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Instructor: C.C. Swan University of Iowa
Period #27: Spring Semester 2008
Textbook: Sections 10.5-10.
Topic: Strain Rosettes and Hooke’s Law
Objectives:
1. Introduce Rosettes for measuring of strains
2. Revisit Hooke’s Law
3. Solve Relevant Example Problems
1. Strain Rosettes
In experimental mechanics measurement of strain states is very common. One way to measure strains on the surface of a mechanical specimen is to bond an electrical strain gage to the surface. The gage shown below responds only to extension or contraction along its axis. To measure the complete state of strain εx , εy, γxy at a point on the surface of a material, a cluster of three gages called a rosette, is typically used.
If the extensional strain can be measured along the three axes with orientations θa, θb , θc then the in in-plane strain components εx , εy, γxy can be determined by solving the system of transformation equations:
xy
y
x
c c c c
b b b b
a a a a
c
b
a
cos sin sin cos
cos sin sin cos
cos sin sin cos
2 2
2 2
2 2
Two specific examples of strain rosettes are the 45° rosette with θa, θb , θc being 0°, 45°, and 90°, respectively, and the 60° rosette which has θa, θb , θc being 0°, 60°, and 120°, respectively.
A single strain gage.
Strain rosette
Instructor: C.C. Swan University of Iowa
For the 45° rosette, the transformation equations yield:
Similarly, for the 60° rosette, the transformation equations give:
2. Material Property Relationships
Here, we will look at the multi-axial form of Hooke’s Law for a linear elastic and isotropic material.
A material point will generally be experiencing multiple different stress and strain components simultaneously as shown below:
( )
xy (^ b c )
y b c a
x a
xy b^ (^ a c )
y c
x a
Instructor: C.C. Swan University of Iowa
For a linear, isotropic elastic material, there are only two independent elastic parameters. Knowing any of these two, the others can be determined.
- If E and ν are known, the shear modulus can be determined by:
E
G
- If E and ν are known, the bulk modulus can be determined by:
E
K
Under a triaxial state of stress σx , σy, σz, the bulk modulus relates the mean normal stress σm to the volumetric strain e of the material:
The mean normal stress σm of the material is the mean of σx , σy, σz or
σ (^) m = 31 ( σ (^) x + σ y + σ z )
The volumetric strain of a material (in small deformation theory) is:
Finally:
( ) ( ν)
E
e (^) x y z m
( )
Ke
Ee
σ m = 31 − 2 ν =
Instructor: C.C. Swan University of Iowa
Example 1 (10.32) The 45 0 strain rosette is mounted on a steel shaft. The following readings are obtained from each gauge:
ε a = 800 ( 10 −^6 ), ε b = 520 ( 10 −^6 ), ε c =− 450 ( 10 −^6 ).
Determine (a) the in-plane principal strains and their orientation.
Instructor: C.C. Swan University of Iowa
Example 3 (10-48): The spherical pressure vessel has an inner diameter of 2m and a thickness of 10mm. A strain gauge having a length of 20mm is attached to it, and it is observed to increase in length by 0.012mm when the vessel is pressurized. Determine the pressure causing this deformation, and find the maximum in-plane shear stress, and the absolute maximum shear stress at a point on the outer surface of the vessel. The material is steel, for which Est =200GPa and νst =0.3.
