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Unformatted text preview: CE 231  Lab #2: Review of Mohr‘s Circle for Stresses Consider the following example: The state of stress in 3 body can be represented by the stress tensor shown
below. Usntg this information and the steps given in cinss constmct Mohr's circle. 5 n3 0
eﬁz—s "3 00m)
0 o 0 Step #1: Draw the State of Stress on the Element. Remember there is no such thing as positive and
negative using the approach described in class. 5 ksi 3 ksi 3 ksi Step #2: Drew the Normal and Sheer Axis. Remember to identify the portion of the x ssis to the right of 0 ns tensile and to the left of 0 as compressive. The portion of the ‘1’ nxis above 0 is for n ciock wise sheer
and the portion of the Y axis below 0 is for counter clock wise sheer. i i Tension I... III
lullIm
III III
I... I...
IIIIIIII
C CW III III .EEEEIEE “CW
IllIll
IIIIII
IIIIII
lullIm
I... I...
IllIII
IIIIIII
IllIII
IllC] Step #3 Draw the Nermal and Shear Stresses fer the ‘X~Faee’ and 'YFaee’. Step #5: Determine the Prineipal Stresses and Maximum Shear Stresses. One way te determine the nrientatinn ef the element and new the orientation needs to be changed will he
presented en Wednesday. Teda},r we will 1 ddk at an alternative called the ‘PnieMethed’ Step #6: Locate the Pole The pole by deﬁnition is one point which can be usedto identify the orientation
(inclination) of the element on which the stress (or strain acts) To ﬁnd the pole construct a line that passes through the point on the graph that corresponds to the stress on
a given face that is parallel to that face as shown below. The point where these lines intersect is called the Pole I. I
UCompressiO El 7 UTBHSEDH
l. I! I'lll'll _
y r ‘ Illu
:55? w
__ h __ I' I at! tea.
‘3 Now what information can he gathered from the ‘Pole’. If we construct a line from the pole to any other
point on the circle the slope of this line represents how the element must he rotated. (Note this is not like
the angles discussed in class where they are doubled on Mohr’s Circle, with the pole method the angles are as they appear.) For example, the element shots'11 in the ﬁgure below et'trresoonds with the XFace and Y
Faee Lines we have drawn. 3 ksi
3 hat 5 ksi ks" as
I. E
8 ksi ""' All E ‘5 OCompression I
s r  v ,4
_ “hall II ‘ ._ __ . —r'I—Il\u _
 H a ""“— Hf” — porn "‘
on TV
[—3 If we desire to know what the orientation of the element is when the maximum principal Stress is reached
we can connect the pole and the position that we went to describe as shown below, . S’IIII
GCompreeeio lulu 4 O.Tnsion
em EIIEEIVIII
II“ E. lam: m m
IlsaI; u .u . i t If we wont to know the orientation where the niexiiiiorn shear stress oeeors we can perform a similar
operation. ugly: * II
GTeneion ﬁ Now you Try this for the following problem 5 2 0
0U = 2 0 0(MPa)
0 0 0 E' P a. __.._....
I ill I I
I=_ I III i G ' III III... 
Compresggn GTBHSan   _._ .i ... . ._._._._ _.,.. .. I  ...u.— .u u." u_..—_..
__— —._._._..,._.— __ ___ .—.u— .— ———
n _._ P New Mnhr‘s Circie fnr nnntnnr normalized prnpnrty (strnin) is very siniilnr, lion'ernr normal strass is
rapinca with nnrrnn] strain :1an shearing strnsa is replaced with shaming strnin f2. Strain wiil bra COVEI'Ed
later this i ruck ricncribing how it (3an be, thought of as n wn}; of viewing how [he elcrnant deforms. 1’s 12‘2 Shear Strain
+1131, down, +vs up Norma! Strain Normal Strain
Compre s on Tendon HE Sh arStrain
sv, down, an: up on will be given enough inform ation that is necessary for the l
in the upcoming classes to describe ah, however details will he provided
the followingmaterial properties:
Stratngm (Change in Length Per Unit Length)  = ~—
i' i L
. . . 1 . G
Modulus of Elastrcrty (Like a stiffness) E 2 1+
gdxio!
, . . . . . . ETmnrverse
Poisson’s Ratio (How Material Deforms in Alternative Direction) v = *————————
East's;
How do we measure strains? Problem Strain is small Le, 10"5 infin  Shout This gage needs to be small, environmentally stahle, cheap, and remote use is encouraged.
of strain measurement have become available including, laser, optical, electrical, and
these gages it is assumed that deformation (strain) is uniform over :1 sn tested and the material is continuous. The most conunon it discovered in 1856 by Lord Kelvin. At the conclusion of his work he concluded that: i) that the electrical resistance of wires increased as they :w'ere pulled and this increase was
proportional to the increase in strain, d develop a ‘strain gage’.
Various types
mechanical. in all tall section of the material being pproach is the electrical resistance strain gages Percent Change
in Resistance Percent Strain 2) different materials respond d For example a typical to sensitivin of 2.1, or a in is Nickle, 20% Chromium, 3%
Aluminum, and 3% Iron, and 3) a "Wheatstone bridge’ to improve the accuracy oi“ these ifferentiy tie, different sensitivity)
atetiai is 45% nickel and 55% Copper (Constantan) which has a
aterial called Karma has 74‘? measurements, (A wheat stone hrid
Is an ru‘rangcment of gages as shown helou' where go
the following erpiatiou is
true KR, = R2 R3) Voltage
Drop V 1930’s Simmons and Ruge independently developed and introduced bonded wire strain gages which could
be directly glued to the surface of specimens to measure volume changes. These were popular between
WW1 and W11 now however mostlyr bonded metal foil gages that were introduced in 1952 by Saunders and Roe. A Metal Foil is etched in this process. If we want to ‘Map Strains‘ on any surface we can go ‘crazy’ putting gages everywhere or we can
detenpine the most critical locations and place the gages there. One common approach is to use a “Strain
Gage Rosette”. This will measure strain in 3 directions, it is preferred that these be the principal directions however this is not always known.
Use the example of lvlohr's Circle for strains from the Book Today’s Experiment: Two Loads will he applied to the Aluminum Member shown. This member is
equipped with two individual strain gages and a strain gage rosette. Look at the Speeirnen to determine the conﬁguration and to see the gages. Thickness : 0.479 inches
Height: 6 inches
Width : 6 inches Vertical " ' C . Horiao ital For this test use the strains measured in legs A, B, and C of the Rosset
plane strain and obtain the principal str te to draw Mohr's Circle for the in
ainsl maximum shearing strain 3, and the angles of orientation. Compare the maximum strain and the vertical strain, Compare the minimum strain and the horizontal
strain. Discuss any differences or similarties. Calculate the normal stress using a book value for the elastic modulus
this to compute the theoretical lo and maximum strain and then u
no. Compare this to the applied load SC: You are to Calculate the Followin List of Items 1» Calculate Possions Ratio for Aluminum using the following equations anal tiescrilie an),r differences
1, _ _ is;
experiment “— ,
(E: if _ ___& mitt
Alohat'f'in‘t'c' — _
:truﬁ;
i (fialeulate the elongation of ‘the plate.
 Also Make. a 'l‘alile for al 1 the strains for the two different l
* Plot One hilohrs Circle on Graph Paper {
it Show Element
1* Show a l (and lot'els
"Using the aoortjipriate scale
orientation on hIolirs Ciroie '
iigu re of the eaperiinent 9 and draws to scale) ...
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 Spring '08
 JasonWeiss
 Force, Shear Stress, Strain, Shear strain

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