Calculating Principal Strains using a Rectangular Strain Gage Rosette

Transcription

1 Clulting Prinipl Strins using Retngulr Strin Gge Rosette Strin gge rosettes re used often in engineering prtie to determine strin sttes t speifi points on struture. Figure illustrtes three ommonly used strin gge rosette onfigurtions. Eh of these onfigurtions is designed with speifi tsk in mind. The fr right rosette in Figure is referred to s tee rosette. Tee rosettes re two element rosettes nd should only e used when the prinipl strin diretions re known in dvne from other onsidertions. The middle rosette in Figure is referred to s retngulr rosette nd the fr left s delt rosette. These three element rosettes re used in pplitions where the prinipl strins re unknown. There re mny other prmeters tht need to e tken into onsidertion when hoosing strin gge. Literture from mnufturers of strin gges provides some guidne in hoosing the proper strin gge for given pplition. Figure : Exmples of ommon Strin gge Rosettes enountered in engineering prtie. loop The primry fous of this tretise is the retngulr rosette. Figure illustrtes the numering sequene nd geometry tht will e used in the following disussion. Figure 3 shows n tul retngulr strin gge rosette instlled on struture with the led wires soldered onto the terminl ts. Figure : Illustrtion of the orienttion of the retngulr strin gge rosette eing evluted in this exmple. In the reminder of this doument the proedures for lulting prinipl strins from the strins tht re red from retngulr strin gge rosette re presented. First the orretion of the rw strin redings for the lods trnsverse to the prinipl xis of the individul gges is presented. Figure 3: Retngulr strin gge rosette This is followed y summry of the lultions instlled on struture with led wires used to ompute prinipl strins diretly from the tthed. orreted strin gge dt. Finlly method used to lulte the prinipl strins from the orreted strin vlues using Mohr s Cirle is disussed. Ronld B. Buinell, Ph.D., P.E. Union College Deprtment of Mehnil Engineering Shenetdy, NY 38

2 Correting Strin Gge Dt Eh of the strin gges in rosette is tthed to seprte ridge iruits through the led wires. The type of ridge iruit used is funtion of the pplition eing onsidered. Under lod, three strin vlues re reorded t eh lod inrement. These rw strin redings re designted s the norml strins ˆ, ˆ, nd ˆ. The ^ designtes tht the strin is rw or unorreted strin. Strins on rosette require orretion euse these gges re used on strutures tht re sujeted to i-xil sttes of stress. Error ours s the loops in the strin gge grids illustrted in Figure re strethed trnsverse to the primry diretion of the gge. The following equtions summrize how the strin redings re orreted for the trnsverse loding of the gges in retngulr rosette []. ˆ ˆ # $ " # K $ K # # $ " # K % $ K # K ˆ ˆ ˆ # $ " # K K #[ # $ " # K # $ K & # $ " # K # $ K ] % $ $ K $ K # K # $ K ˆ ˆ # $ " # K $ K # # $ " # K % $ K # K 3 where,, re the orreted strins; " is Poisson s rtio for the mteril used in lirtion y the strin gge mnufturer typilly.85; nd K, K, K re the trnsverse sensitivity oeffiients for the gges tht re found on the mnufturer s dt sheet. Diret Clultion of Prinipl Strins from Correted Strins The orreted strins re used in the lultion of the prinipl strins. The equtions used to lulte the two prinipl strins from the orreted strins re summrized elow []. % # & & # & # 4 % # & $ # & # 5 where nd re the two in-plne prinipl strins. The lultion of the prinipl ngle ' the ngle etween the xis illustrted in Figure nd the mximum prinipl strin from the orreted strins is given y Eqution 6 []. Ronld B. Buinell, Ph.D., P.E. Union College Deprtment of Mehnil Engineering Shenetdy, NY 38

3 # tn ' % 6 If the smple is frited from mteril tht is liner, isotropi, nd homogeneous the prinipl stresses n e omputed diretly from the orreted strins s follows []. & * % E # + & # & # # $ " # &", -. 7 & * % E # + $ & # # $ " # &", -. 8 where E is the modulus of elstiity of the struturl mteril nd " is Poisson s rtio for the struturl mteril. Clulting Prinipl Strins Using Mohr s Cirle Formuls re often forgotten or rememered inorretly. Although it is lwys possile to look formuls up, when dt is eing olleted in remote lotions referenes re not lwys ville. An lterntive wy of lulting the prinipl strins nd the prinipl ngle from strin dt is through the use of Mohr s irle for strin. This tehnique does not require the use of Equtions 4, 5, or 6. The development of the tehnique strts with the orreted strins,, nd. To help the disussion of this development, n exmple of retngulr strin gge rosette mounted on everge n for the purpose of determining the pressure internl to the sod n is used. Figure 4 illustrtes the orienttion of the retngulr strin gge rosette with respet to the mjor xes of the sod n. The strin gge nomenlture used in Figure 4 is the sme s tht used in Figure the orienttion of the gges in the two figures is different to illustrte tht the lultions re independent of gge orienttion. It is very xil Figure 4: Illustrtion of the orienttion of the retngulr strin gge rosette on pressurized ylindril n hoop Ronld B. Buinell, Ph.D., P.E. Union College 3 Deprtment of Mehnil Engineering Shenetdy, NY 38

4 importnt to note tht the mjor xes prinipl stress nd strin xes xil nd hoop xes of the everge n do not lign with ny of the gge xes. In n experiment retngulr strin gge rosette is mounted on everge n, the n is opened, nd the following strins re reorded. * = in/in 9 * = -7-6 in/in * = -4-6 in/in These strins result from depressurizing the n nd is why they re designted with *. In this exmple the n is pressurized in mnner suh tht the mteril ehves liner-elstilly. If the strin gges were mounted while the n ws opened nd then pressurized, the mgnitude of the strins would e the sme s in Equtions 9,, & ; however, they would ll e positive. Sine the primry interest here is the lultion of the pressure inside the n, the positive strins will e used nd re summrized elow without the * designtion. * = = 37-6 in/in = 37/ * = = 7-6 in/in = 7/ 3 * = = 4-6 in/in = 4/ 4 Strin is often referred to in units of miro-strin / in engineering pplitions. In the reminder of this susetion the determintion of the pressure in this n will e disussed. First the onstrution of Mohr s irle for strin for the purpose of determining the prinipl strins will e desried. This will e followed y disussion of how the prinipl stresses re lulted from the prinipl strins for i-xil stte of stress. Finlly, the pressure in the sod n will e lulted using pressure vessel theory. Constrution of Mohr s Cirle for Strin The onstrution of Mohr s irle for strin strts with drwing horizontl norml strin nd vertil tensor sher strin / xes s illustrted in Figure 5. The orreted strins lulted in Equtions, 3, nd 4 re plotted on the horizontl xis in Figure 5 nd designted with. The tensor sher strins tht ompny these norml strins re unknown nd therefore need to e lulted s prt of the onstrution. Beuse retngulr strin gge rosette is eing used, the enter of Mohr s Cirle is loted on the horizontl xis t the point given y the verge of the two perpendiulr gges, ve ve is plotted on Figure 5. % # & % #37/ & 4 / % 89.5/ 5 ve Ronld B. Buinell, Ph.D., P.E. Union College 4 Deprtment of Mehnil Engineering Shenetdy, NY 38

5 Now irle tht is entered t ve is sried round the three strin vlues plotted on the horizontl xis. It is importnt to mke sure tht the irle ompletely enompsses ll three strins. The dimeter of the irle is not importnt t this point, more ext irle n e drwn fter the onstrution is omplete. After the irle is drwn, vertil lines re projeted in oth diretions from the three lotion of the three sttes of strin for gges,, nd on Mohr s Cirle,,. The proper lotion for eh gge on the irle is determined y understnding tht the ngulr orienttion of the gges on the irle -- lokwise must e the sme s the ngulr orienttion of the gges on the struture -- lokwise nd tht ny ngle mesured on the irle is twie the orresponding ngel on the gge. Sine the gges re 45 prt on the rosette shown in Figure 4, they hve to e 9 prt on Mohr s irle shown in Figure 5. It is therefore pproprite to ssume tht the three gges re in three different qudrnts of the irle lokwise from to. Using the orienttion nd ngulr onstrints disussed in the previous prgrph the three of six points on Mohr s irle tht represents the stte of strin on the surfe n e determined. This proess strts y determining whih of points nd represents the true stte of strin. Consider first point, moving to the next lokwise qudrnt 9 lokwise in Figure 5 point is enountered. Strin gges nd re lokwise 9 prt on the rosette shown in Figure 4 nd therefore nd should e lokwise 8 prt on Mohr s irle. Sine this is not the se, n not represent the stte of strin on the surfe of the n. Now onsider point, moving to the next lokwise qudrnt 9 lokwise point is enountered. Strin gge nd re lokwise 45 prt on the rosette shown in Figure 4 nd therefore nd should e lokwise 9 prt on Mohr s irle. Sine this is the orienttion oserved on Mohr s irle in Figure 5 points nd represent the stte of strin of the n. Continuing from point to the next lokwise qudrnt 9 lokwise from point in Figure 5, point is enountered. Point nd point re lokwise 45 prt on the rosette in Figure 4 nd therefore nd should e lokwise 9 prt on Mohr s irle; this lso is the oserved orienttion in Figure 5. It is therefore onluded tht,, nd on Mohr s irle for strin represent the sttes of strin eing mesured y the retngulr strin gge rosette. These points re designted with the filled irles tht lie on Mohr s irle, ut not on the horizontl xis in Figure 5. The dimeter of the irle needs to e determined. The two tringles sried inside the irle using solid lines will e used to ssist in this lultion. The horizontl distne etween points nd is the horizontl side of the lower tringles illustrted in Figure 5. This horizontl distne is lulted using the results from Equtions nd 4 s follows. - = 37/-4/ =835/ 6 The horizontl dimension of the higher solid tringle in Figure 5 is twie the distne etween nd ve. The lultion of this vlue follows using the results from Equtions 3 nd 5. Ronld B. Buinell, Ph.D., P.E. Union College 5 Deprtment of Mehnil Engineering Shenetdy, NY 38

6 ve - = + - = # 89.5/-7/ = 9/ 7 ve - = + - ve =½ + ve - * #' p p h h #' - Figure 5: Mohr s irle for strin used to illustrte the lultion of prinipl strins from the orreted strins olleted off retngulr strin gge rosette. Ronld B. Buinell, Ph.D., P.E. Union College 6 Deprtment of Mehnil Engineering Shenetdy, NY 38

7 The determintion of the dimeter of Mohr s irle in Figure 5 requires tht either the vertil distne h or h is known. A heuristi pproh is tken to determine the vlues of these vertil distnes. Consider rotting the strin gge rosette illustrted in Figure 4 45 lokwise. This would e the sme s moving point to point on Mohr s irle nd point to point *. In doing so the lower tringle is rotted into the upper tringle, thus onluding tht the two tringles re the sme tringles just rotted 9 on Mohr s irle. Therefore, h = ve - = + - = # 89.5/-7/ = 9/ 8 h = - = 37/-4/ =835/ 9 The dimeter of the tringle n now e lulted using Pythgors s theorem. d % 9/ & 835/ % 8/ r % d % 64/ The prinipl strins p nd p n now e lulted using the results in Equtions 5 nd. p = ve + r = 89.5/ + 64/ = 433.5/ = 434/ p = ve - r = 89.5/ - 64/ = 5.5/ = 6/ 3 The ngle etween the point nd point p on Figure 5, twie the prinipl ngle ', n e lulted using the results in Equtions 8 nd 9. 9/ tn # ' % 3 # ' % ' % / With the prinipl strins lulted, the prinipl stresses n now e determined. Determintion of the Prinipl Stresses from the Prinipl Strins The reltionship etween the prinipl stresses nd nd the prinipl strins p nd p is referred to s Hooke s Lw. For the i-xil stte of stress in this prolem, Hooke s lw redues to the following two equtions. " # E E % $ 5 p " # E E % $ 6 p Ronld B. Buinell, Ph.D., P.E. Union College 7 Deprtment of Mehnil Engineering Shenetdy, NY 38

8 where E is the modulus of elstiity or Young s modulus nd 4 is Poisson s rtio. Equtions 5 nd 6 re solved simultneously in order to form equtions for nd s funtions of p nd p. E % # p &" # p 7 $ " E % # p &" # p 8 $ " For n luminum sod n the modulus of elstiity is pproximtely.4 Msi nd Poisson s rtio is.3. The results in Equtions nd 3 n now e sustituted into Equtions 7 nd 8 in order to determine the prinipl stress in the sod n..4# 6 in $ 6 in $ 6 in 3 l % # # & # # % # % $.3 l [ ] ksi in in in 9.4# 6 in $ 6 in $ 6 in 3 l % # # & # # % # % $.3 l [ ] ksi in in in 3 With the prinipl stresses lulted, pressure vessel theory n e used to lulte the internl pressure of the sod n. Using Thin Wlled Pressure Vessel Theory to Clulte the Pressure in Cn of Sod The prinipl stresses nd orrespond to the irumferentil hoop nd xil stresses, respetively. From the theory of thin wlled pressure vessels these stresses n e lulted in terms of the internl pressure s follows. P # r # t % ur % 3 P % 3 t r P # r # # t % xil % 3 P % 3 # t r where P is the internl pressure in the n, r is the rdius of the n, nd t is the thikness of the n. For this prtiulr sod n the rdius of the n ws mesured to e r=.3in nd the thikness of the n ws mesured to e t=.5in. Equtions 3 nd 3 n oth e used to lulte the internl pressure in the n. 3 l 7.38# #.5in in l P % % in.3in Ronld B. Buinell, Ph.D., P.E. Union College 8 Deprtment of Mehnil Engineering Shenetdy, NY 38

9 3 l 7.7# # #.5in in l P % % in.3in Equtions 33 nd 34 provide independent estimtions of the pressure in the sod n. The differene etween Equtions 33 nd 34 represents the experimentl error. In the sent of experimentl error these two vlues whould e identil. Summry of the Use of Retngulr Strin Gge Rosette The three element strin gge rosette llows the lultion of the prinipl strins when their prinipl diretion is not known hed of time. The lultions of the prinipl strins nd stresses in retngulr strin gge rosette gge orienttion re summrized in Equtions 4, 5, 7, nd 8. The lultion of the prinipl strins from the results of retngulr strin gge rosette n lso omplished with the onstrution of Mohr s irle for strin. This method hs the dvntge tht Equtions 4, 5, 7, nd 8 do not hve to e ommitted to memory. This method lso provides onsiderle more insight into the stte of strin in the sod n. Referenes [] MiroMesurements Group Tehnil Note TN-59, Trnsverse Sensitivity Errors, Mesurements Group, In., 993. [] Dlly, J.W., & Willim, F.R., Chpter 9: Strin-Anlysis Methods, Experimentl Stress Anlysis, 3 rd ed., MGrw-Hill, Boston, 993, p Ronld B. Buinell, Ph.D., P.E. Union College 9 Deprtment of Mehnil Engineering Shenetdy, NY 38