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Resistencia de materiales 1 Gustavo C. Balbastro

RESISTENCIA DE MATERIALESGUA DE ESTUDIO

Gustavo C. Balbastroa

a Facultad Regional Paran, Universidad Tecnolgica Nacional. gbalbastro@yahoo.com, http !!""".#rp.utn.edu.ar

Palabras clave: Resistencia de materiales, mecnica de slidos, tensiones, deformacionesfatiga, fractura, factor de impacto.

Resume . Este documento proporciona una breve gua de estudio para los tpicos cubiertos por elcurso de grado de Resistencia de ateriales de la carrera de !ngeniera Civil. El estudiante encontrarmencionados todos los temas "ue, de acuerdo con nuestro criterio, debe conocer a fin de alcan#ar losob$etivos principales del curso. Este conocimiento ser la base sobre la cual se fundarn los cursossiguientes. %os temas son e&plicados brevemente ' se espera "ue sean ampliados con la lectura de bibliografa seleccionada.

ACLARACI!N" E()E )E*)+ E( ( %+ - R /(+ !0)ER0+ E0 % C )E2R 3 0+E() E% B+R 2+ - R (/ 2!4/(! 0 + RE-R+2/CC! 0 -+R 0!0G50 E2!+.E( /0 REC+-!% C! 0 3 2 -) C! 0 2E 2!6ER( ( 4/E0)E(, % ( C/ %E( (EC!) 0 E0 % B!B%!+GR 4 .

http://www.uncarolina.edu.ar/gmchttp://www.uncarolina.edu.ar/gmc

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Resistencia de materiales 7 Gustavo C. Balbastro

STRENGT# O$ MATERIALSA BRIE$ STUD% GUIDE

Gustavo C. Balbastroa

a Facultad Regional Paran, Universidad Tecnolgica Nacional. gbalbastro@yahoo.com, http !!""".#rp.utn.edu.ar

&e'(or)s : (trengt8 of materials, solid mec8anics, stresses, deformations, fatigue, fracture,impact factor.

Abstract. )8is document provides a brief guide to stud' t8e topics covered b' t8e undergraduatecourse (trengt8 of aterials for Civil Engineering. )8e student 9ill find all t8e t8emes t8at t8e' must

no9 in order to ac8ieve t8e main ob$ectives of t8e course, in our opinion. )8is no9ledge forms t8e basis for t8e follo9ing courses. )8e topics are briefl' e&plained and furt8er reading of selected bibliograp8' is e&pected on t8e part of t8e student.

DISCLAIMER" );!( )E*) !( 4+R !0)ER0 % /(E +0%3

E0)!+0E2 !0 );E B!B%!+GR -;3.

http://www.uncarolina.edu.ar/gmchttp://www.uncarolina.edu.ar/gmc

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Resistencia de materiales = Gustavo C. Balbastro

* INTRODUCTION

)8is document 9as t8oug8t as guidelines for t8e student of t8e course (trengt8 ofaterials, in t8e first semester of t8e t8ird level of Civil Engineering. s a guide, it is notintended to be t8e onl' source but to serve as a list of t8e topics t8at t8e student needs to

no9. Eac8 of t8ese topics is briefl' e&plained but furt8er reading of specific sources ise&pected, in order for students to e&pand t8eir no9ledge. )o do t8is, some good te&t boo sare suggested at t8e end of eac8 c8apter. )8is proposal does not e&clude ot8er te&ts or sourcest8at t8e student ma' find on t8eir o9n.

)8e topics 9ere selected 8aving in mind t8e minimum contents establis8ed as t8eno9ledge base in t8e career curriculum and t8e ones re"uired to successfull' complete t8e

follo9ing courses. )8erefore, some topics be'ond t8e minimum re"uired contents 9ere alsoincluded in t8is selection.

)8e e&pected outcomes of t8e course include ac8ieving some s ills and capabilities, as9ell as some basic no9ledge, ideas and criteria. (pecific ob$ectives are summari#ed at t8e beginning of eac8 c8apter to guide students> reading activit'.

!t ma' be relevant to briefl' e&plain t8e use of Englis8 for t8is guide. )oda', globali#ationis a realit', regardless of 98at 9e t8in about it, and it is imperative to be prepared to deal9it8 t8e needs of t8e 9or 9orld. /ndoubtedl', one of t8ese needs is computer s ill. )8eot8er one is Englis8. an' companies loo for ?or at least prefer? people t8at read, spea and9rite in t8is language@ post?degree studies ta e a proof of Englis8 proficienc', as a minimum,or are directl' presented b' foreign professors t8at use it in t8e classrooms and t8e boo s usedare almost all 9ritten in Englis8, too. great deal of t8e circulating information aboutmac8ines, soft9are, and t8e li e is also 9ritten in Englis8. /nfortunatel', it is "uite commonto see engineers t8at fail an Englis8 test or are appre8ensive about using t8is language.Conse"uentl', 9e believe t8at it is better to begin as earl' as possible.

s regards t8e contents of t8is 9or , since its intention is to ac"uaint t8e reader 9it8tec8nical reading in t8e foreign language, it is not entirel' an original production but acompilation. an' paragrap8s and pictures 9ere ta en or adapted from free sources li e

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Resistencia de materiales Gustavo C. Balbastro

+ INTRODUCTION TO STRENGT# O$ MATERIALS

+b$ectives: t t8e end of t8is c8apter t8e student 9ill 8ave ac"uired basic conceptsconnected 9it8 t8e topic. )8e' 9ill understand basic concepts about mec8anics, elasticmaterials, t8e different failure modes and t8eir relations8ip 9it8 securit'.

+.+ Ge eral co ce,ts)8e bodies t8at civil or mec8anical engineers deal almost ever' da' 9it8 in t8eir 9or

could be vie9ed in practice as formed b' continuum materials, and t8eir molecular or atomicstructure could be ignored for practical applications 9it8out significant errors. )8emat8ematical formulations of t8e mec8anical be8avior of suc8 bodies are t8e field ofcontinuum mec8anics.

2ifferent bodies can be classified as fluids and solids, in a rat8er arbitrar' 9a'. )8e formerare studied b' a branc8 of continuum mec8anics called fluid mec8anics. )8e latter belong tot8e field of solid mec8anics.

)8e materials t8at form solid bodies ma' e&8ibit ver' different properties, but t8ose 98ic8are common in t8e engineering practice can be grouped in a fe9 categories.

s it 9ill be defined later, in certain range of loads, some materials s8o9 an elastic be8avior. )8ose belonging to t8is group are no9n as elastic materials, and t8e' are t8eob$ect of elasticit' t8eor'. )8e same materials at 8ig8er load levels ma' be8ave plasticall'.+t8er materials ma' s8o9 a viscous be8avior, as 9ell, developing deformations t8at dependson time.

!n t8e current course 9e deal 9it8 materials in t8e elastic range, mainl', 9it8 a fe9

references about plastic be8avior.+n some materials, t8e effects of loads are full' developed at t8e moment t8at t8e loads areapplied, but in ot8er cases t8e effects evolve 9it8 time. )8erefore, a brief description ofr8eolog' concepts is made.

)8e t8eor' of elasticit' gives us a set of differential e"uations, 98ic8 must be integrated ineac8 particular problem. (uc8 integration is a 8ard $ob, generall' spea ing, and not ver' practical for dail' engineering problems, e&cept for t8e use of finite elements soft9are, 98ic8appro&imate a solution to t8e differential e"uations under certain boundar' conditions.

!n man' situations, some ideali#ed 8'pot8esis can be considered in order to simplif' t8e problems of elasticit'. )8is 9a' of approac8ing problems is t8e scope of strengt8 ofmaterials, a discipline t8at gives us useful tools for assessing t8e be8avior of bodies 9it8

some c8aracteristics and built 9it8 elastic materials. (ome of t8ose c8aracteristics are t8at onedimension is notoriousl' bigger t8an t8e ot8ers, and t8e geometr' of t8e structure before andafter t8e application of t8e loads is not ver' different.

)8ere are man' te&tboo s about strengt8 of materials, ranging from t8ose 98ic8 areclassics no9 to ne9er ones@ some 9ritten in Englis8 or Russian and man' of t8em translatedinto (panis8. (ome ot8ers 8ave been 9ritten directl' b' (panis8 spea ing aut8ors. )8e titlesinclude t8e classic name of Astrengt8 of materials but it can be found under ot8er names li eAsolid mec8anics , Amaterials mec8anics , Astabilit' , Astructural mec8anics or Atec8nicalmec8anics .

long t8e course 9e 9ill stud' p8'sical p8enomena, developing mat8ematical formulasand doing calculations 9it8 t8em. )8ese calculations impl' "uantities of p8'sical magnitudes,especiall' lengt8s and forces. 4or suc8 operations, dimensional co8erence bet9een t8e"uantities involved in eac8 e"uation is needed, and since a se"uence of operations 9it8

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Resistencia de materiales D Gustavo C. Balbastrodifferent formulas is done before ac8ieving a solution, it is ver' important to c8oose a s'stemof units for measuring t8ose "uantities and, in t8is 9a', avoid mista es. odern codes of

practice are generall' based upon t8e international s'stem of units (!F, so it is convenient becoming accustomed to it. 4or t8is reason, all t8e e&amples s8o9n in t8e course 9ill bee&pressed in (! units, t8us lengt8s 9ill be e&pressed in terms of meters and forces in 0e9tonsand t8eir multiples. 4ollo9ing t8e same rationale and 98enever possible, t8e same notationused b' t8e current rgentinean codes 9ill be used in t8e formulas, noting t8at somete&tboo s ma' use different ones.

+.+.+R- -) a ) )e/ormable bo)-es!n t8e preceding courses t8e e"uilibrium of rigid bodies 9as studied. )8e assumption of

rigidit' 8'pot8esis allo9ed t8e student to calculate reactions on t8e support of an isostaticstructure under loads, and to assess forces on a section of suc8 structure, 9it8out an'

no9ledge of t8e material properties. )8at s ill 9ill be t8e basis for furt8er understanding oft8e be8avior of more comple& structures.

)8is rigidit' means t8at t8e bod' under stud' is capable of bearing loads 9it8out c8angingits geometr', s8ape or dimensions. (uc8 bodies do not e&ist in t8e real 9orld, but in man'cases 9e ma' neglect t8is issue and obtain useful results.

not8er assumption implicitl' made 9as t8e capacit' of t8e bodies to bear arbitraril' largeloads 9it8out collapsing. /nfortunatel', 9e 8ave anot8er piece of bad ne9s: suc8 infinitel'strong bodies neit8er e&ist.

)8e structures conceived b' t8e engineer al9a's c8ange t8eir geometr', s8ape ordimensions 98en sub$ected to loads, including t8eir o9n 9eig8t. 4urt8ermore, loads largeenoug8 9ill al9a's cause t8e failure or collapse of a member or t8e structure as a 98ole. 0evert8eless, t8e art of t8e engineer is to design t8e structures so t8at t8e' accomplis8 t8eirfunction, bearing certain range of loads reliabl' 9it8out failing, collapsing or deformingundesirabl'. ;o9ever, in man' situations a member or a part of t8e structure could beconsidered as rigid in order to simplif' t8e anal'sis of a structure or focus on t8e deformationsof t8e 9ea er parts.

+.+.0$orces a ) stress bod' of an arbitrar' s8ape ma' be sub$ected to different inds of forces. )8ese forces

could be classified as surface forces, 98ic8 act on a part of t8e envelope of t8e bod'.2epending on t8e relative si#e of t8e region over t8ose loads are applied, if t8e area is small

enoug8 t8e forces are calledconcentrated loads . )8e ot8er possibilities areline loads , if t8earea loo s li e a line, or sur#ace loads if t8e loaded region loo s li e an area.)8e ot8er ind of forces are volume forces, 98ic8 arise on ever' point of t8e material.

)8ese inds of forces are t8e result of a potential field, as gravit' or electromagnetic field.)8e loads due to t8e self?9eig8t of t8e structure belong to t8is t'pe.

useful start point for t8e anal'sis of a bod' under loads is setting its free bod' diagram.!f it is in e"uilibrium, t8e s'stem of forces must 8ave null resultant force and moment. !f it isnot t8e case, it means t8at t8e bod' is accelerated and t8e free bod' diagram must include t8einertial forces for completion. !n t8is course 9e deal onl' 9it8 bodies in static e"uilibrium.

!f t8e bod' is sectioned b' a surface ( and one of t8e resulting portions is left, t8e free bod' must be completed 9it8 t8e resultant force and couple due to t8e forces acting on t8eeliminated part, in order to maintain t8e e"uilibrium.

)8e effect of one part over t8e ot8er is transmitted b' point?to?point action, and t8e overall

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Resistencia de materiales Gustavo C. Balbastroeffect of t8e actions in eac8 elemental point of t8e cutting surface is e"uivalent to t8e resultantforce and couple mentioned.

)8e Euler?Cauc8' stress principle states t8at upon an' surface real or imaginar'F t8atdivides t8e bod', t8e action of one part of t8e bod' on t8e ot8er is e"uivalent e"uipollentF tot8e s'stem of distributed forces and couples on t8e surface dividing t8e bod', and it isrepresented b' a vector fieldT1 2, called t8e stress vector, defined on t8e surface ( andassumed to depend continuousl' on t8e surfaces unit vectors.

4igure 1: 4orces acting on a bod' and stresses 9it8in it.

4rom t8e latter concepts, stress vector ma' be defined as

T in = lim

$ H F i$

= dF id$

1F

)8is vector magnitude ma' be t8oug8t as being formed b' t8e sum of t9o components,one normal to t8e surface and t8e ot8er l'ing on it. )8e first component is callednormal

stress % n and t8e second is t8e shear stress & n.

4igure 7: 4orces and stresses in a bod'.

Gree letters% and & are used in4igure 7, as usual in most te&t boo s about strengt8 ofmaterials or elasticit' t8eor', nevert8eless it is also useful to be accustomed 9it8 t8e s'mbolsused on modern design codes suc8 as C!R(+C, based upon t8e (CE, C!, !(C and ot8er

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Resistencia de materiales I Gustavo C. Balbastrodocuments. !n t8ese,f for t8e normal stress andv for t8e s8ear stresses are used. (ince forces8ave dimensions J %)?7K, stresses 8ave J4%?7K L J %?1) ?7K. !n engineering practice 9it8 (!

units, forces are e&pressed in 0 and stresses in -a. !n older te&t boo s and in la'manMsspee ing, ot8er units called Atec8nical are also used, so forces are measured in ilograms,using gf or g as its s'mbol, stresses and pressures in gNcm7. (ome countries use ot8ers'stems based upon Englis8 units, as t8e /( customar' units forces in pounds or ilo? pounds, stresses in psi or siF. (ince different units means means different numerical valuesfor t8e same "uantit', its strongl' advised to use t8e same units s'stem an'time in order toavoid severe errors.

+.0 D-//ere t-al e3uat-o s o/ e3u-l-br-um common approac8 to t8e anal'sis is to pose t8e e"uilibrium of forces on a ver' small or

more specificall', a differential volume of t8e bod'. (uc8 volume is sub$ected to t8e action ofvolume forces, as a distributed vector magnitude times t8e differential volumed'(d).dy.d*,and normal and s8ear stresses acting on eac8 face of t8e volume, bot8 applied to t8e area ofeac8 face. !ts usual to align t8e faces 9it8 a triad of ort8ogonal a&es, so, on opposed sides oft8e volume, one of t8e values is incremented 9it8 t8e variation of suc8 magnitude i.e.:% ) +d% ) !d) F.

4igure =: (tresses on eac8 face.

)8e 98ole process lead to t8e set of differential e"uations referred in t8e title. )8e' can be9ritten in different 9a's, for e&ample using Einstein notation in a compact form as

i , F

i =H

7F!n undergraduate te&t boo s t8e' are often 9rote in Cartesian coordinates more e&pandedas

) )

y) y

*) *

b ) = H

)y )

y y

*y *

b y= H

)* )

y* y

* *

b * = H

=F

)8e former means

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Resistencia de materiales P Gustavo C. BalbastroBarr de (aint?6enant 1IPI?1OO F or simpl'compatibility e1uations . !n a compact form t8e'can be 9ritten as

i ,2m 2m ,i

i2 , m

m ,i2 = H

OFor, using a more common engineering notation, 7 ) y7

7 y ) 7

= 7 7 )y ) y

7 y * 7

7 * y7

= 7 7 y* y *

7 ) * 7

7 * ) 7

= 7 7 *) * )

PaF

7 )

y * =

)

y*

)

*)

y

)y

* 7 y * ) = y y* ) *) y )y * 7 *

) y=

* y* ) *) y )y * PbF

!t must be noted t8at normal stresses are associated to elongation if t8e' are tensile or tos8ortening if t8e' are compressive. 0ormal stresses and t8e trace of t8e stress tensorF are alsoassociated to volume c8anges. (8ear stresses are associated to angular variations, distortions,instead 4igure DF.

4igure D: )9o?dimensional geometric deformation of an infinitesimal material element.

+.5 Co st-tut-ve e3uat-o s" #oo6e7s la((ince real solid materials deforms under loads even 9it8 ver' lo9 loads, alt8oug8

imperceptible, some deformation occursF, a relations8ip bet9een bot8 magnitudes is needed.!ts better to put suc8 relations in terms of specific magnitudes, suc8 as stresses and strains.

Robert ;oo e 1 =D?1IH=F found t8at certain materials follo9 a simple relations8ip bet9een stress and strain. aterials t8at are 9ell described 9it8 t8e so called ;oo eMs %a9 are

no9n as linear elastic materials, or in s8ort, elastic materials. ore precisel', real materials be8ave elasticall' onl' 9it8in a limited range of stresses. +utside t8is range, ot8er be8aviors

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Resistencia de materiales 1H Gustavo C. Balbastroarise.

;oo eMs %a9 is t8e simplest constitutive e"uation, term used to design t8e set of

mat8ematical e&pressions t8at describes some ind of mec8anical be8avior of a materialunder stresses. !t ma' be 9rote as= 3 1HF

!n t8is e&pression, 3 is no9n aselasticity modulus or 4oung5s modulus , named after)8omas 3oung 1II=?1O7PF. !ts values are measured e&perimentall' for most materials andcompiled in tables, design codes, etc.

+f course, t8ere are more complicated constitutive la9s, suc8 as t8ose for 8'perelasticmaterials, GreenMs materials, Cauc8'Ms materials, etc., but t8e' fall be'ond t8e scope of t8e present course.

s presented above, ;oo eMs la9 relates normal stress 9it8 a strain in its direction. Butunder normal stresses acting over a direction, some strains normal to t8at are also found. )8e

amount of t8em is related 9it8 t8e -oissonMs ratio6 (.2. -oisson, 1IO1?1O HF. 4or anisotropic material, 98ose mec8anical properties do not depend upon t8e direction, ;oo eMsla9 is

ii=1

3 [ ii 22 ]

i = 1

7G i

11F

98ere 7 is t8e shear modulus , its dimensions and units are identical to 3 and its related9it8 it b' -oissonMs ratio6 dimensionlessF as

7 = 3 7 1

17F.

+.8 $u )ame tals o/ t9e t9eor' o/ elast-c-t'/pon t8e basis of e"uilibrium, compatibilit' and constitutive e"uations, t8e problem of t8e

deformation of an elastic bod' under prescribed loads and supports could be solved at leastin t8eor'F. But in most real problems, integration of suc8 s'stem of differential e"uations its a8uge effort, or simpl' it couldnMt be solved or found an anal'tical solution. (o, in most casesnumerical solutions are calculated using computer codes b' t8e finite element met8od orot8er.

4ortunatel' for t8e engineer, man' problems ma' be treated in a simpler 9a' b' t8emet8ods of strengt8 of materials, based upon certain 8'pot8esis. )8is ind of problems are t8e

sub$ect of t8e present course.+.: Pr- c-,al stresses

!n a single point of an elastic bod' under loads, a state of stresses and strains is developed.

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Resistencia de materiales 11 Gustavo C. Balbastro

4igure : (tress vector on a plane of normal unit vector.

!tMs possible to find some angle 9it8 no s8ear component, so normal stresses reac8 t8eirma&imum and minimum values and t8e stress vector is normal to t8e surface. (uc8 stressesare called principal stresses and t8eir orientations are t8e principal directions , t8at occur at anangle8 to t8e plane on 9ic8% ) acts, calculated b'

tan 7 =7 )y ) y

lso, rotating t8e plane, ma&imum and minimum values of t8e s8ear stresses are found atsome angle.

-rincipal directions are ort8ogonal, ma&imum and minimum s8ear stresses are located atDT from t8em.4or t8e sa e of simplicit', in a t9o?dimensional case, principal stresses are

1 , 7 = ) y

7 ) y7

7

)y7 1=F

and ma&imum and minimum s8ear stresses are

ma) , min= ) y7 7

)y7 1 F

!n t8e same 9a', principal strains ma' be defined.

+.:.+ Pr- c-,al stresses ra,9-cal calculat-o!n man' problems principal stresses or strains are involved. grap8ical met8od to do so is

t8e o8rMs circle C8ristian +tto o8r, 1O=O?1P1OF. )8e follo9ing figures s8o9 itsapplication to a 7?2 4igure I and4igure OF or =?2 problems 4igure PF.

/sing t8e o8r circle one can find t8e stress components % n,& nF on an' ot8er plane 9it8 adifferent orientation8 passing t8roug8 point P . 4or t8is, t9o approac8es can be used:

U )8e first approac8 relies on t8e fact t8at t8e angle8 bet9een t9o planes passing t8roug8 P is 8alf t8e angle bet9een t8e lines $oining t8eir corresponding stress points % n,& nF on t8e

o8r circle and t8e centre of t8e circle 4igure IF. !n ot8er 9ords, t8e stresses % n,& nF acting ona plane at an angle8 countercloc 9ise to t8e plane on 98ic8% ) acts is determined b' traveling

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Resistencia de materiales 17 Gustavo C. Balbastrocountercloc 9ise around t8e circle from t8e no9n stress point % ),& )yF a distance subtendingan 8 at t8e centre of t8e circle 4igure IF.

U )8e second approac8 involves t8e determination of a point on t8e o8r circle called t8e pole or t8eorigin o# planes . n' straig8t line dra9n from t diminis8es t8e pole 9ill intersectt8e o8r circle at a point t8at represents t8e state of stress on a plane inclined at t8e sameorientation parallelF in space as t8at line. )8erefore, no9ing t8e stress components% and &on an' particular plane, one can dra9 a line parallel to t8at plane t8roug8 t8e particularcoordinates % n,& nF on t8e o8r circle and find t8e pole as t8e intersection of suc8 line 9it8 t8e

o8r circle. s an e&ample, letMs assume 9e 8ave a state of stress 9it8 stress components% ),% y and & )y as s8o9n on4igure O. 4irst, 9e can dra9 a line from point 0 parallel to t8e plane ofaction of% ) or, if 9e c8oose ot8er9ise, a line from point / parallel to t8e plane of action of% y.)8e intersection of an' of t8ese t9o lines 9it8 t8e o8r circle is t8e pole. +nce t8e pole 8as been determined, to find t8e state of stress on a plane ma ing an angle8 9it8 t8e vertical, or

in ot8er 9ords a plane 8aving its normal vector forming an angle8 9it8 t8e 8ori#ontal plane,t8en 9e can dra9 a line from t8e pole parallel to t8at plane (ee4igure OF. )8e normal ands8ear stresses on t8at plane are t8en t8e coordinates of t8e point of intersection bet9een t8eline and t8e o8r circle.

4igure I: o8rMs circle for plane stress and plane strain conditions double angle approac8F.

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Resistencia de materiales 1= Gustavo C. Balbastro

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Resistencia de materiales 1 Gustavo C. Balbastro

4igure P: o8rMs circle for a t8ree?dimensional state of stress.

+.; Duct-le a ) /ra -le mater-als

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Resistencia de materiales 1D Gustavo C. Balbastro)8e lo9er 'ielding stress is denoted asF y at t8e rgentinean C!R(+C code and t8e

ultimate strengt8 asF u.

+t8er materials, suc8 as non?ferrous allo's or cold 9or ed steel s8o9 a diagram 9it8out aa definite 'ield step 4igure 11F, so t8atF y is defined conventionall'.

4igure 11: (tressVstrain curve s8o9ing t'pical 'ield be8avior for nonferrous allo's. 1: )rue elastic limit, 7:-roportionalit' limit, =: Elastic limit, : +ffset 'ield strengt8.

)8ese materials 8ave an elastic period follo9ed b' a non?linear region, 9it8out a definite'ielding. +n t8em, 'ielding strengt8 is defined conventionall'. +n bot8 ind of materials,deformations be'ond t8e 'ielding limit remain permanent, and are called plasticde#ormations.

)8ere are ot8er materials, li e cast iron, concrete 9it8out reinforceF, carbon fiber,

masonr', glass, etc. (uc8 materials do not s8o9 plastic deformations, so, under certain loadt8e' bro e suddenl' 4igure 17F. )8ose are no9n asbrittle materials.

4igure 17: (tressVstrain curve for brittle materials.

!t must be noted t8atductility ma' be refer to propert' of t8e material itself o as a propert'of a structural member. !n t8e first case denotes t8e ratio bet9een total strain up to rupture andelastic strain. !n t8e second, it depends not onl' on material ductilit' and refers to t8e capacit'of t8e member of ta e large plastic deformations.

)emperature, among ot8er factors, 8as big importance in t8e ductilit' or brittleness of a

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Resistencia de materiales 1 Gustavo C. Balbastromaterial. 4or e&ample, steel becomes brittle at temperatures enoug8 lo9 and looses itsresistance at temperatures of HHTC or DHHTC.

+.< %-el) t9eor-es or cr-ter-a 'ield criterion, often e&pressed as 'ield surface, or 'ield locus, is a 8'pot8esis

concerning t8e limit of elasticit' under an' combination of stresses. )8ere are t9ointerpretations of 'ield criterion: one is purel' mat8ematical in ta ing a statistical approac898ile ot8er models attempt to provide a $ustification based on establis8ed p8'sical principles.(ince stress and strain are tensor "ualities t8e' can be described on t8e basis of t8ree principaldirections, in t8e case of stress t8ese are denoted b'% : , % < and% =.

)8e follo9ing represent t8e most common 'ield criterion as applied to an isotropicmaterial uniform properties in all directionsF. +t8er e"uations 8ave been proposed or areused in specialist situations.

+.

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Resistencia de materiales 1I Gustavo C. Balbastro+.

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Resistencia de materiales 1O Gustavo C. Balbastroengine, a 98eel a&le, etc., main loads are variable. /nder c'clic loads, a progressive andlocali#ed damage of t8e material ta es place, 9it8 gro9ing and propagation of crac ing,

microscopic at t8e beginning and leading to failure, even at stresses lo9er t8an 'ield strengt8or ultimate strengt8. (o, in suc8 cases, stress must be limited up to a certain value so t8at t8enumber of load c'cles during t8e memberMs life do not produce failure 4igure 1 F.

)8e () defines fatigue strengt8, $ N# , as t8e value of stress at 98ic8 failure occurs after N # c'cles, and fatigue limit,$ # , as t8e limiting value of stress at 98ic8 failure occurs as N # becomes ver' large.

(ince fatigue is due to propagation of crac s at microscopic scale, itMs closel' related 9it8fracture mec8anics, as 9eMll see in t8e ne&t section.

)8ere are some differences if t8e load c8anges it sign in eac8 c'cle, or if it 8as a base valueand fluctuations around it, etc., but designing against fatigue is basicall' t8e searc8 of a stressfor 98ic8 t8e member could sustain a number of load c'cles 8ig8 enoug8 or indefinitel'F.

)'pical e&amples of fatigue problems are found in mac8ines or ve8icles a&les, elevators, bridges, pressure vessels, offs8ore structures, etc. Even certain t'pes of road pavements aredesigned from fatigue considerations.

4igure 1 : (tress V 0umber of c'cles curve for an aluminium.

)8e interest of stud'ing fatigue and t8e related fracture mec8anics began 9it8 some

catastrop8ic accidents on trains, s8ips and planes, 98ose components 9ere supposed to be9ell designed from a static?load point of vie9. (ome collapses of civil structures 9ere alsocaused b' fatigue of critical components, unions, etc.

+.>.+#- 9?c'cle /at- ue)8is range of fatigue problems includes t8ose t8at re"uire more t8an 1H load c'cles to

failure. )8e stresses are lo9 and deformations remain basicall' elastic. (uc8 t'pe of problemsare c8aracteri#ed b' t8e (?0 curve in semi?logarit8mic scale 4igure 1 F also no9n as>?hler5s curve .

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Resistencia de materiales 1P Gustavo C. Balbastro+.>.0Lo(?c'cle /at- ue

)8is range of problems includes t8ose t8at stresses are 8ig8 enoug8 to produce plasticdeformations, t8at are better described in terms of strains but stresses as t8e former t'pe.+.+*$racture mec9a -cs /u )ame tals

4racture mec8anics is concerned 9it8 t8e stud' of propagation of crac s in materials.)8eoretical and e&perimental investigations are committed in order to asses t8e stress?strainstate in t8e neig8bor8ood of a crac . )8e aim of t8em is to predict 98en a crac 9ill remainstable or gro9. )8is no9ledge is critical in man' situations.

t present da' itMs an active investigation field, since pioneering 9or s of . . Griffit81OP= V 1P =F and G.R. !r9in 1PHI V 1PPOF.

s a general case, t8ree different modes of opening a crac are considered, depending

upon t8e direction of t8e relative movement of crac Ms lips. )8ese modes are called !, !! or !!!4igure 1DF.

4igure 1D: 4racture modes.

Current criteria include plastic deformations influence upon t8e energ' available for cracgro9ing.

4racture 9as t8e cause found in several catastrop8ic failures in s8ips, airplanes, bridges,etc.

+.+*.+ L- ear?elast-c /racture mec9a -cs.)8is approac8 9as developed from t8e 1P7HMs, firstl' b' . . Griffit8 9it8 8is energetic

balance. )8en G.R. !r9in and E. +ro9an 1PH7?1POPF refined and e&tended it up to t8e stressintensit' factor criterion, : , depending on forces, geometr' and crac 9idt8. Because of t8e8'pot8esis used in its deduction, t8e applicabilit' of t8is criterion is limited b' t8e ductilit' of

t8e material, t8e t8ic ness of t8e member, loading speed, etc. !n order to avoid cracinstabilit', must be : Y :A , being t8e later a propert' of t8e material e&perimentall'determined.

+.+*.0 Be'o ) l- ear?elast-c /racture mec9a -cs.)8e limitations of t8e former approac8 are circumvent b' ot8er developments, suc8

elastoplastic fracture mec8anics and criteria C)+2 and W. )8ese and ot8er p8enomena suc8 ascrac gro9ing due to creep, etc., fall be'ond t8e scope of t8is course.

+.++R9eolo ' /u )ame tals

)8e division bet9een fluids and solid materials, alt8oug8 intuitive, is rat8er arbitrar'. !nsome situations t8ere is no doubt, but in ot8er t8e time pla's an important role in

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Resistencia de materiales 7H Gustavo C. Balbastrodeformations under loads.

R8eolog' is t8e term coined b' E.C. Bing8am 1OIO V 1P DF for designate t8e stud' of t8e

be8avior of flo9 of matter. 4rom an epistemological point of vie9, r8eolog' could be bet9een solids and fluid mec8anics branc8es.an' materials, under loads, 8ave a response t8at depends upon time. 4or e&ample,

concrete under a compressive load 8ave an instantaneous deformation, seemingl' elastic. Butif t8e load is sustained enoug8 time, deformation continues as if being a viscous material.Glass, plastic, soil, pol'mers, food, non?ne9tonian fluids in general, are sub$ect of stud' ofr8eolog'.

4or e&plain t8e be8avior of different materials along t8e time, some models 9eredeveloped, suc8 viscoelastic, elastoplastic, etc. Eac8 of t8em lead to a differential e"uationt8at must be integrated to calculate t8e response of certain structural member under loads.

-8enomena li e concrete creep o stress rela&ation must be ta en into account in man'

designs, due t8eir importance in increasing deformations of structures, redistribution offorces, loss of bearing capacit', etc.

+.+0Secur-t' coe//-c-e t@ allo(able stress@ loa) a ) res-sta ce /actors)8e main goal of a structure is to carr' loads successfull'. )8is success 8as different

meanings, li e do not collapse, maintain some s8ape, last some time, etc. ll t8is ob$ectivesmust be accomplis8ed 9it8in a reasonable cost, aest8etic, environment considerations, etc.

!tMs eas' to see t8at all t8e problem ma' be reduced to R n Z (, 98ere R n is t8e nominalresistance of a member to a failure mode and ( is t8e significant internal force i.e. bendingmoment, s8ear force, normal force or torsion coupleF. )8is ine"uation must be satisfied forever' failure mode.

But t8ere are some lac s of our no9ledge t8at are inevitable. 4or e&ample, material properties for t8e real member could be different from t8ose considered or specified, duefabrication issues, aging, etc. )8e effective dimensions of t8e member ma' be different fromt8ose t8eoretical considered b' t8e engineer. )8e actual loads couldnMt be predicted e&actl'.+ur calculation met8ods or 8'pot8esis could left some aspects for t8e sa e of simplicit'. (o,itMs 9ise to c8oose a R greater t8an (. odern design codes, using t8elimit state design p8ilosop8', set t8e condition as [R n Z (u. 0o9, [ is a strengt8 reduction #actor , less t8anunit', t8at diminis8es t8e nominal resistance R n depending upon factors li e failure mode,9arning possibilit' associated 9it8 t8e failure, etc. +n t8e ot8er 8and, (u is t8e solicitation orinternal force obtained b' amplif'ing t8e mandator' loads b' someloads #actors , based upon

probabilistic considerations of t8e variabilit' of eac8 ind of load. Generall' spea ing, loadfactors are greater t8an unit', e&cept 98en t8e associated load 8as a favorable or stabili#ingeffect. )8is approac8 is no9n as limit state design is no9n in European literature or as%R42 in /( .

)8is approac8 8as some advantages over t8e older, called permissible stress design orallo"able stress design. !n t8e last approac8, t8e strengt8 of t8e material usuall' t8e 'ieldstrengt8F is divided b' an uni"ue sa#ety #actor or security coe##icient \, bigger t8an one. )8isreduced value of stress is no9n asallo"able or permissible stress. )8en, t8e engineer mustverif' t8at under service loads, no portion of t8e structure e&ceed t8at stress. (o, t8e conditionmust be ] ^ ] perm, 9it8 ] permL ] 'ield N \. )8is approac8 is no9n as (2 in t8e literature,speciall' in /( .

0o matter 98ic8 approac8 one c8oose, %R42 or (2, material properties must beestablis8ed b' testing samples in a number big enoug8 to 8ave statistical significance,

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Resistencia de materiales 71 Gustavo C. Balbastrocalculating its mean and standard deviation and a c8aracteristic value. )8is value is usuall'

no9n in t8e codes as speci#ied strength . 4or t8e sa e of simplicit', t8e tests used for t8is

purpose is t8e simplest one, t8e a&ial traction test. )8is ind of test is performed b' pulling aspecimen speciall' prepared, loading it slo9l' as s8o9n in figures1H to 17.!s eas' to see t8at in actual structures, eac8 part of t8em are sub$ected to a more

complicated tensional state, as s8o9ed in paragrap81.1.7 and 1. . !n some situations, t8isissue is attended b' calculating a comparison stress using a suitable 'ielding criterion, ass8o9n in paragrap8 1.O. 0o t8eor' is suitable for all materials, so t8e engineer needs to no998ic8 t8eor' or criterion is more representative of t8e be8avior of t8e material used.

+.+4Glossar'$trength : is t8e abilit' to 9it8stand loads 9it8out failure. !f referred to a material, is

e&pressed as t8e ma&imum stress before rupture, ot8er9ise, if referred to a structural memberis t8e ma&imum force before rupture (panis8:resistencia F.

$ti##ness: or rigidit', is t8e amount of load t8at produces a certain deformation in amember. )8e opposite is fle&ibilit'. )8e stiffness of a member or a structure depends on t8ematerial properties, its dimensions, inertia and form. (panis8:rigide* B #le)ibilidad F.

$tability : is to t8e abilit' of a structure or a member to maintain its initial configuration.(panis8:estabilidad F. Cuctility : is t8e abilit' of a material, a member or a 98ole structure of 8old plastic

deformations be'ond t8e elastic limit, 9it8out failure. !s calculated as t8e ratio bet9een totaldeformation and elastic deformation. (panis8:ductilidad F.

Toughness : is t8e abilit' of a material of avoid crac s gro9ing and resist fracture. !s related9it8 t8e abilit' of absorb energ' and deform plasticall'. (panis8:tenacidadD.

+.+5Su este) lectures+. Bellu##i.Aiencia de la Aonstruccin . )omos 1 ' 7. guilar, 1P O.

. Boresi, R. (c8midt, +. (idebottom. /dvanced mechanics o# materials Dt8 ed.

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Resistencia de materiales 77 Gustavo C. Balbastro!0)!. C!R(+C IH1. Aomentarios al Reglamento argentino de estructuras de aluminio.

-ro'ecto en discusin p_blica nacionalF. !0)!, 7HHP.

C. %ipson and R. Wunivall. Iandboo2 o# stress and strenght. Collier, 1P =.W. cCormac. /nlisis estructural . =a. ed. ;arla, 1PO=.). egson. $tructural and stress analysis . Butter9ort8 V ;einemann, 7HHH.! iroliubov et al. Problemas de resistencia de materiales. !R, 1POH.