Period #11 Notes: MECHANICS OF PORTLAND CEMENT CONCRETE (11.1)

Transcription

1 Peiod #11 Notes: MCHANICS OF PORTAND CMNT CONCRT A. Bie Oveview Potland ceent concete is a coposite ateial consisting o aggegate paticles and the hydated ceent paste atix that binds the togethe. The ass density, stiness, and stength popeties o potland ceent concete (pcc) ae thus unctionally dependent on the popeties o both the atix ateial (i.e. the hcp) and the einoceent ateial (the aggegate). In this couse, we study at least thee coposite ateial systes: (1) pcc; (2) asphalt ceent concete (acc); and (3) ibe-einoced plastics (FRPs). Since pcc is the ist coposite ateial on this list, we ll have to begin by intoducing soe basic ideas that apply to all coposite ateials. B. Mass Density and Volue Factions When heavy ateials ae used in coposites, the ass density o the coposite inceases, and vice vesa. Conside the scheatic o the coposite shown in Fig that consists o two ateials: (1) a paticulate einocing phase (); and (2) the continuous atix phase (). The total volue o the coposite saple shown can be decoposed into that o the paticulate einocing phase, and that o the continuous atix phase. V V V (11.1) Fig Repesentative volue eleent o a two-phased coposite ateial. 53:086 Civil ngineeing Mateials, Peiod #11 C.C. Swan The Univesity o Iowa 11.1

2 I one takes the volue equation (11.1) and divides though by the total volue V, the ollowing equation esults: V V V 1 V V V whee : V V (the einoceent volue action) V V (the atix volue action) (11.2a) (11.2b) (11.2c) It is coon to discuss the coposition o coposite ateials in tes o the volue actions o the einocing phase and the atix phase. Geneally in pcc, the aggegate volue action is between 60% and 80%, and the atix volue action between 20% and 40%. The total ass o a coposite ateial is the su o the asses o the einoceent and atix phases. M M M ρ V ρ V The ass density o the coposite is the total ass pe total unit volue: M ρ ρ ρ V (11.3a) (11.3b) (11.4) 53:086 Civil ngineeing Mateials, Peiod #11 C.C. Swan The Univesity o Iowa 11.2

3 C. ective Stiness o Coposites While the eective ass density o a coposite is diectly expessed by (11.4), it is uch less staightowad to expess the eective stiness o a coposite in tes o the stinesses and volue actions o the constituent phases. Fo this eason, thee ae a vaiety o dieent odels and assuptions that can be used to estiate the eective stiness (o elastic oduli) o coposites. Thee odels that will be pesented hee ae: 1. The Voigt isostain ule o ixtues; 2. The Reuss isostess ule o ixtues; and 3. The hybid ule o ixtues. 1. The Voigt ule o ixtues To undestand the isostain ule o ixtues, conside a epesentative volue eleent o the coposite whee the atix and einoceent phases have been sepaated into distinct egions as shown in Fig Now assue that the coposite is subjected to a one-diensional stess loading o agnitude as shown. Aanged and loaded as shown, the atix and einoceent phases ae being loaded in paallel. oaded in paallel like this, the stain in both phases should be the sae. einoceent 53:086 Civil ngineeing Mateials, Peiod #11 C.C. Swan The Univesity o Iowa 11.3 atix Fig Coposite gouped into sepaate ateials and loaded paallel to ateial allignent.

4 53:086 Civil ngineeing Mateials, Peiod # C.C. Swan The Univesity o Iowa I the stain in both the atix and einoceent phases is the sae, then the oveall stain o the coposite is also the sae. Assuing linea elastic behavio, the stess in each phase would be consistent with its Young s odulus: The oces caied in the einocing and atix phases, espectively, ae: The aveage stess in the coposite is the total oce pe goss coss-sectional aea: So, when the ateial phases o a coposite ae oiented in paallel with a one-diensional loading, the stain in both phases is the sae, and the eective stiness o the coposite is the weighted aveage o the two individual phase oduli. (11.5) (11.6b) In the atix phase (11.6a) In the einocing phase (11.7b) atix phase (11.7a) einocing phase A A A A ( ) (11.9c) (11.9b) (11.9a) (11.8) voigt A

5 Restated: voigt (11.10) The Voigt-isostain ule o ixtues povides an uppe-bound on the Young s odulus o a coposite with phases and. 2. The Reuss Isostess Rule o Mixtues To undestand the isostess ule o ixtues, conside a epesentative volue eleent o the coposite whee the atix and einoceent phases have been sepaated into distinct egions as shown in Fig Now assue that the coposite is again Subjected to a one-diensional stess loading o agnitude as shown. oaded in seies like this, the stess in both phases should be the sae and equal to the oveall applied stess. The stain in each phase is as ollows: ; and (11.11) atix Fig Coposite gouped into sepaate ateials and einoceent loaded othogonal to ateial allignent. (11.12) 53:086 Civil ngineeing Mateials, Peiod #11 C.C. Swan The Univesity o Iowa 11.5

6 53:086 Civil ngineeing Mateials, Peiod # C.C. Swan The Univesity o Iowa The total change in length o the specien is as ollows: The oveall stain o the specien is the total change in length divided by the oiginal length: Thus, the eective Young s odulus o a coposite in accodance with the Reuss isostess assuption is: Just as the Voigt isostain ule o ixtues gives an uppe bound on coposite stiness, the Reuss isostess ule o ixtues gives a lowe bound on coposite stiness. (11.13b) atix phase (11.13a) einocing phase Δ Δ (11.15) (11.14b) (11.14a) 1 Δ Δ Δ (11.16) 1 euss

7 3. The Hybid Rule o Mixtues The hybid ule o ixtues is based on the assuption that the einoceent is ebedded within the atix as shown in Fig The eective stiness associated with this aangeent o ateials can be obtained using cobinations o the Voigt and Reuss assuptions. δ δ atix einoceent Fig Coposite gouped into sepaate ateials o the hybid ule o ixtues. The cental egion o the coposite can be teated using the isostess assuption to obtain: id 2δ 1 (11.17) The oveall coposite stiness can then be ound using the isostain assuption: δ 1 δ 1 2 ( ) δ hybid 2δ ( 1 ) id ( 1 ) (11.18a) (11.18b) The odulus pedicted by the hybid ule is always geate than o equal to that o the Reuss ule and less than o equal to that o the Voigt ule. 53:086 Civil ngineeing Mateials, Peiod #11 C.C. Swan The Univesity o Iowa 11.7

8 xaple 11.1: Fo PCC assue that the Young s odulus o the hcp is 20 GPa and that the Young s odulus o the aggegate is 100 GPa. Also, assue that the aggegate volue action in the PCC is 75%. Copute the dieent Young s odulus estiates o the PCC given based on: a) The Voigt isostain ule; b) The Reuss isostess ule; and c) The hybid ule o ixtues. Solution: 0.75; 0.25; 20 GPa; 100 GPa. voigt GPa euss ( 1 ) 1 hybid ( 1 ) GPa 2.7 GPa 56.4GPa 59.1GPa Thus, the odulus o the hybid ule is indeed inteediate to that o the Voigt and Reuss ules, as would be expected.// 53:086 Civil ngineeing Mateials, Peiod #11 C.C. Swan The Univesity o Iowa 11.8

9 D. piical Relations o Stength & Stiness o PCC 1. Realistic values o the Young s odulus o stone that ight be used as aggegate in pcc anges o: 20 GPa 150 GPa, with slates, shales, and sandstones having the lowe stinesses and ganites and liestone having the highe stinesses. 2. Mechanical Popeties o Hydated Ceent Paste: Fo hydated ceent paste, both the elastic stiness and the shea stength tend to decease as the capillay poosity in the hcp inceases. Dieent sei-epiical odels exist that elate the elastic stiness o hcp to the capillay poosity. One in paticula [o Concete 2 nd d., by Mindess, Young and Dawin, Pentice-Hall, (2003)] is : 29 GPa hcp n c whee n c epesents the capillay poosity o the hcp. Thus, the highe the capillay poosity is, the lowe the stiness o the hcp will be. As the lage capillay voids in hcp incease, the actue stess deceases with the squae oot o the typical void size. (The Giith icocack odel pesented in the Peiod #3 notes is the basis o this stateent.) 3. piical Foulae Relating and c 3 ( 1 ) (11.19) Although the shea stength o pcc is typically a unction o the stengths o both the atix phase (hcp) and the einoceent phase (aggegate), the hcp is usually the weakest link that liits the stength o pcc. 53:086 Civil ngineeing Mateials, Peiod #11 C.C. Swan The Univesity o Iowa 11.9

10 This helps to explain why thee ae epiical elations between the unconined copessive stength c o pcc and its Young s odulus. One such odel is as ollows: pcc 4730 MPa c ' whee c' is in units o MPa (11.20) Anothe epiical elation between the static Young s odulus o pcc and c is: pcc 20 GPa 200 c ' whee c' is in units o GPa (11.21) PCC has viscoelastic elastic chaacteistics that ake it stie when loaded apidly and less sti when loaded ove a long peiod o tie. An epiical elation siila to that o (11.21) o the dynaic odulus o PCC is: ( ) pcc 31GPa 160 c' whee c' is in units o dynaic GPa The noal ange o the unconined copessive stength o pcc is oughly: 2000 psi 14 MPa c c ' 8000psi ' 56MPa (11.22) (11.23a) (11.23b) When the unconined copessive stength o pcc exceeds 10,000 psi (70 MPa), the concete is usually called high-peoance concete o hpc. Such high-peoance concete (hpc) is geneally achieved by using blended ceents (i.e. those in which ceent eplaceent has been eployed) togethe with low wate-ceent atios. 53:086 Civil ngineeing Mateials, Peiod #11 C.C. Swan The Univesity o Iowa 11.10

11 The Poisson s atio o pcc usually deceases with an inceasing aggegate volue action: Fo pue hcp (concete w/o any aggegate) ν ; Fo aggegate volue actions in the ange o 70-80%, ν ; I one knows the Young s odulus o pcc and the Poisson s atio, then the shea odulus can be obtained using the ollowing oula appopiate o isotopic, linealy elastic ateials G 2(1 ν ) It is woth entioning that in pcc, the atix phase o the hcp is usually quite a bit weake than the aggegate phase. I one consides the ule o ixtues coposite odels applied to pcc, both the Reuss isostess odel (Fig. 11.3) and the hybid odel (Fig. 11.4) ae such that the stength o pcc is contolled by the weakest constituent (o the weakest link). Thus these odels pedict that: ' The Voigt o iso-stain odel is not ealistic o pcc and thus would ove-estiate the stength o pcc.. Measuing the Stength o PCC It is coon to easue the unconined copessive stength o pcc and less coon, although still soeties done, to easue the tensile stength. 1. Unconined Copessive Stength c ' hcp (11.24) (11.25) This test is usually peoed on cylindes o 6 dia. (150) and 12 height (300). The test can also be peoed on salle cylindes and also on salle cubes o pcc. 53:086 Civil ngineeing Mateials, Peiod #11 C.C. Swan The Univesity o Iowa 11.11

12 It is usually best to use speciens with a height to diaete atio o 2 to 1. With this atio, the conining eects o iction between the specien and the loading platens is educed. With speciens having an aspect atio o 1 (height to width 1), the conining eect tends to ake the easued stength o the pcc highe than it would othewise be. Theeoe, the easued c o cubical speciens would tend to lage than that o 2:1 cylindical speciens. 2. Tensile Stength Testing Thee ae two coon pocedues o easuing the tensile stength o pcc. These ae: (a) the split cylinde test; and (b) the bending test. a) split cylinde test. This test is peoed by taking a 2:1 pcc cylinde, tuning it on its side (Fig. 11.6) and then loading it to ailue. The tensile stength o the pcc o this test is given as ollows: ' t 2P πdh (11.26) whee d is the diaete o the cylinde, h is its height, and P is the agnitude o the load at ailue. It is also woth entioning hee that the load P is applied ove the ull length o the cylinde edge athe than just at a point. Fig Scheatic o unconined copession test. P P Fig Scheatic o the split cylinde test. 53:086 Civil ngineeing Mateials, Peiod #11 C.C. Swan The Univesity o Iowa 11.12

13 b) The bending test: In this test, a pisatic bea like that o Fig is loaded at the one-thid points. When the bea (o depth h, width b, and length ) uptues, the peak tensile bending stess in the iddle thid o the bea is the tensile stength o the pcc: P/2 P/2 Fig Bending test to easue t. M c P h bh 12 P bh ax ax t ' t 2 I (11.27) 53:086 Civil ngineeing Mateials, Peiod #11 C.C. Swan The Univesity o Iowa 11.13