Structural Engineering Handbook

MUSTAFA MAHAMID

EDWIN H. GAYLORD, JR.

CHARLES N. GAYLORD

Fifth Edition

NewYork Chicago San Francisco Athens London Madrid Mexico City Milan New Delhi Singapore Sydney Toronto

viii CONTENTS

3.9 Postprocessing-Solving for Strain, Stress, and Other Quantities....... • • • • • • • • 126 3.10 Dynlimlc Flnlte-ElamentAnalysls........ ••••••••••• ••••••••••• •••••••••••• •••• 126 3.11 NonllnearFlnlte-ElemmtAnalysls............................................. 128 3.12 Verification and Validation • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 128 3.13 Issues and Pitfalls in Finite-Element Analysis..... • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 128 3.14 lntrodudlon to Finite Elements for Thermal, Thennomechanlcal,

and Other Problems • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 130 Ref.ranees • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 131

Chapter 4. Computer Applications in Structural Engineering Raoul Karp, Bulent N. Alemdar, Sam Rubenzer............................ 133

4.1 lntrodudlon...... ••••••••••• ••••••••••• ••••••••••• ••••••••••• •••••••••••• •••• 133 4.2 Computer Structural Analysis Simulation • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 133 4.3 Strudural Finite Elements..................................................... 135 4.4 Foundations.................................................................. 139 4.5 Verifying Analysis Results • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 142 4.6 Building lnfonnation Modeling and Interoperability • • • • • • • • • • • • • • • • • • • • • • • • • • • 142 4.7 Summary..................................................................... 143

RetWrences • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 143

Chapter 5. Earthquake-Resistant Design s. K. Ghosh • • • • • • • • • • • • • • • 145 5.1 Overview..... • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 145 5.2 Nature of Earthquake Motion • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 145 5.3 Design Phllosophy....... • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 146 5.4 Seismic O.slgn Requirements of the 2018 IBC/ASCE 7-16....................... 147

References • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 162

Chapter 6. Fracture and Fatigue Kedar s. Kirane, Zdenllc P. Bablnt, J. Emesto lndacochea, Vineeth Kumar Gattu • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 165 PART A CONCRETE AND COMPOSITES............................................... 165

6.1 lntrodudlon to Quaslbrlttle Fradure • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 165 6.2 Conaete • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 166 6.3 Flber-Relnforced Composites.................................................. 169

Ref.ranees • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 173

PART B STRUCTURAL STEELS • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 175 6.4 Fracture of Structural Steels. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 175 6.5 Fatigue of Structural Steels.................................................... 182

References • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 193

Chapter 7. Soil Mechanics and Foundations Joseph w. Schulenberg, KrishnaR.Reddy.................................................... 195

7.1 Soil Behavior..... •• • • • •• • • • • •• • • • •• • • • • •• • • • • •• • • • •• • • • • •• • • • •• • • • • •• • • • • •• • • 195 7.2 Shallow Foundation Analyses • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 199 7.3 Deep Foundations • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 204 7.4 Retaining Strudures......... ••••••••••• ••••••••••• ••••••••••• •••••••••••• •••• 210 7.S Investigations................................................................ 218 7.11 Soil Improvement............................................................. 219 7.7 Monitoring................................................................... 221

Ref.rences • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 221

Chapter 8. Design of Structural Steel Members Jay Shen, Bulent Akbas, Onur Seirer, Charlies J. Carter. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 225

8.1 Design of Steel and Composite Memben • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 225 8.2 Seismic Design of Steel Members In Moment and Braced Frames • • • • • • • • • • • • • • • 264 8.3 Conduding Remarks..... • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 303

RetWrences • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 303

Chapter 9. Design of Cold-Formed Steel Structural Members Nabil A. Rahman, Helen Chen, Cheng Yu. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 305

9.1 Shapes and Applications. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 305 9.2 Materials..... • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 306

Chapter6

Fracture and Fatigue

PART A CONCRETE AND COM POSIT ES

BY

KEDAR 5. KIRANE Assistant Profeswr of Mechanir.al Engineering. Stony Brook Univmi~ Stony Brook, New York ZDENl!K P. BA2ANT McCormick Institute Professor and Walter P. Murphy Professor of Civil and

Environmental Engineering. Mechanical Engineering and Material Science and Engineering. Northwestern Uni11ersit)I E1/anSton, nlinois

6.1 INTRODUCTION TO QUASIBRITILE FRACTURE

Concrete and fiber-reinforced composite,, are two of the most widely used structural marerials. Both are composite materials consisting of multiple constituents and have a highly heterogeneous microstructure. Concrete typically consists of aggregate bonded by hardened cement paste while fiber composites typically consist of a weaker matrix material reinforced by strong fibers, which may be discontinuous or continuous, consisting of parallel fibers of a fabric. The fracturing behavior of these materials differs from metals. They exhibit almost no plasticity. They fail by propagation of a macrocrack having at its front a fracture process zone (FPZ), which contains microcraclcs and microslips and nonnaDy has a size not negligible compared to structural dimensions.

The fracture propagation starts either from a preexisting macrocrack or by formation of an FPZ in a region with a stress concentration, often at the boundary. The large size of the FPZ, which is dictated by mate­rial hett:rogeneity and is approximately a constant material property, endows the material with a significant capability to dissipate energy, which can provide certain measure of structural ductility. So, even though these materials are colloquiaDy referred to as brittle, in scientific discourse they are properly termed Mquasibrittle:" But note that this term is relative. When the structure becomes so large that the FPZ size is negligible, a quasibrittle structure becomes ~rittle.~ Vice versa, when a brittle structure, made, for example, of fine grained mortar, becomes sufficiently small its behavior becomes quasibrittle. There are many quasibrittle materials-aside from concrete and fiber composites, such as most rocks, coal, wood, sea ice, coarse-grained ceramics, rigid foams, paper, carton, bone, and bio- and bio-inspired materials.1

To assess the strength of structures with cracks, fracture mechanics must be used.l Today there are three types of fracture mechanics: (1) linear elastic fracture mechanics (LEFM), which was originated by in 1921 by Griffith3 and characterizes brittle failure; it deals with sharp craclcs whose FPZ is negligible compared to structure dimension (e.g., the FPZ in fatigued steel haa micrometer dimensions). (2) The ductile fracture mechanics,2 was developed fully during the 1960s, in which there is a long and wide plastic (or plastic-hardening) zone in front of the crack but the FPZ is still negligible; and (3) quasibrittle fracture mechanics,1 there is a large FPZ in front of the crack, with almost no plastic zone surrounding it. The ductile, as well as quasibrittle, fracture mechanics is approximately treated by various adaptations ofLEFM.

In LEFM, the crack propagation criterion can be stated either in terms of the fracture energy, Gf' which represents the energy required to extend the crack by a unit area, or in terms of the critical stress intensity factor, called the fracture toughness, K,. Their relation, due to Irwin, is

(6.1)

where, for plane stress, E' = B = Young's modulus of elasticity and, for plane strain, E' = E/(1 - vl) where v is the Poisson's ratio. The craclc can grow and possibly cause structural failure when K becomes equal to[(,.

For quasibrittle materials, however, due to the FPZ, a significant part of the structure volume acts nonlinearly, making LEFM inapplicable. A simple but effective treatment of this nonlinearity is the equivalent LEFM approach.1.2 In this approach, the nonlinear FPZ is accounted for by assuming that its effect on the compliance and load capacity of the structure is essentially the same as if an LEFM sharp crack were extended, by distance ep roughly into the middle of the FPZ.1 The longer crack, oflength a. = a 0 + ep where 'J is approximately constant, is called the effective or equivalent craclc (see Fig. 6.1). The remaining treatment is similar to LEFM, as used for example, for fatigue-embrittled stt:el. The equivalent LEFM can be applied for crack initiation even when only the FPZ. but no actual crack, is as yet present, which is not the case with LEFM.

Some rules have to be introduced to express how the equivalent craclc extends under increased loads. For instance, the fracture energy G1 for equivalent LEFM (alternately interpreted as the internal material resistance R to crack growth) is not a constant but varies with the craclc length, becoming a constant for larger crack sizes. The plot of this inter­nal resistance R versus the crack extension Ila from the notch is referred to as the R-curve1.2 (or resistance curve). This curve can be empirically dett:rmined from lab rests.

An important aspect that must be considered is the structure size effect on its nominal strength, defined as aN = P ,..JA, where P,..,, is the load capacity (maximum load) and A the structural cross-section area (which can be taken arbitrarily but must be homologous for vari­ous sizes). A host of past studies1-' have established the close interplay between failure load, FPZ size, and the resulting scaling in the structural strength.

115

166 CHAPTER SIX

Figul't! 6. 1 Equivalent LEFM crack.

• x

Size effects of two basic simple types can be distinguished. Consider first the type I size effect, 4 which occurs in unnotched or un-precracked structures that fail when a macrocrack initiates from a smooth surface in the presence of a stress gradient. It is explained by the stress redis­tribution due to FPZ formation, the equivalent LEFM crack becoming nonzero. 5.6 In the absence of the statistical part, which is important for very large structures, the size effect on the nominal structure strength is

(6.2)

Here D is the structure size, CS0 is the asymptotic strength for a struc­ture of infinite size, 10 is a length constant proportional to the material characteristic length scale, and r is a dimensionless empirical constant with typical value 1.45.

For very large sizes (D ~ "") this equation predicts a vanishing size effect. However, in reality, in structures of sizes much larger than the FPZ, the strength in type I failures follows the Weibull statistical size effect.7 A smooth transition to the statistical size effect can be incorpo­rated in Eq. 6.2 by modifying it to a generalized energetic-statistical size effect5 law as follows:

(6.3)

Here n is the number of dimensions of the failure mode (1, 2, or 3) and m is the Weibull modulWJ, typically 24 for concrete and 8 to 10 for fiber composites.8- 10 A best fit of Eq. 6.3 is shown in Fig. 6.2 to a variety of test data on concrete and composites.

The degree ofbrittleneu (opposite of ductility) ia important for the probabilistic distribution function (pdf) of structural strength and of lifetime under cyclic and static fatigue. For brittle materials, the pdf of

4

3

.c a, c: !!!

2

1ij 'C Cl)

.~ 1ij

E 0 z

Taata: <> Nielson 1954 4 3 point Wright 1952 iJ,. 4 point Wright 1952 0 1 Inch Waker & Bloem 1957 :IC 2 Inch Waker & Bloem 1957 a Reagel & Willis 1931 D Sabnis & Mirza 1979 0 Rokugo 1995 0 Rocco 1995 + Lindner 1956 --Staliatical formula, m = 24 -----asymptote-small ---asymptot&-large

Best fit of type I size effect law

10

Normalized size

100 1000

Flguni 6.l Type I size effect (measured fur concrete and composites)' [reproduced with permission from American Concrete Institllte (ACI)].

strength and fatigue lifi:times is of Weibull type,~ while for ductile mate­rials it is Gaussian (or normal). In the central range both pdfs are hard to distinguish, but the very low failure probability point (Pr 1 ~) in the Weibull pdf is almost twice as far from the mean as for the Gaussian [for the same mean and the typical coefficient of variation of errors (CoV)]. For quasibrittle structures, the pdf is a hybrid of both.11- 13 It consist. of a Weibull tail grafted on the left onto a Gaussian core. For structures including one or a few FPZs, the grafting point is at P1 - 0.001. As the structure size increases, the grafting point moves to the right and when the structure volume becomes > 105 times the FPZ, the entire pdf becomes Weibullian. This is important for setting the safety factors for design, which should ensure P1 <10-<S per lifetime (which is about 104

times lower than one's probability of dying in a car accident and about the same as the probability of being killed by a lightning).

In structures with a large traction-free crack at peak load (or a sharp notch) the type II size effect is observed. 1-'11•14-22 This is a purely deter­ministic size effect on the mean strength (while material randomness affects only the Co V). It becomes the strongest for large sizes and corre­sponds to LEFM (ie., aN oc D-112

). As long as the failure modes for differ­ent sizes of geometrically scaled structures are also geometrically similar (which is often the case). the strength follows the Baiant size effect lawl-'11

( J-V2

CSN=Bft' l+~ (6.4)

Here B is an empirical, dimensionless material constant ft.' is the local tensile strength of material, and D0 is a transitional size proportional to the material characteristic length scale, which in tum is proportional to the FPZ size. The size effect factor in Eq. 6.4, proposed to ACI (Ameri­can Concrete Institute) in 1984, was in 2019 incorporated into the ACI design code (ASI Standard 318-2019) for beam shear, punching and strut-and-tie model.

The size effects of types I and II have been observed in concrete and in fiber-reinforced composites, aside from other quasibrittle materials1-'11 (see Figs. 6.2 and 6.3). Both size effect laws consist of a length scale determined by the FPZ size, which is a material property. This length is essential for realistic mathematical prediction of structure strength. The rest of thia chapter describes the fracturing behavior of concrete and fiber composites under various loading conditions which leads to the formation of the FPZ. Most of these failure modes are quasibrittle, exhibit the R curve, transitional size effects on strength, and hybrid Gauss-Weibull probability distributions of strength.

11.2 CONCRETE

Concrete in iu cured form may be treated as isotropic and homoge­neous in the macroscopic sense. On the microstructural levd it is highly heterogeneous, full of cracks of sizes at all scales from the nanoscale up." Iu mechanical response depends highly on the confinement. The uniaxial strength in tension, typically 4-5 MPa, is normally 8-10 times weaker than in compression. This is because the aggregate pieces can withstand large compressive stresses, but under tension the aggregate­cement bond is much weaker . .24.25

11.2.1 Streu-Strain Curw

The typical stress-strain curve of concrete24 under uniaxial compression as well as tension is shown in Fig. 6.4. It consists of two stages-prepeak and postpeak. The initial part is linear elastic. As the load increases, formation of microcracks and frictional microslips induces a prepeak noalinearity in the stress-strain curve, more pronounced in compres­sion than tension.

Beyond the peak, the uniaxial stress decreases gradually at increas­ing strain, which is called postpeak softening and can be observed in a stable manner only in a testing machine with a sufficiently stiff frame. The stability of postpealt testing can be greatly enhanced by controlling some displacement that is monotonically increasing (e.g., the crack opening displacement). Complete failure corresponds to stress reduc­tion to zero. The area under the stress-strain curve represent. the energy dissipated per unit volume.

FRACTURE AND FATIGUE 167

0.25

oTestdata .I:! 0.00 a

~~ -0.25

is = Q -0.50 ~.sz

~ -0.75 -0.35

-0.5 ,_ __ ........_ __ ......._ __ ......._ __ __. -1 00 . 0 0.5 15 2 ~5 3 0 1.5 2 2.5 3

Log (size D)

(a) (b)

Flgure&J 1'ype n me df~ {11) wboD ftberwmpo~ test clataof.2016 aud (b) concme1'teat dmof.2014 [repnidU«d with pcrmill:ian fmm Bltcricr].

Softening curve

Strain

Rguni 6.4 Skekh of a typical lllllaial. llntf-"1'alll cuml of OOllGl'ete.

D

D (a) (b)

In rpedme.ns or structuretlarger than a.few FPZt, thepostpealr.1ofte.n­lng Is inevl.tably accompanied by locallutton of fracturlng damage into a band of a finite width. w Under unl.uia'I tension, this band forms along a. plane normal to the loading dirution. as ahown in Fig. 6.5('1); see also Fig. 6.6(a) for ten.rile fractun: U11der bending. Under uniu:ial amipn:s­llon, various flllure medienlsm• are poalble depending on the boundary conditions. If the ends are sliding, a 8lngle dominmt uial 1plitting crack form• where damage localizes along a normal plane due to temile straiN genmtted. in the latmil dim:tion (perpendicular to the loading direction). The 1plllting .Is cauaed by elaatic Pol88on effect but mainly inelaatic mecla­Dlmls Nch u iDcllned slips and weclglng of one aggregtm: between two, 11 as shown in Fig. 6.s(b) and Fig. 6.6(b). The a:s:ial splitting fracture C'a'll8eS

no lliu effect becauae no strain energy is R1eased. from outaide the band. i.a

But lf one or bod!. en& are fb:ed. a compraslon-mear failure band fonm where the damage localizei along one or two Inclined planes u shown In Fig. 6.5(') and (d).This failure llCtlla1ly comi8ts of paralld small a:da'l 8}llit­ting aac.k.s arranged into an inclined band and does awae a size effect.1

1} D (C) (d)

Rture 6.S {11) Damage lcx;allzat!on a.Dder 1llliu:lal. tutlon. {b) Ami. ~llttiDg fAl!u.re 'Glider 1llllu1al compn!llion with aJidina enda. (c} ComprHrion·1h.ear &mm! under aniaxial compru.lion with one end. fiud.. (d) CompteHloD.-ahear failure Wider lllllulal oomprealon w:lth both ends fixcd.1

1.. CHAPTER SIX

(a)

(b)

fltlure 6.6 (") 'n:b.lllc liallatc In amctetc unda thn:e-poiol bending. (lo) Axial 'Pllllintl comp~m failuze of a~ qllnd.er [rcprodu.ced with pcrmllllon from .!ltmcr].

6.2.2 MuldPl•I fl'MtUN

In majority of ttru.ctura. con~ ii subjected to rmdtiaxtal ltrea. The biuial <:0111ptafion strength of concr= is higher than uniaxial. especially in the prmence of end rellrainb.~ Biaxial tenaion makes little difl'uence for ltmlgth from unluial Under combined. tendoll and compremon, concrdl: ii the -uat. The &!lure under b.luial low also OQ;UQ by loc3lizmon along prckrred orientmom. typi.cally perpendicu­lar to the muimum principll temile .txeN, Z7 and aJ.o abihib li2e eft'ect.t.

niulal c:oinpreadve loading of concrete can cause a major lncreue in ih strength. Under bydromtic a>mpreMon u wdl u uniax1al com­preaive ttrain with virtually rigid con5nement, ccmaete Det'eT RU. and only densifies by pore aill.ap1e. The postpeak bdurrior varies dramati­c:allr depending on the degree of con&ement. Inaeulng confinement from nOJ1e, the .respo!Ue becomes iDcreuiDgly ductile and Cftl1tllally mtchet from aofteniDg to hardenlnt~ u lb.own 1n F1g. 6.7. At the micro«ale, frictional llip becomes dominant cm=r microaacking.

Strain

,..,,. u SdiCl!Qltk beh.&11« of concme 11.Ddcr ClOnftned compm11<m {~ h lndk:aka unluial~).

6.2.J Fetigu9

Ideally, the strength of rdnfor«d cOJ1crete ltrUcturcs depends on steel bar•, and even though luge Qirllne c:racb, tr&vening 5-0 to 80 percent af the 1truc:tun width, typically dewlop. they haTe little effect on

atruc:tural strength under monotcmlc as well u cycl1c load!Jlg. Thua, .reinforced concrete 13 generally much ie.. temltive to filtlgue than metab. N(VCit}ideN, like in all materW., cracb grow under cyclic at well u awitained loada, e&111ed, for eumple, by 1raffic, wind, rotating machinery, or earthquaku.ll This can be quite Important for some situatlorui (plaln concrete, zone• of low reinforcement, development of ac:euive crack width allowing ingreu of corroaive agents, etc.).

ID ~ conaete la markedly differeat from metaJs.1.2'-SO Fatigue occurs by actiYation of pRailtlng .mkroc:raclai and mic:rotllps within the c:ydlc FPZ. Since limilar microcncb allo form during monotonic loading. the &n,ue aack inufu:e1 in conaete have no peculiar topog­raphy unllb metalt, and are huda to dUtinguiah viaually. The only difference la that the cyclic FPZ la mWler in me compared to the m011otonlc: FPZ. '1.Sl

The fatigue beharior of cont.rm: dependt on the strength of the material, the lo~ (mmgnitude, wueform, frequency; etc.) and the environmental conditions. Conaete undergoes &tlgue under both tmllon uid comJ'ftAIOll uid In fa.et compreaton fatigue ba.uds can grvw tranPerlely to compr~ strul.29 At the .microstructure level. fatigue consitta of irrennible microcnckiug ~· The load dil­placement curve ahJbii:. fur each unload-relDld cycle a hylteresla loop (Fig. 6.8) whose area equab the energy dUslpated in that cycle. The elamc modulwi degnde1 gradually with increasing ind.utic; deforma­tion in each cycle. The cyclic load-dilplacement curve is enveloped by the monotonic one.ss-ss

Displacement

Flgu ... 6.1 Schemalic al ~·reload. hyltcre&ls lo0pt Ill COllCfete t.i.tipe. 'o

6.2.4 Dellgn far Altfgue

Widely acapted. lm the fatigue usemnent of concrete (under uniu:ial compression, without stre.n revenal) it Che Au-Jakobsen equation~ for the ao-caI!ed S-N curve (stttas S va. number N of cycles)

"·- =1-PCl-R)logN J.

(6.5}

whidi npmaits a simplified. total life ipproac:h and does not consider the pr:opagalion of damage; a.,,_ it the maximum s!R!1 level of the cyclic loadin.g. f. b the oomp.re11lve strength of concrete. R is the load ratio (a-1a-J, N is the number of cycles to failure and P is an emplrl­<:al material COllnant whose typi<:al value is 0.068S. S« Fif. 6.9 for a typi<:al S-N curve for plain concrete.

There are alternate expre.Wons fur fatigue lif.etima in various design codes:!*

(i) For pure compression

8 IogN=y_

1cs,--1> iflogN<8

logN =8+ Sln(lO)(Y -S,.-)log(s,.....,.-s,,ml!!) iflogN>8 y - I y -s • .-

(6.6)

........ y 0.45+us __ ,_ la .-1 la I ....... ______ ........ =-;_ and S =-· -· S = ....!:!!!:!!.. l+l.8Sc,.m1n -0.3s!..w, •.mla f,;.i. ' • .- f,;.i.

(6.7}

(ii) For compre.Won-tenalon with a.,_ <0.G261cs • .-I. logN =9(1-S,,._) (6.8)

The fatigue reference compressive strength f..At has been inttodw:ed to take into account the increase of &tigue Wltitivity of COllcretc with increasing compnmive llrength. hill given by

J • .r- =0.85,S..,Ct)P • .-.f. (1-~) (6.9)

Here .P .. (t) is an emplrl<:al coeftklent wbic:h. d.epe.ads on the age t of concrete (in days} when htigue loading &ta11s, and P,,... Is another empirical coefficient whic:h takes into account the effi:ct of high mean ~ during loading and lb typical value it 0.85 when all 1he fl:res.1ell are expRMed In .MPa.

(iii) For pure wmion and tension-compression with a-~ 0.026

1°,.-I logN =12(1-Sd-) where S.,,- =l"d;,-1 (6.10)

0.8 OQ;)

Smax

T 0.1

0.6 .._ _______________ _

1.00E+04 1.00E+OS 1.00E+06 Number at cycles to fallure

RguN6.t S·N c.arw fortypical.Flaln C>O'Dctde (camprcl5!011)97 (reptodu.c:cd. with permla!on from Amertca11 Conuete Wtl!Ute (AC!)).

FRACTURE AND FATIGUE 1ff

Here att.- I• the mu:lmwn tenale suus and ft ls the .miD.lmum ~c tensile strength. Thus. Eq. 6.10 pruaibes the fatigue life under tension to depend only on the muimum 11.n!H level. Under vari­able amplitude loading. the Miner rulel'I may be applied.

The total life approach 18 much a!mpler but coneervattve and highly empirical The Jen conse.rvatl.ve, damage-tolm111t approach is U$ually not pursued for cone.me, although it is ponible. This it~ similar to metals, fatigue cracb in a>ncrete aho grow acrording to the Paris law.,. Thie hu long been conaldered u an empirical law relating the rate of crack growth per cycle, daldN, to the amplitude llK. of the stress intensity factor K through a power law (although recently it was derived Crom the activation encqy-am!J:olled breakage rate of intaatomic bonds"'°). The Paria law uadt

: =C(AK)"' {6.11)

The typical value of aponent m for concrete b 8 to 12,41 which is muc:h higher than what I• aeen fur metal• (2 to 4). The codlic:ient C lw been found" to be sensitive to environmental conditions (such as temperat:un), loading frequency a1 well u the load ratio R. Under tension-tension fiiligue, a distlnct llze effect lw !Wo been nported In the Paris law fur concrete which appears to affect the coefficient C but not the aponent m. sui Th1s makes the Par!• law plou of concrete specimens of different sizes appear a1 parallel lines spaced apart on log-log scale. It it expected that tlm tlze effect would vanish for very large mes but providing direct aperimental evidence is difficult aiDce this Is beyond most lab-sc:ale specimen size. (u the cyt1k FPZ is about 1 foot long). Thi• experimental observation also implies a ngnmcant me effect in the fatigue lifetimet of amcrete s!J:uctw:a. The size effect ill fatigue of concrete b a topic of ongoing reaearch.11-'1

6.3 FllER·REINFORCED COMPOSMS

Due to their lti1fne11, atrength. corrosion reslatanc.e, fatigue reatstance. and, moat lmpol'Wlt light weight, ti.ber-reinforced composites are widely used in aerospace, nawl. and automotive engineering. They are aho becoming increuingly prevalent in infrastructure applic:ation1-malnly for structural upgrades such u rapid repail:, rehabilitation, and seismic retrofitting of alrtlng structum.

Thdr basic const!tuems ue sJw or cubon fiben and • polymer malrix. The tibc:n cany me.It of the load and provide most of die stiff. nest and tensile stre.nglh of the material. The matrix and the fiber-matrix Interface bond coll.ttlbute strongly to the composlte's shear strength, tranmne strength. and, also. fU1gue mength. ea The ma1r1x transfers the load between the individual tiben and acts as the agmt holding die 6ber structure in place. Since Che fibers have specific dlrectlonality, a composite la alwayii anlaotrcplc (or orthotropic) at the maaoscale, unlike concrete, which is esse.atl.ally .lsotropl"' In structural applkatl.ons, the mlsotropy Is ~mitigated by unng quasi-iaoll'opic <:Cl!Jlpotitc lay-ups consmmg of individual laymi (or pn:-J?RP) oriented in variollll directiom. Comp~ undergo fracture through a variety of mec:hanWm

depending on the applied load and Its angle with the fl.ber direction. The ~ is progmsivc in nature and occurs at a number oflength scales, from the barely observable phenomenon of tiber-malrix debond­ing on the microacale to the matrix cracldng, flber breaking and delaml.nation43 on the maaoscale. Often one filil'l11'e mechanism trig­ger• some others, depending o.n the specific loading 1<:enario. Numeri­cal modeling and prediction of failure of compo1itll1 under multiuial loads b not a rtraightforward task.

6.11.1 Langltudln.1111 Fl'llc:tuN

When the applied load is tensile, in-plane and aligned with the tibers, the dominant mode of failure ls tlber breaking. The longitudinal temlle strength of the composite lam!na Is governed by the tensile attength of flbers. Ideally. if all the fiben had equal strength. they would break simultaneously when the applied load beame equal to their strength. However, ln reality, the fiben have a strength distribution. So, typlcaI!y, the weakest fil>en fail 1lnt. The stms then gets redimibuted to adjacent

170 CHAPTER SIX

6ber.. whic;h makes them more su.sceptible to failure. This usu.ally aho triggerl other f'ailure modes ruc:h u localiud malrilt shear failure around the Bber breab and Bber-matrh. debond!ng. Eventually the adjacent Bbers break. and the mialler c.raclcs merge to produce one large creck.42 Thl.scrackiseMentiallythelocalludzoneofdunagethatpropa­gares to cawse the Dnal. Wlw:e, u shown in Fig. 6.lO(o). At the end, addi­tional mechanisms, such u Bber pullout, also appear [see Fig. 6.U(o)].

J} (a)

Flber breaks coalesced to form macroaack

Individual fiberbreaks

(b)

Figure f.10 F~of\IDlclirea!on.t compo31Se lamina 11X1clerumam1 locd (a} tm&lon and (b) c:ompmllion.

Under compression, the raponse dlffm vudy from tension, Just like in conaete. However, unlike conaete. the mength of compO&lres in compreuion is 111ually lower than in tension. This is due to the fa.et that under compreulon, composite,, typlcally fall by formatio.n of a kink band~ whose on•et Is triggered by a buckling instability in the Bbers, u shown in Fig. 6.lO(b) and Fig. 6.ll(b). Thus. theultimaie compmslve load capacity of composites depends !en on the comp1ess:ive strength of 6ben than the shear strength of the ma.tm. whose main role is to resist the mic.robuclding of flbe.rs under comprwion.

A compmslon kink band hilJ not simulWleoualy but by propaga­tion of the fiber budding from. cawing a size drcct, similar to tcn.sile fracture.1.» nm size c:ffect is similar to 1ha1 caused by a localized. damage band propagation under tenaJ.on and Is Important to comlder apeclally for de«ign of larger structure~.

Unlike concme and rock. meuuring the pcmpeak softening curve of fiber composites in a sblble fa.Won baa never succeeded until n:cently.51

(a}

In the 19608,, stable measumnent of postpeak in conc:rete and roCk wu made pouible by the disarm:y thll the tating &ame had to be an order of magnitude der 1han previolllly wed, and sblbility of test wa.s further helped by developillg fut aervo-cantrol and by contraDing the test by a crack mouth. or crack tip gauge.51 However, fur composite$ it has not wmk Ulltil 2016 when it wu found that the grips (or 6xt:ures) far tms:ile loading «.fracture tests of~ supplied by manu&.cturet were fu tootoft andllgbt (note that no such grips are needed forconaeteslncethe $'J)CIC1men IJ loaded by contact with 1he loading &ame head or gtued in 1he cue of tension). Upon daigning loading grips that are two orden of mag­nitude st:Ufer and one order of magnih1de heavier than th0.te traditionally supplied by the manufacturer, a stable postpeak in 1atlle composite frac­ture ls now bind to be observed eullf1 (emi under load point control; U.S. Patmt US 10.416.053 B2, 2019).

f.3.2 1\'IMWrM 1ncl ShNr Fl'IC\\lre

When the load ls tramve.rse to the flben, the failure becomes matrix dominated. Streu con<:entrationa arise at the fiber-matrix inter&ce leading to debonding and matrix mic.roa:acking and microalips. This can occur at a vuletyoflocaliom. Eveatually the .miaoc.racka propagate and coalesce be<:oming one large mac:rocrack. which ultimately leadl to failure. aa shown in Fig. 6.12. Interhc;e d.ebonding and ftber splittinf may al.to develop. Under the in-plane shear loading, b:lo, the failure i1 governed by the propertiea of the m.atrh: but in that oue, the cracka often form paralld. to Che flbers.

Ffgur. 6.12 Debo.udlng-llldu«d microc:racb coaleai:ied Into a ~r crad.: under l.ranffcne load.

6.3.3 Multiallll Failure ThHrfa •nd C1lllll1ngu

Compo1ite1 are often subjected to multiuial loads. Besides, due to their anisotropy, off-wa uniulal loadl can lead to a multiu!al state of stren In the material. Muc:h. research effort has been devoted to devel­oping failure theories and model• to predict the multiuial strength envelope for composites. There cxirt a large number of theories

(b)

Figure 6. 11 (a) Tauiile ftu:tun: mowing bmkm fib en inclicatift of fiber pulloul (b} Fibcr lcink band [rqm>duced with pennlaaloll. from Elaeviet].

with varying ciegr«s of empiricism. :Recently, a. world.wide exera.e wu conducted to lfltematically ewlum the prominent one1.S3-S'I

While varying degrte11 of succeae were obtained fur theae theorlee (see Fig. 6.13), overall the predictions were found to dJffer by more than +200 percent and -SO percent from the mean. Also none of the

(a)

8!. :::!! ... t> -100

-200

-1200

FRACTURE AND FATIGUE 171

theories were found to predict the (()lllp!ete envd.ope of the Mure behavi.or for various multidirectional laminates. From a structure dealgner'• perspective cho08lng a failme theory is thua a rather dif­flc:ult task. Another weakness of this aerdse wu that the 8ize effect wu not t«ted.

-<>- Eckolct -- Puck - - Rotem --- Hart-Smhh (2)

• Truncated Max strain

• Experiments

600 1200 1800

o oWolfe

r~=-=-=-=;;ii~~~~t!'.~~::~~!i~-~ --Edge, Zinoviev, and Sun I- a.- l:> Chamls

0 600 1200 O'xMPa

SR= G,/Oy

<v- -<r -

-200

- Tsai ~ Hart-Smith (1)

• Experiments

1800

- - Rotem-A - - Rotem-B - Wolfe-A - Wolfe-B ~ Hart-Smith (3) e Experiments

-300 ..__ ___ ..._ ___ ...._ ___ _._ ___ __._ ___ __. ___ __,

-1800 -1200 -600 0

O'xMPa

1200 1800

Rgure 6.13 Multia:dd ~ envdopes for a.n E·glua/epoxy compo«lt.e-meuuiai a.nd prediaed via a 'fViety of failure dieoriea!n [reproduoed wUfi. permiMIOll from Ebevter).

172 CHAPTER SIX

The simplm lamina. failure theories are based on the muimum stms or maximum strain. They essentially involve detennining the principal streaaea in the in-plane lamina and comparing each 8trU8 component to the cormponding rue.agth. Th'* theorie11 do not account for interac­tions between 8trelJs components and are really p:llll1ic: llmlt andym theorie.1. Tensor polynomial theories such u the Tai-Hill and the Tsai-Wu aiterion" aim to provide one equation fur the entire failure envelope and try to account for the intenc:tions. These are in general slisJitly more effective in pmlicting the strength envelope-eapedally the Tai-Wu criterion. It ii e:qmmed as

(6.12)

where f, and~ are te.nsori conalsting of constants related to the matu:tal ruength and i. j = 1, 2, .. ., 6 (following Voigt notation for the ruess temor CJ, e.g .• CJ1 = CJw o, = ou).

h Jb.ould be noted Chat the "Pl"ic•tinn of 1heae failure theories to structuru must be aa:ampm!ed by sttuctural analym using analytic:al. tools such u the c:1assicaJ. lamlnate 1heory. or munerlc:al. me!hod& suc:h u ~dement analym. This woukl yield the stuc of sa~ at = point within Che lb:uctuR!, on which Che suiblbly chosen failure theory would predict whether :fallure would occur at Chat polnl HOW'IM\4 the approad1ea based ao. the multluial. sUellgth envelope QlD work only fur pnidlct!Dg 1he inilialion of~ Applying 1hem to proiic:t ~or propagating failure leads to a loss of object:ivitJ: In other wonb, c.h.anging the mesh me lea& to dlffermt resuhs.511 This is because 1hese 1heories do not consider the energy n:qulnld fur frac:tme propagation. Consequently. 1hey miss Che quasi br:IUleness of 1he c:omposlte and. moo important die me dl'ect.. Theae problems can be overmme by adapting more advanced dam&F models with a localmtion limiter. which slm.ulate ~ degradalion of 1he moduli and are al1bnted sudi.1hat the pred1cted energy cl!ssJpattao. would be consl.mnt with the fncture enagy of the material. .al.61

6.3.4 O.llmlna'don Fl'lctuN

Out-of-plane messes are generated in a composite laminate, espedally when adjacent iaFS have differing fiber orientations. These s1reMes are oam the highest at the free edges and tend to separate the layer~. lead­ing to another failure mode called delamination. 4:l This can occur under three butc mode.t, v1t. modes I, II, and m ahown in Fig. 6.14.

Unlike iD.tralamiD.ar fruture. the fracture energy fur delamination under each mode <:a11 be meuured in the lab in amble fashion without very stiff grips. This is becaU1e the out of plane stiffness and strength of the compoalte uauaily not very high. There mm a standard teat of clelamination fracture energy, although It needs to be improved to obtain information on the size effect. whidi is lignifiC'allL u.m.~

Model (Opening)

Mode II (Shearing)

Rguni 6.14 Modea of deJam!nat!on

6.3.5 Fadgl.lle

Mode Ill (Tearing)

Composit« in general have a better fatigue resistance than metal•, but the fatigue process u a lot more complica.ted. Unlib metals where one well-de&ed fatJgue crack grow• in a self-similar manner, in composites multiple damege mechanlams c:an occur and progreS6 simultaneously. The fac:tors that infiuence the fatigue of composit« are the tiber type, matrix type. reinforcement architecture (unidi­.rec:tional. fabric., braided, etc.). ccmpoalte lay-up aequence, environ­mental c:o11ditlom (temperature, humidity). stress ratio R of cyclic load, and the frequency.

While in metals the fatigue crack growth ii almost undetectable throughout most of Che fatigue lifetime, until sudden acc.elera.Uon toward the encl, in composites It is detectable evai. In the early Stagei u shown in Fig. 6.15.- The crack growth leadt to progrenivdy reduced stiffness and rtrength. Thwi, the tetb of the residual stiffneu and the residual ruength. provide a elmple meawre of the re.ma1n1ng useful lifetime of the structure (though only the former is nondestruc­tive). Marry phenomenologic:al. modeh for fatisue life pzediction are bued on this ideL ~-e Most design practices rely on phenomeno­logic:al approaches, such as the S-N curve fur the total life, because the complaity of c:rack growth hinders developing a mechanlstic: predic­tive model S-N c:urvu fur variout carbon flber-relnfo.rced pofymer composita IR! ahown in Fig. 6.16."° Whereu many models ai.st,11-1' they lack general applicability in tenm ofloading and material system. Further compllc:atlons arise due to random scatter of strength. and of h1igue l.lfetime., u the scattu I• much higher than that observed in meU!s. A comprehensive probabili.ttic approach to fatigue of compos­ites still awaits developmenl

Failure

Number of cycles

FRACTURE AND FATIGUE 173

110 ~ l

100 ~"' ~ .... ••

""""' ~ ' ,.

90

l """"' ...,.,. . .. ~i loo • ~ ::i •

80

I 70 E :::J E 80 ·~

50 i 'ii 40 ~ 0 z 30

.. .. : . If .. -1111 • II • r-o Cl

~ ... I ~ ·-... UD·- II r-... UD·<illl'llp-'cn • ""~ ._ 1~ .. I> I.a • Of·-

• OT·~ ""'

~ jll

D Of·Y~ ~ -Iii "'• • OT· Z b«ldhg

~ i 0 Cll·Y~

• QI. z b«ldhg Ii' - T...Slcf-andba'ldhg

20 .... TrnlOfOT~ - T...SlcfUO-....-

10

0 111111 1111111 1111111 I II

.... , ... ,

1.E+OO 1.E+01 1.E+02 1.E+03 1.E+04 1.E+OS 1.E+06 1.E+07 1.E+OB 1.E+09 Cycl1J8 to failure [-)

Flgun16.16 S-N curm for varlou.t carbon lib er-reinforced polymer oompOlll!et'° (reproduced with permlaalon from. .Elamerl.

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