Uniformity of Deformation in Element Testing

Transcription

1 Woousng Km, Marc Loen, ruce hadbourn and Joseph Labu Unformt of Deformaton n Element Testng Abstract Unform deformaton s a basc assumpton n element testng, where aal stran tpcall s determned from dsplacement measurements. In applng force or dsplacement, however, some rotaton of the loadng platen ma occur such that the fundamental stress feld s perturbed b bendng. Thus, nonunformt among measures of aal deformaton ma be present, and the response ma consst of a component due to the aal force and a component due to the bendng moment. To estmate ths nonunformt durng an element test, at least three sensors are needed, and for equ-angular placement, t s shown that the mean of the dsplacement readngs s equal to the dsplacement from the aal force; the rotaton does not affect the mean value. Furthermore, the rato of the mamum and mnmum does not provde an obectve evaluaton of unformt, but t s reasonable to lmt the degree of rotaton. Kewords: reslent modulus, homogeneous deformaton, compresson testng Introducton An element test assumes that the materal deforms n a unform manner. For eample, a specmen that s orgnall clndrcal n shape remans a clnder durng testng. Ideall, the nematc boundar condton mposed b a rgd platen means that the loadng platen should not rotate but reman normal to the longtudnal as of the specmen. However, some rotaton s tpcall allowed and when multple dsplacement measurements are compared, non-unformt between readngs s nevtable. In ths paper, the degree of non-unformt due to rotaton s quantfed, and the relaton between the degree of nonunformt and the specmen deformaton s dscussed to evaluate the nfluence of rotaton on the measured dsplacements (Fg. ). Department of vl Engneerng, Unverst of Mnnesota, Mnneapols, MN Mnnesota Department of Transportaton, Offce of Materals, Maplewood, MN 5509

2 The wor s motvated b the reslent modulus (M ) test, conducted to measure the stffness of base and sub-grade sols of the pavement structure []. The reslent modulus can be thought of as Young s modulus based on the recoverable stran under repeatable aal stress. Two test protocols are commonl used: (a) Long Term Pavement Program (LTTP) P46 b the Strategc Hghwa esearch Program (SHP) [], and (b) Natonal ooperatve Hghwa esearch Program (NHP) -8A []. In both protocols, repeated ccles of aal stress are appled to a specmen at a gven confnng pressure wthn a conventonal traal cell. Each ccle s s n duraton, consstng of a 0. or 0. s haversne pulse followed b a 0.9 or 0.8 s rest perod for coarse- or fne-graned sols. Durng a reslent modulus test, force and dsplacement data for each sequence are collected, and the M s calculated from M Δσ a = () Δ ε a where Δσ a = cclc aal (devator) stress and Δε a = recoverable aal stran. In detal, Fcclc Δ σ a = () A F cclc average elastc Δ ε a = () l 0 average elastc where = cclc aal force, A = cross sectonal area, = average elastc (recoverable) aal dsplacement and l 0 = orgnal gage length. Snce aal stran s determned from aal dsplacement, t s mportant to measure t accuratel. Dsplacement readngs are usuall obtaned from lnear varable dfferental transformers or LVDTs [4,5]. For the M test, three LVDTs should be placed at equ-angular postons around two parallel alumnum collars, whch are attached to the specmen (Fg. ). On the lower collar, columns are mounted below the LVDTs as contacts for the sprng-loaded tps of the sensors. Ths arrangement allows the two collars to move ndependentl of each other. Spacers mantan a parallel dstance (gage length) between the collars whle the apparatus s placed on the specmen. Non-unformt of Dsplacement M test data tpcall dspla non-unform dsplacement hstores between three LVDT readngs durng the loadng sequences (Fg. ). ecause the M value s calculated from the aal dsplacement of a specmen durng cclc loadng, t s crtcal to have relable dsplacement values from at least three LVDTs (two LVDTs are not suffcent to evaluate the non-unformt). onsder the boundar condton mposed b a rgd platen that can rotate (Fg. ). The

3 dstrbuton of normal stress vares and the resultant s composed of an aal force and a bendng moment. Thus, the total dsplacement can be decomposed nto () = ()F ()M (4) where () = total dsplacement of LVDT ()F = dsplacement of LVDT due to the aal force ()M = dsplacement of LVDT due to the bendng moment Dsplacement due to the aal force ( F ) wll be the same for the three LVDTs. However, dsplacement due to the bendng moment ( M ) wll depend on the angle of rotaton of the platen (θ) and the poston of the LVDT relatve to the as of rotaton (Fg. 4). To descrbe the rotated plane, consder three LVDTs postoned at equ-angular postons, 0 apart. ecause the as of rotaton s assumed to go through the center of the specmen, dsplacement of each LVDT due to the bendng moment wll be decded b the poston of the LVDT n relaton to the as of rotaton. If an LVDT s on the as of rotaton, dsplacement due to bendng moment s ero, and total dsplacement wll be the same as aal dsplacement. If an LVDT s located on a lne perpendcular to the as of rotaton, dsplacement due to the bendng moment wll be ether mamum ma or mnmum mn (Fg. 4). For a clndrcal specmen of radus, defne angles α, β, and χ as the angles between a lne from the center of the specmen to each LVDT and the as of rotaton such that the locaton of mn s between LVDT and LVDT. Therefore, the dsplacements of the three LVDTs are = F sn(α) sn(θ) (5) = F sn(β) sn(θ) = F sn(χ) sn(θ) and the sum s = F sn(θ) (sn(α)sn(β)-sn(χ)) (6) For equ-angular placement of the three LVDTs, the last term n equaton (6) becomes sn(α) sn(β) sn(χ) = sn(α) sn(60 -α) - sn(0 -α) = 0 (7) From equatons (6) and (7), F = ( )/ = average (8)

4 onsequentl, the dsplacement due to aal force, even f rotaton occurs, s smpl the mean of the dsplacement values from the three LVDTs. Ths means that the angle of rotaton does not affect the value of the aal dsplacement for stffness calculatons. Ths does not mean that the angle of rotaton should not be lmted, as the assumpton of unform deformaton ma be volated as rotaton ncreases. Angle of otaton To estmate the angle of rotaton, note that θ s the angle between the normal vectors of the plane before loadng (the horontal plane) and the rotated plane, defned b the (mnmum) three LVDT dsplacement values. ecallng that a plane s descrbed b A D = 0 (9) the angle between the normals of the two planes s [6] cos A A A A = θ (0) In addton, a plane passng through three ponts P (,, ), P (,, ), P (,, ) s determned b = () The plane before loadng s the horontal plane: = 0 () The plane at a partcular load s defned b the three LVDT readngs: ( ), 0, LVDT = () =,, LVDT (4) =,, LVDT (5) Thus, the equaton of the rotated plane at a partcular load s ( ) ( 0 = ) (6) 4

5 Substtutng equatons () and (6) nto equaton (0), the angle of rotaton θ s cosθ = (7) 9 4 The as of rotaton s the lne of ntersecton of the rotated plane wth the horontal plane, wth The equaton for the ntersecton of two planes n the plane s [6] = (8) A A D D = 0 (9) Substtutng equatons (6) and (8) nto equaton (9) results n the equaton for the as of rotaton: ( ) = 0 (0) In summar, from three sensors placed equ-angular to measure aal dsplacement, the angle of rotaton and the poston of the as of rotaton can be calculated. Unformt ato In NHP -8A, the unformt rato, γ, s gven as γ ' ' ma = () where ' ma, mn are the mamum and mnmum dsplacements measured b two LVDTs; γ. defnes an acceptable test []. The unformt rato γ ma s ntroduced based on the mamum and mnmum dsplacements calculated from three LVDTS. If rotaton occurs durng the load applcaton, γ values can var dependng on where the LVDTs are located wth reference to the as of rotaton. Even f the test result shows that γ s wthn some lmt, the result from the same specmen ma not satsf the condton f γ ma s estmated. Thus, t s more reasonable to lmt the rotaton θ rather than γ. Gven a certan amount of allowable rotaton (sa 0.04 ), the unformt rato wll depend on the amount of recoverable (average) aal dsplacement (Fg. 5). The same value of rotaton could result n dfferent values of γ ma dependng on the stffness of the specmen and the devator stress, both of whch nfluence recoverable aal stran. mn 5

6 oncludng emars In applng load or dsplacement for an element test, some rotaton of the rgd platen ma occur such that the fundamental stress feld s perturbed b bendng. Thus, non-unformt between dsplacement measurements ma be present, and the readngs ma consst of a component due to the aal force and a component due to the bendng moment. To estmate ths non-unformt durng an element test, three LVDTs are needed and for equ-angular placement the sensors, the mean of the three readngs s equal to the dsplacement from the aal stress; the rotaton does not affect the mean value. Furthermore, the amount of rotaton from bendng wth respect to the amount of dsplacement from aal force causes an ncrease n the unformt rato, γ, the value of whch s nfluenced b the locaton of the LVDTs wth reference to the as of rotaton. Therefore, t s more mportant to lmt the degree of rotaton and not set a target value for γ. eferences [] arsdale,.d. and Alba, J. Laborator Determnaton of eslent Modulus for Fleble Pavement Desgn. (Atlanta, GA: Georga Insttute of Technolog, 996). [] Federal Hghwa Admnstraton Pavement Performance Dvson, eslent Modulus of Unbound Granular ase/subbase Materals and Subgrade Sols, Long Term Pavement Performance, Protocol 46 (Federal Hghwa Admnstraton Pavement Performance Dvson, 996). [] Natonal ooperatve Hghwa esearch Program, ecommended Standard Method for outne eslent Modulus Testng of Unbound Granular ase/subbase Materals and Subgrade Sols, Protocol - 8A (Natonal ooperatve Hghwa esearch Program, 00). [4] uccovllo, T. and oop, M.. (997). The measurement of local aal strans n traal tests usng LVDTs. Geotechnque, V47(): [5] Acerle, S. K., Hellngs, J.E. and Jardne,. J. (987). Dscusson on a new devce for measurng local aal strans on traal specmens. Geotechnque, V7(): [6] Tuma, J.J. (987). Engneerng Mathematcs Handboo. McGraw-Hll, Inc, rd Ed. Lst of Fgures. Aal force and bendng moment mposed b rgd platens that rotate.. Apparatus for holdng LVDTs.. Three LVDT dsplacement tme hstores. 4. Geometr of specmen and LVDTs wth respect to the as of rotaton. 5. Influence of rotaton on the unformt rato γ ma at varous levels of aal stran Δε a (gage length = 00 mm). 6

7 Fgure. Aal force and bendng moment mposed b rgd platens that rotate.

8 Fgure. Apparatus for holdng LVDTs.

9 Dsplacement (μm) LVDT LVDT LVDT Tme (s) Fgure. Three LVDT dsplacement tme hstores.

10 Fgure 4. Geometr of specmen and LVDTs wth respect to the as of rotaton.

11 % 0.05% Δε a =0.%.75 γma.50 0.%.5 0.4% Angle of otaton θ ( ) Fgure 5. Influence of rotaton on the unformt rato γ ma at varous levels of aal stran Δε a (gage length = 00 mm).