[mordini] 3d numerical modeling rc behavior
TRANSCRIPT
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Andrea Mordini
Three-Dimensional Numerical Modeling
of Reinforced Concrete Behavior
Twist
T
o r q u e
Tutor: Prof. Roberto Cerioni
Co-tutor: Prof. Ivo Iori
Doctorate Coordinator: Prof. Paolo Mignosa
University of Parma
Doctorate in Civil Engineering – XVIII CycleStructural Mechanics – ICAR/09
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University of Parma
Faculty of Engineering
Doctorate in Civil Engineering – XVIII Cycle
Structural Mechanics – ICAR/09
Andrea Mordini
Three-Dimensional Numerical Modeling
of Reinforced Concrete Behavior
Doctorate Dissertation
Tutor: Prof. Roberto Cerioni
Co-tutor: Prof. Ivo Iori
Doctorate Coordinator: Prof. Paolo Mignosa
March 2006
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8!3 )e5e,%# *r5*%# ,#, "! %r*+!")e:;
Pr#'( I# I#r! +ur!,2 * 3e)%ure(
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TABLE OF CONTENTS
CHAPTER < – THE THREE-DIMENSIONAL PROBLEM (((((((((((((((((((((((((((((((((((((((((((((((((
=(( – C#,)re%e '*!3ure '!e3+" (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((
=((. – Re!,'#r)e5e,% )#,%r!4u%e (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( 9
=(= – SINGLY CRACED MATERIAL (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( .<
– @ –
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T*43e #' )#,%e,%"
=(=(< – M*%er!*3 4e%1ee, %&e )r*)6" (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( .=
=(=( – M*%er!*3 !, %&e )r*)6 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( .>
=(=(= – Cr*)6 "p*)!,2 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( .
=(=(> – A22re2*%e 4r!+2!,2 *,+ !,%er3#)6 )#,%r!4u%e" ((((((((((((((((((((((((((((((((((((((((((((((((((((((( .
=(=( – Re!,'#r)e5e,% *,+ +#1e3 *)%!#, )#,%r!4u%e" (((((((((((((((((((((((((((((((((((((((((((((((((((((((( .9
=(=(. – Te,"!#, "%!''e,!,2 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( @<
=(> – DOUBLY CRACED MATERIAL (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( @@
CHAPTER > – APPLICATIONS ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( @9
>(< – INTRODUCTION (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( @9
>( – NON-LINEAR PROBLEM SOLVING PROCEDURES (((((((((((((((((((((((((((((((((((((((((((((( 0
>((< – Se)*,% 5e%+ ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( <
>(( – Ne1%#,-R*p&"#, 5e%+ (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( <
>((= – M#+!'!e+ Ne1%#,-R*p&"#, 5e%+ ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((
>((> – ?u*"!-Ne1%#, 5e%+ (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((
>(( – I,!%!*3 "%!'',e"" 5e%+ (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( =
>(= – NUMERICAL PROCEDURES IN ABA?US (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((
>(=(< – N#,-3!,e*r pr#43e5" !, ABA?US ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((
>(=( – T&e )#+e =D-PARC !, %&e F!,!%e E3e5e,% 'r*5e1#r6 (((((((((((((((((((((((((((((((((((((((((
>(=(= – Nu5er!)*3 pr#)e+ure '#r "%r*!, +e)#5p#"!%!#, ((((((((((((((((((((((((((((((((((((((((((((((((((((( 9
>(> – PLAIN CONCRETE BIAXIAL TESTS (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( 9<
>(>(< – Eper!5e,%*3 pr#2r*5 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( 9<
>(>( – Re"u3%" *,+ )#5p*r!"#," ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( 9=
>( – PLAIN CONCRETE TRIAXIAL TESTS (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( 9@
>((< – Eper!5e,%*3 pr#2r*5 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( 9@>(( – Re"u3%" *,+ )#5p*r!"#," ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( 9
>(. – VECCHIO-COLLINS SHEAR PANELS (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( (.(< – Eper!5e,%*3 pr#2r*5 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( (.( – Re"u3%" *,+ )#5p*r!"#," (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((
>(@ – PANG-HSU SHEAR PANELS ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( (@(< – Eper!5e,%*3 pr#2r*5 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( (@( – Re"u3%" *,+ )#5p*r!"#," (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((
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T*43e #' )#,%e,%"
>( – REINFORCED CONCRETE TIE (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((
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T*43e #' )#,%e,%"
(( – S%ru)%ur*3 5#+e3!,2 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( 0=
((= – Re"u3%" *,+ )#5p*r!"#," ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( 0>
(. – CONCLUSIONS (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( 0@
CHAPTER . – CONCLUSIONS (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( 09
REFERENCES ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((
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CHAPTER < – THE THREE-DIMENSIONAL PROBLEM
Re!,'#r)e+ )#,)re%e RC "%ru)%ure" e&!4!% * )#5p3e 4e&*!#r ee, '#r 3#1 3#*+ 3ee3"(
N#,-3!,e*r )#5pre""!e "%re""-"%r*!, re3*%!#," %e,"!3e )r*)6!,2 p#"% )r*)6!,2 "#'%e,!,2 *,+
!,%er*)%!#, e''e)%" 4e%1ee, )#,)re%e *,+ re!,'#r)!,2 4*r" *re %&e 5*!, "#ur)e" #' * &!2&37 ,#,-
3!,e*r *,+ )#5p3!)*%e+ re"p#,"e( I, #r+er %# )*p%ure %&e re*3 "%ru)%ur*3 4e&*!#r "#p&!"%!)*%e+
,u5er!)*3 %##3" *re ,e)e""*r7 %# %*6e !,%# *))#u,% *33 %&e re5*r6*43e p&e,#5e,* *,+ %#
per'#r5 %&e %!5e-)#,"u5!,2 ,#,-3!,e*r )*3)u3*%!#,"(
I, %&!" +#)%#r*%e +!""er%*%!#, %&e %&ree-+!5e,"!#,*3 =D )#,"%!%u%!e 5#+e3 '#r ,#,-3!,e*r
*,*37"!" #' RC "%ru)%ure" =D-PARC – T&ree-D!5e,"!#,*3 P&7"!)*3 Appr#*)& '#r Re!,'#r)e+
C#,)re%e !" pre"e,%e+( T&e =D "%u+7 "%*r%e+ "#5e 7e*r" *2# 1!%& %&e *u%r" 2r*+u*%e %&e"!"
1&!)& 3*!+ %&e '#u,+*%!#," #' %&e 5#+e3 =>J( T&*% 1#r6 1*" *, e%e,"!#, #' PARC *
,u5er!)*3 5#+e3 '#r 5e54r*,e e3e5e,%" "u4Ke)%e+ %# p3*,e "%re"" F!2ure
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CHAPTER – STATE OF THE ART
(< – INTRODUCTION
I, %&e '#33#1!,2 )&*p%er *, #er!e1 #, %&e "%*%e #' %&e *r% #' %&e %&ree-+!5e,"!#,*3 =D
5#+e3!,2 #' re!,'#r)e+ )#,)re%e RC !" pre"e,%e+( F!r"% #' *33 %&e 5*!, +!''ere,)e" 4e%1ee,
%&e 4*"!) '#r5u3*%!#," '#r re!,'#r)e5e,%" *,+ )r*)6" *re +!")u""e+ >
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C&*p%er – S%*%e #' %&e *r%
*,+ !,%e2r*%!#, p#!,%" F!2ure (( U"u*337 %&e )#,)re%e 5e"& !" ,#% *''e)%e+ 47
re!,'#r)e5e,%" 4e)*u"e %&e 4*r e3e5e,%" *re ,#% '#r)e+ %# "&*re %&e!r ,#+e" 1!%& )#,)re%e
e3e5e,%"$ T&e re!,'#r)e5e,% ,#+e +!"p3*)e5e,%" *re )#,"%r*!,e+ %# %&e )#,)re%e ,#+e
+!"p3*)e5e,%"(
O,e #' %&e 5*!, +r*14*)6" '#r %&!" *ppr#*)& !" %&e +!''!)u3%7 !, 5#+e3!,2 %&e "%ee3-)#,)re%e
!,%er'*)e p&e,#5e,*( T# "#3e %&!" pr#43e5 !% !" p#""!43e %# u"e "pe)!*3 !,%er'*)e e3e5e,%" %#
)re*%e %&e )#,,e)%!#, 4e%1ee, "%ee3 *,+ )#,)re%e #r %# !,"er% "u!%*43e ,u5er!)*3 )#rre)%!#," !,
)#,)re%e 4e&*!#r !, #r+er %# "!5u3*%e %&e"e e''e)%"( Neer%&e3e"" %&e +!")re%e *ppr#*)& !"
e*"!er %# u"e !, %&e 5#+e3 )re*%!#, p&*"e e"pe)!*337 '#r "%ru)%ur*3 e3e5e,%" !, 1&!)&
re!,'#r)!,2 4*r" &*e * )#5p3e "p*%!*3 +!"%r!4u%!#,(
Figure %.% – An e'aple o& esh superposition.
O, %&e )#,%r*r7 )#,"!+er!,2 "5e*re+ re!,'#r)e5e,%" 5e*," %# 5#+e3 %&e RC 3!6e * "!,23e
)#,%!,uu5 eu!*3e,% 5*%er!*3 1"e pr#per%!e" *re 2!e, 47 %&e "u5 #' +!''ere,% )#,%r!4u%e"
"u)& *" )#,)re%e re!,'#r)!,2 4*r" *,+ !,%er'*)e e''e)%"( I, %&!" '#r5u3*%!#, %&e e''e)%" #' %&e
4*r" *re "5e*re+ *3#,2 %&e!r !,%er*e"( U"u*337 %&!" *ppr#*)& &*" * 2##+ re"p#,"e !,
"%ru)%ure" 1&ere re!,'#r)!,2 4*r" *re "p*%!*337 ,#% %## )#,)e,%r*%e+(
Ge,er*337 %&e +!''ere,% 5e%+" #' 5#+e3!,2 %&e re!,'#r)e5e,%" &*e * 3!%%3e !,'3ue,)e #,
re"u3%"(
2.1.2 – Crack modeling
T&ere *re %1# 5*!, 1*7" #' 5#+e3!,2 %&e )#,)re%e 4e&*!#r *'%er )r*)6!,2 *,+ %&e )!)e
4e%1ee, %&e5 +epe,+" #, %&e !,e"%!2*%e+ pr#43e5 F!2ure (=(I' %&e pr#43e5 !" +#5!,*%e+ 47 * 'e1 ,u54er #' )r*)6" !% !" u"u*337 4e%%er %# *+#p% * +!")re%e
– –
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C&*p%er – S%*%e #' %&e *r%
)r*)6 *ppr#*)&( &e, %&e )r*)6 *r!"e" %&ere !" * "ep*r*%!#, #' %&e 5e"& !, %&e )r*)6e+ #,e
*,+ !, %&!" +!")#,%!,u!%7 ,e1 !,%er'*)e e3e5e,%" 1!%& *r!*43e "%!'',e"" *re !,"er%e+ %#
"!5u3*%e %&e )r*)6 #pe,!,2( T1# 5*K#r +r*14*)6" )*, 4e 5e,%!#,e+$ %&e FE 5e"&
)#,,e)%!!%7 )&*,2e" +ur!,2 %&e *,*37"!" *,+ %&e )r*)6 3!p" *re )#,"%r*!,e+ %# '#33#1 %&e FE
"!+e"( I, #r+er %# *#!+ %&e"e pr#43e5" re5e"&!,2 #per*%!#," )*, 4e !5p3e5e,%e+ *,+ 1!%&
"pe)!*3 pr#)e+ure" %&e )r*)6 )*, 4e *33#1e+ %# pr#p*2*%e *3"# !,"!+e %&e FE( T&e"e pr#per%!e"
eer 5*6e %&!" *ppr#*)& "u!%*43e #,37 !, p*r%!)u3*r )*"e" "u)& *" %&e "%u+7 #' 'r*)%ure
pr#p*2*%!#, !, pre-,#%)&e+ "pe)!5e," "!,)e %&e ,#%)& pr#!+e" 2##+ !,'#r5*%!#, *4#u% %&e
'r*)%ure p#"!%!#, *,+ pr#p*2*%!#,(
Figure %.( – Di&&erent crac) odels in the FE &rae*or).
O, %&e )#,%r*r7 !, pre"e,)e #' * &!2& ,u54er #' )r*)6" * "5e*re+ )r*)6 *ppr#*)& !" 5#re
)#,e,!e,%( T&e 5*%er!*3 !" +e")r!4e+ 3!6e *, eu!*3e,% )#,%!,uu5 1!%& %&e )#,"%!%u%!e
re3*%!#, 5#+!'!e+ 47 )#,"!+er!,2 )r*)6!,2 !,%er'*)e p&e,#5e,* *,+ re!,'#r)!,2 4*r"( &e,
)r*)6!,2 #))ur" * ,e1 3#)*3 )##r+!,*%e "7"%e5 !, 1&!)& !" e*"!er %# +e")r!4e %&e 5*%er!*3 *,+
%&e !,%er'*)e 4e&*!#r !" +e'!,e+ *))#r+!,2 %# )r*)6!,2 +!re)%!#,( A )r*)6e+ 5*%er!*3
)#,"%!%u%!e 5*%r! !" 1r!%%e, !, %&!" "7"%e5 *,+ %&e, !% !" %r*,"'erre+ %# %&e 23#4*3 )##r+!,*%e
"7"%e5 %# )re*%e %&e 23#4*3 "%!'',e"" 5*%r!(
T&!" *ppr#*)& *33#1" %# 5*!,%*!, %&e "*5e 5e"& '#r %&e 1e *,*37"!" *,+ *33#1" %&e )r*)6"
%# #pe, !, *,7 +!re)%!#,( A" +r*14*)6 !% &*" 4ee, ,#%e+ %&*% %&e 5e"& "!e !,'3ue,)e" %&e 3#*+
1&!)& )*u"e" %&e )r*)6 pr#p*2*%!#,( M#re#er %&e "&e*r "%re,2%& #' "#5e "%ru)%ur*3 e3e5e,%")*, 4e #ere"%!5*%e+(
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C&*p%er – S%*%e #' %&e *r%
S5e*re+ )r*)6 5#+e3" )*, *+#p% '!e+ #r r#%*%!,2 )r*)6 '#r5u3*%!#,( I, %&e '#r5er )*"e %&e
)r*)6 +!re)%!#, re5*!," %&e "*5e '#r %&e 1e *,*37"!" ee, !' %&e pr!,)!p*3 +!re)%!#," )&*,2e
1&ere*" !, %&e 3*%%er )*"e %&e )r*)6 #r!e,%*%!#, )&*,2e" *))#r+!,2 %# %&e pr!,)!p*3 +!re)%!#,"(
T&e 5*!, +!''ere,)e 4e%1ee, %&e"e *ppr#*)&e" 3!e" !, %&e ,e)e""!%7 '#r '!e+ )r*)6 %#
)*3)u3*%e %&e )#,%r!4u%e" +ue %# %&e "&e*r %r*,"'erre+ *)r#"" %&e )r*)6 47 %&e *22re2*%e
!,%er3#)6( T&!" )#,%r!4u%!#, !" u"u*337 )*3)u3*%e+ 47 * S&e*r Re%e,%!#, F*)%#r * ,u5er!)*3
)#e''!)!e,% )#5pu%e+ !, *r!#u" 1*7" %*6!,2 !,%# *))#u,% %&e "&e*r "%re""e" *" * 'r*)%!#, #'
%&e "%re""e" !, %&e u,)r*)6e+ )*"e( T&ere'#re %&e r#%*%!,2 *ppr#*)& !" 2e,er*337 e*"!er %# u"e
*,+ %# !,"er% !, )#5pu%er )#+e" 4u% !% )*, 4e )#,"!+ere+ p*r%!*337 8u,p&7"!)*3; "!,)e !, %&e
e5p!r!)*3 re*3!%7 %&e )r*)6 !" ,#% 'ree %# )&*,2e !%" #r!e,%*%!#,( I% &*" %# 4e 5e,%!#,e+
eer %&*% %&e r#%*%!,2 *ppr#*)& +e")r!4e" 4e%%er p*r%!)u3*r "!%u*%!#," 1!%& *r!#u" )r*)6
p*%%er,"( F#r e*5p3e !, * 4e,+e+ 4e*5 %&*% 1!33 &*e "&e*r '*!3ure !' eer7 !,%e2r*%!#, p#!,%
)*, )r*)6 #,37 #,)e %&e '!e+ *ppr#*)& )*,,#% 2!e re3!*43e re"u3%"( I, '*)% %&e '!r"% )r*)6" *re
)*u"e+ 47 %&e 4e,+!,2 5#5e,% er%!)*3 )r*)6( T&e, *" %&e 3#*+ !,)re*"e" "&e*r +!*2#,*3
)r*)6" #pe, #, %&e pre!#u" #,e" +#5!,*%!,2 %&e "%ru)%ur*3 4e&*!#r( I, %&!" )*"e %&e )r*)6
r#%*%!#, *33#1" %# )*p%ure %&e 5#"% !5p#r%*,% )r*)6 p*%%er, 1&!3e %&e '!e+ )r*)6" 5*!,%*!,
%&e!r !,!%!*3 er%!)*3 #r!e,%*%!#, 2!!,2 * 4*+ +e")r!p%!#, #' %&e "u4"eue,% "&e*r )r*)6"(
O,e !5pr#e5e,% #' %&e '!e+ *ppr#*)& !" %&e 5u3%!-+!re)%!#,*3 )r*)6!,2 *ppr#*)& !, 1&!)&%&e 5*%er!*3 !" *33#1e+ %# )r*)6 5#re %&*, #,)e( E*)& )r*)6 !" 'ree %# #pe, !, *,7 +!re)%!#,
*,+ re5*!," '!e+ *'%er %&e '#r5*%!#,( T&!" *ppr#*)& !,#3e" %&e "%r*!, +e)#5p#"!%!#, #'
%#%*3 "%r*!, !,%# )#,)re%e *,+ )r*)6 "%r*!,( T&e "%r*!, #' %&e "u4"eue,% )r*)6" !" *++e+ %# %&e
pre!#u" #,e"(
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C&*p%er – S%*%e #' %&e *r%
( – ABA?US MODEL
T&!" FE )#+e +e")r!4e" %&e RC %&r#u2& %&e '!e+ "5e*re+ )r*)6 *,+ %&e +!")re%e
re!,'#r)e5e,%" *ppr#*)&e"
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C&*p%er – S%*%e #' %&e *r%
5#,#%#,!) 3#*+!,2 )*"e"( Neer%&e3e"" %&e 5#+e3 "u3+ pre+!)% %&e re"p#,"e !, "u)& )*"e"
1!%& * re*"#,*43e *))ur*)7(
A" %&e )#,)re%e )r*)6" !%" "&e*r 5#+u3u" !" re+u)e+ 47 * 5u3%!p37!,2 '*)%#r +e'!,e+ *" *
'u,)%!#, #' %&e #pe,!,2 "%r*!, *)r#"" %&e )r*)6 F!2ure (( I% !" *3"# p#""!43e %# "pe)!'7 *
re+u)e+ "&e*r 5#+u3u" '#r )3#"e+ )r*)6"(
Figure %.9 – #ost &ailure stress-strain relation.
T&e p#"%-'*!3ure 4e&*!#r '#r +!re)% "%r*!,!,2 *)r#"" %&e )r*)6" !" 5#+e3e+ 1!%& 8%e,"!#,
"%!''e,!,2; 1&!)& *33#1" %# +e'!,e %&e "%r*!,-"#'%e,!,2 4e&*!#r '#r )r*)6e+ )#,)re%e( T&!"
'#r5u3*%!#, *3"# *33#1" )#,)re%e-re!,'#r)e5e,% !,%er*)%!#, %# 4e "!5u3*%e+ !, * "!5p3e
5*,,er( I% !" p#""!43e %# "pe)!'7 %e,"!#, "%!''e,!,2 47 5e*," #' * p#"%-'*!3ure "%re""-"%r*!,
re3*%!#, F!2ure (9 #r 47 *pp37!,2 * 'r*)%ure e,er27 )r*)6!,2 )r!%er!#, 4*"e+ #, )r*)6 1!+%&(
I, )*"e" 1!%& 3!%%3e #r ,# re!,'#r)e5e,% %&e "%re""-"%r*!, re3*%!#, #'%e, !,%r#+u)e" 5e"&
"e,"!%!!%7 !, %&e *,*37"!" re"u3%" 1&e, * 'e1 +!")re%e )r*)6" '#r5 !, %&e "%ru)%ure #, %&e
)#,%r*r7 !' )r*)6" *re ee,37 +!"%r!4u%e+ 5e"& "e,"!%!!%7 !" 3e"" #' * )#,)er,( Ge,er*337 *&!2&er %e,"!#, "%!''e,!,2 5*6e" %&e ,u5er!)*3 "#3u%!#, e*"!er(
T&e 'r*)%ure e,er27 )r*)6!,2 )r!%er!#, !" 4*"e+ #, %&e *""u5p%!#, %&*% %&e 'r*)%ure e,er27
reu!re+ %# '#r5 * u,!% *re* #' )r*)6 "ur'*)e !" * 5*%er!*3 pr#per%7( F#33#1!,2 %&!" *ppr#*)&
%&e )#,)re%e 4r!%%3e 4e&*!#r !" )&*r*)%er!e+ 47 * "%re""-+!"p3*)e5e,% re"p#,"e r*%&er %&*, *
"%re""-"%r*!, re"p#,"e( T&e !5p3e5e,%*%!#, #' %&!" "%re""-+!"p3*)e5e,% )#,)ep% !, * FE 5#+e3
reu!re" %&e +e'!,!%!#, #' * )&*r*)%er!"%!) 3e,2%& *""#)!*%e+ 1!%& *, !,%e2r*%!#, p#!,%( T&e
)&*r*)%er!"%!) )r*)6 3e,2%& !" 4*"e+ #, %&e e3e5e,% 2e#5e%r7$ '#r 4e*5" *,+ %ru""e" %&e
!,%e2r*%!#, p#!,% 3e,2%& !" u"e+ '#r "&e33 *,+ p3*,*r e3e5e,%" %&e "u*re r##% #' %&e !,%e2r*%!#,
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C&*p%er – S%*%e #' %&e *r%
p#!,% *re* !" u"e+ '#r "#3!+ e3e5e,%" %&e )u4e r##% #' %&e !,%e2r*%!#, p#!,% #3u5e !" u"e+(
T# *3!+*%e %&e 5#+e3 %&e )#33*p"e #' * RC "3*4 M) Ne!)e "3*4 !" "!5u3*%e+( T&e pr#43e5
2e#5e%r7 !" ", !, F!2ure (
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C&*p%er – S%*%e #' %&e *r%
re%e,%!#, &*" ,# "!2,!'!)*,% !,'3ue,)e #, %&e re"u3%"( S!,)e )#,"!+er*43e ,#,-3!,e*r!%7 !"
epe)%e+ !, %&e re"p#,"e !,)3u+!,2 %&e p#""!4!3!%7 #' u,"%*43e re2!5e" *" %&e )#,)re%e )r*)6"
%&e 5#+!'!e+ R!6" 5e%+ !" u"e+ 1!%& *u%#5*%!) !,)re5e,%*%!#,(
Figure %.11 – 5oad-de&lection response.
T&e ,u5er!)*3 *,+ eper!5e,%*3 re"u3%" *re )#5p*re+ !, F!2ure (
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C&*p%er – S%*%e #' %&e *r%
(= – ADINA MODEL
I, %&e '#33#1!,2 "e)%!#, %&e 5#+e3 '#r RC pre"e,%e+ 47 B*%&e *3)*) e3)& *,+ M!"%r7
.J *,+ !5p3e5e,%e+ !, %&e FE )#+e ADINA !" pre"e,%e+ =J(
T&e p*per +e")r!4e" %&e 5*!, pr#per%!e" #' * 2##+ )#+e$ %&e 5#+e3 "u3+ 4e *" "!5p3e *"
p#""!43e *,+ !% "u3+ 4e 1e33 4u!3% #, %&e#r7 *,+ re3!*43e 'r#5 %&e ,u5er!)*3 p#!,% #' !e1(
T# 'u337 +e")r!4e %&e 5*%er!*3 4e&*!#r %&ree 'e*%ure" *re reue"%e+$
• * "%re""-"%r*!, re3*%!#, %# !,)3u+e %&e &!2& )#,)re%e ,#,-3!,e*r!%7
• * '*!3ure "ur'*)e %# +e'!,e %e,"!3e *,+ )#5pre""!e '*!3ure
• * "u!%*43e %e)&,!ue %# !5p3e5e,% %&e p#"%-)r*)6!,2 *,+ )ru"&!,2 4e&*!#r(
T&e )#,)re%e 5#+e3 )*, 4e e5p3#7e+ 4#%& 1!%& D *,+ =D "#3!+ e3e5e,%" 1!%& "5*33
+!"p3*)e5e,% *" 1e33 *" 3*r2e +!"p3*)e5e,% '#r5u3*%!#, 4u% !, *33 )*"e" "5*33 "%r*!," *re
*""u5e+( M#re#er %&e 5#+e3 )*, 4e *3"# u"e+ '#r #%&er 4r!%%3e 5*%er!*3" 3!6e * 1!+e *r!e%7
#' r#)6"(
Figure %.1( – Copressi!e stress-strain beha!iors.
F!2ure (
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C&*p%er – S%*%e #' %&e *r%
)#,+!%!#, !" 8u,3#*+!,2;( F#r u,3#*+!,2 *,+ re3#*+!,2 )#,+!%!#," up %# %&e "%re"" "%*%e 'r#5
1&!)& u,3#*+!,2 #))urre+ %&e !,!%!*3 e3*"%!) 5#+u3u" !" u"e+( F#r "%r*!, "%*%e" 4e7#,+ %&e
u3%!5*%e )#5pre""!e "%r*!, !% !" *""u5e+ %&*% "%re""e" *re 3!,e*r37 re3e*"e+ %# er#(
Figure %.1+ – Tensile &ailure en!elope.
T&e 5*%er!*3 !" )#,"!+ere+ #r%%r#p!) 1!%& re"pe)% %# %&e pr!,)!p*3 "%re"" +!re)%!#,"( I'
)r*)6!,2 #))ur" !, *,7 +!re)%!#, %&*% +!re)%!#, !" '!e+ 'r#5 %&*% p#!,% #,1*r+ !, )*3)u3*%!,2(
T&e P#!""#, r*%!# !" *""u5e+ %# 4e )#,"%*,% u,+er %e,"!3e "%re""e" *,+ *r!*43e !, %&e
)#5pre""!e re2!#, !, #r+er %# )*p%ure %&e +!3*%*,)7(
Figure %.1 – Tria'ial copressi!e &ailure en!elope.
T&e '*!3ure e,e3#pe" 4*"e+ #, pr!,)!p*3 "%re""e" *re ", !, F!2ure ( %e,"!3e '!e3+"
*,+ F!2ure (
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C&*p%er – S%*%e #' %&e *r%
+!''ere,% *3ue" '#r %&e pr!,)!p*3 "%re""e"( B7 )&*,2!,2 %&e"e *3ue" !% !" p#""!43e %# *+#p% %&e
"*5e 5#+e3 '#r 5*,7 +!''ere,% 5*%er!*3"(
S%*r%!,2 'r#5 %&e )urre,% "%re"" "%*%e !% !" p#""!43e %# e"%*43!"& %&e "%re""-"%r*!, 3*1" %*6!,2
!,%# *))#u,% %&e 5u3%!*!*3 "%re"" )#,+!%!#," *,+ %# )&e)6 1&e%&er %e,"!3e #r )ru"&!,2 '*!3ure
#))ur"( Te,"!3e '*!3ure #))ur" !' * %e,"!3e pr!,)!p*3 "%re"" e)ee+" %&e %e,"!3e "%re,2%& 1&!)&
+epe,+" #, )#5pre""!e "%re""e" !, #%&er pr!,)!p*3 +!re)%!#,"( T&e #%&er %e,"!3e pr!,)!p*3
"%re""e" +# ,#% e''e)% %&e )r*)6!,2(
A'%er )r*)6!,2 !% !" *""u5e+ %&*% * '*!3ure p3*,e +ee3#p" perpe,+!)u3*r37 %# %&e
)#rre"p#,+!,2 pr!,)!p*3 "%re"" +!re)%!#,( T&ere'#re %&e ,#r5*3 *,+ "&e*r "%!'',e""e" *)r#"" %&e
'*!3ure p3*,e *re re+u)e+ *,+ p3*,e "%re"" )#,+!%!#," *re *""u5e+( T&e D '*!3ure e,e3#pe !"
+er!e+ 'r#5 %&e =D #,e( T&e"e "%!'',e"" re+u)%!#," *re )#,"!+ere+ 47 u"!,2 %1# )#,"%*,%" ηn
*,+ η s '#33#1!,2 %&e S&e*r Re%e,%!#, F*)%#r )#,)ep%( T7p!)*337 ηn0(000< *,+ η s0(( T&e
'*)%#r ηn !" ,#% e*)%37 eu*3 %# er# !, #r+er %# *#!+ %&e p#""!4!3!%7 #' * "!,2u3*r "%!'',e""
5*%r!( T&e '*)%#r η s +epe,+" #, * ,u54er #' p&7"!)*3 '*)%#r" *,+ !% 5u"% 4e )"e, )*re'u337(
F#r %&e )#,)re%e 5#+e3 !, ADINA ηn *,+ η s *re 4#%& !,pu% p*r*5e%er"(
Figure %.16 – nia'ial tensile beha!ior.
F!2ure (
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C&*p%er – S%*%e #' %&e *r%
Figure %.1 – #araeter ξ calculation.
A'%er %&e '!r"% )r*)6 #,"e% %&e )##r+!,*%e "7"%e5 !" re+e'!,e+ *,+ %&e "%re""e" !, %&e )r*)6
+!re)%!#," *re u"e+ %# e*3u*%e %&e "%re""-"%r*!, 5*%r! !,"%e*+ #' u"!,2 %&e pr!,)!p*3 "%re""e"
*,+ )#rre"p#,+!,2 +!re)%!#,"( A "u4"eue,% '*!3ure p3*,e !" *""u5e+ %# '#r5 perpe,+!)u3*r37
%# %&e +!re)%!#, #' %&e '!r"% '*!3ure p3*,e 1&e,eer * ,#r5*3 "%re"" *3#,2 %&e #r!2!,*3 '*!3ure
p3*,e re*)&e" %&e %e,"!3e "%re,2%&( T&ere'#re *% *,7 !,%e2r*%!#, p#!,% %&e +!re)%!#, #' %&e %&!r+
%e,"!3e '*!3ure p3*,e !" '!e+ *'%er %&e "e)#,+ )r*)6!,2(
I' %&e 5*%er!*3 )ru"&e" !, )#5pre""!#, !% !" *""u5e+ %&*% %&e 5*%er!*3 4e)#5e" !"#%r#p!) *,+
"%r*!,-"#'%e," !, *33 +!re)%!#," 1!%& er7 "5*33 e3*"%!) 5#+u3u" *3ue"(I, ADINA 5*,7 5*%er!*3 pr#per%!e" )*, 4e +e'!,e+ *" %e5per*%ure-+epe,+e,% !,)3u+!,2 %&e
"%r*!," +ue %# %&e %e5per*%ure e''e)%"(
Figure %.12 – #araeter η & calculation.
T&e 5#+e3 !" *pp3!e+ *% '!r"% %# "!5p3e "%ru)%ure" 1!%& +!''ere,% 3#*+ )#54!,*%!#," !, #r+er %#
er!'7 %&e 4*"!) '#r5u3*%!#,( A'%er1*r+" 5#re )#5p3e "%ru)%ure" *re !,e"%!2*%e+( T&e
"!2,!'!)*,% "%u+7 #' %&e RC )#,%*!,5e,% S*,+!* pre""ure e""e3 !" pre"e,%e+(
T&!" !,e"%!2*%!#, !" p*r% #' %&e US Nu)3e*r Re2u3*%#r7 C#55!""!#, pr#2r*5 #,
)#,%*!,5e,%"( T&e "%ru)%ure !" *,*37e+ pr!#r %# eper!5e,%*3 %e"%" 47 %e, +!''ere,% 2r#up"
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C&*p%er – S%*%e #' %&e *r%
'#ur 'r#5 %&e USA *,+ "! 'r#5 Eur#pe( O,e #' %&e *,*37"!" !" )*rr!e+ #u% 1!%& ADINA 47
u"!,2 *, *!"755e%r!) 5#+e3(
Figure %.19 – 3andia pressure !essel FE esh.
T&e )#,)re%e pr#per%!e" *re +er!e+ 'r#5 pu43!"&e+ re"u3%" 4u% %&e *2ree5e,% 1!%& %&e
ADINA +e'*u3% *3ue" !" "*%!"'*)%#r7 1!%& %&e e)ep%!#, #' %&e &!2& %r!*!*3 )#5pre""!e
re2!#,( T&e *!"755e%r!) *ppr#*)& pre"e,%" "#5e 5#+e3!,2 pr#43e5"$ %&e 5*!, er%!)*3 *,+
#p re!,'#r)e5e,%" )*, 4e 5#+e3e+ e*"!37 4u% *% %&e 4#%%#5 #' %&e 1*33 "#5e +!*2#,*3
"e!"5!) re!,'#r)e5e,%" *re +e"!2,e+( T&ere'#re %&!" re!,'#r)e5e,% "e% !" !,"er%e+ *"
eu!*3e,% e%r* er%!)*3 *,+ #p re!,'#r)e5e,%"( T&e "#!3 !" 5#+e3e+ *" * "e% #' "pr!,2
e3e5e,%"( S!,)e %&e 4#%%#5 p3*%e !" re3*%!e37 %&!)6 %&e "pr!,2 "%!'',e"" !" "upp#"e+ ,#% %#
&*e * 2re*% !,'3ue,)e #, %&e "%ru)%ur*3 4e&*!#r( T&e FE 5e"& !" ", !, F!2ure (
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C&*p%er – S%*%e #' %&e *r%
T&e #,e-"!%& ")*3e "pe)!5e, 1*" 4u!3% *% S*,+!* N*%!#,*3 L*4#r*%#r!e" *,+ %e"%e+ !, Qu37
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C&*p%er – S%*%e #' %&e *r%
(> – ANSYS MODEL
T&e FE )#+e ANSYS 5#+e3" RC 1!%& "5e*re+ )r*)6!,2 *,+ "5e*re+ re!,'#r)e5e,%
*ppr#*)&e" >J(
I, )#,)re%e 5#+e3!,2 %&e =D e3e5e,% SOLID. !" u"e+ F!2ure (( T&!" e3e5e,% *33#1" %&e
pre"e,)e #' '#ur +!''ere,% 5*%er!*3" #,e 5*%r! 5*%er!*3 *,+ * 5*!5u5 #' %&ree !,+epe,+e,%
re!,'#r)!,2 5*%er!*3"( S!,)e SOLID. !" )*p*43e #' )r*)6!,2 !, %e,"!#, *,+ )ru"&!,2 !,
)#5pre""!#, !% )*, 4e u"e+ %# 5#+e3 RC #r #%&er 5*%er!*3" "u)& *" re!,'#r)e+ )#5p#"!%e"
e(2( '!4er23*"" *,+ 2e#3#2!)*3 5*%er!*3" e(2( r#)6( T&e e3e5e,% !" +e'!,e+ 47 e!2&% ,#+e"
&*!,2 %&ree %r*,"3*%!#,*3 +e2ree" #' 'ree+#5 *% e*)& ,#+e(
Figure %.%% – 357D6 geoetric properties.
C#,)re%e !" )*p*43e #' +!re)%!#,*3 !,%e2r*%!#, p#!,% )r*)6!,2 *,+ )ru"&!,2 4e"!+e"
!,)#rp#r*%!,2 p3*"%!) *,+ )reep 4e&*!#r( T&e re!,'#r)e5e,% 1&!)& *3"# !,)#rp#r*%e" )reep
*,+ p3*"%!)!%7 &*" u,!*!*3 "%!'',e"" #,37 *,+ !% !" *""u5e+ %# 4e "5e*re+ %&r#u2u% %&e
e3e5e,%( D!re)%!#,*3 #r!e,%*%!#, !" *))#5p3!"&e+ %&r#u2& u"er "pe)!'!e+ *,23e"(
I, )#,)re%e 1&!)& !" *""u5e+ %# 4e !,!%!*337 !"#%r#p!) )r*)6!,2 !" per5!%%e+ !, %&ree
#r%#,*3 +!re)%!#," *% e*)& !,%e2r*%!#, p#!,% *,+ !% !" 5#+e3e+ %&r#u2& *, *+Ku"%5e,% #'
5*%er!*3 pr#per%!e" 1&!)& e''e)%!e37 %re*%" %&e )r*)6!,2 *" * "5e*re+ )r*)6 4*,+ r*%&er %&*,
+!")re%e )r*)6"(
T&e, * u,!ue 5*%r! !" 1r!%%e, '#r 4#%& 5*%er!*3" *" '#33#1"$
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C&*p%er – S%*%e #' %&e *r%
5*%er!*3 %# %&e %#%*3 #3u5e #' e3e5e,% DcJ !" %&e )#,"%!%u%!e 5*%r! '#r )#,)re%e *,+ DriJ !"
%&e )#,"%!%u%!e 5*%r! '#r %&e ith re!,'#r)e5e,%(
T&e )#,)re%e +e")r!p%!#, !" 4*"e+ #, %&e R*%e I,+epe,+e,% P3*"%!)!%7 1&!)& !" )&*r*)%er!e+ 47
%&e !rreer"!43e "%r*!,!,2 #))urr!,2 !, %&e 5*%er!*3 1&e, * )er%*!, 3ee3 #' "%re"" !" re*)&e+(
T&e p3*"%!) "%r*!," *re *""u5e+ %# +ee3#p !,"%*,%*,e#u"37 !,+epe,+e,%37 #, %!5e(
T&e '*!3ure )r!%er!#, !" +e")r!4e+ 47 %&e !33*5 *,+ *r,6e '#r5u3*%!#, >@J( T&!" 5*%er!*3
5#+e3 pre+!)%" e!%&er e3*"%!) )r*)6!,2 #r )ru"&!,2 4e&*!#r( I' e3*"%!) 4e&*!#r !" pre+!)%e+
%&e )#,)re%e !" %re*%e+ *" * 3!,e*r e3*"%!) !"#%r#p!) 5*%er!*3( I' )r*)6!,2 #r )ru"&!,2 4e&*!#r !"
pre+!)%e+ %&e e3*"%!) "%re""-"%r*!, 5*%r! !" *+Ku"%e+ !, * +!''ere,% 1*7 '#r e*)& '*!3ure 5#+e(
T&e =D '*!3ure "ur'*)e )*, 4e 1r!%%e, *"
0c
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1&ere F !" * 'u,)%!#, #' %&e pr!,)!p*3 "%re"" "%*%e 3 !" * '*!3ure "ur'*)e & c !" %&e u,!*!*3
)#5pre""!e "%re,2%& F!2ure (=(
Figure %.%( – The (D &ailure sur&ace.
F!e !,pu% p*r*5e%er" 1&!)& )*, 4e %e5per*%ure +epe,+e,% *re reue"%e+ %# +e'!,e %&e '*!3ure
"ur'*)e( F#r "5*33 *3ue" #' %&e &7+r#"%*%!) "%re"" !,"%e*+ %&e '*!3ure "ur'*)e )*, 4e "pe)!'!e+
47 u"!,2 "#5e )#,"%*,% re3*%!#," 4e%1ee, p*r*5e%er" 1!%& * 5!,!5u5 #' %1# )#,"%*,%"$ %&e
u,!*!*3 %e,"!3e *,+ )#5pre""!e "%re,2%&( F#r &!2& *3ue" #' %&e &7+r#"%*%!) "%re"" '!e
p*r*5e%er" *re reu!re+(
F#ur +!''ere,% +#5*!," *re !,e"%!2*%e+ 4*"e+ #, %&e pr!,)!p*3 "%re"" "!2," %r!*!*3
)#5pre""!#, %r!*!*3 %e,"!#, *,+ %1# 5!e+ )*"e" *,+ !, e*)& +#5*!, +!''ere,% '#r5u3*%!#,"'#r F *,+ 3 *re 1r!%%e,( T&e '*!3ure "ur'*)e !" )3#"e+ *,+ )*, pre+!)% '*!3ure *3"# u,+er &!2&
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C&*p%er – S%*%e #' %&e *r%
&7+r#"%*%!) pre""ure(
I' %&e 5*%er!*3 *% *, !,%e2r*%!#, p#!,% '*!3" !, u,!*!*3 4!*!*3 #r %r!*!*3 )#5pre""!#, %&e
5*%er!*3 !" *""u5e+ %# )ru"&( Cru"&!,2 !" +e'!,e+ *" %&e )#5p3e%e +e%er!#r*%!#, #' %&e 5*%er!*3
"%ru)%ur*3 !,%e2r!%7( T&ere'#re %&e "%!'',e"" *% %&*% !,%e2r*%!#, p#!,% )*, 4e !2,#re+(
Figure %.%+ – Tensile post crac)ing beha!ior.
T&e pre"e,)e #' * )r*)6 *% *, !,%e2r*%!#, p#!,% !" repre"e,%e+ %&r#u2& 5#+!'!)*%!#, #' %&e
"%re""-"%r*!, re3*%!#," 47 !,%r#+u)!,2 * p3*,e #' 1e*6,e"" !, %&e +!re)%!#, ,#r5*3 %# %&e )r*)6
'*)e( T&e )#,"%!%u%!e 5*%r! '#r * 5*%er!*3 )r*)6e+ !, #,e +!re)%!#, !"
< 0 0 0 0 0
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é ù+ê úê úê úê úê ú
- -ê úê úê úê ú-ê úé ù =ê ú ê úë û + ê úê úê úê úê ú
ê úê úê úê úê úë û
( (=
T&e "%re""-"%r*!, re3*%!#, re'er" ,#1 %# * )##r+!,*%e "7"%e5 p*r*33e3 %# %&e pr!,)!p*3 "%re""
+!re)%!#," 1!%& %&e '!r"% *!" perpe,+!)u3*r %# %&e )r*)6 '*)e( T&e, !, #r+er %# 4u!3+ %&e 23#4*3
"%!'',e"" 5*%r! * )##r+!,*%e %r*,"'#r5*%!#, !" ,ee+e+( I, %&e "*5e 1*7 !% !" p#""!43e %# 1r!%e
%&e "%re""-"%r*!, 5*%r! '#r )r*)6!,2 !, %1# *,+ %&ree +!re)%!#,"(
F!2ure (> "" %&e p#"% )r*)6!,2 4e&*!#r !, %&e +!re)%!#, perpe,+!)u3*r %# %&e )r*)6 *,+
+e'!,e" %&e $t '*)%#r( T&e )r*)6 !" *33#1e+ %# )3#"e$ !, %&!" )*"e *33 %&e )#5pre""!e "%re""e"
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C&*p%er – S%*%e #' %&e *r%
,#r5*3 %# %&e )r*)6 p3*,e *re %r*,"'erre+ 1&ere*" * +!''ere,% "&e*r %r*,"'er )#e''!)!e,% !"
"pe)!'!e+(
A "&e*r %r*,"'er )#e''!)!e,% βt !" !,%r#+u)e+ repre"e,%!,2 * "&e*r "%re,2%& re+u)%!#, '*)%#r '#r
"u4"eue,% 3#*+" !,+u)!,2 "3!+!,2 *)r#"" %&e )r*)6 '*)e(
Figure %.% – $ein&orceent orientation.
Re!,'#r)!,2 4*r" *re *""u5e+ %# 1#r6 !, *!*3 +!re)%!#, #,37( T&e!r #r!e,%*%!#, !" "pe)!'!e+ *"
", !, F!2ure (( A )reep *,+ p3*"%!)!%7 ,#,-3!,e*r 4e&*!#r )*, 4e +e'!,e+(
Figure %.%6 – 8eri&ication bea geoetr".
I, ANSYS +#)u5e,%*%!#, * er!'!)*%!#, e*5p3e %*6e, 'r#5 T!5#"&e,6# >>J !" pre"e,%e+(
A )#,)re%e 4e*5 re!,'#r)e+ 1!%& "%ee3 r#+" !" "u4Ke)%e+ %# pure 4e,+!,2 3#*+ F!2ure (.(
T&e *,*37"!" e*3u*%e" %&e )r*)6 +ep%& 'r#5 %&e 4#%%#5 "ur'*)e %&e 5*!5u5 "%ee3 %e,"!3e
"%re"" *,+ %&e 5*!5u5 )#,)re%e )#5pre""!e "%re"" *""u5!,2 %&*% %&e )#,)re%e %e,"!3e
"%re,2%& !" er#(
I, #r+er %# 5*%)& %&e re'ere,)e *""u5p%!#," %&e "!5u3*%!#, *+#p%" * er# P#!""#, )#e''!)!e,%
*,+ *, !,'!,!%e )ru"&!,2 "%re,2%& '#r )#,)re%e( M#re#er )#,"%r*!,% eu*%!#," *re u"e+ *3#,2
%&e 4e*5 +ep%& %# )#,e,!e,%37 *pp37 %&e 3#*+ *,+ 5*%)& %&e p3*,e )r#""-"e)%!#, &7p#%&e"!"(
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C&*p%er – S%*%e #' %&e *r%
T&e FE +!")re%!*%!#, !" ", !, F!2ure (@(
Figure %.% – FE odel.
Re"u3% )#5p*r!"#, !" rep#r%e+ !, T*43e (
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C&*p%er – S%*%e #' %&e *r%
( – BALAN SPACONE AND ON MODEL
T&!" 5#+e3 1*" +ee3#pe+ *% %&e U,!er"!%7 #' C#3#r*+# *% B#u3+er 47 B*3*, Sp*)#,e *,+
1#, J *,+ *3!+*%e+ %&r#u2& %&e "%u+7 #' "!5p3e )#,)re%e-#,37 "pe)!5e," u,+er +!''ere,%
)#,'!,e5e,%"( L*%er %&e 5#+e3 1*" 5#+!'!e+ 47 1#, *,+ Sp*)#,e J *,+ *pp3!e+ %#
)#,)re%e )#,'!,e+ 47 "%ee3 *,+ '!4er re!,'#r)e+ p#375er K*)6e%" *,+ %# RC )#3u5,"(
T&e 5#+e3 !" 4*"e+ #, &7p#p3*"%!) '#r5u3*%!#, *,+ )*, 4e *pp3!e+ %# )#,)re%e "%ru)%ure" u,+er
5#,#%#,!) #r )7)3!) pr#p#r%!#,*3 #r ,#,-pr#p#r%!#,*3 3#*+!,2(
Figure %.%2 – 3tress-eui!alent unia'ial strain relation.
T&e "%re""-"%r*!, )ure" *re 4*"e+ #, eu!*3e,% u,!*!*3 "%r*!, )#,)ep% *,+ *re +!!+e+ !, %1#
+!''ere,% '!e3+"( I, %&e *")e,+!,2 4r*,)& %&e P#p#!)" '#r5u3*%!#, !" u"e+ =J 1&ere*" *'%er
%&e pe*6 %&e S*e, )ure >J !" pre'erre+ F!2ure (( T&!" )!)e !" +e%er5!,e+ 47 %&e '*)%
%&*% %&e re"u3%" 'r#5 %&e *")e,+!,2 4r*,)& #' %&e S*e, )ure" *re "*%!"'*)%#r7 #,37 !' %&e
!,!%!*3 5#+u3u" !" 2re*%er %&*, %1# %!5e" %&e "e)*,% 5#+u3u" *% pe*6( I, %&e )*3)u3*%!#, %&e
%#%*3 "e)*,% 5#+u3u" !" u"e+(
T&e "%re""-"%r*!, )ure" +epe,+ #, %&e )urre,% "%re"" "%*%e %&r#u2& %&e "%re"" *,+ "%r*!, re3*%!e
%# %&e pe*6( I, #r+er %# '!,+ %&e5 * '*!3ure "ur'*)e !" reu!re+( T&e '*!3ure "ur'*)e #' %&!" "%u+7
!" +er!e+ 47 %&e !33*5 *,+ *r,6e #,e >@J *,+ !% !" * )#54!,*%!#, #' %&e R*,6!,e *,+ %&e
M#&r-C#u3#54 )r!%er!* *))#r+!,2 %# %&e Me,e%re7 *,+ !33*5 '#r5u3*%!#, ==J( M#re#er *
)*p "ur'*)e !" *++e+ %# )*p%ure %&e '*!3ure ,e*r %&e &7+r#"%*%!) *!" F!2ure (9(
T&e "%r*!, p*r*5e%er" '#r %&e "%re""-"%r*!, )ure" *re #4%*!,e+ 47 u"!,2 * "!5!3*r '*!3ure
"ur'*)e !, %&e "%r*!, "p*)e 1&ere %&e "%r*!, p*r*5e%er" rep3*)e %&e "%re"" #,e"(
S!,)e %&e 5#+e3 !" )*p*43e #' +e")r!4!,2 *3"# %&e u,3#*+!,2 * "u!%*43e 3#*+ 'u,)%!#, 1r!%%e,1!%& "%r*!, *r!*43e" !" +e'!,e+(
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C&*p%er – S%*%e #' %&e *r%
Figure %.%9 – Failure sur&ace in $endulic plane.
T&e 5#+e3 !" !5p3e5e,%e+ !, %&e FE pr#2r*5 FEAP *" * "%*,+ *3#,e r#u%!,e( I, %&e '!r"% "%*2e
%&e 5#+e3 !" *pp3!e+ '#r !%" *3!+*%!#, %# "!5p3e u,!*!*3 4!*!*3 *,+ %r!*!*3 )#,)re%e-#,37
%e"%" *33 )*rr!e+ #u% *% %&e U,!er"!%7 #' C#3#r*+# *% B#u3+er(
T&e '!r"% *3!+*%!#, !" per'#r5e+ #, u,!*!*3 %e"%" #' )#,)re%e "pe)!5e," u,+er )#,"%*,% 3*%er*3
)#,'!,e5e,%( T&e '!,*3 *!5 #' %&e"e %e"%" !" %# " %&e %r*,"!%!#, 'r#5 4r!%%3e-"#'%e,!,2 %#
+u)%!3e-&*r+e,!,2 4e&*!#r *" %&e 3*%er*3 )#,'!,e5e,% !,)re*"e"( !%& *, *))ur*%e e"%!5*%!#,
#' %&e )*p *,+ p#"% pe*6 p*r*5e%er" * 2##+ *2ree5e,% 4e%1ee, eper!5e,%*3 *,+ ,u5er!)*3re"u3%" !" *)&!ee+ *,+ %&e %r*,"!%!#, !" )3e*r37 !"!43e F!2ure (=0(
Figure %.(0 – Constant con&ineent copression tests.
T&e "e)#,+ *pp3!)*%!#, #' %&e 5#+e3 !" %&e "%u+7 #' )#,)re%e "pe)!5e," u,+er )7)3!) *!*3 3#*+
*,+ 3*%er*3 )#,'!,e5e,%( T&e )#5p*r!"#, 4e%1ee, eper!5e,%*3 *,+ ,u5er!)*3 re"u3%" '#r%&ree +!''ere,% )*"e" !" ", !, F!2ure (=
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C&*p%er – S%*%e #' %&e *r%
"%re"" *,+ "%r*!, %&e p#"% pe*6 re"p#,"e *,+ %&e 3#*+!,2-u,3#*+!,2 4e&*!#r( A ,u5er!)*3
5!e+ )#,%r#3 !" u"e+ %# %r*)e %&e "%re"" p*%& 1!%& *, !5p#"e+ *!*3 +!"p3*)e5e,% *,+ )#,"%*,%
3*%er*3 )#,'!,!,2 "%re""(
Figure %.(1 – C"clic con&ined copression test results.
T&e 5#+e3 !" *3"# *pp3!e+ %# %&e )*"e" #' 4!*!*3 pr#p#r%!#,*3 *,+ %r!*!*3 ,#,-pr#p#r%!#,*3
3#*+!,2( T&e re"u3%" '#r %&e 3*%%er )*"e *re rep#r%e+ !, F!2ure (=( T&!" !" *, !5p#r%*,% %e"%
"!,)e %&e 5#"% +e5*,+!,2 "%re"" &!"%#r!e" !, )#,)re%e *,*37"!" *re %"e 1!%& ,#,-pr#p#r%!#,*3
3#*+!,2( T&e "pe)!5e, !" *% '!r"% 3#*+e+ *3#,2 %&e &7+r#"%*%!) *!" up %# . 6"! *,+ %&e, %&e
"%re""e" 1ere 5#+!'!e+ 47 )7)3!) 3#*+!,2 #, %&e )#rre"p#,+!,2 +e!*%#r!) p3*,e( T&e
*2ree5e,% 4e%1ee, eper!5e,%*3 *,+ ,u5er!)*3 re"u3%" !" 2##+(
I, %&e "e)#,+ "%*2e %&e 5#+e3 !" 5#+!'!e+ 47 1#, *,+ Sp*)#,e 1!%& "#5e %&e#re%!)*3e,&*,)e5e,%" *,+ !,"er%!,2 %&e e''e)% #' "%ee3 re!,'#r)!,2 4*r"(
I, %&e pre!#u" er"!#, #' %&e 5#+e3 %&e e))e,%r!)!%7 #' %&e '*!3ure "ur'*)e +e'!,!,2 %&e #u% #'
r#u,+,e"" #' %&e +e!*%#r!) "e)%!#, 1*" +e'!,e+ *" e0( 1&ere*" !, %&!" er"!#, %&e
e))e,%r!)!%7 !" 1r!%%e, *" * 'u,)%!#, #' %&e 4r!%%3e,e""(
M#re#er !, #r+er %# +e")r!4e "!5p3e "&e*r #,37 "&e*r "%r*!, *pp3!e+ *,+ pure "&e*r #,37
"&e*r "%re""e" *pp3!e+ *, *++!%!#,*3 %er5 )#,%*!,!,2 %&e #)%*&e+r*3 "%re"" !" *++e+ %# %&e
#3u5e%r!) "%re"" +e'!,!%!#,( T&!" 5#+!'!)*%!#, )#up3!,2 ,#r5*3 *,+ "&e*r re"p#,"e
repre"e,%" %&e #3u5e%r!) "%re"" !,+u)e+ 47 %&e +e!*%#r!) "%re""(
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C&*p%er – S%*%e #' %&e *r%
F!,*337 %&e e''e)%" #' re!,'#r)!,2 4*r" *re !,%r#+u)e+ u"!,2 %ru"" e3e5e,%" "uperp#"e+ #,
"#3!+ e3e5e,% )#,)re%e 5e"& 1!%& * "!5p3e u,!*!*3 4!3!,e*r "%r*!,-&*r+e,!,2 )#,"%!%u%!e
3*1( Per'e)% 4#,+ !" *""u5e+(
Figure %.(% – Tria'ial non-proportional loading test results.
T&e "5e*re+ )r*)6 *ppr#*)& !" u"e+ !, %&!" 5#+e3 "!,)e !% )*, 4e%%er +e")r!4e %&e )r*)6 p*%%er,
#' "%ru)%ure" 1!%& &e*!37 +!"%r!4u%e+ re!,'#r)e5e,%"( I, %&!" )*"e %&e "&e*r "%!'',e"" #' %&e
"%ru)%ur*3 e3e5e,% )*, 4e #ere"%!5*%e+ +ue %# %&e "%re"" 3#)6!,2 !, )#55#,37 u"e+ e3e5e,%"
4u% %&e pr#43e5 )*, 4e *#!+e+ u"!,2 '!,e 5e"&e"( T&e )r*)6 +!re)%!#," *re *33#1e+ %# r#%*%e
1!%& %&e pr!,)!p*3 "%r*!, +!re)%!#," +ur!,2 %&e *,*37"!"( T&e pr!,)!p*3 "%re"" *e" *re ,#%
)#*!*3 1!%& %&e pr!,)!p*3 "%r*!, #,e" *,+ %&e )r*)6" *re "upp#"e+ %# #pe, ,#r5*337 %# %&e
pr!,)!p*3 "%r*!, +!re)%!#,"(
T&e 5#+e3 !" *pp3!e+ *% '!r"% %# *!*337 3#*+e+ )73!,+r!) )#,)re%e "pe)!5e," )#,'!,e+ 47 "%ee3
*,+ '!4er re!,'#r)e+ p#375er FRP K*)6e%" !, #r+er %# &!2&3!2&% %&e +!''ere,% )#,'!,e5e,%
5e)&*,!"5( T&e "%ee3 )#,'!,e5e,% !,)re*"e" 1!%& %&e er%!)*3 3#*+ up %# 7!e3+!,2 *,+ %&e, %&e
)#,'!,!,2 "%re"" re5*!," )#,"%*,%( O, %&e )#,%r*r7 %&e FRP K*)6e% !" e3*"%!) up %# %&e '*!3ure
p#!,% *,+ %&e, %&e )#,'!,!,2 "%re"" !,)re*"e" 1!%& %&e er%!)*3 3#*+( T&e 2##+ *2ree5e,%
4e%1ee, eper!5e,%*3 *,+ ,u5er!)*3 re"u3%" "" %&*% %&e 5#+e3 !" *43e %# )*p%ure %&!"
+!''ere,)e F!2ure (==(
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C&*p%er – S%*%e #' %&e *r%
Figure %.(( – E'periental le&t: and nuerical right: results.
F!,*337 %&ree RC )#3u5," 'r#5 *, eper!5e,%*3 pr#2r*5 #' %&e U,!er"!%7 #' C*3!'#r,!* *%
S*, D!e2# *re !,e"%!2*%e+( T&e "pe)!5e," * #,e-%&!r+ rep3!)* #' re*3 4r!+2e p!er" 4u!3+ !, %&e
5!+ .0" *re )3*5pe+ *% 4#%& e,+" *,+ "u4Ke)%e+ %# * )#,"%*,% *!*3 3#*+ *,+ %# * )7)3!) 3*%er*3
+!"p3*)e5e,% F!2ure (=>(
Figure %.(+ – Test con&iguration and FE esh.
I, #r+er %# re+u)e %&e )#5pu%*%!#,*3 %!5e *,+ "!,)e %&e '!,*3 #4Ke)%!e #' %&e "%u+7 !" %#
)*p%ure %&e '*!3ure 5#+e #' %&e )#3u5," #,37 %&e 5#,#%#,!) 3#*+!,2 e,e3#pe !" '#33#1e+ !,
%&e "!5u3*%!#,"( T&e ,u5er!)*3 #u%)#5e" *re !, er7 2##+ *2ree5e,% 1!%& %&e eper!5e,%*3
%e"%" *" ", !, F!2ure (=(
T&e '!r"% )#3u5, R< !" +e"!2,e+ !, #r+er %# *#!+ %&e "&e*r '*!3ure$ !% &*" &!2&er #p "%re,2%&1!%& re"pe)% %# %&e 3#,2!%u+!,*3 re!,'#r)e5e,% "%re,2%&( T&e '*!3ure 5#+e *,+ %&e u3%!5*%e 3#*+
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C&*p%er – S%*%e #' %&e *r%
*re )*p%ure+ 47 %&e 5#+e3 *" 1e33 *" %&e 3#,2!%u+!,*3 "%ee3 4*r" 7!e3+!,2 %#2e%&er 1!%& %&e
pre"e,)e #' %&e )#5pre""!e "%ru% !, )#,)re%e(
T&e )#3u5, R= !" +e"!2,e+ 1!%& * )#,)re%e 1!%& 3#1er pr#per%!e" 1&!)& 3e*+" %# * "&e*r
'*!3ure 1&e, %&e '3eur*3 "%re,2%& !" "%!33 ,#% *)&!ee+( T&e '*!3ure "eue,)e pre"e,%" *% '!r"%
%&e )ru"&!,2 #' %&e 5*!, )#,)re%e "%ru% p#!,% A *,+ %&e, %&e "u++e, "&e*r '*!3ure p#!,% B(
I, 4#%& )*"e" %&e 5#+e3 !" *43e %# )*p%ure %&e "%ru)%ur*3 4e&*!#r(
Figure %.( – Colun $1 le&t: and $( right: result coparisons.
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CHAPTER = – THE PROPOSED MODEL =D-PARC
=(< – BASIC HYPOTHESES
T&e pr#p#"e+ 5#+e3 =D-PARC – T&ree-D!5e,"!#,*3 P&7"!)*3 Appr#*)& '#r Re!,'#r)e+
C#,)re%e +e")r!4e" %&e 5e)&*,!)*3 4e&*!#r #' * %&ree-+!5e,"!#,*3 =D re!,'#r)e+ )#,)re%e
RC e3e5e,% "u4Ke)%e+ %# 2e,er*3 "%re""e"( I% !" 4*"e+ #, '!e+ 5u3%!-+!re)%!#,*3 )r*)6!,2 *,+
"5e*re+ re!,'#r)e5e,% *ppr#*)&e" *,+ !% !" '#r5u3*%e+ !, %er5" #' 23#4*3 5*%er!*3 "%!'',e""
5*%r!(
I, re3*%!#, %# %&e RC p&7"!)*3 )#,+!%!#," %&e pr#p#"e+ 5#+e3 !" *43e %# "!5u3*%e %&ree
+!''ere,% p&*"e"$
• u,)r*)6e+ 5*%er!*3
• "!,237 )r*)6e+ 5*%er!*3
• +#u437 #r 5u3%!-)r*)6e+ 5*%er!*3(
I, %&e u,)r*)6e+ p&*"e per'e)% 4#,+ !" *""u5e+ 4e%1ee, )#,)re%e *,+ re!,'#r)e5e,%"(
T&ere'#re %&e %1# 5*%er!*3" 4e&*e !, p*r*33e3 *,+ %&e!r "%!'',e"" )#,%r!4u%e" *re *++e+(
I, %&e "!,237 )r*)6e+ p&*"e %&e 5*%er!*3" !, %&e )r*)6 *,+ 4e%1ee, %&e )r*)6" *re )#,"!+ere+
*" 1#r6!,2 !, "er!e"( T&e 5#+e3!,2 pr#)e+ure !,#3e" * "%r*!, +e)#5p#"!%!#,$ %&e %#%*3 "%r*!,
!" +!!+e+ !,%# %&e "%r*!, #' %&e 5*%er!*3 4e%1ee, %&e )r*)6" *,+ !, %&e )r*)6( &!3e %&e RC
4e%1ee, %&e )r*)6" !" 5#+e3e+ *" u,)r*)6e+ 4u% +*5*2e+ 47 %&e pre"e,)e #' %&e )r*)6 %&e
5*%er!*3 !, %&e )r*)6 !,)3u+e" *33 %&e p&e,#5e,* 2e,er*%e+ 47 %&e )r*)6!,2 "u)& *" *22re2*%e
4r!+2!,2 *,+ !,%er3#)6 %e,"!#, "%!''e,!,2 *,+ +#1e3 *)%!#,( F!,*337 %&e '3e!4!3!%!e" #' %&e %1#
5*%er!*3" *re *++e+ %#2e%&er(
I, %&e +#u437 #r 5u3%!-)r*)6e+ p&*"e e*)& )r*)6 !" *""u5e+ %# 1#r6 !, "er!e" 1!%& %&e5*%er!*3 4e%1ee, %&e )r*)6"(
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
Cr*)6!,2 !" *""u5e+ %# *r!"e 1&e, %&e pr!,)!p*3 %e,"!3e "%r*!, e)ee+" %&e "%r*!, %e,"!3e 3!5!%
*,+ %&e )r*)6 p*%%er, !" )#,"!+ere+ *" !55e+!*%e37 'u337 +ee3#pe+ 1!%& * )r*)6 "p*)!,2 a
)#,"%*,% +ur!,2 %&e 3#*+!,2 pr#)e""(
T&e pr#p#"e+ 5#+e3 !" "%ru)%ure+ !, * 5#+u3*r 'r*5e1#r6( A33 %&e 5e)&*,!)*3 p&e,#5e,* *re
*,*37e+ "ep*r*%e37 #, %&e 4*"!" #' %&e!r pr#per%!e" *,+ p&7"!)*3 "%*%e( A'%er1*r+" *33 %&e
)#,%r!4u%e" *re *""e543e+ %# )re*%e %&e eu!*3e,% ,#,-3!,e*r )#,%!,uu5 5*%er!*3 1&!)&
e&!4!%" !, %&e 5e*, "e,"e %&e 4e&*!#r #' %&e "u5 #' %&e )#,%r!4u%e"( I, %&!" 1*7 e*)& p*r%
#' %&e 5#+e3 )*, 4e 'ree37 5#+!'!e+ *,+ up+*%e+(
F#r e*)& p&*"e %&e "%re""-"%r*!, re3*%!#, *+#p%e+ !, %&e 23#4*3 )##r+!,*%e "7"%e5 '- "- ; !"$
{ } { } '"; '"; '";
Dσ εé ù= ê úë û =(
-
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
T&e =D-PARC %&e#r7 '#r5u3*%!#, re'er" %# * u,!%*r7 "#3!+ RC e3e5e,% re!,'#r)e+ 47 #r+!,*r7
"%ee3 4*r" *rr*,2e+ !, 3*7er" F!2ure =(
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
=( – UNCRACED MATERIAL
I, %&e u,)r*)6e+ p&*"e )#,)re%e *,+ "%ee3 *re "upp#"e+ %# 4e&*e *" %1# 5*%er!*3" !, p*r*33e3(
S!,)e per'e)% 4#,+ !" *""u5e+ 4#%& )#,)re%e *,+ "%ee3 "%r*!," *re eu*3 %# %&e %#%*3 "%r*!,$
{ } { } { } '"; '"; '"; c sε ε ε= = ( =(>
A'%er1*r+" %&e eu!3!4r!u5 !" !5p#"e+( T&e %#%*3 "%re"" !" %&e "u5 #' %&e "%re""e" !, %&e
)#,)re%e *,+ !, %&e "%ee3$
{ } { } { } { } { } { }
'"; '"; '"; '"; '"; '"; '"; '"; '";
c s c c s s
D D Dσ σ σ ε ε εé ù é ù é ù= + = + =ê ú ê ú ê úë û ë û ë û
=(
1&ere
<
r 4
'"; '";
s si
i
D D=
é ù é ù=ê ú ê úë û ë ûå =(.
4e!,2 '";
c Dé ùê úë û %&e "%!'',e"" 5*%r! #' %&e )#,)re%e
'";
si Dé ùê úë û %&e "%!'',e"" 5*%r! #' %&e i
th "%ee3 3*7er
*,+ 4 r %&e %#%*3 ,u54er #' re!,'#r)!,2 3*7er"(T&ere'#re %&e 23#4*3 5*%er!*3 "%!'',e"" 5*%r! !"
'"; '"; '";
c s D D Dé ù é ù é ù= +ê ú ê ú ê úë û ë û ë û ( =(@
I, %&e '#33#1!,2 p*2e" %&e pr#)e+ure" *+#p%e+ %# 5#+e3 %&e %1# )#,%r!4u%e" *re +e")r!4e+ !,
+e%*!3(
3.2.1 – Concrete contribute
C#,)re%e !" 5#+e3e+ *" *, !,)re5e,%*337 3!,e*r 5*%er!*3 #r%%r#p!) 1!%& re"pe)% %# %&e
pr!,)!p*3 "%r*!, +!re)%!#," 1-%-(( T&e #r!e,%*%!#, #' %&e )r*)6 "7"%e5 1-%-( 1!%& re"pe)% %# %&e
23#4*3 "7"%e5 '- "- ; !" +e'!,e+ 47 %&e > i *,23e" F!2ure =(=(
T&e '#33#1!,2 5*%er!*3 "%!'',e"" 5*%r! !" *+#p%e+
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
1&ere
( ) ( ) ( )
( ) ( )
( )
< = <
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
4e!,2 E c0 %&e !,!%!*3 e3*"%!) 5#+u3u" σ ic *,+ εic %&e pe*6 "%re"" *,+ %&e re3*%e+ "%r*!, !, %&e ith
+!re)%!#, σ i& *,+ εi& %&e )##r+!,*%e" #' * p#!,% #, %&e +e")e,+!,2 4r*,)&( T&e '#33#1!,2 *3ue"
*re )"e,$ σ i& σ ic /> *,+ εi& > εic(
Figure (. – Concrete copressi!e la*.
T&e P#!""#, )#e''!)!e,% @i !" )#5pu%e+ !, e*)& +!re)%!#, *" * 'u,)%!#, #' %&e r*%!# 4e%1ee,
%&e pr!,)!p*3 "%re"" *,+ %&e pe*6 "%re"" =J$
( )
0
0
1&e,
< 1&e,<
li
ili
& & li
li
$ $
$ $ $ $
$
ν
ν ν ν ν
ì £ïïïïï= æ öí - ÷çï ÷- - - >çï ÷çï ÷ç -è øïïî
=(ili &
ic
$ $σ
ν σ
= = = ( =(
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
* ,e1 )*p "ur'*)e !" pr#p#"e+ !, #r+er %# )3#"e %&e "ur'*)e !, %&e &!2& &7+r#"%*%!) "%re""
re2!#,(
T&e B*3*,-Sp*)#,e-1#, "ur'*)e F!2ure =(. pre"e,%" "#5e 5*K#r *+*,%*2e" 'r#5 %&e
,u5er!)*3 p#!,% #' !e1 "!,)e !% )*, 4e 1r!%%e, *" * p*r*4#3* *,+ %&ere'#re !% pr#!+e" %&e
reu!re+ *3ue" !, )3#"e+ '#r5( O, %&e )#,%r*r7 '#r %&e pre!#u" !33*5-*r,6e "ur'*)e *,
!%er*%!e Ne1%#,-R*p&"#, "#3!,2 pr#)e+ure 1*" reu!re+ %# '!,+ %&e )#,)re%e "%re,2%& *3ue"(
Figure (.6 – The (D &ailure sur&ace.
I, %&e pr!,)!p*3 "%re"" "p*)e %&e )#,)re%e "%re,2%& !+e,%!'!e+ 1!%& %&e e)%#r σ c )*, 4e
e*3u*%e+ *" !% '#33#1"( S%*r%!,2 'r#5 %&e )urre,% pr!,)!p*3 "%re"" *3ue" %&e )#rre"p#,+!,2
p#!,% !, %&e pr!,)!p*3 "%re"" "p*)e !" '#u,+( T&e, * "%r*!2&% 3!,e 47 %&e #r!2!, *,+ 47 %&!" p#!,%
!" +r*1, u,%!3 !% !,%er"e)%" %&e '*!3ure "ur'*)e$ %&e )##r+!,*%e" #' %&e !,%er"e)%!#, *re σ 1c , σ %c ,
σ (c(
T&e "ur'*)e !" 1r!%%e, *" * 'u,)%!#, #' %&ree !,*r!*,%"$ %&e #)%*&e+r*3 "%re""e" σ o *,+ τ o *,+%&e L#+e *,23e θ$
( ) ( ) ( )
< =
0(
< = = <
< =
0(
=
<
=
)#"
=
o
o
σ σ σ σ
τ σ σ σ σ σ σ
σ σ σ θ
+ +=
é ù= - + - + -ê úë û- -
=
=(0
1&ere
– >9 –
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
( ) ( ) ( )0(
< = = <
<
. σ σ σ σ σ σ é ù= - + - + -ê úë û ( =(
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
( )( )
<
t o
& B &
A α α= - =
- +=(@
*,+ pr#!+e" * 4!*!*3 )#5pre""!e "%re,2%& eu*3 %#
( ) ( )
< <
ccc
& & α α
é ùê ú= - + - +ê úë û
( =(
I, #r+er %# )*p%ure %&e 5*%er!*3 '*!3ure '#r %&e "%re"" "%*%e" ,e*r %&e &7+r#"%*%!) *!" * )*p
"ur'*)e !" '#r5u3*%e+( B*3*, Sp*)#,e *,+ 1#, J pr#p#"e * )*p "ur'*)e 1r!%%e, !, %&e
"%re"" "p*)e *" * 'u,)%!#, #' %&e !,*r!*,%"( I, =D-PARC * +!''ere,% "#3u%!#, !" pr#p#"e+$ %&e
)*p "ur'*)e !" +e'!,e+ +!re)%37 !, %&e Re,+u3!) p3*,e 47 * p*r*4#3*( T&!" )ure &*" %&e 5*!,
*!" p*r*33e3 %# %&e τ o-*!" !% !" %*,2e,% %# %&e "ur'*)e !, %&e p#!,% 1!%& σ o- & h *,+ !% p*""e" 47
%&e eu!-%r!*!*3 )#5pre""!#, p#!,% 1!%& σ o- & ccc F!2ure =(@( I, %&!" 1*7 1!%& *, e*"7
,u5er!)*3 '#r5u3*%!#, %&e )#,%!,u!%7 *,+ %&e "5##%&,e"" #' %&e =D +#5*!, !" *""ure+(
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
σ o ! f c
τ o
! f c
pr#p#"e+
)*p "ur'*)e
Me,e%re7-!33*5
"ur'*)e
Figure (. – The t*o parabolas de&ining the &ailure sur&ace in the $endulic plane.
T&e )*p !" +e'!,e+ *"
o o o D E F τ σ σ = + + =(9
1&ere
– < –
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
u,!*!*3 )#5pre""!e "%re,2%& & c *,+ %&e e))e,%r!)!%7 e( T&e !,'3ue,)e #' %&e e))e,%r!)!%7 #, %&e
+e!*%#r!) "e)%!#, )*, 4e "ee, !, F!2ure =((
I, 2e,er*3 %&e u,!*!*3 p*r*5e%er" & t *,+ & c *re 1e33 6,#1, #r e*"7 %# e"%!5*%e( O, %&e
)#,%r*r7 %&e e''e)% #' %&e e))e,%r!)!%7 !" 3e"" !,e"%!2*%e+ ee, !' !% "%r#,237 !,'3ue,)e" %&e
"ur'*)e( T&!" e''e)% )*, 4e )3e*r37 +e%e)%e+ 47 p3#%%!,2 %&e !,%er"e)%!#, #' %&e '*!3ure "ur'*)e
1!%& %&e σ (0 p3*,e #4%*!,!,2 * 4!*!*3 '*!3ure +#5*!,(
Figure (.9 – Bia'ial &ailure doain as a &unction o& e.
I, #r+er %# pre"ere %&e )#,e!%7 *,+ "5##%&,e"" #' %&e e33!p%!) +e!*%#r!) "e)%!#, #' %&e
"ur'*)e %&e e))e,%r!)!%7 5u"% 4e 6ep% !, %&e r*,2e 0(e
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
T&ere'#re !% !" )3e*r %&*% %&e e))e,%r!)!%7 "u3+ 4e )"e, )*re'u337( T&e +!*2r*5 !, F!2ure
=(
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
"%re,2%& & c *,+ %&e re3*%e+ pe*6 "%r*!, εcp( D*r1!, *,+ Pe)6,#3+ 0J pr#p#"e *'%er 4!*!*3
"%u+!e"$
( )
=
ïïïî
=(=.
1&ere %&e )#e''!)!e,% $ !" +er!e+ 'r#5 eper!5e,%*3 !,e"%!2*%!#,"( T&e "u22e"%e+ *3ue !"
$=( T&e eu*%!#, =(=. !" p3#%%e+ !, F!2ure =(
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
T&e '*!3ure !" )#5pre""!e( F#r *33 %&e +!re)%!#," %&e σ ic !" )#5pu%e+ 1!%u% 5#+!'!)*%!#,(
T&e "%re""-"%r*!, 3*1 '#r +!re)%!#, ( !" u"e+ '#r *33 +!re)%!#,"(
A "u55*r7 !" rep#r%e+ !, T*43e =(ïïïî
=(>=
– 9 –
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
1&ere E s0 !" %&e !,!%!*3 e3*"%!) 5#+u3u" E sp !" %&e p3*"%!) 5#+u3u" ε s !" %&e )urre,% 4*r "%r*!,
*,+ ε s" !" %&e 7!e3+!,2 "%r*!,(
T&e )urre,% 4*r "%r*!, !, %&e 3#)*3 "7"%e5 '= - "= - ;= !" )#5pu%e+ 'r#5 %&e %#%*3 "%r*!, !, %&e 23#4*3
"7"%e5 '- "- ; %&r#u2& %&e %r*,"'#r5*%!#, >
1&ere T εJ !" %&e "%r*!, %r*,"'#r5*%!#, 5*%r!(
S!,)e %&e "%ee3 4*r" e&!4!% "%!'',e"" *3#,2 %&e!r *e" #,37 %&e!r )#,"%!%u%!e 5*%r! '#r %&e ith
"%ee3 3*7er )*, 4e 1r!%%e, !, %&e 3#)*3 "7"%e5 *"
H H H
0 0 0 0 0
0 0 0 0 0
0 0 0 0
0 0 0
( 0 0
0
si
' " ;
si i
E
D
s"
ρ
é ùê úê úê úê úê úé ù =ê ú ê úë ûê úê úê úê úê úë û
( =(>
Su4"eue,%37 %&e "%ee3 5*%r! !" 1r!%%e, '#r e*)& 3*7er !, %&e 23#4*3 "7"%e5 %&r#u2& %&e
%r*,"'#r5*%!#,
[ ] [ ]H H HT '"; ' " ;
si si D T D T
ε εé ù é ù=ê ú ê úë û ë û ( =(>.
– .0 –
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
re3*%e+ p*r% #' %&e )#+e( M#re#er !% !" p#""!43e %# )#,"!+er %&e e''e)% #' "u4"eue,% )r*)6!,2
p&e,#5e,* 47 "!5p37 *++!,2 %&e e''e)% #' %&e ,e1 )r*)6"( Fur%&er5#re %&e "u4"eue,% )r*)6"
*re ,#% '#r)e+ %# +ee3#p !, * '!e+ +!re)%!#, 4u% %&e7 *re 'ree %# *r!"e !, %&e re*3 5*!5u5
%e,"!3e "%r*!, +!re)%!#, *'%er %&e '!r"% '*!3ure(
Figure (.16 – #rincipal strain s"ste a: and crac) displaceent b: de&inition.
T&ree +!''ere,% )##r+!,*%e "7"%e5" *re u"e+$ %&e 23#4*3 "7"%e5 '- "- ; %&e 3#)*3 "%ee3 "7"%e5
'
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
T&e e''e)% #' %&e )#e''!)!e,% #, %&e )#5pre""!e "%re""-"%r*!, )ure !" ", !, F!2ure =(
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
4e!,2
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
Figure (.12 – Aggregate bridging odeling in the plane 1-%.
0.0
0.2
0.4
0.6
0.8
1.0
0 0.2 0.4 0.6 0.8 1
u ! u c
σ c t
! f t
Figure (.19 – CEB Model Code 90 σ -u la*.
T&e *,*37%!)*3 '#r5 #' %&e 3*1 !"
( )
<
<
<
<
0(< 1&e, 0
0(
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
4e!,2 / F %&e )#,)re%e 'r*)%ure e,er27 u
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
<
ct t ct
ac
u
σ σ
ε= = ( =(..
Figure (.%1 – Aggregate interloc) odeling in the plane 1-%.
I, =D-PARC %&e *22re2*%e !,%er3#)6 ,*5e37 %&e "&e*r %*,2e,%!*3 "%re""e" %r*,"'erre+ 4e%1ee,
%&e )r*)6 3!p" F!2ure =( = >
> >
5*E 5*E
> >
< <
T < *,+ T <
< <
< *,+ <
! *
u! ! u*
! *a a a a
a au uu uc c
D u D u! *a a
u u
a a a a! *c c ! c
u u u u
τ τ
-
+ +æ ö æ ö÷ ÷ç ç÷ ÷= ç - = ç -÷ ÷ç ç÷ ÷÷ ÷ç çæ ö æ öè ø è ø÷ ÷ç ç+ +÷ ÷ç ç÷ ÷ç çè ø è ø
æ ö æ öæ ö æ ö÷ ÷ç ç÷ ÷ç ç÷ ÷ç ç= + = +÷ ÷÷ ÷ç çç ç÷ ÷ç ç÷è ø è ø÷ç çè ø è ø
*c *
-
÷÷
=(.
1&ere
– . –
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
< = >
(> >0(. (>> < 0(@ *,+ 0(
c
a a a a & τ τ τ
æ ö÷ç= = = - = =÷ç ÷çè ø =(.9
4e!,2 Da' %&e 5*!5u5 *22re2*%e "!e(F!,*337 %&e )#,)re%e 5*%r! !, %&e )r*)6 "7"%e5 !" #4%*!,e+$
-
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
I, eu*%!#, =(@ %&e "%r*!, *r!*43e" *re 1r!%%e, *" 'u,)%!#," #' %&e )r*)6 +!"p3*)e5e,%" !,
%&e "%ee3 "7"%e5 '= - "= - ;= ( I, p*r%!)u3*r δ !" %&e +!"p3*)e5e,% *3#,2 %&e '= -*!" η.J( I, %&e
'#33#1!,2 eu*%!#," %&e !,+e i !,+!)*%e" %&e ith "%ee3 3*7er 1&!3e %&e !,+e ?G1,% !,+!)*%e" %&e
– @0 –
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
+!re)%!#, #' %&e %*,2e,%!*3 "%re""e"(
F#r %&e ith "%ee3 3*7er %&e +#1e3 '#r)e #r%#,*3 %# %&e 4*r" !" "5e*re+ *3#,2 %&e !,%er*e" *,+
%&e re3*%e+ "%re""e" *re #4%*!,e+$
< <
< <
*,+i i ' " i ' ; i
i i i i
3 3
p p p pτ τ τ τ = = = = ( =(@
T&e '#r)e 3 i? !" )*3)u3*%e+ *" * 'u,)%!#, #' %&e )r*)6 +!"p3*)e5e,%" δ i ηi
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
!,)re*"!,2 %&e *3ue #' %&e 5e*, "%ee3 "%r*!,(
Figure (.%+ – Euilibriu conditions o& a $C portion a:, o& a steel bar b: and o& concrete c:.
T&e eu!3!4r!u5 eu*%!#," 2#er,!,2 %&e pr#43e5 *re$
• "e)%!#, eu!3!4r!u5 F!2ure =(>-*$
0c sd d
d' d'
σ σ ρ+ = =(0
• "%ee3 4*r eu!3!4r!u5 F!2ure =(>-4$
( )( )> sd s '
d'
σ τ
Φ= =(-)$
( )( )>cd s '
d'
σ ρ τ Φ
= - ( =(
T&e )#5p*%!4!3!%7 eu*%!#, !"
s c s u u= - =(=
1&ere s !" %&e "3!p *,+ u s *,+ uc *re %&e "%ee3 *,+ )#,)re%e +!"p3*)e5e,% re"pe)%!e37( A'%er %&e
+!''ere,%!*%!#, %&e =(= 4e)#5e"
s cdsd'
ε ε= - ( =(>
– @ –
-
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
C#54!,!,2 %&e pre!#u" eu*%!#," %&e "#3!,2 eu*%!#, !" #4%*!,e+$
( )( )( )
>< s
s c
d s ' E s '
d' E E
ρ τ
Φ
æ ö÷ç ÷= +ç ÷ç ÷çè ø
( =(
Figure (.% – CEB Model Code 90 bond-slip la*.
T&e 4#,+-"3!p 3*1 !" %*6e, 'r#5 %&e CEB M#+e3 C#+e 90
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
I, =D-PARC %&e pr#43e5 !" "#3e+ !, * ,u5er!)*3 1*7 47 u"!,2 %&e F!,!%e D!''ere,)e Me%+
FDM( F#33#1!,2 %&!" *ppr#*)& %&e "#3u%!#, !" )*3)u3*%e+ !, * '!,!%e ,u54er #' p#!,%" *,+ %&e
*))ur*)7 )*, 4e !5pr#e+ 47 !,)re*"!,2 %&e ,u54er #' p#!,%" F!2ure =(.(
F!r"% #' *33 * "e% #' p#!,%" u"u*337 eu!-"p*)e+ !" )"e,( T&e '-*!" !" !,%e,+e+ %# 4e %&e
"%ee3 4*r *!" &*!,2 !%" #r!2!, !, %&e 5!++3e p#!,% 4e%1ee, %&e )r*)6" *,+ %&e p#!,% 'l s/
)#!,)!+!,2 1!%& %&e 5!++3e #' %&e )r*)6( T&e +er!*%!e" #' %&e "#3!,2 'u,)%!#, !, %&e ith p#!,%
*re 1r!%%e, 47 u"!,2 %&e *3ue" #' %&e 'u,)%!#, !, %&e "*5e p#!,% *,+ !, "#5e #%&er p#!,%" ,e%
%# !%( I, p*r%!)u3*r %&e "e)#,+ +er!*%!e #' s ' !, %&e ith p#!,% )*, 4e 1r!%%e, *"
( ) < <
i i i
d s ' s s s
d' 'Δ
- +- += ( =(@
T&e eu*%!#, =( 1r!%%e, !, %&e ith p#!,% !"
< –
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
<
< <
< 0 0 ((( 0 0
< < ((( 0
(((((( (((
0 ((( < <
0 ((( 0 0 <
n n
n
s
s ) '
s ) ' s
τ
τ δ
Δ
Δ- -
ì üé ù ì üï ï ï ïï ï ï ïê ú ï ï ï ïê ú ï ï ï ï- ï ï ï ïê ú ï ï ï ïï ï ï ïê ú =í ý í ýê ú ï ï ï ïï ï ï ïê ú- ï ï ï ïê ú ï ï ï ïï ï ï ïê ú ï ï ï ïê ú ï ï ï ïë û î î
( =(90
S!,)e %&e %*,2e,%!*3 "%re"" !, * p#!,% !" * 'u,)%!#, #' %&e "3!p !, %&*% p#!,% *, !%er*%!e
pr#)e+ure !" reu!re+( T&e )#,er2e,)e r*%e !" u!%e '*"%(
A'%er )#5pu%!,2 %&e s ' *33 %&e #%&er u,6,#1, 'u,)%!#," )*, 4e )*3)u3*%e+ !, * "!5!3*r
5*,,er( B7 *pp37!,2 %&e FDM *ppr#!5*%!#, %&e =( )*, 4e 1r!%%e, *"
<
>c i c i i
'σ σ ρ τ ΔΦ
+- = ( =(9
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
< < <
< < <
0 0 ((( 0
< 0 < ((( 0<
((( ((( ((((((
0 ((( < 0 <
0 ((( 0 0
s c
s c
s n c n n
s n c n n
s
s
' s
s
ε ε
ε ε
ε ε
ε ε
Δ
- - -
ì ü ì ü ìé ùï ï ï ï ïï ï ï ï ïê úï ï ï ï ïï ï ï ï ê ú ï-ï ï ï ï ïê úï ï ï ï ïï ï ï ï ïê ú= +í ý í ý íê úï ï ï ï ïï ï ï ï ïê ú-ï ï ï ï ïê úï ï ï ï ïï ï ï ï ïê úï ï ï ï -ê úï ï ï ï ë û îî î
üïïïïïïïýïïïïïï ïï ï
( =(9
T&e "%ee3 "%r*!, +!"%r!4u%!#, *3#,2 %&e 4*r !" ,#1 +e'!,e+( A'%er1*r+" %&e 5e*, "%ee3 "%r*!,
'r#5 %&e %e,"!#, "%!''e,!,2 '#r5u3*%!#, !" )#5p*re+ 1!%& %&e 23#4*3 5e*, "%ee3 "%r*!,
)#5pu%e+ *" %&e 23#4*3 "%r*!, !, %&e 4*r +!re)%!#,( T&e %e,"!#, "%!''e,!,2 +!"%r!4u%!#, !"
)#rre)%e+ %# *""ure %&e 5e*, "%r*!, eu*3!%7$
global ts
s ean s ean sε ε εΔ= + ( =(9.
T&e 5e*, *3ue #' %&e %e,"!#, "%!''e,!,2 "%r*!, +!"%r!4u%!#, !" )#5pu%e+ 47
<<
<
<
n s s nts
s ean s i
in
ε εε ε
-
=
æ ö÷ç= + + ÷ç ÷ç ÷- è øå ( =(9@
A'%er %&e"e pr#)e+ure" %&e "%ee3 "%r*!, *3#,2 %&e 4*r !" #4%*!,e+ F!2ure =(@( ε s,nε scri !" %&e
"%ee3 "%r*!, #' %&e ith re!,'#r)!,2 3*7er !, %&e )r*)6( T&ere'#re %&e %e,"!#, "%!''e,!,2 )#e''!)!e,%
g i !" )#5pu%e+ *))#r+!,2 %# =(@>(
Figure (.% – 3teel strain distribution.
– @. –
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
=(> – DOUBLY CRACED MATERIAL
A'%er %&e '!r"% )r*)6 '#r5*%!#, %&e 5*%er!*3 4e%1ee, %&e )r*)6" "%!33 1#r6" *" *, u,)r*)6e+
5*%er!*3( T&e pr!,)!p*3 "%r*!, +!re)%!#," *re re)#5pu%e+ !,+epe,+e,%37 #, %&e '!r"% )r*)6
#r!e,%*%!#,( Due %# %&e 5#+u3*r 'r*5e1#r6 #' %&e 5#+e3 1&e, * ,e1 )r*)6 *r!"e" !, %&e
pr!,)!p*3 "%r*!, "7"%e5 !%" e''e)% )*, 4e e*"!37 !,"er%e+(
A'%er %&e "e)#,+ )r*)6!,2 %&e "%r*!, !" +e)#5p#"e+ !,%# %&ree )#,%r!4u%e" %&e '!r"% re3*%e+ %#
%&e )#,)re%e 4e%1ee, %&e )r*)6" *,+ %&e #%&er" re3*%e+ %# %&e )r*)6"$
{ } { } { } { }< '"; '"; '"; '"; c cr cr ε ε ε ε= + + ( =(9
A" pre!#u"37 ep3*!,e+ '#r %&e "!,237 )r*)6e+ 5*%er!*3 %&e "%*r%!,2 p#!,% !" %&e eu!3!4r!u5(
T&e eu!3!4r!u5 !, %&e '!r"% )r*)6 )*, 4e 1r!%%e, *"
{ } { } { } { } ( ){ } { }< < < < < < <
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C&*p%er = – T&e pr#p#"e+ 5#+e3 =D-PARC
{ } { } { }( )<
'"; '"; '"; '"; '";
c c s s D Dε σ ε
-é ù é ù= -ê ú ê úë û ë û ( =(
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CHAPTER > – APPLICATIONS
>(< – INTRODUCTION
S!,)e %&e 4e2!,,!,2 %&e pr#p#"e+ 5#+e3 =D-PARC – T&ree-D!5e,"!#,*3 P&7"!)*3 Appr#*)&
'#r Re!,'#r)e+ C#,)re%e 1*" '#r5u3*%e+ !, #r+er %# 4e !5p3e5e,%e+ 'r#5 %&e )#5pu%*%!#,*3
p#!,% #' !e1( T&e '!r"% er"!#, #' %&e 5#+e3 +ee3#pe+ !, %&e *u%r" 2r*+u*%e %&e"!" 1*"
!5p3e5e,%e+ !, %&e )#+e TRE =>J * FORTRAN )#+e *43e %# e*3u*%e %&e 4e&*!#r #' *
5*%er!*3 p#r%!#,( T&ere'#re #,37 er7 "!5p3e "%ru)%ure" "u4Ke)%e+ %# u,!'#r5 "%re"" "%*%e" )*,
4e *,*37e+ 1!%& TRE(
Su4"eue,%37 !, #r+er %# pr#!+e %&e 5#+e3 1!%& 5#re '3e!4!3!%7 %&e 1#r6 1*" '#)u"e+ #,
%&e F!,!%e E3e5e,% FE !5p3e5e,%*%!#,( T&e )#55er)!*3 FE )#+e ABA?US
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C&*p%er > – App3!)*%!#,"
>( – NON-LINEAR PROBLEM SOLVING PROCEDURES
Re!,'#r)e+ C#,)re%e RC "%ru)%ure" e&!4!% * )#5p3e 4e&*!#r +ue %# %&e &!2&37 ,#,-3!,e*r
re"p#,"e #' 4#%& u,)r*)6e+ *,+ )r*)6e+ 5*%er!*3( T&e "%ru)%ur*3 eu*%!#, "7"%e5 )*, 4e
1r!%%e, *"
[ ]{ } { } L u F = >(
-
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C&*p%er > – App3!)*%!#,"
#' %&e +!"p3*)e5e,% u *,+ *, e%er,*3 3#*+ # !" *pp3!e+ F!2ure >(( I, %&!" "%u+7 %&e #p%!#,
#' "#'%e,!,2 4e&*!#r !" )"e, *" !% !" %&e 5#"% )#55#, '#r RC "%ru)%ure"(
".2.1 – Secant met+od
S%*r%!,2 'r#5 %&e #r!2!, %&e 3#*+ # A !" *pp3!e+ *,+ %&e )#rre"p#,+!,2 )#,'!2ur*%!#, u A !"
"e*r)&e+ F!2ure >(=( !%& %&e !,!%!*3 "%!'',e"" ) 0 %&e p#!,% a re3*%e+ %# %&e 3#*+ # A *,+ %# %&e
)#,'!2ur*%!#, u
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C&*p%er > – App3!)*%!#,"
Figure +.+ – 4e*ton-$aphson techniue.
".2.3 – )odified Ne*tonRa-+son met+od
T&!" 5e%+ +!''er" 'r#5 %&e pre!#u" #,e 4e)*u"e %&e %*,2e,% "%!'',e"" !" ,#% up+*%e+ '#r
e*)& !%er*%!#, 4u% re5*!," )#,"%*,% +ur!,2 %&e 1e !,)re5e,% #r *3%er,*%!e37 !% !" up+*%e+
#,37 'r#5 %!5e %# %!5e +ur!,2 %&e !,)re5e,% F!2ure >(( I, %&!" 1*7 %&e )#5pu%*%!#,*3 )#"%
!" "%r#,237 re+u)e+( I, '*)% %&e )*3)u3*%!#, #' %&e %*,2e,% "%!'',e"" '#r e*)& !%er*%!#, )*, 4e *
4!2 ,u5er!)*3 e''#r%( T&e )#,er2e,)e r*%e +epe,+" #, %&e up+*%!,2 'reue,)7 4u% #4!#u"37
!% +e)re*"e" 1!%& re"pe)% %# %&e Ne1%#,-R*p&"#, 5e%+(
Figure +. – Modi&ied 4e*ton-$aphson techniue.
".2." – uasiNe*ton met+od
T&!" 5e%+ !5p3e5e,%" %&e '!r"% %1# !%er*%!#," 1!%& %&e Ne1%#,-R*p&"#, *ppr#*)& u"!,2 %&e
!,!%!*3 %*,2e,% F!2ure >(.( I, %&!" 1*7 %&e p#!,%" < *,+ #, %&e )ure *re '#u,+(Su4"eue,%37 "%*r%!,2 'r#5 %&e"e %1# p#!,%" %&e "e)*,% "%!'',e"" 3!,e p*""!,2 47 < *,+ !"
– –
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C&*p%er > – App3!)*%!#,"
e*3u*%e+( Fr#5 %&*% p#!,% #,1*r+ %&e "*5e "e)*,% "%!'',e"" !" u"e+( T&e )#,er2e,)e r*%e !"
"uper3!,e*r(
Figure +.6 – Nuasi-4e*ton techniue.
".2.% – 4nitial stiffness met+od
U"!,2 %&e *4#e 5e,%!#,e+ 5e%+" ,u5er!)*3 pr#43e5" )*, *r!"e !, p*r%!)u3*r "!%u*%!#,"( I,
%&e Ne1%#,-R*p&"#, 5e%+ +e"p!%e * er7 '*"% )#,er2e,)e r*%e pr#43e5" )*, #))ur ,e*r
%&e 5*%er!*3 )ure pe*6" 1&e, %&e %*,2e,% !" )3#"e %# er#( T&e "e)*,% '#r5u3*%!#, #, %&e
)#,%r*r7 &*" * "3#1er )#,er2e,)e r*%e 4u% !, 2e,er*3 !% !" 5#re "%*43e 'r#5 %&e ,u5er!)*3
p#!,% #' !e1( Neer%&e3e"" pr#43e5" )*, 4e '#u,+ !, %&e p#"%-)r*)6!,2 p&*"e 1&e, "#5ee3e5e,%" #' %&e "%!'',e"" 5*%r! )*, 4e er#(
Figure +. – 7nitial sti&&ness techniue.
I, #r+er %# *#!+ %&e"e ,u5er!)*3 pr#43e5" * +!''ere,% *ppr#*)& )*, 4e !5p3e5e,%e+( I, %&e
FE '#r5u3*%!#, '#r e*)& !,%e2r*%!#, p#!,% %&e "%re"" e)%#r '#r %&e !,%er,*3 '#r)e )*3)u3*%!#,
*,+ %&e Q*)#4!*, 5*%r! '#r %&e "%!'',e"" )#5pu%*%!#, *re reu!re+( I, #%&er 1#r+" %&eQ*)#4!*, 5*%r! +e%er5!,e" #,37 %&e "3#pe 1&!)& !" u"e+ %# *ppr#*)& %&e "#3u%!#,( T&ere'#re
– = –
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C&*p%er > – App3!)*%!#,"
%&e Q*)#4!*, 5*%r! )*, 4e )#,"!+ere+ #,37 * ,u5er!)*3 %##3 1&!)& )*, 4e +e'!,e+ 47 %&e u"er(
F#r "#'%e,!,2 "%ru)%ure" eer !% "u3+ 4e *31*7" "%!''er %&*, %&e #,e re3*%e+ %# %&e
)urre,% "%ru)%ur*3 )#,'!2ur*%!#, #%&er1!"e %&e eu!3!4r!u5 p#!,% )#u3+ 4e u,*%%*!,*43e(
I, %&e !,!%!*3 "%!'',e"" 5e%+ %&e Q*)#4!*, 5*%r! )#5pu%e+ *% %&e 4e2!,,!,2 #' %&e *,*37"!"
!" 5*!,%*!,e+ +ur!,2 %&e 1e )*3)u3*%!#, F!2ure >(@( I, %&!" )*"e %&e Q*)#4!*, 5*%r! !" %&e
3!,e*r e3*"%!) 5*%r!(
O4!#u"37 +ue %# %&e '*)% %&*% %&e 5*%r! !" "%!''er %&*, %&e "e)*,% %&e )#5pu%*%!#,*3 %!5e !"
&!2&37 !,)re*"e+ "!,)e 5u)& 5#re !%er*%!#," *re ,e)e""*r7 %# *)&!ee %&e )#,er2e,)e(
T&ere'#re %&!" ,u5er!)*3 %##3 "u3+ 4e u"e+ )*re'u337 *,+ #,37 1&e, %&e #%&er 5e%+" *re
u,"u!%*43e(
– > –
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C&*p%er > – App3!)*%!#,"
>(= – NUMERICAL PROCEDURES IN ABA?US
".3.1 – Nonlinear -roblems in A5AS
T&e '!,*3 *!5 #' ,#,-3!,e*r *,*37"e" !" %# e*3u*%e * re*3!"%!) re"p#,"e up %# '*!3ure #'
"%ru)%ure" u,+er 2e,er!) 3#*+" %*6!,2 !,%# *))#u,% %&e p&7"!)*3 re*3!%7 #' %&e 5*%er!*3
)#,"%!%u%!e 3*1" *,+ #' %&e 2e#5e%r!) )&*r*)%er!"%!)" F!2ure >((
Figure +.2 – 4on-linear response e'aple.
S!,)e %&e pr#43e5 !" ,#,-3!,e*r *, !,)re5e,%*3 *,+ !%er*%!e pr#)e+ure !" reu!re+
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C&*p%er > – App3!)*%!#,"
I' %&e )#,er2e,)e )r!%er!* *re ,#% "*%!"'!e+ *,#%&er !,)re5e,% !" reu!re+(
F!2ure >(9 "" %&e e%er,*3 *,+ !,%er,*3 '#r)e" *)%!,2 #, * 4#+7( T&e !,%er,*3 '#r)e" *% *
,#+e *re )*u"e+ 47 %&e "%re""e" !, %&e e3e5e,%" )#,,e)%e+ %# %&e ,#+e( T&e eu!3!4r!u5 !"
"*%!"'!e+ 1&e, e%er,*3 '#r)e" # *,+ !,%er,*3 '#r)e" 7 4*3*,)e e*)& #%&er *% eer7 ,#+e$
0 # 7 - = ( >(
S%*r%!,2 'r#5 %&e )#,'!2ur*%!#, u0 * 3#*+ !,)re5e,% Δ # !" *pp3!e+( I, %&e '!r"% !%er*%!#, %&e
"%ru)%ure "%!'',e"" 5*%r! L 0 re3*%e+ %# %&e )#,'!2ur*%!#, u0 *,+ %&e !,)re5e,% Δ # *re u"e+ %#
)#5pu%e * +!"p3*)e5e,% )#rre)%!#, ca( T&e "%ru)%ure )#,'!2ur*%!#, !" up+*%e+ %# ua 47 u"!,2 ca
F!2ure >(
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C&*p%er > – App3!)*%!#,"
0a au u uΔ = - ( >(>
B#%& %&e )#,er2e,)e )r!%er!* 5u"% 4e "*%!"'!e+ %# )#,"!+er %&e eu!3!4r!u5 *)&!ee+( T&e
%#3er*,)e *3ue" )*, 4e "pe)!'!e+ 47 %&e u"er(
Figure +.11 – 3econd iteration.
O, %&e )#,%r*r7 !' %&e )#,er2e,)e !" ,#% *%%*!,e+ *,#%&er !%er*%!#, !" )*rr!e+ #u% !, #r+er %#
re+u)e %&e re"!+ue F!2ure >((.
I' ,e)e""*r7 * 'ur%&er !%er*%!#, !" per'#r5e+ 1!%& %&e "*5e pr#)e+ure(
I% )*, 4e )#,)3u+e+ %&*% '#r eer7 !,)re5e,% * 23#4*3 "7"%e5 #' eu*%!#," !" *""e543e+ *,+
"#3e+( T&ere'#re %&e )#5pu%*%!#,*3 )#"% #' e*)& !%er*%!#, !" )3#"e %# %&e )#"% #' * )#5p3e%e
3!,e*r *,*37"!"( F#r %&!" re*"#, %&e )#5pu%*%!#,*3 epe,"e #' * ,#,-3!,e*r *,*37"!" !"
p#%e,%!*337 5*,7 %!5e" 2re*%er %&*, %&e )#"% #' * 3!,e*r *,*37"!"(
– @ –
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C&*p%er > – App3!)*%!#,"
".3.2 – T+e code 3'ARC in t+e &inite Element frame*ork
T&e FE )#+e ABA?US *33#1" %&e u"er %# +e'!,e * )#,"%!%u%!e 5#+e3 47 !,"er%!,2 *
FORTRAN "u4r#u%!,e )*33e+ UMAT U"er MATer!*3 +e")r!4!,2 %&e !,%e2r*%!#, p#!,%
4e&*!#r(
T&e *r!*43e" p*""e+ !, *re %&e 5*%er!*3 pr#per%!e" *,+ %&e *r!*43e" )#rre"p#,+!,2 %# %&e
)urre,% "%*%e #' %&e "%ru)%ure( T&e #u%pu% *r!*43e" *re %&e "%re""e" u"e+ %# )#5pu%e %&e
!,%er,*3 '#r)e" *,+ %&e Q*)#4!*, 5*%r! u"e+ %# )*3)u3*%e %&e e3e5e,% "%!'',e"" 5*%r!(
=D-PARC reu!re" %&e '#33#1!,2 !,pu% 5*%er!*3 pr#per%!e" '#r )#,)re%e$ %&e u,!*!*3 %e,"!3e
"%re,2%& & t %&e u,!*!*3 )#5pre""!e "%re,2%& & c %&e "%r*!, εcp )#rre"p#,+!,2 %# & c !, %&e u,!*!*3
"%re""-"%r*!, )ure %&e !,!%!*3 e3*"%!) 5#+u3u" E c0 *,+ %&e !,!%!*3 P#!""#, )#e''!)!e,% ν 0( F#r
e*)& "%ee3 3*7er %&e '#33#1!,2 pr#per%!e" *re reu!re+$ %&e e3*"%!) 5#+u3u" E s0 %&e p3*"%!)
5#+u3u" E sp %&e 7!e3+!,2 "%re,2%& & s" %&e u3%!5*%e "%re,2%& & su %&e 4*r +!*5e%er Φ *,+ %&e
*,23e" +e'!,!,2 %&e 4*r #r!e,%*%!#, θ
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C&*p%er > – App3!)*%!#,"
4e )#5pu%e+( T&e 1e pr#)e+ure )*, 4e repe*%e+ up %# %&e )#,er2e,)e(
Seer*3 +!''ere,% FE )*, 4e u"e+ 1!%& =D-PARC( I, %&!" 1#r6 %&e 0-,#+e u*+r*%!) re+u)e+
!,%e2r*%!#, "#3!+ e3e5e,% C=D0R !" pre'erre+(
{ } { }ε εΔ+
=D-PARC
{ }σ [ ] D
{ } [ ] { }i
T
int i
8
& B d8 σ = ! [ ] [ ] [ ][ ]i
T
i i
8
L B D B d8 = !
{ } [ ]{ }<
el 4
int i int i
i
& 5 & =
=
å [ ] [ ] [ ][ ]
<
el 4 T
i i i
i
L 5 L 5=
=
å
[ ]{ } { } { }e't int L . & & Δ = -
{ }.Δ and { }εΔ
Figure +.1% – The code (D-#A$C in the FE &rae*or).
".3.3 – Numerical -rocedure for strain decom-osition
A'%er %&e )r*)6 '#r5*%!#, *, !%er*%!e pr#)e+ure !" reu!re+ 1!%&!, %&e )#+e =D-PARC !,
#r+er %# *)&!ee *, e*)% "%r*!, +e)#5p#"!%!#, 'u3'!33!,2 4#%& )#5p*%!4!3!%7 *,+ eu!3!4r!u5
F!2ure >(
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C&*p%er > – App3!)*%!#,"
1*7 %&e )#5p*%!4!3!%7 !" !5p#"e+( Fr#5 %&e"e "%r*!," %&ree )#,"%!%u%!e 5*%r!)e" '#r "%ee3
)#,)re%e *,+ )r*)6 *re )#5pu%e+( T&e, %&e %&ree 5*%r!)e" *re u"e+ %# 4u!3+ %&e %#%*3
)#,"%!%u%!e 5*%r! DJ *,+ %# )#5pu%e %&e "%re""e" 4e%1ee, %&e )r*)6" "%ee3 *,+ )#,)re%e
*,+ !, %&e )r*)6( M#re#er %&r#u2& %&e DJ 5*%r! %&e %#%*3 "%re"" σ !" #4%*!,e+(
A'%er1*r+" %&e eu!3!4r!u5 #' %&e "%re"" !, %&e )r*)6 *,+ 4e%1ee, %&e )r*)6" !" )&e)6e+( I' %&e
eu!3!4r!u5 !" "*%!"'!e+ %&e DJ 5*%r! *,+ %&e %#%*3 "%re"" σ *re %*6e, *" #u%pu%( I' ,#% *
,e1 )r*)6 "%r*!, εcr !" +e'!,e+ 47 u"!,2 %&e )r*)6 "%!'',e"" 5*%r! *,+ %&e %#%*3 "%re""( I, '*)%
%&e "%re"" !, %&e )r*)6 *" 1e33 *" %&e "%re"" 4e%1ee, %&e )r*)6" 5u"% 4e eu*3 %# %&e %#%*3 "%re""(
{ }ε as input
{ } { } { }c cr ε ε ε= - { }cr ε
[ ] s D [ ]c D [ ]cr D [ ] D
{ } sσ { }cσ { }cr σ { } [ ]{ } Dσ ε=
{ } { } { }Z
c s cr σ σ σ + = { } [ ] { }
<
cr cr Dε σ
-=
[ ] D and
{ }σ as output
Figure +.1( – 3train decoposition procedure.
– 90 –
Copatibilit"
Euilibriucon!ergence test
4
OE3
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C&*p%er > – App3!)*%!#,"
>(> – PLAIN CONCRETE BIAXIAL TESTS
".".1 – E6-erimental -rogram
A" '!r"% *3!+*%!#, %&e pr#p#"e+ 5#+e3 =D-PARC !" *pp3!e+ %# p3*!, )#,)re%e "pe)!5e,"
"u4Ke)%e+ %# 4!*!*3 "%re""e"( T&e '!,*3 *!5 #' %&!" e*3u*%!#, !" * +eep !,e"%!2*%!#, #' %&e
)#,)re%e )#,"%!%u%!e 5#+e3 47 u"!,2 1e33 6,#1, *,+ re3!*43e eper!5e,%*3 +*%* @J(
S!,)e %&e +ee3#p5e,% #' *, u,!er"*3 '*!3ure +#5*!, !" * &!2&37 !,%ere"%!,2 %#p!) 5*,7
+!''ere,% eper!5e,%*3 "%u+!e" 1ere per'#r5e+ #, )#,)re%e "u4Ke)%e+ %# 4!*!*3 "%re""e"(
Figure +.1+ – 3uar" o& pre!ious e'periental results.
I, 2e,er*3 %&e #4%*!,e+ re"u3%" pre"e,% * 2re*% +!")rep*,)7 "!,)e 5*,7 +!''!)u3%!e" 1ere
e,)#u,%ere+ !, #4%*!,!,2 * "u!%*43e %e"% "e%up( I, %&!" 6!,+ #' %e"%" %&e eper!5e,%*3 "e%up
"u3+ 4e )"e, )*re'u337 +ue %# %&e '*)% %&*% !% !" er7 !5p#r%*,% %# #4%*!, * 1e33 +e'!,e+ *,+
u,!'#r5 4!*!*3 "%re"" "%*%e !,"!+e %&e "pe)!5e,( O,e #' %&e 5*K#r pr#43e5" !" %&e e''e)% #' %&e
'r!)%!#, 4e%1ee, %&e 4e*r!,2 p3*%e" *,+ %&e "pe)!5e, 1&!)& )#u3+ pr#+u)e * )#,'!,e5e,%
5#+!'7!,2 %&e "%re"" "%*%e *3#,2 %&e )#,)re%e e+2e" *,+ !,)re*"!,2 %&e u3%!5*%e "%re,2%&( M*,7
+!''ere,% "#3u%!#," 1ere %r!e+ %# *#!+ %&!" pr#43e5 3!6e 3u4r!)*,%" "ur'*)e %re*%5e,%" #r "#'%
p*)6!,2 4e%1ee, %&e "ur'*)e" 4u% !, 2e,er*3 %&e re"u3%" 1ere ,#% re3!*43e$ '#r eu*3 4!*!*3
)#5pre""!e "%re""e" %&e u3%!5*%e "%re,2%& *r!e" 'r#5 0 up %# =0 #' %&e u,!*!*3
)#5pre""!e "%re,2%& F!2ure >(( O, %&e )#,%r*r7 !, %&e 5!e+ %e,"!3e-)#5pre""!e '!e3+"%&e *2ree5e,% 1*" 2##+( F!,*337 %&e )#,)re%e "pe)!5e, "!e *,+ "&*pe &*+ * "%r#,2 !,'3ue,)e
– 9< –
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C&*p%er > – App3!)*%!#,"
#, %&e re"u3%"(
Figure +.1 – E'periental setup.
I, %&!" eper!5e,%*3 "%u+7 "u*re p3*%e" 3#*+e+ !, %&e!r p3*,e 1ere u"e+ "!,)e !% "ee5" %# 4e *
"u!%*43e "pe)!5e, "&*pe *,+ 4ru"& 4e*r!,2 p3*%e," 1ere *+#p%e+ !, #r+er %# re+u)e %&e
)#,'!,e5e,% e''e)%( T&e"e p3*%e," )#,"!"%e+ #' * "er!e" #' )3#"e37 "p*)e+ "5*33 "%ee3 4*r"
1&!)& +ue %# %&e!r '3e!4!3!%7 )*, '#33#1 %&e )#,)re%e +e'#r5*%!#, 1!%u% 2!!,2 *ppre)!*43e
re"%r*!,!,2 e''e)%"( I, #r+er %# *#!+ 4u)63!,2 !,"%*4!3!%7 "r%er "%ee3 4*r" 1ere u"e+ 1!%&
&!2&er )#,)re%e "%re,2%& 1&!3e !, )*"e #' %e,"!3e "%re"" %&e 4*r" 1ere 23ue+ %# %&e )#,)re%e
e+2e"(
Figure +.16 – ntreated and brush bearing platens results.
00-)5 p3*!, )#,)re%e "pe)!5e," 1ere "u4Ke)%e+ %# "%re"" )#54!,*%!#," !, 4!*!*3
%e,"!3e %e,"!3e-)#5pre""!e *,+ 4!*!*3 )#5pre""!e '!e3+" !, #r+er %# +r*1 * )#5p3e%e '*!3ure
– 9 –
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C&*p%er > – App3!)*%!#,"
+#5*!,( F#r e*)& '!e3+ '#ur +!''ere,% "%re"" r*%!#" 1ere )"e, *,+ "! "pe)!5e, 1ere %e"%e+(
T&ree +!''ere,% %7pe" #' )#,)re%e &*!,2 +!''ere,% u,!*!*3 )#5pre""!e "%re,2%&" 1ere u"e+(
T&e 5*!5u5 *22re2*%e "!e 1*"
-
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C&*p%er > – App3!)*%!#,"
perpe,+!)u3*r %# %&e 'ree "ur'*)e(
I, 2e,er*3 %&e "%re,2%& +*%* 1ere "3!2&%37 *''e)%e+ 47 %&e +!''ere,% u,!*!*3 )#5pre""!e
"%re,2%& 4u% '#r e*5p3e !, %&e 5!e+ %e,"!3e-)#5pre""!e '!e3+ %&e )#,)re%e "%re,2%&
+e)re*"e+ *" %&e u,!*!*3 "%re,2%& !,)re*"e+(
T&e )#,)re%e "%re,2%& u,+er 4!*!*3 )#5pre""!#, 1*" ( –
-
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C&*p%er > – App3!)*%!#,"
%&e 5#"% 3#*+e+ +!re)%!#,( O, %&e )#,%r*r7 %&e "%r*!," !, %&e perpe,+!)u3*r +!re)%!#," *re
+e%e)%e+ 1!%& 3e"" *))ur*)7(
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
-4-3-2-10123
ε x 10-3
eper!5e,%*3
=D-PARC
ε = ε
-
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-
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C&*p%er > – App3!)*%!#,"
>( – PLAIN CONCRETE TRIAXIAL TESTS
".%.1 – E6-erimental -rogram
I,
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C&*p%er > – App3!)*%!#,"
".%.2 – Results and com-arisons
F!2ure" 'r#5 >(= %# >( " %&e eper!5e,%*3 re"u3%" re3*%e+ %# %&e =D '*!3ure "ur'*)e 47
*r7!,2 %&e σ 1 "%re"" )#5p#,e,%( T&e !,%er"e)%!#," 4e%1ee, %&e '*!3ure "ur'*)e *,+ "eer*3
p3*,e" σ 1/ & c *re rep#r%e+ !, %&e p3*,e σ %/ & c-σ (/ & c ( A33 %&e +!*2r*5" *re "755e%r!)*3 1!%& re"pe)%
%# %&e 4!"e)%!,2 3!,e( T1# +!''ere,% "er!e" *re p3#%%e+ '#r %&e eper!5e,%*3 re"u3%"$ %&e
8eper!5e,%*3
-
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C&*p%er > – App3!)*%!#,"
%*3) *,+ %&e *3u5!,u5 p*+" *re ,#% "u''!)!e,% %# pree,% %&e )#,'!,e5e,% e''e)% *,+ %&e re3*%e+
"%re,2%& !,)re5e,%(
0
1
2
3
4
0 1 2 3 4
σ 2 ! f c
σ 3 !
f c
eper!5e,%*3 <
eper!5e,%*3
=D-PARC
Figure +.%+ – E'periental and nuerical eG0.6: &ailure doain &or σ 1G0.% & c.
0
1
2
3
4
5
0 1 2 3 4 5
σ 2 ! f c
σ
3 !
f c
eper!5e,%*3 <
eper!5e,%*3
=D-PARC
Figure +.% – E'periental and nuerical eG0.: &ailure doain &or σ 1G0.+ & c.
– 99 –
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C&*p%er > – App3!)*%!#,"
0
1
2
3
4
5
6
0 1 2 3 4 5 6
σ 2 ! f c
σ 3 ! f c
eper!5e,%*3 <
eper!5e,%*3
=D-PARC
F