑥𝑦
2
𝑠𝑖𝑛2𝜃
𝜖 𝑦𝑦
′
=
𝜖 𝑥𝑥+𝜖 𝑦𝑦
2
+
𝜖 𝑦𝑦−𝜖 𝑥𝑥
2
𝑐𝑜𝑠2𝜃 −
𝜸 𝑥𝑦
2
𝑠𝑖𝑛2𝜃
𝜖 𝑥𝑦
′
=
𝜸 𝑥𝑦
2
′
=
𝜖 𝑦𝑦−𝜖 𝑥𝑥
2
𝑠𝑖𝑛2𝜃 +
𝜸 𝑥𝑦
2
𝑐𝑜𝑠2𝜃
BIOE 3200 - Fall 2015
Image from
http://www.efunda.com/
formulae/solid_mechanic
s/mat_mechanics/calc_st
rain_transform.cfm

7. Strain Gauges
Function: Measure normal strain ()
Design: Stretching of wires in gauge during
deformation results in a change in electrical
resistance

8. Strain Gauge Rosettes
• Function: Measure normal strain () in three
directions; use these to find x, y, and  xy
• Design: Two or three arranged at known
angular positions
45° Strain Rosette
x
45
°
45
° 0
90
45
x = 0
y = 90
 xy = 2 45 – (0 + 90)
measured

9. Strain gauge applications
 Measuring strains in devices or
structures
◦ Strain gauges for snow board testing:
https://www.youtube.com/watch?v=h1CvL
ZQABjE
◦ Strain gauges in human motion research:
http://file.scirp.org/Html/7-
4000046_33327.htm
 Validating computer models and
predicted strains
BIOE 3200 - Fall 2015