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M E C H A N I K A 
T E O R E T Y C Z N A 
!  S T O S O W A N A 

2/3,  21  (1983) 

O N T H E O R Y O F L A T T I C E - R E I N F O R C E D S H E L L S 

K R Z Y S Z T O F  H .  Ż M I J Ё W S К I 

Politechnika Warszawska 

1. Introduction 

In the present paper a continuum approach for analysing elastic shells with lattice-type 

reinforcement is proposed. 

The shell structures are widely used in engineering practice. In many cases (especially 

in the c i v i l engineering) these structures require to be reinforced. Thus, from the theoretical 

point of view material of the shell ought to be treated as nonhomogeneous mixture o f 

two components: reinforcement and matrix. Even i f additional, simplifing assumptions 

of homogenity and isotropy of both components are being utilised, the known composite 

and mixture theories lead to the complex mathematical models, which cannot be recom­

mended for engineering practice (analysis). That is why in the majority of papers, authors 

-do not apply the theories mentioned above and assume stronger simplifications: in' most 

cases material of the shell is supposed to be homogeneus and anisotropic (or even iso­

tropic); the crucial point is to determine effective moduli for the hipothetic material o f 

the shell. 

The purpose of this work is to generalize the energy functional for continuum shell 

by adding a term concerning elastic reinforcement energy and then deriving the equations 

of equilibrium as well as appropriate (in particular: natural) boundray conditions. 

In the course of the paper, materials of both components are supposed to be linear-

-elastic, homogeneus and isotropic. Considerations are confined to the case o f small 

strains and displacements. The state of strain of the reinforcement is described according 

to Wozniak's lattice-type shell theory [1]. The state of strain of the matrix is assumed 

according to generalised Reissner's hypothesis. Thus, both models belong to the six-

-parameter classes of surface structures theories and no additional constraints on the rein­

forcement are imposed. 

Considerations concern the lattice reinforcement constructed of two or three families 

of intersecting bars, lying on a surface parallel to the middle surface of the shell. C o m ­

patibility conditions o f matrix displacements and approximated displacements of lattice 

nodes are supposed to be satisfied. 



256  К .  Н .  ŻMIJEWSKI 

2. Geometry of  shell 

The region o f the shell is parametrised by two convected normal coordinate systems 

{л- } and  {x'}. A t every point  x\ of the fundamental surface  л ,  x3 = 0 a natural reference 

triplet  (ga) is fixed. Similarly, at all points  x' i n the shell region marks (#,) can be determi­

ned. Particularly, the surface  л ,  x3  =  x3 = const, which includes the axes o f the rein­

forcement bars. The bases (#,) refere to the point o f this surface. 

Base vectors  qt can be expressed by means of qa vectors; the same can be stated about 
the reciprocal bases  q' and  q". The mentioned relations have the forms 

(2.1)  g,  =  Vfga,  g'  =  /tg". 

Eqs (2.1) yield from Weingarten formulae and make it possible to shift an arbitrary tensor 

object from the point  x1' on  л , along the normal to the point  xa on the reference funda­

mental surface. 

The shifters  Vg,,Al (cf. eg [2] [3]) are defined as follows 

(2.2)  V? =  V  + dl'gbX3,  VfA{ -  6{,  VfAla=oba, 

hence 

(2.3) Л* =  Л 6кь[{\­2Hx
3)уba  + x

3'gba  +  (x
3)2уh3,9aK], 

where 

(2.4) .  Л  =  V~\  V'm det(K*) =  \­2x3H+(x3)2K, 

and 

(2.5)  'gab=  ­g3.bg
a  =  g3g%^ 

н  =  ± е г ( 'Щ A - = detftj?). 

It is easy to prove, that the relation 

(2.6)  <k =  Vdn, 

hold true. 

3 .  Tensor  fields  and  their  derivatives 

Values of any vector field  v are elements of linear spaces. Thus they can be represented 

by their components either in the basis g{ or in the basis ga as follows: 

(3.1)  v  =  v,gl  = vlgi  =  vag°  =  v
aga. 

F r o m (2.1) it yields that different components of the same object are related by the 

formulae 

(3.2)  V,  m  V?va,  v>  =  A\v
a. 

Per analogiam one can obtain similar relation for any tensor o f  (p,  q) valency. The ap­

propriate formula takes the form 

(3.3)  t\\­± -  Ali  ...Ai>y4i...  v?<tl\::p. 



LATTICE-REINFORCED SHELLS 257 

A p p l y i n g spatial gradient operation to the tensor field  t, and taking into account (2.1) 

one arrives at the following formulae for the components o f tensor derivatives: 

(3-4) ftrjfo =  K6*A°\ - •  AW  ­  v b • ś & f c wr 
where 

Vi  =  d'Vf. 

4 .  Displacements 

The state of displacements in the shell region is assumed to be compatible with the 

generalised Rcissner's kinematic hipothesis 

(4.1)  u(x",  x3)  = .°«(xx) +  'u(x*)x3 

Additionally vector functions (cf. [1]) approximating displacements й =  u(x") and rota­

tions  9 = ,9(A-*) o f reinforcement lattice nodes are introduced. The meaning o f  йк compo­

nents is the same as in the continuum surface structure theory, wheras  ё „ components 

describe the node rotations i n the planes perpendicular to parametric lines = const 

and # з  — its rotation i n the plane tangent to  ж . The vector functions  и and  9 are supposed 

to be of C 1  — class o f continuuity. Obviously they can be interpreted as displacements 

° n l y in the lattice nodes. 

It is assumed, that interactions between the reinforcement and the material o f the 

matrix (in which it is embedded) are caused by the ideal adherence; and therefore the 

compatibility displacement conditions are supposed to be v a lid . Moreover the latter 

relations are assumed to  be weakened by substituting into them the approximated lattice 

displacements, instead of the real ones. Such procedure leads to the following formulae 

(4.2)  й  =  u\xtm~l and  9 = ft&jT, 

where  9 denotes an infinitesimal rotation vector. Hence 

f 4 - 3 ) 5 = T r 6 t n = T e * ' " i r m M , f t 

where  eklm — R i c c i tensor on  h. 

B y virtue o f (2.1) (3.3) (3.4) and (4.1) the components of lattice displacements can 

be referred to the basis on the fundamental, reference surface 

(4-4)  &„ = ^ C « e + ' « « *
3 ) =  Viua, 

(4.5) 



258  К .  H .  ŻMIJEWSKI 

5.  The  state  of  strain 

The state of strain of the lattice can be determined by means of the functions  exl(x''), 

'exl(x'') as follows 

(5.1) ?„, =  n,~x  + clxJr\  'e,j  =  #,П х. 

By virtue of (4.2) and (4.3) we have 

(5.2)  %a = -j("iifc+«Kfi«).,  %i  =  ~  е \т пйп?т к. 

T a k i n g into account (3.4) and (2.1) and making some simple rearrangements the following 

formulae expressing the strains (5.2) can be obtained: 

(5.3) 

with the denotations 

2*1 =  \­ Ф Й Йа :, 0 , 

Ј  — ~2  Ql'Zl0"a'  be — ®xlUal\b) , 

(5.4)  =  К е ?У ?Льа, 

6.  Potential  energy  of  shell 

The total potential energy of the shell is expressed by the formula 

(6.1)  J= f  5с 1л + I'  (a­q)dn­  jxd(8n), 

where  a and  a denotes surface densities of the strain energy o f the matrix and reinforce­

ment, respectively; QJ and r denotes densities of potential energy due to external surface 

and boundary loads. 

Methods o f determining the quantities  a,  <p, and т can be found in papers devoted 

to so called six-parameter shell theories (cf. [4] [5]) 

In this paper attention is confined to the method of the derivation of the function 

(6.2)  j  ­ j  odn = J  oVdii, 

and — the stationary condition  dl = 0 o f the variational theorem coupled with (6.1). 

The density of reinforcement strain energy can be written in the form 

•  (6.3)  a  =  I  ­(А ^"ёхГ е ,п  +  С ^'"'ех/ё ^. 



LATTICE-RUN FORCED SHELLS 259 

Where, for the lattices under consideration 

r , , > . = 2 '  fw  ąA){tU  ą n Щ +nU<, *<• ,>+у3 ą  , 

(6.4) 

The vectors r ( i 1 ) and are referred to the plane  л and are tangent and normal to the 

axis of the bar (belonged to the A — family of bars), respectively. 

The quantities  R\A],  S{^,  T = 0 , 1 , 2 are stiffneses of reinforcement rods. 

A p p l y i n g the Gauss-Stokes theorem in the form 

(6.5) J V , a c 7 7 r = - J 2 Ш ; Ч л+  fv4ad(8n), 

when stationary condition  dJ  = 0 is being examined, we arrive to the following set o f 

e q u i l i b r i u m equations 

( 6 6 )  • ^ 3 ' d t + ( 2 ^ 3 _ 1 ) ' d 3  + ' Ј ^ Z ' i = ° -

with boundary conditions 

д й „п л:  r**lB  =  0, 

(6.7)  d°ud:  ­(tdp­°X
d%  =  atd, 

Underlined  terms  in  (6.6) and  (6.7)  appear,  when  variation  of  appropiate  terms  i n 

(6.1)  is  considered. 

The  vector  lp  is tangent  to  л  and is exteriorly  normal  to  the  boundary  line  8л . 

Moreover  following  auxiliary  quantities  are  introduced 

(6.8)  fb  _  r«b*_sab_pab_ 

and 

tf*  =  А  У А *1'^>%Ф $йа А Ь, 

r * »  =  0 , 

7.  Conclusions 

In  the present  paper  the energy  functional  for the shells  with  lattice­type  reinforcement 
l s  obtained.  In  the  variational  way equilibrium  equations  as  well  as  natural  boundary 

f 



260  К .  H .  ŻMIJEWSKI 

conditions are derived. The assumed mathematical model makes it possible to consider 

an influence of reinforcement stiffness on resultant shell response in more systematic 

way then in the hitherto used approaches in which homogenity of the structure is postula­

ted. In the proposed model a geometry, directions, and full set o f elastic features o f the 

fibrous is taken into account. 

In the case o f slender reinforcement rods a formal resemblance of the proposed theory 

to the anisotropic model o f Reissner's shell is worth mentioning. 

Equations obtained in the paper can be applied in several other special cases. 

Presented variational approach, in particular the energy functional (6.1) can be used 

for the finite element formulation of the problem considered. 

References 

1.  C z .  W O Ż N I A K,  Siatkowe  dź wigary  powierzchniowe [Lattice­type  surface  structures,  in  Polish),  P W N 

Warszawa  (1970) 

2.  C z .  W O Ź N I A K,  Nieliniowa  teoria powłok  [Nonlinear  theory  of  shells,  in  Polish],  P W N , Warszawa (1966) 

3.  P.  M .  N A G H D I ,  Foundations of elastic shell theory,  in  Progress  in Solid  Mechanics vol. 4,  North — Holland 

P.  C ,  Amsterdam  (1963) 

4.  W .  PIETRASZKIEWICZ,  Finite rotations  and  Lagrangean description  in the non­linear theory  of shells,  P W N , 

W a r s z a w a ­ P o z n a ń  (1979) 

5.  И .  H .  ВЕ К У А,  Н е к о т о р ы е  о б щ и е  м е т о д ы  п о с т р о е н и я  р а д л и ч н ы х  в а р и а н т о в  т е о р и и  о б о л о ч е к . 

[Some  general  m:thods  of  constructing  different  variants  of  shell  theories,  in  Russian),  Н а у к а,  Mockba 

(1982) 

P  e  3   ю  M  e 

О  Т Е О Р ИИ  О Б О Л О Ч ЕК  А Р М И Р О В А Н Н Б 1Х  С Е Т К А МИ  

В  р а б о те  и с п о л ь з уя  в а р и а ц и о н н ый  м е т о д,  п о с т р о е но  у р а в н е н ия  т е о р ии  у п р у г их  о б о л о ч ек  а р­

м и р о в а н н ых  с е т к а м и. 

И з г иб  с п л о ш н ой  о б о л о ч ки  ( м а т р и ц ы)  о п и с а н о,  п р и н и м ая  о б щ ую  г и п о т е зу  Р е й с с н е р а.  Д е ф о р­

м а ц ия  с е т ч а т ой  а р м а т у ры  с о г л а с но  т е о р ии  с е т ч а т ых  о б о л о ч ек  В о з ь н я к а. 

П р е д с т а в л ен  в  р а б о те  п р и ем  о п р е д е л е н ия  в л и я н ия  а р м а т у ры  н а  д е ф о р м а ц ию  о б о л о ч ки  п о  

х а р а к т е ру  „ т е х н и ч е с к и й ".  Э то  с л е д у ет  и з  п р и н я т ой  и д е а л и з а ц ии  с т р у к т у ры  с е т ч а т ой  а р м а т у ры  

к ак  и  е е  н е п р е р ы в н о го  о п и с а н и я. 

S t r e s z c z e n i e 

О  T E O R I I  P O W Ł O K  Z B R O J O N Y C H  S I A T K A M I 
ч  

W  pracy  na  drodze  wariacyjnej  uzyskano  r у w n a n i a  teorii  p o w ł o k  s p r ę ż y s t y ch  zbrojonych  siatkami. 

Zginanie  kontynualnej  p o w ł o k i  (matrycy)  opisano  p r z y j m u j ą c  u o g у l n i o n ą  h i p o t e z ę  Reissnera, a  defor­

mację  zbrojenia  o k r e ś l o no  zgodnie  z  r у w n a n i a m i  teorii  p o w ł o k  siatkowych  W o ź n i a ka  [1]. 

Przedstawiona  w  pracy  p r у b a  u w z g l ę d n i e n ia  zbrojenia  na  stan  deformacji  d ź w i g a ra  ma  charakter 

..techniczny"  co  wynika  z a r у w n o  z  z a ł o ż eń  o d n o ś n ie  struktury  zbrojenia  jak  i  z  zastosowanego  konty­

nualnego  opisu  siatki. 

Praca  została  złoż ona  w  Redakcji  dnia  14  kwietnia  1983  roku 

i