Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

pRoBLEmS
p13.16–p13.17 The principal strains are given for a point in a
body subjected to plane strain. Construct Mohr’s circle, and use it to
(a) determine the strains ε x , ε y , and γ xy . (Assume that ε x > ε y .)
(b) determine the maximum in-plane shear strain and the absolute
maximum shear strain.
(c) draw a sketch showing the angle θ p , the principal strain deformations,
and the maximum in-plane shear strain distortions.
Problem ε p1 ε p2 θ p
P13.16 780 µε 590 µε 35.66°
P13.17 −350 µε −890 µε −19.50°
p13.18–p13.24 The strain components ε x , ε y , and γ xy are
given for a point in a body subjected to plane strain. Using Mohr’s
circle, determine the principal strains, the maximum in-plane shear
strain, and the absolute maximum shear strain at the point. Show
the angle θ p , the principal strain deformations, and the maximum
in-plane shear strain distortion in a sketch.
Problem e x e y γ xy
P13.18 380 µε −770 µε −650 µrad
P13.19 760 µε 590 µε −360 µrad
P13.20 −1,570 µε −430 µε −950 µrad
P13.21 475 µε 685 µε −150 µrad
P13.22 670 µε 455 µε −900 µrad
P13.23 0 µε 320 µε 260 µrad
P13.24 −180 µε −1,480 µε 425 µrad
13.7 Strain measurement and Strain Rosettes
Many engineered components are subjected to a combination of axial, torsion, and bending
effects. Theories and procedures for calculating the stresses caused by each of these effects
have been developed throughout this book. There are situations, however, in which the
combination of effects is too complicated or uncertain to be confidently assessed with
theoretical analysis alone. In these instances, an experimental analysis of component
stresses is desired, either as an absolute determination of actual stresses or as validation for
a numerical model that will be used for subsequent analyses. Stress cannot be measured.
Strains, by contrast, can be measured directly through well-established experimental procedures.
Once the strains in a component have been measured, the corresponding stresses
can be calculated from stress–strain relationships, such as Hooke’s law.
Strain Gages
Strains can be measured by using a simple component called a strain gage. The strain gage
is a type of electrical resistor. Most commonly, strain gages are thin metal-foil grids that are
bonded to the surface of a machine part or a structural element. When loads are applied, the
object being tested elongates or contracts, creating normal strains. Since the strain gage is
bonded to the object, it undergoes the same strain as the object. The electrical resistance of
the metal-foil grid changes in proportion to its strain. Consequently, precise measurement
of resistance change in the gage serves as an indirect measure of strain. The resistance
change in a strain gage is very small—too small to be measured accurately with an ordinary
ohmmeter; however, it can be measured accurately with a specific type of electrical circuit
known as a Wheatstone bridge. For each type of gage, the relationship between strain and
resistance change is determined through a calibration procedure performed by the manufacturer.
Gage manufacturers report this property as a gage factor GF, which is defined as
the ratio between the unit change in gage resistance R to the unit change in length L:
RR / RR /
GF = ∆ = ∆
∆LL
/
ε avg
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