MODIFICATON AND RESEARCH ON BENDING RIGIDITY
Foundation item :Suppo rted by the Comm unication Scientific R earch Pr oject of Jiang su P ro vince(06Y 21). Received date :2010-03-05;revision received date :2010-04-01 E -mail :aijun 0822@nuaa .edu
FORMULA OF REINFORCED CONCRETE BEAM
A i J un ,Chen J ue ,Zhang L if ang ,L iu F ang
(Colleg e of A er ospace Eng ineering,NU AA ,29Yudao Str eet,N anjing,210016,P.R.China )
Abstract :Bea m stiffness deg rades w ith its age in rv ice and its r vice per for mance is w eakened.A cco rding to the fundamental cha racter istics o f the reinfo rced concrete,the influence of stiffness degr adat ion caud by different kinds o f damag e is obtained .A mong them ,the crack is the most dir ect and obv ious factor .Fur therm or e ,acco rding t o the analysis of bending r ig idity for mula pr ented in cur rent standa rd ,an influence paramet er of crack dev elo pm ent o n the st iffness deg ra ,no nunifo rmity coefficient o f tensile st eel st rain ,is ext racted.A ver age cr ack distance and cr ack depth ar e t aken as cr ack stat istic pa rameter s.Bad on analy sis and m
o deling with A NSY S ,t he m odified bending rig idity for mula r elated t o cr ack is o btained .Key words :reinfo r ced concrete ;beam and g ir der s;stiffness;cra ck;AN SY S
CLC number :U 446 Document code :A Article ID :1005-1120(2010)04-0299-07
INTRODUCTION
At prent ,simple discounted estimatio n metho d is o ften ud to char acterize br idge girder stiffness in home and abr oad standards .Such metho ds ar e valid in general bridg e design,but fo r rvice br idge asssment they ar e limited .With the incr easing of rvice ag es,cr acks and dam ag e,
such
methods
g radually
cannot
characterize the actual stiffness,and can cau unnecessar y inter ference and error to the bridg e inspection.So rearch o f the method that can accurately characterize actual stiffness of the bridge beam by the damage param eters is necessary.
1 INFLUENCE OF DAMAGE ON
CROSS -SECTION STIFFNESS
Reinforced concrete is composite m aterial.There are thr ee commo n damag es in u which are co ncrete cracks ,concrete corr osion and steel co rrosio n .The formation mechanisms of the dam ag es are co mplex ,diver and differ ent ,but all o f them hav e an influence o n bending r ig idity ,
and directly o r indirectly lead to the deg radation of structur al stiffness.Amo ng them ,concrete cracks play the most direct and obvious role in stiffness deg radation .
Accor ding to the basic theory of stiffness analytic method [1]
注册会计师教材,a form ula of beam stiffness in norm al ca is
B s =
E g A g h 02
1.15 +0.2+6n
(1)
w her e E g ,A g ,h 0,n , are all inv ar iant ,and tho not keep in pace with the changing o f the str ucture external fo rce.No nuniform ity coefficient of tensile steel strain is the only variable about structure external loads.T he law of specific variatio n can be given by
= 1.11-M cr
M
(2)
w her e M cr is the cracking mo ment of member ,and M the mom ent v alue o f current ex ter nal loads.
Eq .(2)show s the nonuniformity coefficient of tensile steel strain gr adually becom es larg er from ze
ro w ith the increasing of the ex ternal load after crack.And the value of the member stiffness B s will also gradually degrade from the
top w ith the increasing of until reaching a certain stable v alue.It sho ws a r elationship betw een the nonuniform ity co efficient of tensile steel strain and stiffness deg radatio n pr ocess. On the o ther hand,fr om a larg e number o f exper im ents,it can be en that r einfor ced co ncrete member s have tw o major chang es in the stiffness degr adatio n pro cess,which are the developments of cracks and plastic deform ation o f co ncrete co mpression zone.When the developments tend to be stable,the beam stiffness w ill also remain r elatively stable.This indicates that there is a clo relationship betw een the developments of cracks and plastic deformation of concrete compressio n zone,and the member stiffness B s.Therefore,the nonuniform ity coefficient of tensile steel strain , and the developm ents of cracks and plastic deformation of co ncrete compressio n zone have a clo connection.
Nonunifo rmity coefficient of tensile steel strain characterizes the influence deg ree of the tensile concrete betw een the cracks on the longitudinal tension steel strain.After first crack appear es,the steel in the un-cracked zone and co ncrete still keep a cer tain relationship o f deformation coo rdination becau of co hesiv e action betw een the concrete and the steel.But the capacity of tensile
concrete defor matio n is limited,so steel strain in the un-cracked zone is much smaller than that in the cracked zo ne. Then,deformation energy caud by ex ter nal fo rces in the tensile zone is m ainly co nsum ed by steel strain.With the load increasing,tensile co ncrete betw een the cracks gradually w ithdraw. Large strain appears in the o riginal un-cracked region of r einfo rced concr ete and new cracks appear,which means that the crack spacing decr eas.T hen,mor e concretes are out o f w ork,resulting in the incr easing of the coefficient of tension steel strain.On the o ther hand,the beam cur vature co rresponding ly becom es larger w ith the further increas o f the load.Tip str ess of crack on the action o f bending tensile stress o f its tw o-sided concrete gradually increas.In additio n,the pr oblem s of crack tip stress concentration still ex ist,which make the tip concrete constantly fracture and w ithdraw from w orking and lead to the increa in heig ht of crack and nonunifo rmity coefficient o f tensile steel str ain .So it is sure that ther e is a clo relationship between the height of crack and nonunifo rmity coefficient of tensile steel strain , and the developm ent of cr ack is the main factor for .
T he structure destr oyed by concrete corrosion is m ainly concrete cracking and spalling,so its influence o n the stiffness deg radation can be studied together w ith concrete cracks.
T he m ain influence of steel cor rosio n o n the stiffness degradation has three aspects,and they are
the w eakening of steel ctions,the dam age of agg lutinate fo rce between co ncrete and steel,and the cracking caud by concrete cor rosion.The first factor can be considered by using the practical reinforcing steel area.T he cond is the m ost important factor that the co rrosio n effects on stiffness deg radatio n.Becau the influence is relatively independent and does not inter fere w ith each other,it can be solved by the independent coefficient of stiffness degradation.T he third factor is an indirect influence on stiffness thro ug h crack,so it can also be analyzed and rearched tog ether w ith cracks[2-8].
T hus,the damag e influence on stiffness is bad on the cracks,and it is also the chief content of this paper.Scholars in Dalian U niversity of T echnology car ried out m any experiments about beam crack dev elo pm ent for indicating crack characteristics[9-10].In this paper, finite element simulation m ethod is ud to find the relationship betw een beam stiffness and cracks.
2 MODEL ANALYSIS OF REINFORCED CONC RETE BEAM
2.1 Basic ideas of model analysis
T he paper is to study the relationship betw een crack dev elo pment and deg radatio n of str uctural stiffness through adjusting the cr acking parameters of concrete unit and co ntrolling artificiall
y the specific location and height of the cracks.Specifically,concr ete unit is permitted to
the crack in the area that is allow ed to the crack ,w hile in other areas,it is no t permitted to the cr ack.The two kinds of co ncrete units have the sam e param eter v alues except the optio ns o f cr acks .The model according to this way is an artificial idealized one.Although the larger w orkload
of
modeling ,
it
can
suppr ess
interference o f the plastic m aterial development to the cracks,and it also can analyze and discuss the influence o f
differ ent
crack
识别英文parameters
independently .
2.2 ANSYS finite element modeling
The beam model is analyzed by using ANSYS.Solid65and Link8element is ud to sim ulate concrete and steel .The constitutive law of concrete in China Standard GBJ 0010-2002is adopted .A nd bilinear ideal elastic -plastic curve is ud as steel bar constitutive law .
Geom etric parameter s and material param eters of concr ete m odel are show n in T able 1and Fig.1.
Tw o crack parameter s of average crack developed height h cr and av erage crack spacing l cr are chon to study .At the sam e tim e ,according to refer ence,the variation r ange o f av erag e developed heig ht h cr is about 0.1h —0.8h (h is the depth of ction)and the verag e crack spacing l cr
isabout 50—4cm in the process of reinforced concrete fr om cracking to destructing completely .T herefo re,the values of the tw o variable par am eters ar e am ong the tw o ranges.In addition,the paper mainly rearches the str ucture r elatio nship betw een the cr ack development and stiffness deg radation under the normal rvice condition .And it is generally thoug ht that averag e crack height reaches only abo ut 0.5h in this state.So the lection of par am eter betw een 0.1h and 0.5h can be encrypted.The specific values of the variable par am eters are show n in Table 2.
In T able 2,the v alues of averag e crack height h cr are the relative height of ction .It is the same in T ables 3, 4.Becau o f the influence of reinforcem ent cover,it appears small unit of 5mm leng th of side according to 0.1h of modeling,and it is difficult for the model to constringe .So the mo del o f 0.1h is ud to r eplace the m odel of 0.08h .In order to check the accuracy of theor etical analy sis ,tw o beams o f L -crack and L -non -crack are ud to be checked in addition .According to the principle of reinforced concrete structur e desig n and "Str ength Evaluation of Ex isting Concrete Bridg e ",the proof test value of mid-span concentrated load of mo del beam P s is 25547.98N.
Table 1 Geometric and material parameters of beam model
Geo metr ic par ameter
M ater ial paramet er
Item P ara meter v alue /mm
pst timeItem
P aramet er v alue
Beam depth 250Concrete g ra de C20
Bea m leng th 2500
蒋在怡Beam brea dth 150Calculated length o f beam
2300
T ensile steel HRB 335(2 12)Hanger stir rup
R 235(2 8)Steel
R 235( 6@era是什么意思
150)
Fig.1 Geo metr ic pa rameter and a rr angement o f reinfo rcement o f simulated beam
N o 301
.4A i Jun,et a l.M odificat on and R ear ch on Bending R ig idity F or mula of …
Table2 Values of two variable parameters and numbers of simulated beam
h cr
一级消防工程师考试时间
l cr/cm
451015202550
0.08L4-0.08L5-0.08L10-0.08L15-0.08L20-0.08L25-0.08L50-0.08 0.20L4-0.20L5-0.20L10-0.20L15-0.20L20-0.20L25-0.20L50-0.20 0.25L4-0.25L5-0.25L10-0.25L15-0.25L20-0.25L25-0.25L50-0.25 0.30L4-0.30L5-0.30L10-0.30L15-0.30L20-0.30L25-0.30L50-0.30 0.35L4-0.35L5-0.35L10-0.35L15-0.35L20-0.35L25-0.35L50-0.35 0.40L4-0.40L5-0.40L10-0.40L15-0.40L20-0.40L25-0.40L50-0.40 0.45L4-0.45L5-0.45L10-0.45L15-0.45L20-0.45L25-0.45L50-0.45 0.50L4-0.50L5-0.50L10-0.50L15-0.50L20-0.50L25-0.50L50-0.50 0.60L4-0.60L5-0.60L10-0.60L15-0.60L20-0.60L25-0.60L50-0.60 0.70L4-0.70L5-0.70L10-0.70L15-0.70L20-0.70L25-0.70L50-0.70 0.80L4-0.80L5-0.80L10-0.80L15-0.80L20-0.80L25-0.80L50-0.80 Collat ion o f additional
beam
L-cr ack L-no n-cr ack
Table3 Calculated results of mid-span deflection of model beam mm
h cr
l cr/cm
451015202550
0.08 2.59E-03 2.56E-03 2.45E-03 2.38E-03 2.34E-03 2.31E-03 2.27E-03 0.20 3.26E-03 3.06E-03 2.85E-03 2.70E-03 2.59E-03 2.52E-03 2.37E-03 0.25 3.57E-03 3.26E-03 3.01E-03 2.83E-03 2.70E-03 2.60E-03 2.42E-03 0.30 3.84E-03 3.48E-03 3.18E-03 2.96E-03 2.79E-03 2.68E-03 2.46E-03 0.35 4.12E-03 3.71E-03 3.39E-03 3.15E-03 2.94E-03 2.80E-03 2.51E-03 0.40 4.37E-03 3.93E-03 3.57E-03 3.31E-03 3.04E-03 2.89E-03 2.57E-03 0.45 4.60E-03 4.07E-03 3.69E-03 3.39E-03 3.12E-03 2.94E-03 2.60E-03 0.50 4.78E-03 4.22E-03 3.83E-03 3.51E-03 3.19E-03 3.01E-03 2.64E-03 0.60 4.96E-03 4.37E-03 3.93E-03 3.61E-03 3.28E-03 3.08E-03 2.68E-03 0.70 5.02E-03 4.40E-03 3.95E-03 3.64E-03 3.31E-03 3.09E-03 2.69E-03 0.80 5.02E-03 4.40E-03 3.95E-03 3.64E-03 3.31E-03 3.10E-03 2.69E-03 Collat ion o f additional
beam
L-cr ack L-no n-cr ack Deflectio n 5.09E-03 2.25E-03
Table4 Nonunif ormity coeff icient of tensile steel strain of simulated beam
h cr
l cr/cm
451015202550
0.080.10940.09970.06340.04170.02710.01930.0044
0.200.32550.26070.19400.14540.10870.08610.0384
0.250.42590.32580.24480.18750.14340.11200.0527
0.300.51370.39610.29960.22730.17450.13790.0656
0.350.60580.47190.36830.28890.22340.17680.0831
0.400.68550.54450.42690.34070.25620.20500.1023
0.450.75970.58890.46450.36890.28050.22380.1123
0.500.81830.63560.51020.40680.30440.24450.1253
0.600.87600.68420.54290.43890.33330.26780.1385
0.700.89610.69450.55040.44960.34270.27240.1418
0.800.89610.69450.55040.44960.34270.27330.1418
Collat ion o f additional
beam
L-cr ack L-no n-cr ack
0.91970.0002
302T r ansactions of N anjing U niv ersit y o f Aer onautics&A str onautics V ol.27
2.3 Calculated results of ANSYS finite element T hr oug h the ANSYS finite element analysis, w e can obtain calculated results of m id-span deflectio n in each group beam,as show n in T able
3.
According to the geometric parameter s and material parameters of the simulated beam,as w ell as Eq.(1),w e can o btain the v alue o f nonuniform ity coefficient o f tensile steel strain of the cor respo nding sim ulated beam through mid-span deflection.Concrete results are sho w n in Table4.
Table4show s that w hen the beam deform s according to the w ay of naturally cracking co mpletely,the v alue of is0.9197.Bad o n the for mulas g iven by the specification.
specificat ion= 1.1M cr
M
=0.9006
The relativ e er ror betw een them is only 2.12%.Considering the influence of convergent error caud by non-linear calculatio n,the er ror value meets the requirements o f the limited value of error co mpletely.At the same time,it also show s that the sim ulated degree and sim ulated results o f the model are higher and better in the paper.
3 DATA ANALYSIS OF FINITE ELEMENT AND RIGIDITY
MODIFIED FORMULA
3.1 Relationship between average crack height
and
T able4sho ws that the calculated values o f nonuniform ity coefficient o f tensile steel strain thr oug h finite elem ent,and w e can obtain h cr- curve in different crack spacing situations.T he specific curv e is show n in Fig.2.
error是什么
Fig.2show s that when h cr≤0.5h,the curv e is linear.And w hen h cr>0.5h,the cur ve slo pes gently.Especially,w hen h cr>0.6h,the value o f has barely chang ed.That is to say,the crack average heig hts ar e o ften about0.5h—0.6h under
load.So,w e can pick up the splattering
of≤0.5h,and linearly fit them.A fter fitting,
partofspeechFig.2 Scatter diag ram of h cr- o f differ ent crack
heig ht s
linear co rrelatio n coefficients o f the cur ve ar e ov er
0.9946o n av er ag e,and linear relativity is sky-
high.It sho ws that the value o f chang es linearly
w ith the increasing of h cr.In other w ords,h cr and
hav e a linear relationship and can be ex presd
by =kh cr+C.
3.2 Relationship between average crack spacing
and
From the calculated values thr oug h finite2012年5月
element in T able4,we can obtain l cr- curv e at
different crack heights.Besides,fro m the abo ve
analysis w e can e that the cr ack average heig hts
are about0.5h—0.6h under checking load.So w e
can only pick up splatters of h cr≤0.5h to analy ze
在线学习英语的网站and fit.The specific cur ve is show n in Fig.3.
Fig.3show s that decreas w ith the
increasing of l cr,but it is hard to get the concrete
curv e from this figure.So this paper makes a
equivalent change
N=100/l cr(3)
T he physical meaning of N is av er ag e
quantity of cracks per meter alo ng with the beam.
T hr oug h the equivalent change of Eq.(3),
w e can obtain scatter diagram of N- ,as show n
in Fig.4.
N- curve sho ws that N o btains larger w ith
the increasing of .Its increasing speed slow s
dow n at first,and then g oes up,and has a clear
inflection po int in the rang e o f N∈(10,20).That
is to say,N- is a ty pical cubic curve.So,the
paper us cubic po lynomial to fit.It is discov ered
that the fitting degr ee of curve is over0.9984on
the aver ag e under cubic polynom ial,and the
fitting effects ar e ex cellent.T her efore,the N o303 .4A i Jun,et a l.M odificat on and R ear ch on Bending R ig idity F or mula of…
