Mathematical theory of stressed skin action in profiled sheeting with various edge conditions

Mathematical theory of stressed skin action in profiled

sheeting with various edge conditions

Citation for published version (APA):

Bogaard, van den, A. W. A. M. J. (1987). Mathematical theory of stressed skin action in profiled sheeting with

various edge conditions. (Bouwstenen; Vol. 5). Technische Universiteit Eindhoven.

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Published: 01/01/1987

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F . M

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7

0

bouwstenen

5

MATHEMATICAL THEORYOF

STRESSED SKIN ACTION IN

PROF I LED SHEETING WITH

·

VARIOUS EDGE CONDITIONS

IR.A.W.A.M.J.v.d.BOGAARD

faculteit

t(iJ

bouwkunde

F

C" .

·

-

~.

3

7

a o s

J

J

'STENEN" is een publikatiereeks van de it Bouwkunde, Technische Universiteit

'en.

;enteert resultaten van onderzoek en aktiviteiten op het vakgebied der

.. ,unde, uitgevoerd in het kader van deze Faculteit.

Kernredaktie Prof.drs. G.A. Bekaart Prof. dr.dipl.ing. H. Fassbinder Prof.ir. J.W.B. Star!<

Prof.dr. H.J.P. Timmermans International Advlsory Board Dr. G. Haaijer PhD

American lnstltute of Steel Constructions, lnc. Chicago, U.S.A.

Prof. ir. N.J. Habraken

Massachusetts lnstitute of Technology Cambridge U.S.A.

Prof. H.Harms

Technische Universität Hamburg-Harburg Hamburg, Duitsland

Pof. dr. G. Helmberg Universität lnnsbruck lnnsbruck, Oostenrijk Prof. dr. H. Hens

Katholieke Universiteit Leuven Leuven , België

Prof. dr. S. van Moos Universität Zürich Zürich, Zwitserland Dr. M. Smets

Katholieke Universiteit Leuven Leuven, België

Prof. ir. D. Vandepitte Rijksuniversiteit Gent Gent, België Prof. dr. F.H.Wittmann Universiteit van Lausanne Lausanne, Zwitserland

Bibliotheek

Technische Universiteit Eindhoven

8802878

Mathematica! theory of stressed skin action in profiled sheeting with various edge conditions

A.W.A.M.J. van den Bogaard

manuscript beëindigd november 1985

uitgegeven: juni 1987

FACULTEIT BOUWKUNUE

BOUWSTëNEN

publikaties van bouwkundig onderzoek, verricht aan de

Faculteit bouwkunde van de Technische Universiteit Eindhoven

publications of building research at the

Faculty of Building and Architrecture of the Eindhoven University of Technology

Uitgave:

Technische Universiteit Eindhoven Faculteit Bouwkunde

Postbus 513 5600 MB Eindhoven

CIP-gegevens Koninklijke Bibliotheek, 's-Gravenhage van den Bogaard, A.W.A.M.J.

Mathematical theory of stressed skin action in profiled sheeting with various edge conditions I A.W.A.M.J. van den Bogaard; (ill. van de auteur).

Eindhoven: Technische Universiteit Eindhoven. -111-(bouwstenen: dl. 5)

Uitgave van de Faculteit Bouwkunde, vakgroep Konstruktief Ontwerpen - met lit. opg.

ISBN: 90-6814-505-3

Trefw.: geprofileerde plaat I schijfwerking Copyright T.U.E. Faculteit Bouwkunde, 1987

Zonder voorafgaande schriftelijke toestemming van de uitgever is verveelvoudiging niet toegestaan.

SUMMARY

In this r-epor-t a gener-al mathematica! basis is pr-esented for- the cal·· culation of defor-mation and str-esses of trapezoidally pr-ofiled sheeting

in shear.

Explicit expr-esslons ar-e given for the calculation of war-ping due to shear-str-ain and cur-vatur-e of foldlines.

Var-ious boundary condit ions a long transverse edges can be treated, such as limited war-ping in sheeting with inter-mediate pur-lins and differ-ent types of attachment to pur-lins, pr-ovided that conditions of antisymmetry are satisfied.

Str-esses can be found in any point.

The analysis is based on two-dimensional theor-y of elasticity. Explicit formulae ar-e given for the two wavelengths, belonging to a profiled sheet with all corr-ugations equally attached. For shearpanels much shorter- or-much longer than wavelengths defor-mation is simply obtained by means of constant factors independent of length and thickness. Thus also str-ess

-distr-ibution in long continuous sheets with inter-mediate pul"lins can be found.

SAMENVATTING

In dit rapport wordt een algemene mathematische basis gepr-esenteerd voor-de ber-ekening van voor-de ver-vor-ming en spanningen in een als schijf belaste tr-apeziumvormig gepr-ofileerde beplating.

Er- wor-den expliciete uitdr-ukkingen gegeven voor de ber-ekening van welving als gevolg van zowel schuifrek als kr-omming van de vouwlijnen.

Verschillende r-andvoorwaar-den aan de dwarsranden kunnen worden verwerkt, zoals beperkte welving in beplating met tussengelegen gordingen en verschillende typen plaat-gording-verbindingen, vooropgesteld dat er wordt voldaan aan de voorwaarden voor keersymmetrie.

Spanningen kunnen worden bepaald in elk punt. De analyse is gebaseerd op de tweedimensionale elasticiteitstheorie. Er worden expliciete formules gegeven voor de twee golflengten, die behoren bij een in elk golfdal identiek bevestigde geprofileerde plaat. Voor- schuifpanelen die hetzij korter hetzij langer dan beide golflengten zijn, wordt de vervorming eenvoudig verkregen met behulp van konstante faktoren, onafhankelijk van plaatlengte en -dikte.

CONTENTS

1. LIS'!' OF SYMBOLS 2. PREFACE

3. PRIOR WORK

4. PRESENT WORK AND SCOPE 5. INTRODUCTION

6. DEFINIT !ONS

7. SHEETING CONTINUOUSLY ATTACHED TO PURLINS 7 .1. General remarks

7 .2. Profiled sheet as or thotropic membrane 7.3. Straightness of foldlines

7.4. Shearpanel without purl ins

7.5. Shearpanel with inextensional purlins

2 4 8 13 22 27 32 32 32 35 36 38

7.6. In-plane curvature of basic shearpanel 39

7.7. Excentricity of global farces 41

7.8. Shear deflection of panel without attachments to purlins 43

8. SHEE'l'ING DISCRETELY ATTACHED TO PURLINS 8.1. General remarks

8.2. Assumptions in constitutive equations

8.3. Equilibrium and compatibility along foldlines 8.4. Other interesting vector equations

8.5. Boundary conditions

8.6. Shear deflection of basic shearpanel

9. APPLICATION OF THEORY 47 47 50 50 56 58 63 64 9.1. Cantilever diaphragm beam. transverse edge built-in 64 9.2. Cantilever diaphragm beam, longitudinal edge supported 69

9.3. various bounda~y conditions 74

9.4. Shear diaphragm with tntermediate purlins 76

10. NUMERICAL EXAMPLE 81

ll. REFKHENCES 90

1. LIST OF SYMBOLS 1.1 Indices 0 w u xy

planar, refering to topflarige planar, refering to web

planar, refering to bottomflange

acting parallel to y-axis in cross-section x-axis

function of x' or y' in-planar system function of x' and y' in-planar system

normal to

1.2 variabie numbers and functions A,B,C,D,E,F,G,H,K,L,M A' ,B' ,C' constants of integration constants of integration variabie functions of x' or y' variabie numbers a' ,b' ,c' ,d' ,e' ,f' ,g• ,h' i. j

IJ. finite difference; mathematica! operator

1.3 Material constants and properties of shearpanel

D E G As,Ar Is,Ir Es,Er a,b bd bo,bu 2-w•bw eo,eu

flexural rigidity of sheet

modulus of elasticity in tension and compression modulus of elasticity in shear

cross sectional area of sheet, purlin resp. second moment of area of sheet, purlin resp.

effective modulus of elasticity of sheet, purlin resp length and width of shearpanel

pitch of corrugation

width of topflange, bottomflange resp. width, projected width of web resp

distance between plane of gravity and topflange, bottomflange resp.

height of corrugation thickness of sheet

smallest and largest wavelength respectively arclength of one corrugation divided by pitch ratio between h, t, u

angle between flanges and webs Poisson's ratio

width of topflange, bottomflange resp divided by pitch two times width of web divided by pitch

non-linear combination of

à

0,aw,au

1.4 Forces and displacements

F Airy's stressfunction

L longitudinal force, acting in and parallel to foldline N normalforce

V shearforce; strainenergy M bending moment or couple

T transverse force, acting in strip perpendicular to foldlines n membrane force

m unit bending moment in plate

p membrane force along and perpendicular to foldline s shearforce in membrane

t transverse shearforce in plate

u in-plane displacement perpendicular to foldline v in-plane displacement parallel to foldline w out-of-plane displacement

c normal strain y shearstrain

K" curvature

't shearstress

~ rotation with respect to foldlines

1.5 Matrices and vector functions

K shapematrix for different width of flanges

J in-plane rigidity matrix, out-of-plane flexibility matrix H geometry matrix of corrugation

L quotientmatrix of J and K

m

vector function for in-plane couples in strips

st vector function for in·-plane shearforces in strips E vector function for planar membraneforces along and

perpendicular to foldlines

! vector function for planar transverse shearforces along and perpendlcular to foldlines

~ vector function for planar in-plane displacements perpendicular to foldlines

~ vector function for planar out-of-plane displacements along foldlines

2. PREFACE

Trapezoidally corrugated sheets are frequently applied in practice as structural elements in roof sheeting, floor decking or wall cladding. In order to achleve an acceptable flexural rigidity and load bearing capacity flat sheet is corrugated and for manufacturing reasens corrugation is con-stant. In practice spans in order of 10 m are possible. Generally profiled sheets are used in combination with purlins and rafters, thus forming a compos it e p l ane .

Connected to rigid walls or end gables these composite planes may well be used for stabilising portal frames. Sufficient rigidity against tn-p lane deformation must be present for that purpose.

figure 1: roof sheeting used as diaphragm beam

Profiled sheets are also perfectly suited for the creation of spatlal curved planes; if applied in an appropriate way then lateral loads will be carried mainly by in-plane forces. A basic use of sheeting in this sense has been made in the design of a double curved, suspended shellroof for an indoor skating center [19].

figure 2: stressed-skin-design of corrugated sheeting in the form of a double curved, suspended shellroof for an indoor

skating center.

A shallow shell of 4.50 m depth is suspended to a large rectangular edge frame of 54.00 m x 86.40 m.

The edge frame is made of beams with I-shaped cross sectien and placed on top of 26 supporting frames, 10.80 m from each other. Besides the roof, the supporting frames also carry cladding of inclined walls and, together with smaller intermediate frames, the concrete steps of the galleries. The structural roof is made of cold formed arches with radius of curva-ture 81.00 m. spanning 54.00 m in transverse direction, and trapezoidally corrugated sheets, spanning 86.40 m in longitudinal direction. Arches are suspended 1. 80 m from each other to the edge frame and consist of 5 curved rafters each, with z - cross sectien of 300 rnrn height and 3.5 rnrn thickness.

_j:_ _{- Jj}

-i

;

_{_ d ... o•sdoor sn•d•

u·· ....y

roof drain leaders

longsdoorsnt-dP I ~~ ; ,, i

86.400 m

il:

_

Cl _

figure 3: cross sections of skating center and detail of the structural roof

OI

w

Sheeting is 40 mm high and l mm thick and is attached to rafters through the wide flanges. The average weight of the structural roof is lï5 N/m2; including the edge frame, it is 275 N/m2.

The most important types of lateral loading are continuously distributed loads, both synunetrical (such as dead loads and snow) and antisymmetrical (such as simultaneous windpressure and windsuclc over the roof). In case of symmetr ical loading the suspended arches and, of less amount, sheeting carry lateral loads by means of tensile forces. Thus permanent loads guarantee tension to a certain extent. In case of antisymmetrical loading the middle line over the roof tends to deEleet in a horizontal way. In this case sheeting has the function of a diaphragm beam.

Especially the latter has urged the need of more understanding of previously described composite planes.

The combination of in· plane forces and deformations is well expressed by the term "stressed-slcin-action".

3. PRIOR WORK

During the 1960's several analyses of trapezoidally corrugated sheets in shear have been publisbed [1-7]. During the next decade application of corrugated sheeting as shear diaphragms was allowed in Germany and the

Netherlands 1f sufficient in-plane strength and rigidity could be proved

by means of a registered certificate of the product. With respect to flexibility of sheeting in shear the certificate provided an effective shearmodulus G*, which was basedon theoretica! results of a strongly

sim-plified deformation model by Steinhardt and Einsfeld (8], publisbed in

1970 .

The first general theory for stressed skin design of steel buildings was given by Bryan in 1973 [11]. I t was based on a simple model for shear deformation, that was supported by testresults from the preceding decade. In this work the total deformation of a shear diaphragm is dividecl in parts, representing shear deformation of the folded sheet, extension and compression of purlins and slip of fasteners. Strain energy methods are used in order to determine flexibility components for the separate aspects. In both simplified models the most radical assumption, regarding defor-mation of a single shearpanel, is that foldlines remain straight and un-strained, though they are permitted to move as rigid bodies. Another im-portant assumption. regarding the deformation of a diaphragm with inter-mediate purlins, concerns the distribution over several purlins of normal-Eorees due to the in-plane bending moment: a linear distribut ion is as-sumed where the edge purlins supply the greatest normalfarces in order to

resist in-plane curvature. Nyberg showed in 1976 [15], that extreme

forces do not necessarily occur in edge purlins, but may occur in adjacent purlins.

Furthermore Bryan's theory is restricted to shapes of corrugation, where webs are perpendicular to flanges. Flexural rigidity of the sheet is taken Et 3 I 12, whereas longitudinal bending and twist ing moments are neglected. Prevention of downward movement of webs by purlins is not taken into con-sideration (see figure 5).

However, the assumption of straightness of foldlines was felt to have the most serious influence on accuracy of the theoretica! approach, since this might only be acceptable for corrugations of limited length.

figure 4: topview of panel with

a) straight foldlines

b) curved foldlines

+ -

-

-

·

----

+

f

figure 5: sideview of panel with movement of webs:

Libove presented in 1971 [9] a more appropriate model. that also in-cludes the effect of prevented downward movement of webs by purlins (see figure 5). Foldlines are free to curve. thus yielding warpof the cross section, which is non-linear over the length of the shearpanel (see figure 4). In-plane displacementsof flanges and webs perpendicular to the origi-nal direction of foldlines are taken as hyperbolle functions.

Minimisation of total potentlal energy yields the parallel shift of fold

-lines with respect toeach other.

The share of shearstrains in the tot al shear deformat ion is re_pres~nted by a ratio Q , which is plotted as a function of the sheet thickness and the panel length for a variety of shapes of corrugation. Originally only flanges of equal width were considered, but in 1973 [12] tables are presented for more accurate calculation of Q and a wider variety of shapes.

The effective shearmodulus Geff is only a fraction Q of the shear

-rlgidity G/a of a corrugated sheet with rigid cross section. Libove also mentiones warp caused by shear strains. but no further attention is paid to this phenomenon.

In 1971 Horne and Raslan proposed two solutions of the problem [10]. The first one is an energy solution, where in-plane displacement functions of flanges and webs are taken in the form of one linear term and one term of Fourrierseries. For a number of given diaphragms shear deEleetion is computed by mlnimisation of total strain energy. The alternative is an equilibrium solution yielding three simultaneous differentlal equations of the fourth order. Since no exact solution is available a numerical method with finite diEferences is used in computing shear deEleetion of a number of panels. Results according to both methods show, that the latter is a better approximation in the case of long shearpanels, where distortion of the corrugations is concentrated near the edges.

Davles continued the work of Bryan and presented in 1975 [13] a modi·

-fied flexibility component with respect to distortion of corrugations. An energymethod is used where displacement functions are taken in the form of one linear termand three terms of a Fourrierseries.

The size effect of long corrugations leads to a modification of th~ first flexibility component proposed by Bryan. It also includes the restraint by purlins preventing downward movement of webs. Although the energy solution includes Bryan' s resul t for short length the modified flexibility compo-

11

A complete equilibrium solution was presented by Schardt and Strehl [14] in 1976. In fact it is a continuatien of an earlier published study of folded plate structures [4], where in this case the number of foldlines per pitch is restricted to four. conditions of equilibrium and compatibility along foldlines lead to two simultaneous differentlal equations of the fourth order. Solving the problem of eigenvalues two different wavelengths are found, giving the distance where the edge conditions have influence. Characteristic displacement functions prove to be two periodic functions with hyperbolle amplitude. By virtue of this work, the German certificate mentioned before was revised by the end of the 1970's.

In an energy solution the choice of displacement functions is cruelal to the process. In order to improve earlier results Davles produced in 1982 a report [18] with a general solution for the problem in question, which is based on energy methods, where displacement functions are taken in the form of a linear function and two periodic functions with hyperbolle amplitude. The number of foldlines per pitch is unlimited and results are obtained for trapezoidally corrugated sheets with flange- and

webstiffeners as well as are and tangent profiles. Results of Oavies and schardt/Strehl are very much alike.

THI!ORY

- Principle: energy methad energy methad energy methad energy method equlllbrlum meth. energy methad equlllbrlum meth.

- displacement func- llnear llnear hyperbolle llnear • three perlodlc wlth periodlc hyperb. perlodlc wlth

tions for distors ton: Fourler terms hyperbolle ampl. • llnear part hyperbolle ampl.

- num.her of

corruga-tlons/pltc h: I 1,2,3,4 I I 1 any number

- tntermediate purll.ns no ye• no no no no ye•

- consldered aspects

of rlgldlty flexural rlgldlty flexural rlgldlty; flexural, torslonat flexural and flexural. torslonat flexural,torsional extenslonal of sheetlng; slip of fastenerst and warplng warping rlgldlty and warping and warping rlgldlty of purlJns; shear rlgldlty shear rlgldlty rlgldlty of sheet; of sheetlngi regidlty of sheet; regidlty of sheet; flexurai and warping

shear rlgldlty shear rlgldlty shear rlgldlty shear rlgldlty rigidlty of sheetlng;

shear rlgldlty CORRUGAT!ON

-shape of cross flanges of rectangular trapezoidally; trapezoidally trapezoidally trapezoldally; trapezoidally

sectlon eq.al wldth are and tangent are and tangent

- length of corru- small small large large any length any length any length gatlans

- longltudlnal

sUffeners no no no no no yes no ...

N

BOUNDA!!Y COND!T!ONS

- Introduetion of bottorn flanges bot tomflanges bottomflanges; bottornHang es bottomflanges; bottornflanges bottomflanges;

transverse forèes topflanges topflanges;webs topflanges;webs

- vertical movement free free free;prevented; free; prevented free; prevented; free; prevented free; prevented;

of webs prev.downward downward prev.downward downward

- warping restralnt no no no no no no ye•

RBSULTS

- global results shear deflectlon shear deflection; shear deflectlon shear deneetion shear deflection shear deflection shear deflectlon,

tn-plane curvature; tn-plari.e curvature,

purlln farces purlln farces;

-detaUed results model wlth spring!

wavelengthst waveleng:ths; wavelength.s;

stresses and streues and

stralns stralns

4. PRESENT WORK AND SCOPE

All previously given reEerences have in common, that results are obtained for a single shearpanel only, where no reactive forces are possible paral-lel to foldlines at the ends of corrugations, which means that warping is free along both transverseedges (see fig. 4).

This is a severe restrietion in the case, that long sheets continue over several intermediate purlins. Substantial longitudinal stresses may arise in a transverse cross section, where the average shear force suddenly changes.

Present theory provides a general solution for various boundary conditions along transverse edges, taking account for several types of attachments to purlins (topflange, bottomflange or webs in any combination) together with free or limited warp of the cross section (free ends, continuous sheeting, built-in edges). A comparison with priorworkis given in table 1.

V

a) free movement of webs b) vertical movement of webs prevented figure 6: several types of attachments to purlins

c) prevented distortion

a) free warping b) limited warping along intermediate purlin

rt

-

I

,,

I

.

-I

-

_I

-u

c) prevented warping

figure 7: warping and warping restraint along a transverse edge of a corrugated sheet

Present work is still restricted to cases, where all corrugations are attached to purlins in the same way, and where vertical movement of webs is antisymmetrical (see figure 6). However theory can be expanded to practical cases where only downward movement of webs is prevented by purlins (see figure 8).

a) free vertical movement of webs b, (antisymm.)

b) upward movement of webs b, (symmetr.)

c) zero downward movement of webs

figure 8: prevented downward movement of webs divided into an antisymmetrical and a symmetrical part.

Chapter l starts with a shearpanel. subjected to in-plane forces along longitudinal edges only. Expresslons are given for the distribution of forces to sheeting and purlins, analogue to the theory of slender beams with non-homogeneous cross section. Also solutions are given for in-plane cuevature and extension of the entire panel as well as sheardeformation in case that sheeting is attached to purlins continuously.

In this chapter foldlines are assumed to remain straight and warping of the cross section is merely the result of parallel shift of foldlines, due to shear strains.

Chapter 8 continues with a shearpanel, where sheeting is discretely attached to purlins and only subjected to a uniform shearforce. From equations for behaviour on planar level (individual strips), showing an exact conformity between membrane-action and plate-action, equations are derived for in-plane displacements of separate strips. conditions of equilibrium and compatibility along foldlines yield a double wavelength problem. Foldlines no longer remain straight nor unstrained and a second mode of warping occurs as a result of in-plane cuevature of each

individual strip.

In-·plane cuevature of strips can be described by a combination of two periodic functions with hyperbolle amplitude. Explicit expresslons are derived for the two wavelengths (i.e. length of a half period) of a profiled sheet with attachments in every corrugation.

1

The wavelengths are proportional to ~tand they are important parameters in estimating the area influenced by boundary conditions.

\ \ +€ -ny/9.

'

'

' '

_{i..;' ....}

::~ve

leng

th

/

wave ,fength

figure 9: periodic function with hyperbolle amplitude

In the case of short shearpanels ( length of corrugations smaller than smallest wavelength) foldlines almest remain straight. If the distance between two successive purlins is sufficient ly large ( larger than the greatest wavelength), the effects of loc.al introduetion of forces do not interfere with each ether.

The force-displacement-ratio of each purlin then becomes constant, as if

purlins were connected to a flat sheet by springs. NUmerical values of spring constants {iepend upon the shape of corrugations and the nature of boundary conditions: sheeting may be simply supported by o-r continuous over a purlin, vertical 1110vement of webs aud warping of the cross section may be free or prevented.

1'he original conformity of equations on planar level is maintained

throughout and this explains why only one matrix J is needed for many

express i ons; i t is used as a st iffness matrix for in-p lane curvature and

as a flexibility matrix for lateral displacements of the strips. Symmetry

of matrices J and K in the wavelength problem reflects Maxwell's

reelprocal theorem.

A single shearpanel, shown in figure 10, that is the basis of all given references, is used in chapter 9 as an example for application of present theory.

Shear rigidity of the corrugations against distortion can be expressed by:

(1)

where Geff is a function of the dimensions and the modulus of elasticity

of the folded sheet.

1ta

For panels with length of corrugations b smaller than the smallest wave length a suitable expression proves to be:

where gs is an asymptot ie funct ion that becomes equal to a constant

of

-1

c

_s decreasing length of corrugations (see

(2)

the dimensions of a corrugation, (independent of b and t) with figure ll). Cs is a flexibility component for short corrugations of given shape (see figures 12 and 13). If Poissons ratio 1> is taken zero, Bryan's solution is found; if all flanges are of equal width also the solution of Steinhardt/Einsfeld is in accordance with this result.

f"or panels with length of corrugations b larger than the 1argest wave length a more suitab1e expression proves to be:

(3)

where g!l. is an asymptotic function of the dimensions of a corrugation,

-1

that becomes equal to a constant cl!. (independent of b and t) with increasing 1ength of corrugations (see figure 11). cl!. is a flexibil ity component for long corrugations of given shape (see figures 12 and 13). Here present theory is in accordance with so1utions proposed by Libove and Schardt/Streh1.

~

19 bd/4

.J

panel:

b

corrugation:

bd/2 I

--

_{~bd/4, bd/2}

a

" '1 _bd ~· >t V

=

0.3 -1 0.25

r-____",:::;:---=:::::::::::~~==--==-:::...~l

__

___

___

c

f

STEINHARDT /EINSFELD s BRYAN - - -

-0.20 {

I

t

0.15l 0.10

1

i

I

0. 05

t

0.3

t02

g

Q.

0.1 figure 11: /h/t = 20 lh/t

=

10 /h/t = 5

-+-2 3 4 5 6 7 _{b 8 9}

tJ.o

11 12 13 14 15 TT bd -1 - - - -~ - - - CQ. LIBOVE-20 SCHAROT /ST RE HL V.D. BOGAARD 1 2 3 4 5 6 7 b 8 9

p;o

11 12 13 14 15 TT bd rigidity functions g

5 and g~ for rectangular corrugations

t

c

_s

t

c,Q,

corrugation:

]

h

=

_bi2

I

bo

tiJ.J

"

bd

"

>

"

"

u ' 9 8

STEINHARDT/EINSFELD

7 6 5 4 3 2 8 7 6 5 4 3 2 1 0.0 0.1 0.2 0.3 b 0 bd

BRYAN

SCHARDT/STREHL}

V.D. BOGAARD

(

v

=

0)

t

0.5 0.6 0.7 0.8 0.9 1.0

p

LIBOVE

SCHARDT/STREHL

V.D. BOGAARD

prevented

downw rd

movement

of web

0. 0.1 0.2 0.3 b 0 0.5 0.6 0.7 0.8 0.9 1.0

Dd

:p.

figure 12: flexibility constants es and ei fór rectangular

t

Cs

t

figure 13:

b =b

0

corrugation:

>!'

bd

8 7 6 5 4

i

3 ..,.. I

I

2 8 7 6 5 4 3 2

u

_nv

211

h -

_bd/2

ö

2-~,u~' A'

STEINHARDT/EINSFELD

SCHAROT/STREHLl(

_ O)

V.O. BOGAARD

f

v

-0.4 0.5 0.6 0.7 0.8 0.9 1.0 bo

bd

LIBOVE

SC

HAROT/

STR

EHL

V

.

O.

BOGAARD

t

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 bo

p

lid

flexibility constants es and c~ for corrugations with

5. INTRODUCTION

A state of in-plane forces in roof- or wall structures as a result of in-·

plane deformation is described as stressed-skin-action. Since profiled sheeting in that situation only resists shear deformation, in this context also is spoken of shear diaphragms or shearpanels; actually a shearpanel is only a part of the total structure bounded by supporting members. Normal forces, perpendicular to foldlines, are taken by supporting purlins. Deformation of the total structure and distribution of forces to several panels and members is different for each building, for they depend on the configuration of shearpanels, loading combinations. boundary conditions etc. The smallest characteristic element of a structure. that contains all essential properties. is defined as a single shearpanel.

lH

I

TI

I:

i

I

I

i

i

I

]

!

11 Jlll

!

i IJ ' I I

il

I

i

:

i

i I I

i

i

i

I

I

figure 14: roof structure and diaphragm beam, each consisting of four shearpanels.

As a starting point for present work a shearpanel will serve, that con

-sists of a trapezoidally corrugated sheet, continuously fastened a long longitudinal edges (parallel to foldlines of the sheet) to inextensional rafters and attached along both transverse edges to purlins, as shown in figures 15 and 16.

figure 15: basic shearpanel in practice a

+

-

-

. t

--

!

--

.

-

.

--

1-b I

+

I

--

..

--

.

--

.

--

.

i_

--;J-v-~

·

­

f=--~+

' '

figure 16: basic shearpanel schematical1y

(top view and sideview)

a

shear

connector

This shearpanel will be subjected to a system of external in-·plane forces along the edges only, as shown below.

a a

b M

s.a

figure 17: possible external in-plane forces

In-plane deformation of a shearpanel, indicated as behaviour on global level, is devided in two parts: in-plane curvature of the entire panel, as a result of extension and compression of purlins and sheet perpendicular to foldlines, and shear deEleetion of the corrugated sheet, yielding a parallel shift of edge merobers with respect toeach other.

In-·plane curvature of the entire panel is caused by an in--plane bending moment, that is distributed partly to purlins and partly to sheeting. The ratio of both parts depends upon the extensional rigidities of purlins and sheeting respectively. For this aspect of deformation the panel may be treated like a slender beam with non--homogeneous cross section, as if sheeting were attached to purlins continuously (see figure 18).

A shearforce parallel to foldlines will be taken by sheeting only. One part of the shearforce, corresponding to the forces transmitted by faste

-ners from sheeting to purlins, is uniform over the panel from a global point of view; the other part correspondends to the in-plane bending moment taken by sheeting, which is related to in-plane curvature of the panel.

r

~

;-a)

;-~

I I

Mx

~ - __ _r--i _f--_::J- --=·~

~

~····(T-·

r·

J

·

__

y

+

figure 18: deEleetion of shearpanel due to a) in-plane curvature b) shear deformation I I I

:

-.___ I

I _y __ I I I

'

'

I 1 -I T

V

x

+

1-

- ·

-

·

--·

---~ IV U1

Shear deEleetion by the latter may be n.eglected, since its contributton to displacements will be small in comparison to in-plane curvature.

Shear deflection. caused by the uniform part of the total shearforce, is independant of the fact whether there is in-plane curvature or not and may not be neglected. In the first place a parallel shift of foldlines, caused by shear strains.

and can be calculated,

continuously.

is responsible for this aspect of deformation

as if sheeting were attached to purlins

Near purlins however the uniform shearforce will be concentrated dis-cretely around fasteners and from a planar point of view (flanges and webs individually) local in-plane bending will be the result.

From a global point of view this will be seen as distortien of corru-gations, which is responsible for sheardeflection in the second place.

This local effect may even cover the entire span of corrugations in ex-·

treme cases and then this kind of shear deformation is in fact the same as in-plane curvature of the corrugated sheet alone.

27

6. DEFINITIONS

6.1 Shape of corrugations

A profiled sheet in this report is meant to be a regularly folded plate of tiny thickness and parallel flanges, that can be described by the following five parameters:

' '

b

0

figure 19: cross-section of a corrugation

t plate thickness h height of corrugation bd pitch of corrugation bo width of topflange b _u width of bottomflange I I I h

From these data other relevant parameters can be derived in the following manner:

(b 2 h - b ) e angle between webs and flanges

arctan _{- b} :

d 0 u

h I sine !1. width of web w

b - !:> - b

d 0 u

b projected width of web

28

-b

+

'd _'I ~

tb

~

w

~ 1 1

figure 20: variabie angles between webs and flanges

6.2 Dimensionless shapefactors

+

A number of shape factors, that will be used frequent ly for short notation, is denoted in the following manner:

b _Q a bd 0 Tlo a a _{+ a a}2 _{- y,} aaa 0 wo 0 w u 2 2. w bd a w b a a t 2 - Y. Tl u _u a a w u aaa 0 w u

e

a a _{+ a a} 2 + Y. aaa 0 0 wo 0 w u _..!:! a bd u +

e

3 + a a 2 + y, a aaa u u w u 0 w u b + 2 2. + b 0 w u bd a Notice T]o - ~

=

8o -- 8u a 0 + ~ = a - aw 1 - awcosS

29

6.3 Geometrical planes and coordinate systems

In order to describe displacements and forces of a profiled sheet, followi.ng concepts and notations are introduced:

global system with coordinate axes x-y-z:

y-axis is parallel to foldlines, x-axis is paralle 1 to flanges and perpendicular to y-axis, z-axis is perpendicular to both other coordinate axes 1.n such a way, that a right-handed system is formed.

planar systems with coordinate axes x'-y'-z':

every flange and web has its own planar system in such a way, that y'-axis is parallel to foldlines (and also to y-axis), z'-axis is perpendicular to the relevant surface and x'-axis is perpendicular to both other coordinate axes, forming a right-·

handed system.

plane of reEerenee is the plane formed by x- and y-axes:

for the time being the origin of the global system will not be fixed and therefore the plane of reEerenee may be at any distance

of the profiled sheet.

plane of gravity is the collection of lines of gravity in cross

sections perpendicular to foldlines; on that account this plane

is parallel to the plane of reference.

y

z

figure 21: global planes and coordinate system

plane of

gravi ty

plane of topflange

plane of web

z'

figure 22: planar planes and coordinate systems

6.4 Conditions for stressed-skin-action

Stressed-skin-action of profiled sheeting will be defined as the be-haviour of a trapezoidally profiled sheet, of which the original cur-vature of the plane of gravity does not change in a state of stress or strain. This definition implies, that all planar forces and ben-ding moments can be substituted by global forces alone, acting in the plane of gravity. The position of the plane of gravity can readily be expressed by the shapefactors of the corrugation:

h

l"rom the cross-sectional moment of area with respect to flanges 1t can be concluded: e ~ + b a + y, a _2. w u u w ₍₄₎ h _bo + 2 ~ _w + b a u eu b 0 + ~ w a 0 + y,

_"w

(5) h b + 2 ~ + b a 0 w u

However, the definition of stressed-skin-action will be interpreted in a less rigorous way by stating, that the average curvature of the plane of gravity, perpendicular to foldlines, between the parallel edges of a shearpanel must be zero (see figure 24).

Parallel edges remain in-plane dur1ng stressed-skin-action.

7. SHEETING CONTINUOUSLY ATTACHED TO PURLINS

7.1 General remarks

Two modes of in-plane deformation. mentioned already in the introduc-tion, will be treated in this chapter, namely in-plane curvature of a shearpanel and shear deEleetion due to shear strains. Bath aspects concern global deformation of the entire panel caused by a global distribution of farces, where continuous boundary conditions along transverse edges may be assumed. The influence of actual discontinuity will be treated separately in the next chapter.

In this chapter a trapezoidally corrugated sheet is treated as an elastic continuum, although it consists actually of rnany discrete strips connected to eachother in a regular way; constitutive proper-ties of corrugations are smoothly distributed over the entire plane, leaving a combination of an orthotropic membrane and an orthotropic plate. I t is assumed implicitly, that stress and displacement functions, which are discontinuous in the direction perpendicular to foldlines, may be equalised to continuous functions over thc entire panel. This simplification is only justified i f global farces and displacements gradually decrease or increase, hence for continuous boundary conditions only.

7.2 Profiled sheet as orthotropic membrane

The smallest characteristic element in a profiled sheet of a single

shearpanel with all distinctive properties has dimensions bd

perpendicular to foldlines and dy pärallel to foldlines; thus the dimension is finite in x-direction and infinitely small in y·-direction.

... /

figure 25: characteristic profiled element

lt will be assumed now, that average changes ~ of global forces or

dis-placements in x-direction over one pitch are small compared to actual

forces or displacements; therefore the finite diEferenee quotiënt

~/bd may be replaced by the differentlal quotient a/ax.

Variatiens of in-plane forces or displacements yielding a global

resultant over one pitch equal to zero will be considered to be

disturbances and treated in the next chapter.

In order to satisfy internal equilibrium it is sufficient to satisfy

equilibrium of global resultant forces. Hence equilibrium equations of a flat membrane are obtained.

n dy

XX

tm

an ___g + ~ 0 (6) bd

ay

an tl.n __.I:l_ + ___El_ 0 (7) ay _bd n n (8) xy yx where: ~ a bd a x

However, it should be noted clearly, that in every web also a vertical component

constant

of the shearforce exists shearforce can only exist

by 1E a at magnitude of n h. yx A

both transverse edges equilibrium is satisfied for each individual strip, in horizontal as well as in vertical direction. If in few strips per corrugation these conditions cannot be satisfied, substantial disturbances will arise; next chapter deals with this problem.

In appendix Bl the elastic properties of a characteristic element have been determined, yielding following constitutive equations:

tl.u _<I> a.n _{___g}

_{- \) ...n}

n

(9) bd Et a. Et

av

n

_...n

n ~ ₍₁₀₎

ay

a. Et

-

\) a. Et au tl.v n

ay

+ _bd

a.~

(11) where: ~ Q_ bd a x (!:!.) 2 (1-\) 2 ) a. + a. a. - a. 2 <I> {1 + 2( ...2...__) - 3(~) (12) t _{a. a.}

In the case of constant or slightly varying global forces in the plane of gravity a corrugated sheet can be treated as an orthotropic membrane with elastic properties given above.

35 7.3 Straightnessof foldlines

In absence of forces n parallel to foldlines, deformation of yy

profiled sheeting perpendicular to foldlines depends entirely upon forces nxx· Consequentlal large diEferences in displacements of the flanges arise, mainly because of plate bending of separate strips rather than normal strain. The ratio of these two flexibility components is at least ~.

Even for a low and comparatively thick sheet as given in figure 27 the ratio ~ is well over 1000.

\

I

'<: )) a 0

<1w

a u a b 0

bd

0.6503 0.4564 0. 2186 1.3253 figure 27: thick and low profile

119

mm

183

mm

+

\

1. 25

mm

I

h "'bu ))

40

~m /1 h/t 32 u 0.3 ~ 1857

For this reason it is allowed to imagine the strips as rigid bodies mutually connected by non-rigid springs as presented in figure 28.

\~/

\~!

figure 28: strips as rigid bodies connected by springs

Shear deformation of profiled sheeting, due to a constant shear force n

xy is merely the result of shear strain in separate strips without any plate bending. consequently a cross section does not change of shape perpendicular to foldlines, but parallel. on the contrary. it does.

The result is a parallel shift of foldlines with respect to each other, giving warp of the cross section.

"'J

x

~

z

/ 7

I

I

I

l

/

(

I

j

I

f

·

t

4Y

-

·

-

·

·

-top view of unfolded sheet; -top view and cross sections of folded sheet

figure 29: shear strain yielding warp of profiled sheet

If warp of the cross section is prevented, a certa1.n restraint will be obtained at the cost of disturbing longitudinal forces in combination with plate bending of the strips; this is another disturbing aspect that will be treated in the following chapter.

Now for both modes of deformation treated in this chapter the assumption is justified, that foldlines remain straight and unstrained during stressed-skin-action.

7.4 shear panel without purlins

The shear panel, shown in figure 30, can beregardedas an extreme example of the basic shear panel, mentioned in the introduction.

~

I

·

-~

p

I

r.

N

w

_.

_M

Î

figure 30: shear panel with free transverse edges

External forces are applied to longitudinal edges in the plane of gravity.

Both transverse edges are free and deEleetion of the panel depends entirely upon extension and compression perpendicular to foldlines.

This fact and the assumption, that foldlines remain straight, show exact resemblance to the principle of Bernouilli in the theory of slender beams. This analogy has been worked out in detail in appendix B2.

For any cross sectien parallel to foldlines the global forces n

XX and

internal that:

where

can be substituted by an internal normal force Nx' an shear force Vx and an internal couple Mx in such a way

n _g t n __ !Y_ t y + M .y _x _ Is ]_ vx {1 (l,y )2} 2 A ·- b s tb and Is

N /A

x

s

+-+

nxx -t -M /I x s V /A x s (13) (14) (15) b

Using a modified modulus of elasticity Es for the profiled sheet the internal forces of a cross section can be expressed by a global unit elongation ex and in-plane cuevature Kx of that cross section:

M a2v _JL I(" E I x _{a x} 2 s s (16) V dK a3v _ _JL _ __! E I dx _{a x} 3 s s (17} N au a2v _ __!__ ( a x + y

a?

E A x s s (18)

where: E _I]; and au + ~ 0

s a<l>

äy

a x (19)

7.5 Shear panel with inextensional purlins

The shear panel. shown in figure 32. is also an eX'treme example of the basic shear· panel.

b {. /

inextènsional purlin

v

ll

. -·

-~

-111

:u:

I I:E:

i

-

·

=

·

tJ-tt=

·

-a - t ---~

39

Extension of the panel is prevented by inextensional members.

Distortion of the corrugations along transverse edges is prevented by purlins, but warping is still possible, since flexural rigidity of purlins may be neglected.

Deformation of this panel exists only by a parallel shift of

foldlines. caused by a uniform shear force s:

where: s

=

V/b

<lS

Gt (20)

(21)

Detailed stress- and displacement functions are given in appendix B3.

7.6 In-plane curvature of basic shear panel

A shear panel continuously connected to extensional members along

transverse edges. as shown below, can be regarded as a combination of both previous examples.

i

y

I

L ~ ~ ~

-IT

I

b

_-

_·

-rx

M

I

+

p

-~

-a

figure 33: sheeting continuously attached to purlins

External forces are applied along a longitudinal edge in the plane of

The total panel may curve in-plane to a certain extent, meanwhile purlins and sheet together taking account for the in-plane bending moment Mx. Simultaneously a shear force s exists between transverse edge merobers and sheet, which may be considerable and causes shear deformation. The distribution of forces to sheet and edge merobers is determined in appendix B4.

Denoting properties of transverse edge merobers by:

A : area of cross section of one member. r

I _{r. second moment of area of one member,} .

E • _{r. modulus of elasticity of edge members.}

internal forces of a cross section over the panel parallel to foldlines are: M .EEI a 2 v EEI IC _{- ax} 2 x x .EEA

au

a 2 v EEA N _x c _{x <ax} + y ax2 ) diC V sb + ____.! (E I + 2 E I ) x dx s s r r where: EEI E I + 2E I + 2E A b2/4 s s r r r r EEA E A + 2E A s s r r

figure 34: global forces over a parallel cross section

( 7.2) (23) (24) (25) (26) n xy - t -b

One part of the total shear force Vx is distrl.buted parabolically over the cross se ct ion and causes shear de format ion which may be neglected in comparison to in-plane curvature.

The second part of vx is constant over the entire panel (sb) and causes a shear deformation, which may be considerable and therefore may not be neglected. This kind of deformation is treated in the next chapter. The magnitude of the uniform shear force s is found Erom:

E I + 2 E I s y ( 1 _ s s r r

b EEI (27)

I t should be noted here, that in most practice the terms E I and

E 1 are small compared to E A b 2 I 4 and may be

r r r r

s s

neglected; then i t is obvious, that in-plane curvature entirely depends upon extension and compression of transverse edge merobers and shear deformation is caused by the total shear force V = sb.

7.7 Excentricity of global forces

ln practical applications global forces usually are not applied in the plane of gravlty, but in a plane of supporting members, at a distance z below

u the plane of gravity. Then a normal force n XX

(perpendicular to foldlines) causes a bending moment mxx plane of gravity, that yields a deEleetion wout of its plane.

in the plane of grav1ty

4---z

u plane of

~ference

n XX

figure 35: profiled sheet in bending by excentric forces

n XX

The bending moment mxx in the plane of gravity is:

m

XK z u n XX (28)

Furthermore displacements of the supporting members are important, rather than displacements in the plane of gravity. For this reason the plane of reference will be chosen at a distance zu below the plane of gravity. A global displacement u (perpendicular to fold I i nes) as a resul t of deflect ion w of the plane of gravi ty is found by: u plane of reference

·

-

.

-...

(29)

l

!

u

j

+

'

·

"

figure 36: excentric displacements by deflection out of plane

curvature of the plane of gravity as a result of bending moments mxx is:

a

2

w am _ __!.!

ar

D (30)

combination of ( 28). (29) and (30) yields:

9.!!.

2 an __ XX

a

x z u D (31)

The latter represents normal strains, perpendicular to foldlines, in the plane of reference as a r.esult of curvatu·re of the plane of gravity. This effect must be added to the flexibility of a profiled sheet as an orthotropic membrane. Hence (9) can be completed to:

(<IJ+ 'V} where: an ____g Et - \) n

...:n

a Et (32} (33}

All foregoing formulae concerning in-plane bending of a profiled

sheet still hold, if the modified modulus of elasticity Es is taken:

E

(34}

a(<!l+'V}

If boundary conditions along tranverse edges prevent deEleetion out

of plane. then IV must be taken zero.

I f global forces are applied in bottomflanges only, and vertical

movement of webs is free. z must be taken e , hence:

and: z _I! h a(<!l+'V) y, (1 + a 0 u u 2 h 2 12 (1-1) } (t) (a 0 + ~/3)

7.8 Shear deEleetion of panel without attachments to purlins

(35}

(36)

A special case of shear deEleetion is obtained when (several)

corrugations are not attached to purlins, as shown in figure 37.

~ ~

-r

-I

·-

~ x

p

I _-

--

-

- ---

Forces in purlins must be constant over the entire length a. In-plane bending moments are applied to sheeting by means of the longitudinal

edge members. For symmetry reasons these bending moments at

longitudinal edges must be equal. Thus the situation of figure 38 is obtained.

Va

2b

va

2b

purlin in tension

purlin in

compression

figure 38: distribution of forces

Va

~

Va

zo

In-·plane curvature of the panel is the result of extension and

compression of purlins (where the rigidity of sheeting is neglected), which is simple:

Va _1_ ~

2b E A b r r

(37)

Between the longitudinal edges sheeting is not attached to purlins and may deEleet as in the case of a shear panel without purlins. Foldlines may be assumed to remain straight and constitutive equations obtained before still hold. Using (15) and (17):

3 d V - dx3 V __ x_ E I s s

the displacement function v can be written:

V 2 A + B.x + C.x 2 V x 3 E t ("b) s (38)

Boundary conditions are: for x = 0: V

=

o,

dv/dx 0 for x

=

a: dv/dx 0 hence: A

o.

B 0,

c

3 Va (39) E tb3 s

The parallel shift of longitudinal edges with respect to each other, or in one word the shear deflection, becomes:

/:::N (40)

The average shear deEleetion per unit length can be written:

~.Y

a

where s = V/b according to (27) and k is the number of corrugations

between longltudinal edges. Since forces are applied to bottorn flanges

along the longitudinal edges E must be taken according to (34) and

s

(36), hence:

~.Y (41)

a

Rigidity of sheeting against shear deEleetion can be expressed

according to (1); taking' = s/t and y =~v/a:

Taking I y l

i"2"ï?(

l-u ) (a. + a. /3) 0 w (42)

c

=

12 k2(l-u2}(a. +a. /3) this expression is similar to (2},

s 0 w

though the latter is for sheeting discretely attached through every

I t must be realized, that (42) is based on a continuurn model for sheeting with discrete foldlines; a reliable approximation oE shear

rigidity by (42) is only possible for a reasonable number of

corrugations k between attachments to supporting members.

Cs (BOGIV\RD] (u = 0 ) Cs [BRYAN] k

=

4 k

=

3 k

= 2

k 4 k 3 k - 2 0.0 64.0 36.0 16.0 0.1 83.2 46.8 20.8 0.1 24.4 21.8 16.6 0.2 102.4 57.6 25.6 0.2 31.7 28.4 22.1 0.3 121.6 68.4 30.4 0.3 38.6 34.9 27.2 0.4 140.8 79.2 35.2 0.4 44.6 40.2 31.4 0.5 160.0 90.0 40.0 0.5 49.5 44.5 34.5 0.6 179.2 100.8 44.8 0.6 53.9 48.3 37.0 0.7 198.4 111.6 49.6 0.7 58.7 52.5 40.2 0.8 217.6 122.4 54.4 0.8 65.7 58.8 45.4 0.9 236.8 133.2 59.2 0.9 77.2 69.7 55.1 1.0 256.0 144.0 64.0

table 2: flexibility component Cs for rectangular corrugations according

to eq. (42) and according to Bryan '73 [11].

e

90o, aw

47

8. SHEETING DISCRETELY ATTACHED TO PURLINS 8.1 General remarks

In this chapter only shear deformation will be treated of sheeting, that is discretely attached to purlins through every corrugation. From a global point of view sheeting is strained by a uniform shear force s only. In the foregoing chapter it has already been mentioned, that in profiled sheeting two types of disturbance may occur.

The first and in practice always present type of disturbance is due to the fact, that reactive forces a long transverse edges, produced by purlins, are applied to bottomflanges only. Therefore the uniform shearforce over the corrugations will be concentrated in the bottorn flanges in the area around fasteners and consequently the assumption, that foldlines remain straight, is no longer justified.

figure 39: profiled sheet fastened through every corrugation

The second and in practice also frequently occurring type of distur

-bance is due to prevented or limited warping: prevention of warping because of built-in edges and lirnited warping, when two shear panels with different uniform shear force and consequently different warp are connected toeach other along a transverse edge (see figure 40).

For both oE these types of disturbance, which may appear independently, but rnostly coïncide, solutions will be given in this chapter . .

L

P<•v•n!:d wa<p

1'

limit•d w«p

___j

figure 40: example of warping restraint

average shear force zero

Consictering the fact, that for each corrugation the internal shear force s and boundary conditions are identical, also deformation of all corrugations must be identical; noticing further more. that each corrugation is symmetrical where the shear force s is antisymmetrical, also the deformation must be antisymmetrical. For this reason only six displacement functions are distincted in order

displacements of middle-lines of separate strips: _{u '} 0

to describe

u

w u u for

in-plane displacements perpendicular to foldlines and v

0, vw' vu for in-plane displacements parallel to foldlines of topflanges, webs and bottorn flanges respectively (see figure 41).

Since each corrugation is symmetrical where internal forces and displacements are antisymmetrical with respect to middle lines of flanges, displacement diEferences parallel to foldlines over a range bd/2 left and right of a line of symffietry must be equal but opposite of sign; thus a diEferenee t!. over one pitch of displacements v must be equal to twice the diEferenee between middle lines of topflange and bottorn flange, hence:

t!.v 2 (v 0 - V u )

bd bd (43)

where: !à_ g_ bd dx

I I I

t::.v

71'1---:---=-~ 2+----'---+---1----+-- ---,----+/:::,V

dy

I

s.dy

I I I

-·-Eigure 41: cross section and top view oE proEiled element beEore and aEter distortion

2

\

\

_\

s.dy

*

.

-·-

·

8.2 Assumptions in constitutive equations

I t has been stated al ready, that foldlines may not remain straight, but still 1t is reasonable to expect, that in-plane curvature of a strip will be small compared to out-of-plane curvature of the same strip. Th is will be assumed throughout and in each strip transverse shearforces in longitudinal direction as well as twisting moments will be disregarded (see appendix Cl).

Furthermore following assumptions are made for in-plane behaviour of each strip:

- the principle of Bernouilli holds, thus straight lines perpendicular to foldlines remain straight

- shear strains. due to the uniform shearforce s. may not be neglected (see figure 18); influence of shearstrains. due tonon-uniform shear forces, is small compared to the effect of in-plane curvature and will be disregarded

- normal strains, perpendicular to foldlines. are small compared to a rigid-body-displacement and will be neglected (inextensional bending of the cross section of a corrugation).

These assumptions incorporated in general two-dimensional theory of elasticity yield for each strip one general function for in-plane forces and one general function for out-·of-plane forces and displacements; both functions. given in appendix Cl, are of the same form and depend only upon curvature of foldlines.

8.3 Equilibrium and compatibility along foldlines

Analogue to theory of slender beams, in-plane stresses of a strip can be substituted by cross-sectional forces and bending moments (appendix Cl). This yields in-plane forces for topflange, web and bottorn flange as indicated by figure 42.

dy

figure 42: sectional forces and bending moments, in-plane displacements

I

+

Along foldlines flanges and webs must satisfy the conditions of equilibrium and compatibility. Therefore all forces and displacements

of a web can be expressed by components of adjacent Elanges (see

equilibrium of

cross sec t ion

bu

k

Ir

1

'1

+

b.,...

~

1

figure 43: transverse forces and displacements

+

bo

compat

i

bi

l

it

y

of

c

ro

ss

section

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