WOOD & ARMER
Method of calculating plate and shell reinforcement - Wood&Armer You can use use the Wood&Armer method (the European code supplement [ENV 199-1-1 E! "esign of !oncrete #tructures - Appendi$ % point A' einforcement in #la)s*+ ,he conception of determining eui.alent moment is authored )/ Wood and Armer "etails concerning the method can )e found% for instance% in 0Wood% ,he reinforcement of sla)s in accordance 2ith a pre-determined field of moments% !oncrete% 3e)ruar/ 194'% August 194' (correspondence+*
!alculation 5rocedure When calculating reinforcement of a plate structure or s2itching on the option of panel design for simple )ending in a shell structure% design moments are calculated according to the method )/ Wood and Armer 3or a selected directions $ and /% t2o t/pes of design moments M6 are calculated7 the lo2er ones (positi.e% causing mainl/ tension in the )ottom parts+ and the upper ones (negati.e% causing tension in the upper parts+ ,he general procedure ta8es the follo2ing form7 Determination of the 'lower' moments M xd *, M yd *.
M xd 6 M x : ;M xy ; M yd 6 M y : ;M xy ; 0o2e.er% if M x < -;M xy ; (ie the calculated M xd 6 < =+ M xd 6 = M yd 6 M y : ;M xy 6M xy >M x ; #imilarl/% 2hen M y < -;M xy ; (ie the calculated M yd 6 < =+ ? (6+ M xd 6 M x : ;M xy 6M xy >M y ; ? (6+ M yd 6 = ? (6+
@f an/ of thus o)tained moments M xd 6% M yd 6 is smaller than ero% /ou should assume ero (the design moments for tension in the upper la/ers are determined further on in the te$t+ Determination of the 'upper' moments M xg *, M yg *.
M xg 6 M x - ;M xy ; M yg 6 M y - ;M xy ; @f M x B ;M xy ; (ie the calculated M xg 6 B =+ ? (6+ M xg 6 = ? (6+ M yg 6 M y - ;M xy 6M xy >M x ; ? (6+ #imilarl/% 2hen M y B ;M xy ; (ie the calculated M yg 6 B =+ M xg 6 M x - ;M xy 6M xy >M y ; M yg 6 = @f an/ of thus o)tained moments M xg 6% M yg 6 is )igger than ero% /ou should assume ero (such moments 2ould design the lo2er reinforcements% 2hich is alread/ guaranteed )/ the formerl/ calculated Clo2erC moments M xd 6% M yd 6+ Analogousl/% design forces are calculated from the formulas gi.en )elo2 for a plane stress structure or for the acti.ated option of panel design for compression> tension in a shell structure 3or the selected directions $ and /% t2o t/pes of design forces N6 are calculated7 the tensile (positi.e% causing main tension in a section+ and the compressi.e (negati.e% causing section compression+ ,he general procedure ta8es the follo2ing form7 Calculation of 'tensile' forces N xr *, N yr *.
N xr 6 N x : ;N xy ; N yr 6 N y : ;N xy ; 0o2e.er if N x < -;N xy ; (ie calculated N xd 6 < =+ N xr 6 = N yr 6 N y : ;N xy 6N xy >N x ;
#imilarl/% if N y < -;N xy ; (ie calculated N yr 6 < =+ ? (6+ N xr 6 N x : ;N xy 6N xy >N y ; ? (6+ N yr 6 = ? (6+ @f an/ of thus o)tained forces N xd 6% N yd 6 is less than ero% one should assume the ero .alue (forces designing a section )/ reinforcement compression are determined further on+ Calculation of 'compressive' forces N xs *, N ys *.
N xs 6 N x - ;N xy ; N ys 6 N y - ;N xy ; 0o2e.er% if N x B ;N xy ; (ie calculated N xs 6 B =+ ? (6+ N xs 6 = ? (6+ N ys 6 N y - ;N xy 6N xy >N x ; ? (6+ #imilarl/% if N y B ;N xy ; (ie calculated N ys 6 B =+ N xs 6 N x - ;N xy 6N xy >N y ; N ys 6 = @f an/ of thus o)tained forces N xs 6% N ys 6 is greater than ero% one should assume the ero .alue (such forces design a section )/ reinforcement tension% 2hich is alread/ guaranteed )/ the tensile forces N xr 6% N yr 6 calculated earlier+ 3or comple$ stresses (shells 2ith the acti.ated option of panel design for )ending : compression> tension+ 2ith )ending moments (M xx % M xy % M yy + and mem)rane forces (N xx N xy % N yy+ acting simultaneousl/% there is no simplified algorithm de.ised #ince it is often the case that the modeled shells 2or8 almost as plates (2ith slight mem)rane forces acting+% therefore the possi)ilit/ to calculate moments M xd 6% M yd 6 according to the method presented still remains and these design moments are superimposed 2ith longitudinal forces N xx % N yy
!rac8ing of 5lates and #hells !alculations
,he 2idth of crac8ing is calculated independentl/ for t2o directions ,he/ are defined )/ a$es of reinforcement and analogous to the simplified methods presented in rele.ant studies @mplementation of the method not related to codes results from lac8 of rele.ant recommendations concerning plates 2ith cross reinforcement ,he algorithm of calculations is )ased on the formulas ena)ling calculation of crac8ing 2idth for )eam elements !alculations are carried out on the cross-section 2ith reinforcement resulting from the Dltimate imit #tate (see7 einforcement of plates and shells - calculations+ Moments recognied in calculations of #er.icea)ilit/ imit #tate are eui.alent moments calculated according to the selected calculation method7 anal/tical% NEN or Wood&Armer ,he anal/tical method for #er.icea)ilit/ imit #tate does not recognie actions of m xy moments "ue to the implementation of NEN or Wood&Armer method% one ma/ recognie the m xy moments in calculations )/ increasing the moments m xx and myy ,he Wood&Armer method is recommended for calculations of plates 2ith cross reinforcement among others )/ EN199-1-17==F A!7==' [Anne$ 3 (@nformati.e+ einforcement e$pressions for in-plane stress conditions* ,he calculated crac8ing 2idth .alue 2hich is presented in the ta)le of results is the ma$imum .alue o)tained from all the anal/ed load cases When einforcement adGustment is selected for calculations% the area of reinforcement undergoing tension increases% reducing the crac8ing 2idth 2hen it is not possi)le to fulfill the user defined condition of the ma$imum crac8ing 2idth% the ta)le of results 2ill highlight the result cell in red ,here are no non-code limits set on the reinforcement ratio% so /ou should pa/ attention to the economic aspect of the solution pro.ided
5late and #hell "eflections - !alculations ,he algorithm for deflections of ! plates is )ased on the use of calculations of an isotropic elastic plate made of an elastic material !hanges of material stiffness due to crac8ing are considered "isplacements are calculated appl/ing the 3inite Element Method (3EM+% then the/ are modified !alculations are performed separatel/ for each panel #uch an assumption is correct if a panel can )e identified 2ith a structural element (span% floor segment+% other2ise stiffness .alues a.eraged 2ithin a panel ma/ )e distorted ,his ma/ result in the influence of .er/ distant elements on displacements of an anal/ed 3E @nfluence of such distur)ances on e$treme .alues is not .er/ significant% ho2e.er deformation (deflection+ maps should )e treated 2ith great caution
@t is recommended to model each floor segment 2here local e$tremes of deflections ma/ occur as a separate panel ,hat panel should )e defined 2ithin the limits set )/ the supports around it (similarl/ as spans 2hich are limited )/ supports in the case of )eams+ 3or a panel% these supports do not need to )e continuous throughout the 2hole panel contour "i.ision to panels does not affect the results of .erification using the With stiffness update (3EM+ method pro.ided that loads% geometr/ and calculated reinforcement are the same !alculations are performed for a selected com)ination (separate for the lo2er and upper displacements+ or a com)ination group% if that is reuired )/ the code (freuent% rare and uasi-permanent com)inations+ ,hat com)ination is chosen for calculations for 2hich there are ma$imum elastic displacements (positi.e and negati.e separatel/+ @f a panel ma/ not )e treated as a structural element (it comprises more structure elements+% deformation (deflection+ maps should )e treated 2ith great caution @t has% ho2e.er% no significant influence on e$treme deflection .alues for a gi.en panel "eflections ma/ )e identified 2ith displacements onl/ for non-deformed supports While calculating ! plate deflection in the shell module (H"+% the displacement of the least displaced support is su)tracted from displacements of each element "eflections are measured from the plane parallel to the surface of the non-deformed plate that passes through one support point of the deformed plate Note7 ,his is done onl/ for supports )eing .ertical )ar elements of for nodal supports 2ith ad.anced attri)utes that define their real sies Ine should pa/ attention to displacements of the remaining supported corners of a plate ,he calculation algorithm used in the o)ot program is )ased on the assumption that the total (real+ deflections of an ! sla) eual the product of its elastic deflections and the stiffness coefficient " > J
2here7 - eal displacements of i-th calculation point of a sla) 2hich ta8e account of crac8ing and calculated reinforcement - Elastic displacements of i-th calculation point of a sla) " - #la) stiffness assuming elasticit/ of the material (as in 3EM calculations+
J - Eui.alent stiffness of an ! plate% calculated considering element crac8ing% rheological effects% adherence of calculated reinforcement% etc and a.eraged for )oth directions #uch an approach comes do2n to linear scaling of indi.idual elastic displacements )/ the glo)al coefficient of stiffness reduction The calculation algorithm for the euivalent stiffness !elastic" method is as follows#
After performing the structure anal/sis using 3EM and calculating the reuired reinforcement area for the ultimate limit state% the ser.icea)ilit/ limit state (as regards crac8ing calculations% stress limits% or other issues that ma/ )e considered locall/+ and the accidental limit state% stiffness for each finite element (3E+ is e.aluated in the program #tiffness calculations are carried out for t2o directions of reinforcement ,he scope and method of calculating these stiffness .alues depends on detailed reuirements of a gi.en code As a result of these calculations% t2o stiffness .alues (different in most cases+ are o)tained for each finite element 3or further calculations% a 2eighted a.erage of component stiffness .alues is used ,he 2eight for a.eraging is the ratio of moments acting on a gi.en element in )oth directions
2here7 J$% J/ - eal stiffness .alues calculated for t2o directions of reinforcement cf - Weight coefficient calculated according to the formula 1
@f ; M$$ ; > ; M// ; B F% then cf 1
H
@f =K L ; M$$ ; > ; M// ; L F% then @f ; M$$ ; > ; M// ; < =K% then cf =
As a result of appl/ing these formulas% in the case of large disproportion of moments (the ratio of the larger to the smaller moment is greater or eual to F=+% the stiffness from the direction of action of the larger of the moments is ta8en into account When .alues of moments are similar% the thic8ness from a gi.en direction is ascri)ed in proportion to the moment ratio ,he ne$t step is to e.aluate the ratio of the elastic stiffness to the 2eighted a.erage of real stiffness .alues o)tained as mentioned a)o.e #uch calculation is performed for each finite element
,he sla) coefficient (1 - n6n+ is considered in calculations of the stiffnesses J and " eal stiffness .alues o)tained in calculations ma/ )e .ie2ed )/ s2itching on maps of #tiffness factor Note7 #tiffness factor and #tiffness factor Y correspond to ">J$ and ">J/% respecti.el/ @f properties of materials used during design are identical to those used in a model% then the .alue of the coefficient " > J B 1= ,his coefficient can )e interpreted (mainl/ for sla)s su)Gected to unidirectional )ending+ as an elastic deflection multiplier @f different materials are used in a model and calculations (for e$ample% 2ith different classes such as concretes 2ith different YoungCs modulus or 5oissonCs ratio+% the coefficient .alue is corrected automaticall/ @t ma/ result in distur)ing the earlier mentioned ineualit/ ,he su)seuent step is to calculate the a.erage of the stiffness ratios e.aluated earlier ,he final glo)al stiffness ratio% used for calculation of real displacements of a sla) (such as linear scaling of elastic displacements+ is an a.erage of the a.erage of stiffness ratios (2ith the 2eight eual to =K+ and the stiffness ratio recorded for an element in 2hich there is the e$tremum of the )ending moment acting in an/ direction (2ith the 2eight eual to =K+
,he eui.alent stiffness (elastic+ method algorithm assumes a.eraging the stiffness for all finite elementsO a shape of the deflection line is% therefore% identical to the deflection line multiplied )/ the stiffness coefficient
,he algorithm of the method 2ith stiffness update (inelastic+ is identical to the algorithm of the eui.alent stiffness (elastic+ method until the calculated stiffness is ascri)ed independentl/ to each finite element (different stiffness for the direction and the Y direction+ An anisotropic sla) of .arious rigidit/ is o)tained 3or thus-determined stiffness .alues the sla) deflection is calculated @n the method 2ith stiffness update% stiffness of each element is calculated independentl/% thus deflection lines ma/ differ A different stiffness is o)tained for each finite element for each direction
@f the einforcement correction option is selected on the ## 5arameters ta) in the 5late and #hell einforcement 5arameters dialog during calculations% reinforcement area increases (to increase the element stiffness+% 2hich reduces sla) deflections einforcement in )oth directions is distri)uted in in.erse proportion to stiffness When it is not possi)le to limit deflections )elo2 the user defined admissi)le deflection .alue (2hen further correction of reinforcement due to the allo2a)le reinforcement ratio is
impossi)le+% then 2hen calculations of the reuired reinforcemement area are completed% a 2arning prompts that the admissi)le deflection .alue has )een e$ceeded for the panel ,here are no predefined limits set on reinforcement other than those in codes Note the economic aspect of the solution pro.ided and that the more a deflection differs from reuirements% the less effecti.e the used method is
