Relationship of time-dependent parameters from destructive and non-destructive tests of structural concrete

  Ci a ion: Lehne , P.; H abo á, K. Rela ionship o Time-Dependen Pa ame e s om Des uc i e and Non-Des uc i e Tes s o S uc u al Conc e e. Ma hema ics 2022,10, 460. h ps://doi.o g/10.3390/ ma h10030460 Academic Edi o s: Theodo e E. Simos and Cha alampos Tsi ou as Recei ed: 31 Decembe 2021 Accep ed: 29 Janua y 2022 Published: 30 Janua y 2022 Publishe ’s No e: MDPI s ays neu al wi h ega d o ju isdic ional claims in published maps and ins i u ional a il- ia ions. Copy igh : © 2022 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h ps:// c ea i ecommons.o g/licenses/by/ 4.0/). ma hema ics A icle Rela ionship o Time-Dependen Pa ame e s om Des uc i e and Non-Des uc i e Tes s o S uc u al Conc e e Pe Lehne 1,* and K is ýna H abo á2 1Depa men o S uc u al Mechanics, Facul y o Ci il Enginee ing, VSB—Technical Uni e si y o Os a a, L. Podéš ˇe 1875, Po uba, 70800 Os a a, Czech Republic 2Ins i u e o Building Tes ing, Facul y o Ci il Enginee ing, B no Uni e si y o Technology, Ve eˇ í331/95, 60200 B no, Czech Republic; k is yna.h abo a@ u b .cz *Co espondence: pe [email p o ec ed] Abs ac : Rein o ced conc e e s uc u es a e ypically exposed o a combina ion o agg essi e sub- s ances and mechanical s esses, which con ibu e o as deg ada ion. The p esen esea ch was conduc ed o e alua e i e ime-dependen pa ame e s om se e al di e en es s, namely com- p essi e s eng h, s a ic modulus, dynamic modulus, su ace, and bulk elec ical esis ance. Some pa ame e s we e ob ained using des uc i e es ing (DT) and some using non-des uc i e es ing (NDT). Due o he co ela ion and calcula ion o eg ession cu es, i was possible o compa e he co ela ion o pa ame e s impo an o es ima ing he du abili y o ein o ced conc e e s uc u es in ela ion o deg ada ion and co osion. Conc e e o C40/50 g ade was examined in se e al ime pe iods, and he pa ame e ela ionships we e analysed. A he same ime, a s a is ical e alua ion was ca ied ou , and he e o e he s udy con ains he a e age alues and s anda d de ia ions o all measu ed pa ame e s. The esul s show ha he comp essi e s eng h and he elec ical esis i i y o he su ace and bulk ha e a high co ela ion. In con as , he dynamic modulus and elec ical esis i i y ha e low linea co ela ion, bu i was possible o apply a quad a ic cu e wi h a high deg ee o i . Fo he compa ison o s a ic elas ic modulus and elec ical esis ance, he quali y o he quad a ic eg ession model was low bu su icien . The esul s show ha , o s uc u al conc e e, he p esen ed NDT me hods can be used o es ima e o he pa ame e s ob ained om he DT me hods. Keywo ds: conc e e; NDT; quad a ic eg ession; du abili y; modulus o elas ici y; s eng h; co ela ion analysis 1. In oduc ion Building s uc u es a e a ec ed h oughou hei li e ime by mechanical, dynamic, clima ic, and deg ada ion ac o s [ 1 , 2 ]. All hese aspec s can, o e ime, cause he pa o he s uc u e o become unsa is ac o y in e ms o load-bea ing capaci y o se iceabili y. Due o his, egula inspec ions and quali y con ol a e some o he basic equi emen s o he cons uc ion indus y [ 3 ]. In he e alua ion o building s uc u es, hei cu en condi ion is assessed based on app op ia ely selec ed diagnos ic me hods. Cu en ly, he diagnos ics o building s uc u es a e mainly based on des uc i e me hods. Thei use is no always op imal p ecisely because o he in asi e in e en ion in o he s uc u e. Non-des uc i e es s a e a sui able choice o s uc u es diagnos ics because hey a e as , accu a e, and can be epea ed [4]. The comp essi e s eng h o conc e e is one o he mos impo an a iables in assess- ing he quali y o a s uc u e and should be iewed as a a iable pa ame e . Thus, he e alua ion o he condi ion o exis ing s uc u es is pe o med s anda dly in co e wells and hei de e mina ion o he p esen alue o s eng h [ 5 ]. The e a e also many s udies dealing wi h he analysis o he ela ionship be ween des uc i e and non-des uc i e me hods o de e mining comp essi e s eng h and o he impo an pa ame e s. The mos Ma hema ics 2022,10, 460. h ps://doi.o g/10.3390/ma h10030460 h ps://www.mdpi.com/jou nal/ma hema ics Ma hema ics 2022,10, 460 2 o 13 commonly used non-des uc i e me hod o e i y he quali y o conc e e and de e mine i s comp essi e s eng h is he ebound ha dness es me hod [6]. I is also possible o use he ul asonic e alua ion me hod, which wo ks acco ding o he p inciple o ansmi ing ul asonic wa es o he su ace o he s uc u e by an exci e and hen moni o ing he speed o he ansmi ed pulses [7,8]. The gene al ad an ages o non-des uc i e es ing o s uc u es a e well known, bu he mos impo an a e speed and economy. On he con a y, he accu acy o non-des uc i e es s can be a ec ed by many pa ame e s such as agg ega e g ain size, conc e e age, and wa e coe icien [ 9 – 12 ]. T ykoz e al. [ 13 ] p esen ed e idence o he impo ance o non-des uc i e es ing (NDT) measu emen s in p ac ice in he e alua ion o exis ing conc e e s uc u es. This pape p esen s a co ela ion and eg ession analysis o se e al conc e e pa ame- e s ob ained du ing he ini ial cu ing s age om 1 o 128 days. The selec ed pa ame e s can be used as inpu pa ame e s o he nume ical analysis o he du abili y o ein o ced conc e e s uc u es subjec ed o common o ce and chemical loads [ 14 – 17 ]. These nume ical models a e a ailable in bo h de e minis ic and s ochas ic o ms, and since hey a e com- pu a ionally and ime-consuming, an adequa e desc ip ion o he inpu pa ame e s o he ime a iables is equi ed. The ime-dependen pa ame e s men ioned include comp essi e s eng hs, s a ic modulus o elas ici y, dynamic modulus o elas ici y de i ed om ul a- sonic eloci y [ 18 , 19 ], and su ace and bulk elec ical esis i i y [ 20 – 22 ], which can be used o de e mine he chlo ide di usion pa ame e o conc e e [ 23 ]. In s udies, a compa ison o he esul s o he esis i i y o he su ace and he eloci y o he ul asonic pulse was p esen ed [ 24 , 25 ]. Con en ional conc e e shows he same alues om bo h me hods, e en in agg essi e en i onmen s. The objec i e was o e alua e he simples possible ela ionship be ween wo selec ed pa ame e s, ha is, a linea eg ession equa ion o apply a quad a ic eg ession cu e. Due o he in oduc ion o hese equa ions in u he calcula ions, he linea o m is p e e able, bu i i is no accu a e, a mo e complex polynomial mus be applied. This pape p esen s a basic desc ip ion o heexpe imen al e alua ion, s a is ical and co ela ion analyses, and he esul ing equa ions, accompanied by he deg ee o ag eemen . All o his suppo s he long- e m goals o imp o ing non-des uc i e es ing p ocedu es, using non- adi ional ins umen a ion, and analysing ela ionships be ween he esul s o s anda dised des uc i e me hods. 2. Ma e ials and Me hods Fo he analysis o he ela ionship be ween des uc i e and non-des uc i e me hods o conc e e es ing, a s anda d conc e e based on Po land cemen was chosen, wi h an expec ed s eng h class o C40/50. This conc e e is used in he Czech Republic, among o he hings, o he load-bea ing elemen s o ein o ced conc e e b idge s uc u es [ 26 ]. This ma e ial was chosen o pu ely p agma ic easons, as i is widely used in mode n imes and is no bu dened wi h any chemical addi i es o non-s anda d ing edien s. A he same ime, his conc e e composi ion is ideal o u he ex ension o esea ch whe e he e ec o change in composi ion, non-s anda d admix u es, and o he s can be analysed. 2.1. P ope ies o Conc e e The conc e e mix u e was p epa ed acco ding o he p ac ice o EN 206 + A1 [ 27 ]. The composi ion is shown in Figu e 1. The equi emen was a wa e coe icien w/c = 0.4 and a s eng h class o a leas C40/50. Fo he needs o he expe imen al p og am, conside ing he cu ing o conc e e a he ime, a la ge numbe o samples we e p epa ed. Cubes wi h an edge leng h o 150 mm, la ge cylinde s wi h a diame e o 150 mm and a leng h o 300 mm, and small cylinde s wi h a diame e o 100 mm and a leng h o 200 mm we e p epa ed. The uses o each sample o speci ic es s a e gi en below. The esh conc e e mix was pou ed in o he moulds, and a e an ini ial cu ing pe iod o 24 h, he samples we e demoulded. The s o age o all samples was managed unde labo a o y condi ions o a oid undesi able deg ada ion. The es s we e pe o med 7, 28, 63, and 126 days a e conc e ing. Acco ding o s anda ds, i was necessa y o use 3 samples o each ype o es and each Ma hema ics 2022,10, 460 3 o 13 ime. Due o his, a leas 12 samples o e e y me hod we e p epa ed a he beginning. Fo he non-des uc i e es s, his need was no p esen , and he same 3 samples we e es ed a each ime. Figu e 1. Conc e e mix u e p ope ies; amoun pe 1 m2[kg]. 2.2. Des uc i e Tes ing Me hods Fo he p esen esea ch, wo s anda d me hods o labo a o y es ing o conc e es we e selec ed, namely a es o de e mine he comp essi e s eng h on cubes and a es o de e mine he s a ic modulus o elas ici y om ul asonic measu emen s. The comp essi e s eng h was es ed acco ding o EN 12390-3 [ 28 ] on h ee cubes se e al imes-7, 28, 63, and 126 days a e conc e ing. The es was ca ied ou on a s anda d hyd aulic p ess machine, and he cubic s eng hs o he conc e e we e calcula ed om he maximum o ce and he load a ea eco ded. The s a ic modulus o elas ici y in he comp ession o ha dened conc e e exp esses he dependence be ween he s ess and he s ain ha a gi en elemen o s uc u e exhibi s unde s ess. The p ocedu e o de e mine he s a ic comp essi e modulus o elas ici y o conc e e is de ined in ISO 1920-10 [ 29 ]. The de e mina ion o he s a ic modulus o elas ici y was ca ied ou by cyclic loading o he es pieces in a p ess, which was ca ied ou acco ding o EN 12390-13 [30]. 2.3. Non-Des uc i e Tes ing Me hods Th ee small cylinde s wi h a diame e o 100 mm and a leng h o 200 mm and h ee la ge cylinde s wi h a diame e o 150 mm and a leng h o 300 mm we e p epa ed o he NDT me hods. Two ypes o cylinde s we e chosen o analyse he e ec o he shape and size o he sample on he esis i i y o he su ace. A Wenne p obe [ 31 ] (see Figu e 2) was used o measu e he elec ical esis i i y o he su ace. The es was ca ied ou on six cylind ical samples ( h ee small and h ee la ge) acco ding o he AASHTO T358 s anda d [ 32 ]. Su ace esis ance was measu ed om ou longi udinal sides, and hen he mean alue and s anda d de ia ion we e de e mined. The su ace esis i i y was hen con e ed o an adequa e bulk esis i i y using a calib a ion ela ionship dependen on he dimensions o he sample acco ding o p e iously published p ocedu es [33]. Ma hema ics 2022,10, 460 4 o 13 Figu e 2. A Wenne p obe be o e he measu emen . An CON me e [ 34 ] was used o measu e he bulk elec ical esis i i y. This es was also ca ied ou on six cylind ical samples ( h ee small and h ee la ge) acco ding o ASTM C1876 [ 35 ]. Fu he mo e, an ul asonic pulse me hod was used o de e mine he dynamic modulus o elas ici y acco ding o EN 12504-4 [ 7 ]. The dynamic modulus o elas ici y was de e mined om he bulk densi y o conc e e and he ime o ul asound passage h ough he sample. The es was pe o med on h ee la ge cylind ical samples. 2.4. Co ela ion and Reg ession Analysis Since all es s we e pe o med on specimens a he same ime pe iods since conc e ing, he ela ionship be ween he esul s o des uc i e and non-des uc i e me hods could be analysed. When looking o a ela ionship be ween wo se s o esul s, he co ela ion coe - icien can be used a he i s le el. The p esen ed esea ch shows he Pea son co ela ion coe icien (PCC), which is one o he mos widely used [ 36 ]. The PCC shows he deg ee o linea co ela ion be ween wo se s o da a. I he alue is close o 1 (o − 1) he e is a possible high linea co ela ion (posi i e o +1 and nega i e o − 1). On he o he hand, i i app oaches 0, i is no possible o alk abou a linea co ela ion. PCC p o ides a basic iew o he da a se s bu can de ini ely help o u he analysis. The nex s ep in assessing he ela ionship be ween he selec ed ou comes was o in oduce a poin se consis ing o one se o da a on he X-axis and ano he se o da a ela ed o he same ime on he Y-axis. The poin s depic ing he measu ed alues we e hen i ed wi h a line segmen showing he s anda d de ia ion. This p ocedu e hen allows he sea ch o a linea eg ession equa ion o , i he i is low, a quad a ic eg ession equa ion ha uses a pa abola [ 37 ]. In ei he case, i is necessa y o e alua e g aphically and ma hema ically how closely he line o pa abola i s he se o measu ed alues. The coe icien is use ul o de e mine he bes i [ 38 ]. The coe icien o de e mina ion shows he i be ween he measu ed alues and he chosen cu e. The limi s o R 2 a e a ied. Fo example, a alue o 0.82 o 1 is epo ed o indica e a e y s ong i , a alue o 0.5 o 0.81 is a s ong i , and lowe alues a e mode a e o weak i s [ 39 ]. The e o e, he alue o 0.8 was chosen as he bounda y whe e linea co ela ion is su icien , and he e is no need o look o a quad a ic equa ion. 3. Resul s The ollowing sec ions show he esul s o all measu emen s and hen he indi idual co ela ions and eg essions on he 10 pai s o esul s. The g aphs con ain a ma hema ical desc ip ion o he linea o quad a ic eg ession cu e and he alues o R 2 . Fu he analysis o he esul s was pe o med a he le el o applica ion o linea eg ession and bes - i anal- Ma hema ics 2022,10, 460 5 o 13 ysis using he coe icien o de e mina ion. In case o low ag eemen , quad a ic eg ession was applied. 3.1. Time-Dependen Resul s Due o he chosen p ocedu e o measu ing all pa ame e s in se e al ime s eps, i is possible o display he esul s oge he on one g aph and look o p ima y ela ionships. Figu e 3shows he g aph o he elec ical esis i i y measu ed in small and la ge cylinde s om bo h su ace and bulk ins umen s. The same g aph in he igu e also shows a plo o he inc ease in comp essi e s eng h alues on he cubes. The i s iew o eshadows a ela ionship ha could be u he e alua ed. Figu e 3. Time-dependen esul s o elec ical esis i i y and comp essi e s eng h. The nex g aph (Figu e 4) again shows he same elec ical esis i i y alues bu adds he s a ic and dynamic modulus alues a he gi en imes. The li e a u e indica es ha he s a ic modulus co esponds o app oxima ely 60% o he dynamic modulus [ 40 ]. This ac is also obse ed he e. Figu e 4. Time-dependen esul s o elec ical esis i i y and modulus o elas ici y. The Pea son co ela ion coe icien desc ibed abo e was chosen as he i s nume ical esul o he analysis. Table 1shows he esul s o his coe icien o each pai o esul s. I can be seen ha he elec ical esis i i y has an almos 100% co ela ion when compa - ing su ace and bulk measu emen s. The co ela ion be ween comp essi e s eng h and elec ical esis ance anges om 0.94 o 0.96. The o he co ela ions a e a o below 0.8. Ma hema ics 2022,10, 460 6 o 13 Table 1. Resul s o he nume ical analysis based on Pea son’s co ela ion coe icien (PCC). Fi s Pa ame e Second Pa ame e PCC Small cylinde su ace elec ical esis i i y Small cylinde bulk elec ical esis i i y 0.99 La ge cylinde su ace elec ical esis i i y La ge cylinde bulk elec ical esis i i y 0.99 Comp essi e s eng h Small cylinde bulk elec ical esis i i y 0.96 Comp essi e s eng h La ge cylinde bulk elec ical esis i i y 0.94 Comp essi e s eng h Dynamic modulus o elas ici y 0.85 Comp essi e s eng h S a ic modulus o elas ici y 0.78 Dynamic modulus o elas ici y Small cylinde bulk elec ical esis i i y 0.77 Dynamic modulus o elas ici y La ge cylinde bulk elec ical esis i i y 0.68 S a ic modulus o elas ici y Small cylinde bulk elec ical esis i i y 0.75 S a ic modulus o elas ici y La ge cylinde bulk elec ical esis i i y 0.65 3.2. Bulk and Su ace Elec ical Resis i i y Fo he i s esul s, i.e., su ace and bulk elec ical esis i i y on small and la ge cylinde s, a high ag eemen o linea dependence can be obse ed (see Figu e 5). The esul s show ha i is no necessa y o look o disc epancies in he Wenne p obe and Con measu emen s o he p epa ed conc e e ype. The e o e, only he bulk esis i i ies we e used o u he e alua ion. Figu e 5. Linea eg ession analysis o he ela ionship be ween su ace and bulk elec ical esis i i y on la ge and small cylind ical samples (LC = La ge Cylinde s, SC = Small Cylinde s). 3.3. Comp essi e S eng h s. Elec ical Resis i i y The ela ionship be ween he conc e e s eng h ob ained om he cubic es and he elec ical esis i i y measu ed in la ge and small cylinde s using linea eg ession is shown in Figu e 6. The coe icien o de e mina ion alues o bo h se s a e high, 0.89 o he la ge cylinde s and up o 0.93 o he small cylinde s. Thus, i can be concluded ha he ime dependence o bo h pa ame e s is almos linea , which suppo s he elec ical esis i i y measu emen as an al e na i e NDT me hod o de e mining he comp essi e s eng h o conc e e. I is possible o use linea eg ession cu es o nume ical modelling. Ma hema ics 2022,10, 460 7 o 13 Figu e 6. Linea eg ession analysis o he ela ionship be ween bulk elec ical esis i i y and com- p essi e s eng h (LC = La ge Cylinde s, SC = Small Cylinde s). 3.4. Comp essi e S eng h s. Modulus o Elas ici y The ela ionship be ween comp essi e s eng h and modulus o elas ici y was also in es iga ed. Bo h he s a ic modulus ob ained om des uc i e es ing (DT) and he dy- namic modulus ob ained om NDT ul asonic p opaga ion eloci y es ing we e e alua ed. Figu e 7shows he plo s o he linea eg ession be ween he pa ame e s. The coe icien o de e mina ion has medium alues. Figu e 7. Linea eg ession analysis o he ela ionship be ween modulus o elas ici y and comp essi e s eng h (D = Dynamic, S = S a ic). The e o e, he quad a ic eg ession and i s i o bo h se s o assessed esul s we e analysed (see Figu e 8). The ela ionship be ween he s a ic modulus and he comp essi e s eng h o he conc e e also shows he di e gence in his case, which is p obably due o he non-uni o m beha iou o he s a ic modulus o e ime. On he o he hand, when he quad a ic cu e is applied o he pai o comp essi e s eng h and dynamic modulus, he alue o he coe icien o de e mina ion is 0.85, again showing high ag eemen . Ma hema ics 2022,10, 460 8 o 13 Figu e 8. Quad a ic eg ession analysis o he ela ionship be ween modulus o elas ici y and comp essi e s eng h (D = Dynamic, S = S a ic). 3.5. Dynamic Modulus o Elas ici y s. Elec ical Resis i i y The o e all e alua ion included he analysis o wo se s o esul s om NDT mea- su emen s, namely, dynamic modulus o elas ici y de i ed om ul asonic eloci y and elec ical esis i i y. Linea eg ession is p epa ed in Figu e 9 o he combina ion o la ge and small cylinde s ha we e used o he elec ical esis ance measu emen s. The alues o he coe icien s o de e mina ion a e low, below 0.6, so i can be concluded ha he e is a small linea ela ionship. Figu e 9. Linea eg ession analysis o he ela ionship be ween bulk elec ical esis i i y and dynamic modulus o elas ici y (LC = La ge Cylinde s, SC = Small Cylinde s). As in he p e ious case, he nex s ep was o apply a quad a ic eg ession cu e, which could show a highe i . Thus, in Figu e 10, i can be seen ha , indeed, he quad a ic eg ession p o ides unexpec edly high goodness o i o bo h cases. Thus, he change in he measu ed elec ical esis i i y and ul asonic eloci y pa ame e s is highly simila o e ime and can be al e na ed o he conc e e ype e alua ed he e. Ma hema ics 2022,10, 460 9 o 13 Figu e 10. Quad a ic eg ession analysis o he ela ionship be ween bulk elec ical esis i i y and dynamic modulus o elas ici y (LC = La ge Cylinde s, SC = Small Cylinde s). 3.6. S a ic Modulus o Elas ici y s. Elec ical Resis i i y Ano he pai , his ime om DT and NDT measu emen s, is s a ic modulus e sus elec ical esis i i y on small and la ge cylinde s. In he i s s age, he linea cu e was applied again (see Figu e 11). Howe e , i showed a low ag eemen , al hough he coe icien o de e mina ion o he small cylinde s was almos 0.73. Figu e 11. Linea eg ession analysis o he ela ionship be ween bulk elec ical esis i i y and dynamic modulus o elas ici y (LC = La ge Cylinde s, SC = Small Cylinde s). I was expec ed ha a quad a ic eg ession, which was app op ia e o some o he p e ious me hods, would gi e much be e esul s. The quad a ic cu e analysis o he s a ic modulus and elec ical esis i i y is shown in Figu e 12. He e again, di e en alues can be seen o la ge and small cylinde s. Howe e , in his case, he quad a ic eg ession imp o ed he alue o he coe icien o de e mina ion o small cylinde s o only 0.87, bu o la ge cylinde s, his alue was inc eased o 0.78. Bo h alues a e signi ican ly highe han o linea eg ession and demons a e some ag eemen be ween he esul s o he wo me hods.