Orthotropy Theory Enu

8/13/2019 Orthotropy Theory Enu

1/39

TheoryPhysical and Shape Orthotropy

of Plates


2/39


3/39

2

PHYSICAL AND SHAPE ORTHOTROPY OF PLATES

A guide to the definition of input physical data for bridge, floor and foundation slabs with different cross-sections in twoperpendicular directions

Prof. Ing. Dr. Vladimr Kol, DrSc.

Copyright: FEM consulting s.r.o.

1993 - 2006


4/39

Physical and Shape Orthotropy of Plates

3

Purpose of the guide................................................................................................................4

Method......................................................................................................................................5

Core principal of solution........................................................................................................6

Stress components in physically orthotropic plates..............................................................7

Internal forces in physically orthotropic plates....................................................................10

Technical theory of plates with the effect of transverse shear taken into account ..............................................10

Plates with the effect of transverse shear taken into account .................................................................................12

Shape orthotropy of plates....................................................................................................14

Main principles of the transformation into physical orthotropy ..............................................................................14

Simple types of orthotropic plates ...............................................................................................................................14

Energetic equivalence of sectional characteristics....................................................................................................14

Bending and torsional moments ................................................................................................................................18Plates without transverse contraction .................................................................................................................18Plates with transverse contraction ......................................................................................................................22

Shear forces and reactions ........................................................................................................................................26

Plates with the effect of transverse shear taken into account .................................................................................27Box-sections....................................................................................................................................................................30

Thick-walled box-section non-solid slabs ...............................................................................................................30Thin-walled box-sections ............................................................................................................................................32

Comparative example ................................................................................................................................................35

Multi-cell slabs with linear hinges in longitudinal direction .....................................................................................36

Other plate types.............................................................................................................................................................37


5/39


4

Purpose of the guide

The purpose of this guide is to provide instructions for the calculation of those input data for orthotropic plates that characterisetheir physical properties, i.e. flexural, torsional and shear stiffness. In the program, these properties are described by the matrixof physical constants D = [Dik]. It follows from the said that that elements Dik of this matrix are the very data that must becalculated manually.


6/39


5

Method

It is assumed that the solution is obtained by means of the finite element method in its most often used deformation variant withunknown deformation parameters of geometric nature. Matrix D combines (i) mutually determined static quantities (stresscomponents or, in case of plates, their resultant over a section i.e. internal forces in the plate) with (ii) corresponding geometricquantities (deformation components or, in case of plates, those derivatives of deflection area w on which deformationcomponents depend):

(1) = D .

Matrix D can be defined separately for (i) each element or (ii) a specific group of elements. This allows for the analysis of platesthat are non-homogenous over the elements or that change their shape over the elements. Consequently, it is possible toexpress, rather approximately, even the gradual change of geometry in a haunch or in a similar detail.

As the text should serve as a fast and easy-to-read reference, it is practically without any l iterature references.


7/39


6

Core principal of solution

Present-day programs and solution methods for orthotropic plates deal, in fact, only with what we call physically orthotropicplanar(two-dimensional) continuum filled with points in the mid-plane (x, y) of the plate, i.e. in the plane z = 0. The body of a real

slab is limited by the upper and lower surface z =

h/2, where h is the thickness of the plate. Alternatively, the height may bevariable, i.e. h (x, y). It is assumed that a normal to the mid-plane (e.g. points with coordinates (x1, y1, z), - h/2 z h/2) remainsstraight, i.e. undistorted, even in the deformed plate. In the standard Kirchhof theory, it remains even perpendicular to thedeflected surface of the plate w (x, y). If we take into account the effect of transverse shear xz, yz on angular changes xz, yz,this normal is generally rotated around x- and y-axis by angles x (x, y) and y (x, y). Even after such a generalisation, we haveonly three functions of two variables to describe the deformation of the plate continuum (Boltzmann continuum for Kirchhof platesand Cosserat continuum for Mindlin plates)

(2) w (x, y), x (x, y), y (x, y).

In a real three-dimensional body of a given plated structure, e.g. box-section, the displacement can be fully described by meansof three functions of three variables

(3) u (x, y, z), v (x, y, z), w (x, y, z),

from which the full description of its stress-state can be derived. In order to be able to solve such a body as a plate, we mustdefine certain geometric assumptions that make it possible to apply functions (2) to find functions (3). For example, the simplestKirchhof assumption mentioned above is of the following form (Fig. 1):

(4)

y/)y,x(wz)z,y,x(v

x/)y,x(wz)z,y,x(u

y/)y,x(w)y,x(

x/)y,x(w)y,x(

)o,y,x(w)y,x(w)z,y,x(w

y

x

Just one function of two variables w (x, y) is sufficient to describe functions (3).

This is the basis of the core principal of the solution, i.e. the transformation of the structure into a plate, and, conversely, theutilisation of the results obtained from the plate-calculation in the design and checks of the analysed structure. An unambiguousrelation must exist between functions (2) and (3) - with the simplest one being of type (4). Under such conditions, it is no moreproblematic to establish similar relations between internal forces or stress in (i) the structure and (ii) its plated-model.

In general, we thus reduce the three-dimensional problem into a two-dimensional one.

x x, u y

w/x > 0y, v

w/y > 0

z, w

Fig. 1


8/39


7

Stress components in physically orthotropicplates

For brevity, we will limit ourselves to the most frequent examples with the coordinate axes (x and y) put into the axes oforthotropy. This approach is generally recommended, as it simplifies the notation.

The approach is based on general relations between (i) deformation components and (ii) stress components in the primary form= D

-10:

(5)

xy

zx

yz

z

y

x

66

55

44

333231

232221

131211

xy

zx

yz

z

y

x

a

a

a

0

||

|

|

|

|

|

0

aaa

aaa

aaa

Let us define the following constants known as technical constants:

() Three (Young) moduli of elasticity:

(6) .a

1E,

a

1E,

a

1E

33

3

22

2

11

1

() Three (Lam) shear moduli of elasticity:

(7) .a

1G,

a

1G,

a

1G

66

12

55

31

44

23

() Six (Poisson) coefficients of transverse contraction ik using the following formulas:

(8)

3323222323

3313111313

2212111212

E/aE/a

E/aE/a

E/aE/a

.

The identities () follow from the symmetry aik = aki, which means that from the nine E and constants, only six are independent.Thus, together with three G constants, we get nine constants E, G, and that are necessary to describe the physical propertiesof the analysed type of an orthotropic substance. Axes x, y, and z are marked with subscripts 1, 2, and 3. The coefficient ikequals to the relative transverse contraction in the i-direction with tension k = E2 in the k-direction. The subscripts of shearmoduli can be swapped: Gik = Gki.

The physical law (5) with technical constants can be, for clarity, re-written separately for normal and shear components (thisdecomposition appears only in orthotropy):


9/39


8

(5a)

1

z

y

x

33

32

3

31

2

23

22

21

1

13

1

12

1

z

y

x

D,

E1

EE

EE

1

E

EEE

1

.

Each line of matrix D-1 contains the same E modulus. The matrix is symmetrical and, therefore, it remains identical aftertransposition. Consequently, we can write the formula with the same E moduli in columns:

(5b) .D,

E

1

EE

EE

1

E

EEE

1

T1

z

y

x

32

23

1

13

3

32

21

12

3

31

2

21

1

z

y

x

In considerations and in calculations we always use the form that is more suitable under the given circumstances. The form (5a)is more frequent.

The matrix of physical constants is diagonal for shear components:

(5c) .D,

G

100

0G

1

0

00G

1

1

xy

zx

yz

12

31

23

xy

zy

yz

It is thus simple to write a reverse formula:

(5d)

12

31

23

G00

0G0

00G

D,D

The reverse formula to (5a) is less clear in terms of technical constants. It is however significantly simpler for orthotropic plates,as it is based on the main static assumption of the theory of plates:

(5e) z (x, y, z) 0 .

Here, the formula (5a) splits into two simpler formulas:

(5f)

y

x

21

21

1

12

1

y

x

E1

E

EE

1

and


10/39


9

(5g) .EE y

x

3

32

3

31z

Component z can be considered insignificant and can be omitted in further considerations, which applies also to orthotropy.Contrary to isotropy, we can even define a material with non-zero transverse contraction in the (x, y) plane, but with zero 31, 32or with E3 , and meet the condition that z 0 at z 0. It is however of no practical meaning.

The formula (5f) can be easily inverted, and thus we get to the full relation = D of type (5.1) = (5.5) (5.1) that takes form (9) forplates, if we arrange the components in a way that is practical for plates:

(9)

yz

xz

xy

y

x

23

13

12

2112

2

2112

212

2112

121

2112

1

yz

xz

xy

y

x

.

G0

0G______________________

G00

01

E

1

E

01

E

1

E

0

0


11/39


10

Internal forces in physically orthotropic plates

Technical theory of plates with the effect of transverse shear

taken into accountThis is what is termed as the Kirchhof theory of thin plates, based on formulas (4) and valid approximately within the range of

(10) ,100C,c

Lw m

(11)5

Lh ,

where:

wm - the maximal plate deflection,

h - the plate thickness,

L - the characteristic plate dimension in plan-view,

L = diameter of a circular plate, L = the shorter side of a rectangular or rhomboid plate, etc.

The consequences of failing to comply with (10) are:

Stress of significant intensity appears in the mid-plane of the plate, plane stress is added to the plate stress. We talk aboutgeometrically non-linear plates with large deflections. Only tensile stress appears in the limit point h 0, the plate becomes amembrane under tension (film of pneumatic structures, three-dimensional geometrically non-linear problems).

The consequences of failing to comply with (11) are:

For h > L/5, we get thick plates with a significant influence of transverse shear on the overall energy, deformation and stress-state of the plate, see paragraphs 5.2 and 6.3.

When conditions (10) and (11) are met, we define the following internal forces: Bending moment (subscript = the direction ofreinforcement) in [Nm/m]:

(12) .zdzM,zdzM yyxx

Torsional moment in [Nm/m]:

(13)

.zdzMM xyyxxy

Shear forces in [N/m]:

(14) .zdT,zdT yzyxzx

They are ignored over the thickness of the plate in the interval h/2 z h/2. Considering the hypothesis (4), geometricconditions

(15) z/vy/u,y/v,x/u xyyx ,

physical relation (9) and conditions of moment equilibrium of a plate element around x- and y- axis, we get formula (1) in thefollowing form:


12/39


11

(16)

xxy

yyy

yyx

xxx

xy

yy

xx

322

311

33

2221

1211

y

x

xy

y

x

w

w

ww

w2

w

w

.

DD00

00DDD00

0DD

0DD

T

TM

M

M

0

0

with the following elements of stiffness matrix D:

() Flexural stiffness:

(17) .)1(12hE

D,)1(12

hED

2112

3

222

2112

3

111

() Contraction stiffness:

(18) ,DDDD 221221112112

() Torsional stiffness:

(19) .12

hGD

3

1233

() Mixed stiffness:

(20) .DD2DD2D 213312333

In total, we need five entries for matrix D: h, E1, E2, G12, 12. The second coefficient of transverse contraction in the x-directionwith the elongation in the y-direction is

(21) .D

D

E

E

11

2212

1

21221

This number of entries can be reduced to four if we accept the assumption that a relationship similar to that valid for isotropy canbe applied to shear modulus G12, even though for geometric averages, see (39) below (another derivation was already presentedby M. T. Huber):

(22) .)1(2 2112

2112

EEG

Therefore, torsional stiffness D33 is no longer an independent constant, but, following from (19) and (17), it can be expressed as

(23) ,DD)1(2

1D 2211211233


13/39


12

which can be written using (21) in the form:

(24).DD)

D

D1(

2

1D

,DD)D

D1(

2

1D

2211

22

112133

2211

11

221233

Formula (20) represents a coefficient at mixed derivation in the main equation for plates

(25) pwDwD2wD yyyy22xxyy3xxxx11

and is decisive for the type of orthotropy defined by constant

(26) .DD

D2211

3

In the civil engineering practice, also the type (0 < 1) is usual. Sometimes we can meet the second type ( = 1) that can bereduced to an isotropic solution. The third type ( > 1) can appear only rarely in steel plates with closed ribs that are rigid intorsion.

Orthotropic constant can be, in certain types of plates, considered a primary piece of data that has been verified in practice and

that represents the mixed stiffness of the plate .DDD 22113 Thus, using (20) and (23), we can obtain thecontraction stiffness in the form

(27) .DD)1(D2DD 2211211233312

Usually, for = 1 we get

(28) .)DDisothropy(DDD 11122211211212

Plate reactions Qx, Qy are equal to shear forces Tx, Ty (similarly to beams) only if torsional moments Mxy at the edge are equal tozero (e.g. fully fixed edge). Generally, e.g. in a simply supported edge, the complement due to torsion must be added:

(29) ,/MTQ nmnmm

where m n is either x y or y x or any other direction of the edge n with normal m. If the reactions are not calculated, theycan be obtained from presented results with the derivatives calculated approximately by means of two adjacent values, e.g. in asequence of equidistant border points 1, 2, 3 with step d:

(30) d2:)]1(M)3(M[)2(T)2(Q xyxyxx

Plates with the effect of transverse shear taken into account

In the classical theory of plates that was discussed in the previous paragraph, shear moduli G13, G23 from (9) have no effect as

the shear forces (14) are determined from the condition of moment equilibrium of the element, which leads to the last two lines ofmatrix (16). In thick plates and within the range of approximately


14/39


13

(31) L / 5 < h < L ,

the right angle between the normal and the mid-plane of the plate is skewed by due to transverse shear xz, yz and, followingfrom (4), the second and third line are eliminated. Then, according to (2) we have three independent functions w, x, y. Undersuch conditions, the following formulas are considered

(32)xyyz

yxxz

w

w

together with the fourth and fifth line of matrix (9). Therefore, instead of matrix (16) we get the matrix:

(33)

y

yx

xy

yy

xx

55

44

33

2221

1211

y

x

xy

y

x

ww

w2

w

w

D0

0D

D00

0DD

0DD

TT

M

M

M

0

0

with new elements that, for the simplest configuration of a constant transverse shear across the plate thickness h, are:

(34) ,hGD,hGD 23551344

and consequently, the five entries h, E1, E2, G12, 12 are extended by two more: G13, G23, which means in total seven entries forthe calculation of input data for D.


15/39


14

Shape orthotropy of plates

Main principles of the transformation into physical orthotropy

Certain bridge, floor, foundation and other structures are similar to plates in terms of the hypothesis (11), i.e. their totalthickness h is small in comparison with plan-dimensions L. On the other hand, they represent a body of a more general shape,e.g. ribbed plates, hollow core slabs, plates with both weak and stiff reinforcement, e.g. with I-beams embedded in the concrete,corrugated plates, double-layer braced plates, etc. Only occasionally is the total flexural stiffness of such shapes identical in both(i) longitudinal x-direction and (ii) transverse y-direction. Usually, the stiffness in the two directions can differ even by a factor often. This is not due to different modulus E1 and E2, but due to a varying section in planes x=constant. It is therefore the shape(not physical) orthotropy of the plate in terms of shape or technical orthotropy. Global behaviourof such plates (without adetailed stress analysis in the vicinity of statically, geometrically or physically singular points and without other irregularities dueto the real shape of the body) can be analysed by means of methods and programs developed for physically orthotropic plates,as long as the following assumptions are taken into account:

Displacement components u, v, w (3) must be unequivocally derivable from the plate deflection surface w (or from the threefunctions (2) in case of plates with the effect of shear taken into account) across the whole analysed body, which means also itsdeformation and stress components, or the internal forces acting in the section across any part of the body. This requiresestablishment of suitable geometrical hypotheses, such as (4), verified in terms of accurateness through various reasoning,

experiments, practical experience, etc.Based on the hypotheses (a), the following must be unequivocally specified for every type of shape orthotropy in plates:

b1) INPUT: Constants Dik in the matrix of physical constants D for physically orthotropic plate that is going tosubstitute the analysed plate in the calculation.

OUTPUT: Further utilisation of outputs of internal forces in the substitute physically orthotropic plate for the needs of design andchecking of the analysed plate with shape orthotropy, i.e. what internal forces or what stresses arise in the analysed plate.

An accurate analysis of meeting the assumptions (a) and accurateness or technical applicability of results would require acomparison with the exact three-dimensional solution of a real structure at least in several characteristic or limit states, or withreliable experiments and tests. This is available only for certain examples, e.g. for steel ribbed plates, etc. Then, more precisedata about the composition of the slab section appear in the input (b1). Mostly however, such analysis can not be performed and

only an approximate method can be used for the comparison, e.g. for box-section plates, which, however, is not a proof, as thevariation from the exact solution is not known. In view of numerous factors that influence the properties of civil engineeringstructures, some well-tried formulas extended by up-to-date knowledge about, in particular, torsional stiffness can beaccepted for the approximate calculations.

Simple types of orthotropic plates

Energetic equivalence of sectional characteristics

In paragraph 6.2 we will discuss plates in which the effect of transverse shear can be neglected and in which the classicalKirchhof hypothesis (4) is satisfied within the whole range. Practically speaking, these are common, not-too-thick plates that areribbed, corrugated or stiffened in two directions x y identical with the selected direction of coordinates (e.g. [1], p.37, Fig.9).

The principle of potential energy equivalence of internal forces in the real and substitute body must be observed in the

transformation into a physically orthotropic plate, i.e. in the calculation of constants D ik in the stiffness matrixi

D

(16). For the

plates in question, it is the following formula

(35) dxdy)w(M)w(M)w(M)w(M2

1yxyxxyxyyyyxxxi

which is the integral of the products that represent only work of moments (similarly to a slim beam subjected to bending andtorsion) on curvatures that can be, for small deflections w, expressed by second derivatives. In the calculation of the substituteplate by means of the finite element method, the second mixed derivative will be continuous everywhere (or at least everywherewith the exception of shapes with zero surface area) and the following condition of equivalence is met


16/39


15

(36) wxy = wyx .

Similarly, the theorem of reciprocity of torsional moments will be valid in the substitute plate:

(37) Mxy = Myx ,

and thus (29) can be simplified to:

(38) )w(M2)w(M)w(M21

xyxyyyyxxxi dxdy.

Myx

Ty

My

xMx+ dxMxTx

Mxy+ dxMxy

Mx Mxy

Tx

yMyx+ dyMyx

My+ dyMy

Ty

z, w

wxx

wyy

wxy

wyx

Fig. 2

The positive direction of all quantities is shown in Fig. 2. The comparative level (i = 0) is the primary non-deformed shape. i isalways positive.

Let us have a system of two beam skeletons that are parallel to x- and y- axes and have flexural stiffness (EJ)x, (EJ)y, torsionalstiffness (GJk)x, (GJk)y, flexural curvature x = wxx,, y = wyy and relative twisting x, y. The potential energy of bending andtorsional moments of such a system is

(39)

.ds)JG()JG()JE()JE(

2

1 2yyk

2xxk

2yy

2xxi

Comparing (39) with (35), we can get only formally at the moment the following formulas for plate strips of unit width dx = 1 ordy = 1:

(40)yykkyyxxxkkxxy

yyyyxxxx

)GJ(MM,)GJ(MM

w)JE(M,w)JE(M

As the transverse contraction and elongation of element sections cannot occur freely in a compact plate (Fig. 3),


17/39


16

PLATE BEAM

x, u

z, w

M

w

M

Fig. 3

the plate strips are rather stiffer in bending than the beams. This can be included into the modulus of elasticity, and thus e.g. inthe case of an isotropic plate we have the module

(41) ,1

EE

2

i

which follows from the exact relation between and . In addition, it can be seen in Fig. 3 that the transverse contraction of thesection is prevented in plates by certain transverse moments M. E.g., the transverse moments occurring in isotropic platessubjected to bending moment My and deformed to a cylindrical surface w(y) are

(42) M = Mx = My

and similarly, in plates subjected to bending moment Mx and bent to the surface w(x) we have:

(43) M = My = Mx .

This effect vanishes in materials without transverse contraction ( = 0).

Let us examine the consequences of formal equations (40) to (43) in isotropic plates with the thickness h


18/39


17

(47))ww(DM)1

,)ww(DM)2

yyxxx

xxyyy

that are in full compliance with formulas derived from the hypothesis (4). In addition, using Fig. 4, let us consider that the totaltwisting (more precisely the torsional curvature) of the plate element is influenced by both pairs of torsional moments, so that

(48) .yx

In plates the continuous mixed derivative wxy = wyx = (see also Fig. 2, the element is deformed into a warped line surface of a

hyperbolic paraboloid type yxd

2)y,x(w in local coordinates x, y with the origin in the centre of the element, so that in

the corners2

dy,

2

dx we get w = w0. A general formula then follows from (40)

(49)

yk

yx

xk

xy

xy)GJ(

M

)GJ(

Mw

and in the case of isotropic plates it is

(50).wGJ

2

1M,

GJ

M2w

:GJ)GJ()GJ(,MMMM

xykxy

k

xy

xy

kykxkyxxykykx

Myx

w0=w1+w2Myx

d

w2 0

MxyMxy

yz

x

Mzy

w1

Mxy

d

w2 1x

d Myx

w2

Myx

d

w2 2y

Fig. 4


19/39


18

After substitution from (44) and (45), multiplication by one in the form of

1

1and use of

(1+) (1-) = (1-2) we obtain

(51) ,w)1(Nw)1()1(12Eh

)1(

)1(w

3

h

)1(2

E

2

1M xyxy2

3

xy

3

xy

which is the same formula that can be derived by an accurate procedure (integration of xy) from hypothesis (4). Therefore, thebeam-based theorization about physical constants of a plate leads to the correct stiffness matrix D of an isotropic plate. Using(16) or (33), (47) and (51), we can write it in the form usual in FEM:

(52) .

w2

w

w

2

100

01

01

D

M

M

M

xy

yy

xx

xy

y

x

It can be therefore expected that such a theorization will not be principally (i.e. as far as equilibrium and continuity conditions areconsidered) defective even in simple plates with shape orthotropy, which is also supported by the existing experience obtained inexperiments and in real practice.

Bending and torsional moments

The dimension of these quantities is force, or more clearly, force length per unit width of a plate section. Using the main SIunits, it is N (Newton) or Nm/m (Newton meter per 1 meter of width).

Conversion to previously used units:

1 kp = 9,80665 N = 10 N

1 Mp=10 4N = 10 kN

All sectional characteristics Jx, Jy, Jkx, Jky must be calculated either (i) for a unit width of a section or (ii) for another width of thesection b, e.g. for the distance between ribs or the size of the finite element and then the result must be divided by this b. Thedimension of J is therefore m3.

Plates without transverse contraction

This group includes practically all ribbed plates with open ribs (not hollow core slabs), because the flexural stiffness of suchplates is derived mainly from ribs that do not influence each other in transverse direction as there is no continuous contraction inthis direction. The value termed as effective value of is very small (e.g. 0.02) also in reinforced concrete plates with thin ribs,especially after the formation of cracks in the tensile concrete, and calculations of such plates for = 0 are accurate. The matrixof physical constants is diagonal as D12 = D21 = 0, see (28). What remains is to determine D11, D22, and D33.

The first two bending constants are quite clear and they can be calculated using (40) from

(53) D11 = (EJ)x, D22 = (EJ)y, [Nm].

The formulas include also possible diverseness Ex Ey [Nm-2]. Jx, Jz [m

3] are related to the unit width of plate sections in planes x= constant and y = constant. For the most common situation of Ex = Ey = E and for the calculation of Jxb, Jyb [m

4] for ribs with an

effective width of the plate bx, by we have


20/39


19

(54) .b

EJD,

b

EJD

y

yb

22

x

xb11

If the plate of the height h is ribbed only in its x-direction, then

(55) D22 =12

Eh3

For the needs of this calculation, in common concrete plates with ribs in both x- and y- direction, the full distance shown in Fig.5a can be taken as the effective width bx, by.

a)

b)

I I I I

bx bx bx bx

bx bxbx bx

Fig. 5

Only for thin plates, e.g. steel orthotropic deck slabs, the reduction of b x, by according to technical standards is more significant(Fig. 5b). It is however necessary to considered the real loading conditions of the plate and other circumstances and not onlyapply the formula given in the standard, as it is valid for stress rather than for the substitute flexural stiffness. When b x, by areuncertain, we recommend a consultation with the author of this guide.

The third constant, torsional D33, is more problematic, but in most situation we can get satisfactory result using the formula thatfollows from (24) for = 0.

(56) D33 = 2211DD2

1

For isotropy and = 0, this formula transforms into the correct value of2

1D, see (52). This is then represented by one torsional

moment MF

xy = MF

yx of the substitute physically orthotropic slab.

(57) .xy2211xy33Fyx

Fxy wDDw2DMM

The equality MFxy = MF

yx follows from the theorem of reciprocity of shear stresses xy = yx (Fig. 6a) that must be valid on thevertical edge of the plate element also in plates with shape orthotropy, but that does not imply the equality of torsional moments,which clearly follows from Fig. 6b. Torsional moments can be obtained by the integration of the shear stress flow over the sectionthrough planes x=constant and y=constant:

(58) Fx

xyx33xxnxy ,w2.DdFrM


21/39


20

(59) Myx= Fy

yn r d Fy = D33y . 2 wyx .

In order to apply formulas (58 and 59) accurately, we would have to know the exact distribution of shear flow over the sections ofthe given plate with shape orthotropy. It is quite a difficult three-dimensional problem that would require rather challengingapplication of the finite element method. Therefore, let us perform first an approximate technical calculation based on theestimate of torsional stiffness strips, like beams in a grate, without continuous dependencies. As wxy is a continuous function in

plates, we havewxy = wyx (Fig. 6c) and the comparative beams have the same relative twisting angle, and thus the ratio of moments (58) and (59)is

(60) .)GJ(

)GJ(

M

M

yk

xk

yx

xy

Let us define a sum-relation between moment (57) and moments (58), (59) that is not in contrast with the isotropy. If we comparethe formulas for energy (35) and (38) where there is a continuous mixed derivation of the function w, in which wxy = wyx, it follows:

a) x

y

z

MyxMxy

yx= xyMyx= Mxy

b)

yn

Fy

yx= xy

xn

Fx

Myx Mxy

c)

wxy= wyx

GJkx

GJky

Fig. 6


22/39


21

(61) .M2MM Fxyyxxy

This leads us to the following values:

(62) .M2)GJ()GJ(

)GJ(M,M2

)GJ()GJ(

)GJ(M Fxy

ykxk

yk

yxFxy

ykxk

xkxy

Substituting (62) into the formula (49) we get

ykxk

Fxy

xy)GJ()GJ(

M4w

Comparing with the formula (57)

33

Fxy

xyD2

Mw

we obtain the formula that will be used to calculate the input value D 33, i.e. the torsional stiffness of the substitute physicallyorthotropic plate.

(63) .])GJ()GJ[(8

1D ykxk33

Quantities Jx [m3

] are related to a unit width of the section. Usually, they are calculated for another suitable width (beam-likesection) b, so that Jk = Jkb [m4]: b [m].

The drawback of this approach is that a pure Saint-Venant torsion is prevented in the plate in the x- and y- direction, as:

the surface shear stress is not zero ( 0 on vertical sides of the surfaces ) over the whole surface of these strips,

free warping is prevented in these strips.

The deviation is not too large in rather thin plates with rather thick ribs. Under different configuration this approach can be appliedif everywhere Mxy Mx ,My. If we calculate quantities Jk from the accurate shear flow (58) and (59), the approach will bealways acceptable. Exceptions, such as plates with encased I-beam etc, should be consulted.

Approximate calculations of common structures can be performed with D33 determined from the formula (56) that does notrequire identification of torsional stiffnesses. It can be proved that both the formula (56) and the future formula (72) for plates withtransverse contraction are in the case of isotropic plates identical with the formula (63), see the reasoning in paragraph 6.2.1

following the formula (44). It generally follows from (61) that

(63a) D33 = )DD(2

1y33x33

and in our simplification we have

D33x= .)GJ(4

1D,)GJ(

4

1yky33xk


23/39


22

Let us remind that the first subscript of and M always denotes the area (section x = constant) on which or M acts. Themoment Mxy thus shortens the plate strips parallel to the x-axis, and the moment Myx the strips parallel to the y-axis. See also Fig.4 where the positive direction of these moments can be clearly seen.

Two special situations for formulas (62):

a) A plate with the same torsional stiffness in the x-direction and y-direction, i.e. J kx=Jky and Mxy= Myx=FxyM is the printed value

of the torsional moment.

b) A plate with a predominant torsional stiffness in one direction, caused e.g. by thick ribs that are rigid in torsion in one direction let us mark it x. It is characterised by a strong inequali ty Jky Jkx. In such a plate the formulas (61) and (63) give

Mxy 2 FxyM , Myx 0 ,

and thus the torsional moment in the x-direction is the double of the printed moment and it is zero in the y-direction.

The real value is between the limits of type (a) and (b). If we deal with what is called design moments represented in outputs byvalues

(64)F

dimxM = (sign. F

x

F

x MM ,MF

xy

F

dimyM = (sign. Fy

Fy MM ,M

Fxy

it must be considered that Mx=Fyy

Fx MM,M (the superscript F still means the physically orthotropic plate for which the

whole calculation is done), butFxyM is only a formal value. The application of values (64) is therefore useful for situations

approaching the limit of type (a). Otherwise, it would be reasonable to use a correction in the meaning of the formulas (61) or(63).

Plates with transverse contraction

Coefficients of transverse contraction in plates with shape orthotropy can be, following from (16) to (18), assigned the followingvisual meaning (Fig. 7):


24/39


23

Fig. 7

a)

b)

y

x

z, w

Mx

My Mx

w (x)

My = 21Mx

My

Mx

Mx

w (x, y)

My = 0

c)My

My

Mxw (y)

Mx= 12My


25/39


24

Let us deform the plate element as in Fig. 7a into the shape corresponding to a cylindrical surface w(x) with a constant curvaturewxx. Then wyy = 0 and, according to (16), the following moments are necessary to cause such deformation:

(65) Mx = -D11wxx , My = -D21wxx = 21Mx.

If we subject the element only to moment Mx, it would lead to a state shown in Fig. 7b, because the following condition follows

from the second line of (16) with My = 0:

D21wxx+D22wyy = 0 ,

and therefore, the element would be distorted also in the y-direction with the curvature wyy = -D21wxx / D22 = -12wxx. To producethe same curvature wxx, smaller moment would be sufficient:

Mx = -D11(1-2112) wxx .

Similarly, the following moments are necessary to produce a cylindrical deflection w(y) of the element in the y-direction (Fig. 7c):

(66) My = -D22wyy , Mx = -D12wyy = 12My .

If the load is formed only by moments My, i.e. if Mx = 0, it leads, according to the first line of (16), to a non-zero curvature w xx = -D12wyy/ D11 = -21wyy. To produce the same wyy, a smaller moment is necessary:

My = -D22(1-1221) wyy.

Therefore, for plates with shape orthotropy we can introduce the following definition of the coefficients of transverse contraction:

The coefficient 21 is numerically equal to the moment My that must be applied to y=constant-edges of the element that issubjected to moment Mx = 1 on edges where x=constant, in order to bend the element into a cylindrical surface w(x). Similarly,the coefficient 12 is the value of Mx on x=constant-edges of the element subjected to moment My = 1 on edges where y=constant and bent to a cylindrical surface w(y). It can be easily verified that in the case of physically orthotropic plates thisdefinition is equivalent to the original definition (8) and in the case of isotropic plates it results in the known relation 12 = 21 = .

At the same time, with this definition i t is clear that if the structure resembles a strong grate with a thin plate, it is practically truethat 12 = 21 =0 and formulas from paragraph 6.2.2.1 are applicable.

The Maxwell-Betti theorem for plates with shape orthotropy:

If a plate element is subjected only to moment M x = 1, it leads to curvatures

wxx = -1 / D11(1 - 1221),

(67) wyy = + 12 / D11(1-1221).

If it is subjected only to moment My = 1, the curvatures are

wyy = -1 / D22(1-1221),

(68)wxx = + 21 / D22(1-1221).

If the deflection is small, the radii of curvature are R x = 1 / wxx , Ry = 1 / wyy. If we relate the moment to a unit width, i.e. if we thinkabout an element with sides bx = by = 1, then the angles of mutual twisting of the originally parallel vertical sides of the elementare

b x / Rx = wXX, yb / Ry = wyy.

The Maxwell-Betti theorem results in the equality (67) = (68), i.e.


26/39


25

(69)22

21

11

12

DD

which is identical with the equality (18) for physically orthotropic plates. However, in that case it was a result of a generalsymmetry of physical constants (8) that follows directly from the requirement that the potential energy of internal forces of thebody be a homogenous quadratic function of stress components or deformation components.

Also in plates with shape orthotropy it must be ensured that the relation (69) is satisfied. If we use a technical reasoning or an

experiment to determine e.g. coefficient 21 for the situation shown in Fig. 7a, also the other coefficient (for the situation in Fig.7c) is determined.

(70) .D

D21

22

1112

Satisfying this relation does not result in large values of for technical materials that behave like physically orthotropicmaterials (plywood, fibreglass, etc.). For example one specific type of pressed plywood has

;7,46

305

D

D

22

11 ;13,0;02,0 1221

or another type of cross-glued plywood gives

;60

120

D

D

22

11 .071,0;0355,0 1221

In practice, there may be a big ratio D11/D22 in shape-orthotropic plates, which may be in the range of 10 to 20.

Usually however, it is found that, at the same time, the coefficient21

is very small, and therefore12

does not exceed 0.5 or

1.0. With regard to the definition of 2112 , by means of bending moments (Fig. 7), also values exceeding 0.5 or even 1.0 arenot, in technical point of view, faulty. After all, they do not represent physical coefficients of contraction, which would be limited byvolume changes of the substance. In real situations and with proper thinking about transverse moments, such values are reallyrare.

As the determination of the actual coefficient 12 or 21 can itself represent quite a difficult problem, simple approximateformulas are used in practice for what is called non-diagonal stiffness element D12 that does not include these coefficients, butonly the coefficient of the isotropic material that the plate is made of, for example = 0.15 for concrete or 0.30 for steel. Inribbed plates and hollow core slabs, we can use the simplified formula (28):

(71)

D12 = ,DD 2211

or a similar formula (27) that, however, gives positive D 12 only for 1 , which e.g. limits its application in concrete tosituations when 0.85 1 that are however quite frequent.

Similarly, instead of (56) the simplified formula (23) can be used:

(72) 221133 DD2

1D

and the increased modulus of elasticity (41) is substituted into the formulas (53) to (55), which represents a broadly smallincrease of 2.25% in concrete and 9% in steel.

Conclusions from the previous paragraph 6.2.2.1 are applicable for the utilisation of values of M xy.

Note concerning the plate mid-plane:


27/39


26

It can be seen in the previous figures that, generally, the centre of gravity of sections x=constant is located in a different distance

xe from the top fibre than the centre of gravity of sections y=constant ye . Therefore, we may ask a question about wherethe mid-plane of the plate is located. This question disappears if we consider that we calculate with a two-dimensional plate

continuum where yx e,e belong in fact among the physical properties. More serious error, however, occurs in plates with

different ribs as a result of neglecting the planar stiffness

20

1

EtD

of the top plate, or the area of thickness t. What

proved useful for such situations is the Giencke formula for mixed stiffness (20)

(73) ,D4

1eeDeeCD 0

2

yx0yx3

that is based on the total torsional stiffness of the plate

(74)

.GJGJ112

EtC

ykxk2

3

And this can be used to determine the orthotropy constant (26); see [1], p. 38

Shear forces and reactions

Shear forces result from the requirement of moment equilibrium of a plate element around the y-axis and x-axis (Fig. 2):

(75) ,y

M

x

MT

xyxx

(76) .x

M

y

MT

xyYy

After substitution of (16), (58), (59) we have

(77) ,wD2DwDw2.DwDwDT xyyy3312xxx11yyxy33xyy12xx11x

(78) .wD2DwDw2.DwDwDT xxyx3321yyy22xxyx33yyy22xx21y

If we compare this formula with the fourth and fifth line of matrix (16) for physically orthotropic plates, we can see that instead ofthe mixed stiffness D3 according to (20) the fourth line now contains the element

(79) D3x = D12 + 2D33y

and the fifth line contains

(80) D3y = D12 + 2D33x .


28/39


27

In the basic plate equation (25) - which is the condition of vertical equilibrium of a plate element

y

y

x

xTT

+ p = 0

and simultaneously the Euler's differential equation of variational plate problem - the application of (79) and (80) changes theexpression 2D3 into the expression (D3x + D3y), so that we have the following formula for the determination of the value of D 3

(81) .DD2

1D y3x33

Approximately (see 63a) we have .GJ4

1D,GJ

4

1D

yky33xkx33

However, programs for physically orthotropic plates calculate Tx and Ty according to matrix (16). Values applicable for plates withshape orthotropy can be derived from these values only by means of a rather complex calculation. If we already know torsionalmoments Myx and Mxy, this calculation can be inspected directly by (75) and (76). In both situations, the derivatives aresubstituted by differences of values in adjacent finite element nodes of the analysed plate and, therefore, we cannot get the fullconformity.

The first members (75) and (76) prevail over the second ones in the larger part of the plan-area of common plates and, inaddition, the difference between (79), (80) and (20) is not too big. Therefore, values Tx and Ty can be used for an approximateshear design.

The reactions of the plate are calculated from (29) or (30) also for the plates with shape orthotropy.

Plates with the effect of transverse shear taken into account

This is an analogy to short, high, etc. beams in which it is not possible to neglect the effect of shear forces T on the deformation,as that effect is comparable to the effect of moments M. This influences the shape of deflection line w(x) and thus also the values

of all statically determined quantities, e.g. hogging moments that are decisive for the design.Current programs have been developed for physically orthotropic plates following the paragraph 5.2 with the matrix of physicalconstants (33) and optional expansion into a full matrix of (5, 5) type in the case of a general anisotropy. Therefore, it is firstnecessary to transform a shape-orthotropic plate into a physically orthotropic plate, i.e. to determine constants Dik in matrix (33).

Instructions from paragraph 6.2.2 apply to constants D11, D12, D22, D33. What remains is to determine constants D44 and D55 informulas

(82) Tx = D44 yz55yxz DT,

that specify the relation between shear forces and transverse shear components of deformation , i.e. the change of rightangles between the normal of the mid-plane of the plate after its transformation into the flexural surface w(x, y), see Fig. 8. These

deformations (32) are zero only if wx = xyy w, (see the sign convention, Fig. 1 and 8), i.e. the Kirchhof hypothesis (4)is valid.

The most important formulas are obtained if the transverse shear stress xz, yz is assumed distributed uniformly across thesectional area Fx, Fy of sections with the width of b = 1 constructed though planes x=constant and y = constant. For a ribbedplate in Fig. 8 we have

(83) .b

FF,

b

FF

y

1y

y

x

1x

x


29/39


28

Fig. 8

In that case Tx = xzFx , Ty = yzFy , and with identical shear modulus G in both directions we have

(84) D44 = GFx , D55 = GFy

If there are no ribs in one or both directions and if the plate thickness is h, then in that direction

(85) Fx = h , Fy = h , and thus in the solution of an isotropic thick plate

(86) D44 = D55 = Gh.

These formulas are sufficient for the estimate of the magnitude of shear stress and for the estimate of required shearreinforcement in concrete.

More detailed analysis however requires that the actual distribution of shear stress xz(z), yz(z) in interval -h1 z h2 be taken

into account, where h1 + h2 = h is the plate thickness and h1, h2 the distance of extreme fibre from the centroid of the section. Inplates with shape orthotropy, it is possible to use the well known Grashof Zuravky formula (with plate indexes)

(87) xz(z) =

)z(2

zS

J

T

x

x

and similarly for yz where 2(z) is the width of a section in the point specified by the coordinate z, and S(z) is the first moment ofthe section above this width related to the horizontal centroidal axis (Fig. 9). In rectangular section of width = 1 this formula

gives the distribution xz (z) that follows a parabola of the second order with its maximumh

T

2

3in the centroid.

The same distribution can be obtained in isotropic plates in the Kirchhof theory (4) from Cauchy equations of equilibrium, andtherefore, we can assume that the application of (87) in plates with shape orthotropy will be fairly accurate.

An uneven distribution of shear stress xz (z), yz (z) results in shear deformations xz (z), yz (z), distributed non-uniformly acrossthe plate thickness.


30/39


29

If we introduce the assumption of a rigid normal (Fig. 8), the consequence is that we get constant xz , yz and thus also xz , yz ,and therefore the formulas (83) (86) are justified. If we apply the more accurate distribution (87) and want to stick to theprocedure of the finite element method, we have to find out the relation between Tx, Ty and values xz , yz, that are independenton z and represent angular changes in the sense of the equivalence of the potential energy of internal forces. For plates withshape orthotropy let us again proceed on the assumption of beam theorization: In a beam element of length l, in which aconstant force Tx is acting, the accumulated potential energy is:

Fig. 9

i = Fxl2

1(xzxz + xyxy)dFx

If we substitute into this formula from the Grashof hypothesis xz (87) and

xy (y,z) = xz (z)

zzytg

and from Hooks law

G

,

G

xy

xyxzxz

,

we get

.dF

ztgy1

z4

zS

GJ2

lTxz2

22

2

2

Fx2x

2x

i

Let us introduce the usual formula for the energy with a corrective coefficient that expresses the variation of across thesection

(88) i = ,GF

Tl

2

1

x

2x


31/39


30

(89)

.dydz

z

ztgy1

z

zS

J4

F2

22

2

2

Fx2x

x

Let us write the energy (88) in the form of a half the product of the force Tx and path wT (Fig. 9):

,wT2

1Txi

.llGF

Tw xz

x

xT

Then it is clear that a constant angular change, equivalent in terms of energy to variables xz (z), is

x

xxz

GF

T

.GF1

T xzxx

Instead of formulas (84) that are valid for a constant xz across the whole section, we may use the following elements of thematrix of physical constants:

(90) D44 = ,GF1

D,GF1

y

Y

55x

X

where subscripts indicate that can be different for sections x=constant and y=constant.

Box-sections

Thick-walled box-section non-solid slabs

A non-solid plate with continuous hollow cores in one direction (let us denote it x) can be calculated as a shape-orthotropic platewith transverse contraction and thus formulas from paragraph 6.2.2 apply to it.

The assumption is that the webs of the box-sections are thick enough. Their thickness ti (it can be even variable) should roughlysatisfy the inequality

(91) ti ,10

h

where h is the total plate thickness. If the vertical webs are thick enough (ratio b s / b 1/10 ), the influence of shear forces on thedeformation of the plate can be neglected and the following formulas from paragraph 6.2.2 can be used to determine the matrixof physical constants (Fig. 10):

2y

222

xb11

1

EJD,

1b

EJD


32/39


31

.DD2

1D,DDD 221133221112

Section:

Rectangle (plate without ribs)

Solid circle and approximately also

solid n-angle (n 6)

Circle thin-walled circular

rings and approximately n-angles (n 6)

Steel I-beam no. 8 to 45

Concrete profiles: I, square, T, etc.

with sectional area F and web area F s,

approximately

I-sections, squares, etc. with

the sum of vertical webs thickness t1and radius of gyration to the neutral axis r:

= t1

t2

e1

e2

6/5

32/27 (TP 3)

10/ 9 (ROARK)

2

2.8 to 2.1

F/Fs

2

22

1

2

r10

er41

t

tk1

32

121

22

e2

eee3k

The value Jx is calculated for section y=constant with a unit width that goes through the thinnest part of horizontal webs of thebox-sections.


33/39


32

Fig. 10

Values Jxb are calculated for an I-section where the flange width b is always the distance

between the vertical axes of the boxes. In case of unequal boxes also the I-sections areasymmetrical. Jxb is always related to the horizontal centroidal y-axis. The difference in theheight of the centroids of sections x=constant and y=constant (see the note at the end of 6.2.2)is not significant. To determine the torsional element D33, also formula (63) can be used:

,GJGJ8

1D

ykxk33

on condition that we take into account the remarks stated in paragraph 6.2.2 that are given after that formula. It means that thedifferences between (i) values Jkx, Jky that follow from the Saint Venant torsion of beams and (ii) values J kx ,Jky relating to theactual shear flow by (58), (59) are in box-sections more significant than in plates with open ribs (Fig. 6b). In Fig. 10 the arrows

and circles show the shear flows that approximately occur due to the necessity to keep .yxxy In closed box-sections withvertical webs thick enough, shear circulation may develop and the corresponding Jxk from the beam-based calculation will bebroadly accurate. The circulation in the top and bottom part of the section y=constant will be however analogous to Fig. 6b, it willbe strongly suppressed and the torsional moment will be approximately defined by the pair of horizontal forces Vx with the leverarm h0. What is important is whether there are transverse diaphragms in the plate and what the distance between them is, as theshear flow is affected in their vicinity. If the diaphragms are distributed close to each other, box-sections are again in fact formedin sections y=constant and their Jky, relating to a unit width, will be decisive for the calculation.

Thin-walled box-sections

They can be approximately analysed as plates with the influence of transverse shear taken account. In this analysis we use input

data based on the comparison with modified formulas of V. Kstek:


34/39


33

(92) ,D2

1D,DD,

a

J

1

EDD 11331112

xa

22211

where Jxa is the moment of inertia of the I-section of width a (Fig. 11) between box axes. It may however vary across elements,i.e. different boxes. The stiffness D22 is not too overestimated as the influence of the web J xa is small. Similarly, D33 is quiteaccurate especially in internal boxes, as the amount of shear flow that gets into the internal thin webs is very small. We may

approximately calculate with a proximate valuea

Jxa = ,th2

1 2 where t is the thickness of horizontal plates and h is their centre-

to-centre distance. For unequal thicknesses this value may be approximately determined from formula (96) presented later.

The shear stiffness D55 in transverse direction y (see the fifth line of the matrix (33)) can be found through the comparison of the

formula Ty = D55 yz with the formula defining a relation between the transverse force

Ty = Va + Vb

in the highlighted I-shape frame of unit width and total skewing ,21yz where 21 are beam deflections of websand flanges. Taking dimension as in Fig. 11 we get

Fig. 11

(93) Vb = ,Va ,

t6

a

t

h

t6

a

t

h

3b

3h

3a

3h


35/39


34

.2

t

h

t

a

1E

aT23h

3a

y

yz

Therefore:

(94) D55 =

.Nm,

2t

h

t

aa2

1E 1

3h

3a

.

The shear modulus G23 (paragraph 5) was not needed for this calculation, but it could be determined for the substitute physicalplate continuum e.g. on the assumption of (34) from the formula G 23 = D55 / h without introducing the conception of a sandwichplate with a soft core.

Stiffness D55 is substantially increased in the location of transverse diaphragms. If they are very rigid and if they provide for non-deformability of the section projection into the vertical plane, then yz can be neglected in elements located in the vicinity of thesediaphragms (see the procedure for D44). This should happen with reasonably designed diaphragms, the web stiffness of which ishigher by a factor of ten in comparison with the previously stated stiffness. The variation of D 55 over elements can be easily takeninto account in FEM. In case of densely located diaphragms, if one diaphragm relates to each finite strip, we can calculate withthe average value of D55, but the effect of shear yz will be small, otherwise the diaphragm would not meet one of its mainpurposes.

The stiffness D44 (4th line of the matrix (33)) is according to (95) larger than D 55 by a factor of ten and the shear changes xz can

be neglected. In the input it can be expressed by the value

.32,10 4455 orDD

It could be also possible to modify the program for plates with the effect of shear in one direction that nullifies xz beforehand.

Processing of printed internal forces:

Mx ....N per one I-section according to Fig. 11we have M = a Mx ,stress x= Mz / J ,

extremes x = M / W .

Tx ....Nm 1 similarly, per one I-section, we have T = aTx , stress xz according to (87), approximately forvery thin webs xz = T / h th only in the web.

....NMy In the top and bottom plate, axial forces in the transverse y-direction will become apparent Ny = My / h and stress yb = -My / h tb, ya = My / h ta.

....NmT 1y In the top and bottom plate, the transverse shear forces V b andVa will occur following from (94),i.e. Va = Ty / (1+), Vb = Va , with identical plate thickness.

ba VV yT2

1 .

The transverse shear stress is approximately:

byzb V .t/V,t/ aayzab

....NMxy In the top and bottom plate, the horizontal shear forces T xy = Mxy / h occur and stress xyb = Txy /tb , xya = Txy / ta .

These values can be used in steel plates to calculate principal stresses that can be used for the assessment of their safety. It issimilar in reinforced concrete structures (thin-walled), where the tension is carried by normal or prestressing reinforcement and

the effect of Mxy is reflected in the design moments, i.e. in the substitution of M x , My by Mxdim, Mydim .


36/39


35

Comparative example

In order to compare the input data D ik, let us discuss an example of a deck slab box-section of the overbridge C 201 in Vsetn,

Czech Republic, made as a prefabricated lamellar structure.

Fig. 12

The division into finite elements was selected to coincide with the plan-view of the boxes, which means that the stiffness ofelements was based on the stiffness of individual boxes. The transverse section is schematically shown in Fig. 12 that would bemore complex due to the varying thickness of webs. The calculation according to paragraph 6.4.1 thick-walled structures (92): Moments of inertia of individual boxes Jxb were found by means of a program for the calculation of geometric characteristicsof planar shapes. Moments Jy can be easily calculated using the formula

(95) ,tt12

1h

tt

ttJ

3b

3a

2

ba

bay

where ba t2

1t

2

1vh is the centre-to-centre distance of horizontal webs.

Vertical webs are of unified thickness 30 cm and their centre-to-centre distance is 300 cm. Further, let us give the approximatedimensions of the box b3 in [cm]: tb = 18, v = 96, ta = 14, h = 112 and the same for box b4: tb = 19.5, v = 101.5, ta = 14, h = 118.

Physical constants E = 385000 daN/cm2 (kp/cm2), =0,15.

The stiffness matrix of box b4 (higher numbers by (92) thick-walled, lower numbers by (93) thin-walled neglecting the verticalwebs):

.10kNm

(Mpm)

509

809

70 316

0448

000

67 100

D = 70 316 431 03

80

67 100 448

000

199

228

0 0 224

000


37/39


36

For box b3 :

475

046

64

7200

389

000

58

300

D = 64720

391887

058

300

389

000

183

374

0 0 194

500

The reduction of the flexural stiffness D11 by neglecting the vertical web is apparent; the stiffnesses D22, D33 will be larger, whichfollows from the thin-walled theory. The differences in moments (multiples of type D11wxx etc.) will be smaller, as the larger D11gives smaller deflection w and also smaller derivative wxx. It is known that moments in isotropic plates do not depend on D at all ifwe consider only the action of forces. This independence is of different nature in orthotropic plates: the moments do not dependon the absolute values of D ik, but on their ratios. The most typical ratio for this relation is

,DD/D2 221133

see (26). Examples of the influence of on moments can be found in [ 1 ], pp. 48., table 2 and in diagrams in Fig. 13, p. 52.

Multi-cell slabs with linear hinges in longitudinal direction

This means perpendicular or oblique plates assembled from prefabricated blocks, in which we assume that no reliable monolithicconnection exists in the transverse direction (e.g. through transverse prestressing, which would cause that they would be treatedas monolithic plates in accordance with the previous paragraphs).

Fig. 13


38/39


37

Longitudinal joints transfer only shear forces Ty and no bending moments My.

There are many types of multi-cell slabs with linear hinges in longitudinal direction. In terms of input data D ik, all of them fallbetween two limit situations (Fig. 12):

a) The vertical webs of the box-sections are so thin that practically no shear flows gets into them and the torsional moment M xy

transfers only the shear in the top and bottom web. Then we may consider the equality kkykxyxxy JJJandMM ,and therefore, following from (63), the torsional stiffness is

(96) Nm,b

GJ

4

1D kb33

if we calculate Jkb [m4] for one prefabricated block of width b [m].

Similarly to box-sections, the formula (92) applies to other stiffnesses.

b) The vertical webs of the sections are so thick that a continuous circulation of shear flow occurs in sections x=constant, which

(considering the theorem of reciprocity yxxy ) influences also the shear flow in sections y = constant. Then, we apply theformula (63) with varying torsional stiffnesses Jkx , Jky [m

3] and constant shear modulus G:

kykx33 JJG8

1D

c) A special situation arises when the longitudinal joints cannot transfer any torsional moment. Then we have J ky = 0 ,

.GJ8

1D kx33

Formulas (62) generally apply to the evaluation of printed Mxy.

If the webs in plates in Fig. 13a are so slim that transverse skewing occurs, the appropriate stiffness D 55 from paragraph 6.3 canbe found on the basis of the formula (95) from paragraph 6.42.

Other plate types

Reinforced concrete plates with different reinforcement Fax, Fay in x- and y- direction behave in the first phase practically likeisotropic plates until cracks form in the tensile part of the concrete. In the second phase, we have to find moments of inertia J x, Jyof the non-homogenous sections of unit width composed of (i) steel and (ii) concrete in compression. The ratio Jx/ Jy isapproximately equal (plus or minus a few per cent) to the ratio F ax/ Fay. As the zone of concrete in compression, where sometransverse contraction (or dilatation) still occurs, is small, the effective value is lower than 0.15. The value 0.02 was measured.The stiffnesses are calculated from the formulas stated earlier

(97) ,DD2

1D,DDD,JED,JED 221133221112y22x11

(98) .1

EE,1,Gh

1DD

25544

The effect of shear is taken into account only in plates whose thickness h L / 5, see paragraph 5.2 and 6.3. Deviations that areof practical significance occur only if h L/3.

Corrugated plates are calculated using the formula (98), and for sinusoidal shape of sections x = constant we take:

(99) ,

l4H5,21

81,01hH

2

1J

22

2x


39/39


,12

h

s

lJ

3

y

where the shape of the corrugation is z = H sin x / l, the thickness of the corrugated plate is h, the amplitude of waves is H, andthe length of the chord of one half-wave is l. This can be used approximately also for non-sinusoidal shape of the waves.Doubled corrugated plates (system BEHLEN, PUMS, etc.) require a special calculation of J x. In extra-thin corrugated plates, Jxcan be calculated on the curve z = f(x) and then multiplied by the thickness h. The ratio Jy / Jx is almost zero.Double-layer braced plates are beam systems used for roofing of extensive areas (stadium, etc.). For the needs of a preliminarycalculation, they can be considered as orthotropic plates, with moments of inertia derived from sectional areas F ax , Fay of the

beams in strips in x- and y- direction and lever arms rx , ry to the centroids of these areas, approximately rx . ry = .H2

1It is

possible to establish the effect of transverse shear similarly to paragraph 6.3. The torsional stiffness is practically zero (D33 = 0, = 0 ) in common right-angled systems of strips without diagonals in horizontal planes; in other systems it must be determined bymeans of comparative calculations. The printed values Mx, My, Tx, Ty can be used for the calculation of axial forces in beams bymeans of the procedure that is common for lattice girders (intersection method). The procedure can be extended to generallyanisotropic plates and is suitable for a preliminary design or for the determination of an optimal variant, etc. After the final designof the system, the accurate assessment can be performed following the calculation of axial forces in beams of the given system.The beams are, depending on the nature of joints, considered as a three-dimensional lattice girder or as a frame.

Orthotropy Theory Enu

Documents