Question

A rosette strain gage is an electromechanical device that can measure relative surface elongations in three directions. Bonding such a device to the surface of a structure allows determination of elongational strains in particular directions. A schematic of one such gage is shown in the following figure, and the output of the device will provide data on the strains along the gage arms a, b, and c. During one application, it is found that ea = 0.001, ep = 0.002, and ec = 0.004. Using the two-dimensional strain transformation relations, calculate the surface strain components ey ey, and exy. 60" 2VVVVVVVV

Answer 1

We know that

$e_{\Theta }=\frac{e_{x}+e_{y}}{2}+\frac{e_{x}-e_{y}}{2}cos2\Theta +e_{xy}sin2\Theta$

when $\Theta_{a} =30^{o}$ then   $e_{a}=.001$

$\Theta_{a} =90^{o}$ then   $e_{a}=.002$

$\Theta_{a} =150^{o}$ then   $e_{a}=.004$

by substituting given value we get three equation and three unknown

after solving these equation we get required answer.

& De=1508 26=80 X Ca-0.001 when Da=30° 26=0.002 whes 06 = 90° @= 0.004 when Oc = 150 we know that @ge extes + - Ey cos 20 + evt ne inze when Qa= 30 faz extes + @q- G Cosco + Boxy sin60° 90.00).3 Cztett er der x (0-8) + egy > 2(exter) + (Cx-en) + 13 Exy = 0.004 3 3 le+ er + 13 Czy = 0.004 -

When Ob=90 = @xt By + Cacer cos180° + entry siniso es = exter - ex-er + exy xo 9 0·002 = 92 ter – etter - @y= 0.002 - ® when c= 150 ecz Czter + ex-er Cos 300° + Bxy Sin300 De z extex + extes + Catetxea +0.004= [2 (Cx+ @r) + (@z-er) – VB Exy Jx4 $ 32eter-v3 @xy = 0.016 — © Substitute the value of er in equation © 3) 3(x+ 0.002 + V3 Cxy = 0.004 9 32x+v3 @xy = 0.002 0 again Substitute the value of er in equation ☺ 3 le + 0.002-13 lxy = 0.016

- 3 ex-Bexy = 0.014 from equation ® + ☺ 32x+ V3 Czy = 0.002 +30x - Brexy = 0.014 6C2= 0.016 : ex = L = 0.002667 again from equation. ☺ 3x373 + 13 Czy = 0.002 S2 Cryo -0.003464 * Required Sumsface strain component me Ex= 0.002667 (Ans) er = 0.002 (ANS) Cryo -0.0034641. Gans)