Equivalent orthotropic properties of corrugated sheets

EQUIVALENT ORTHOTROPIC PROPERTIES CORRUGATED SHEETS Department

Drix1r.1~1:~ BKMSOIII.IS+ Engineering. University of Illinois.

of Agricultural

OF

Urbana-Champaign.

Illinois.

U.S.A.

.Ahstract-The analq>is of corrugated shells Iplales) i> based on the aswmpLion Ihat they can be analyzed as thin. equivalent orthotropic shells of uniform thicl\nesh. The anal) tical expressions fog the equivalent rigidities of orthotropic thin shells given in the literature are reviewed. The results of a finite element analysis of a corru_eated sheet subjected to constant strain states reveals an inadequacy in some of the classical expressions in use today. These equivalent orthotropic properrics are improved with the derivation ofne\v expressions. In addition. expressions for the localized ztresz wncentT;ltions developed

Corrugated phragms

plates

and

of standard

alysis

mainly

in a wide

shells

barrel

problems

dia-

of structures.

shells steel

are derived.

as shear

variety

corrugated

and stability

ally examined

of the corrugations

functioning

are found

Cylindrical built

at the ridges

are also sheets.

often

The

an-

of these shells are usu-

using the theory

oforthotropic

shells

(plates)[l-6]. The approach shells

has been

to be valid tures

of using the theory considered

ness and ultimate

strength

are [3]: (i) design

analytically,

element

Analytical fastener

for

ultimate

shear

In finite

loads

is critical.

term

which

elements.

0.3 Nu,

where

elements.

used

effective

special

by an equivalent

form

thickness. detailed of internal

this

is not

+ Visiting

replace

analysis

information forces

possible

Assistant

through

with

the

regarding

small

of

corruga-

material

of uni-

the

pro-

distribu-

a diaphragm[J] the approximate

Professor.

finite

appropriate

of diaphragms

within

mais the

the geometry using

orthotropic

element

tion

modulus difficulties.

IMang er (I/.[ IO] consider

tions

formulations

shells

of the shell

ortho-

elasticity

shear

corrugated

and

diaphragms.

In such

of the orthotropic

doubly

finite

dia-

Nu is the

simple

not

while ana-

Referring

among

by practical

However.

of which

depend

bolts

(silo

do not

structures).

The

theoretical

are

related

to

the

many

factors

themselves.

Thus.

used in dense arrangements present

analysis

is based

analyzed

problems

of pre-

of uniform

The

of a corrugated

on the assumption

as a thin.

(plate).

concept

equivalent

of corrugated

element.

In this

configuration

cant

compared

to the structure’s states

within

the modeled elastic

tutive

relations

sheet.

relating

overall

stress

derivation elasticity

associated

found

In this paper, (plate)

be determined.

of the elastic

some assumptions

to obtain

a better

shell (plate).

constants

of

ofthe

simulation

In addition,

for the nonuniform

veloped

within

the corrugated stress

a given

orthotropic is based on

orthotropic

section. for the thin

in certain

shell cases.

of the corrugated

analytical forces shell.

concentrations

lothe

analytical

expressions

and modified

derived

corresponding

the

that only

with The

in the next

the analytical

are reviewed

to the corcorrugated

but

in the literature

discussed

rigidities

the

has the meaning

resultants

matrix

equivalent

of

consti-

orthotropic

are equal

determined.

can

apply ortho-

in the

not

of strain

and

to

The equivalent

properties

are

of

insignifi-

dimensions.

the equivalent

“average”

effects

the geometry

(rigidities)

and strains.

average

Here.

calized

of the

by an equivalent

be assumed

element.

properties

stresses

responding

dimensions

is rather

can

in

is very

bray.

corrugations

tropic

is presented

and modeled

the

(nominal)

shell

sheet which

to the important

is isolated

strain

shell

it can be

orthotropic

of the equivalence

small compared

constant

that

thickness.

I. An element

state

offas-

failure.

(plate)

Fig.

factors

on

are the fasteners

e.g. high strength

an-

that in a

computations

problems

fasteners

element

I ] states

the long and costly

justified.

presence

to the finite

El-Dakhakhnifl

that is governed

details

orthotropic

of the

of shear

of shear

91 have

The

elements.

The

values

problem tener

structure,

distributions,

predict

analyses

the big corrugations curved

force

deflections

has presented

Analyzing

vides

may

below

stress

the determination trix

in this

are determined

strength[8].

element

plane

design:

strength

and shear

investigators[3.

tropic

requires

analysis.

formulas

forces

phragms

most

and

based on assumed

(iii) finite

the stiff-

this approach

(ii) approximate

flexibility

shellj7).

of the shear diaphragms

by testing:

testing.

procedure

the main fea-

used to determine

approach. procedure.

mature

investigators

predict

of the corrugated

The main procedures

full-scale

by many

and to adequately

of the behavior

of orthotropic

lytical alysis

thin

expressions

are

or moments

de-

from

which

the

are obtained.

sheil. The ~arne concept i\ used b> >e\eral invrstipators dealing with anaiy\is of orthotropic cylindrical shcll$i. IO. i?]. The prclssnt anal>>is does not include nonlinearities but a general rev& of the causes of nonlinear behavior in the case of corrugted sheets is included for the sake of completeness.

2. RIGIDITIES

--_j--(Cli

l -

-3

?

( bi

Fig. I. Model ofao element ofcorrugated sheet by equivz11en1 orthotropic plane stress element (-1 normal stresses: (a) xtual orthotropic element: Ibl equivalent orthotropic element.

To achieve the above objectives. a representative element of corrugated shell (plate) of a structure is exactly simulated with Lagrangian degenerated shell finite elements and the corresponding rigidity expressions of an equivalent orthotropic shell are derived by applying states of constant nominal strain to it and calculating the stresses developed. The resulting rigidities are compared with those obtained by the analytical expressions and improved expressions are derived using Castigliano’s second theorem when the old analytical expressions are shown to be inadequate. For simplicity. a flat shell (plate) is considered on the assumption that the radius of the structure (e.g. silo) is very large compared to the corrugations of the

The analytical expressions for the equivalent orthotropic rigidities of corrugated sheets presented in the literature were obtained analytically. in most cases. by energy methods. Thus. the strain energy due to the actual (or assumed) stress resultants developed within the corrugated element was determined for a given state of constant strain and then. the corresponding rigidities were derived. Refering to Fig. 1. assume a standard corrugated sheet with modulus of elasticity E. material’s Poisson’s ratio p. thickness f. rise of the corrugations f. developed length 1. and projected length c. Then. the relation between the nominal stress and strain components for the case of plane stress of an orthotropic shell is given by [I31

9 node quadratic ~agrangian shell element

Fig. 2. Finite element mesh and corrugated

sheet standard profile

where

E,. E,. and E,, are the nominal

E,.

properties

of the equivalent

orthotropic

t.~, and kI are the nominal x. respectively. perscript

(The

indicate

actual

the corrugated

The extensional cial

stress

wave

in .V and

without

appear

later

stresses-not

suin the

on

the

form)

sheets

or bearing 01 tr/.[9]

failure

assumption

of a sine-

diaphragms

tlexibility

by 55%.

= E,r

D,

= LIE. (’

where. with

the sheets dicate

= kzE., is considered

= k,E,

E,

no influence

on the mechanical

by many

authors[7].

[3. 7. I?.

of

(11 ~/.[9].

in-

Nilson

analytical

rigidity

Et

is given

where

p =

factor

is shown

reduction

factor

C’

tic buckling

connections). detail

13. 7. I?,

rigidity.

of several

This param-

of the importance

arrangements

wishes

to utilize

fully

in the literature.

it is discussed

of a diaphragm.

silos

high

where

are used

expressions

are given

by

141

0.E

profile

to I ). A more exact

of inertia

can be found

161. Mang

element

(4)

of

(assuming formula

in some

of a toroidal

of constant

The analytical

from

full

needed shear

if one stiffness

in dense

arrange-

continuity

along

the

f/c

for this old

pub-

B, = 0 in

section

with

depth.

expression

rigid-

by (3, 7. 12. 141

forces

effects

long

B,,. =

According

to Bryan

of a diaphragm

Et’

1 (’ 131

cr o/.[?].

local

2.) Since

axial

strain

of the internal

forces

values

for

(Note

that

it

forces “parto axis

is reasonably

characteristic

may

of the redistribution

to be investigated[-l].

linear

bolts

are used. as

the behavior

after

at a substantially In such

to corso that

as parallel

high strength

steel silos,

is rather

place.

behavior

the nature

when

is the case with

seam.

always

parallel

takes

is defined

to allow

of the con-

the initial

slip which

low load (bearing

a case.

type fas-

the flexibility

of the con-

has to be added to the flexibility

of the sheet

behavior

source

of nonlinearity

comes

either

higher

from

plastic

loads.

distortions[j]. theory

for

of the structure. then

to be markedly

tropic

of the edge members.

of forces

average

fastener

profile

strain

and redistri-

it is experimentally

elastic-plastic

in

shear

failure

in a given

a simple

shown

that

to consider

at seam and edge

profile:

deformations

due to:

before

Y. Fig.

At

the shear flexibility

is the sum of the flexibilities

of the corrugation

the sheeting:

+ p*) .

out

redistribution

to corrugations

However,

at the

to equalize

Screw-fastened

ductility

in any line of fasteners

be sufficient

effects

tend

failure.

substantial

full

ductile,

teners).

of non-

Plastic

connectors

points

in the fasteners

nection

40% of

source

may be important.

that

place

is about

the main

before

show

confirmed

nection

diaphragms

and that

stressed

diaphragms

occurs

for the twisting

many tests that the limit of elas-

is the connectors[9].

highly

for the elastic

fasteners:

bolts

to ensure

of shear

is sufficient

er a/.[101 assume

analysis

corrugations

bending

are

This is the case in steel

strength

load,

rugations

for the moment

I, of the corrugation

is given

most

allel”

is an approximation

lications[lj,

linearity

takes

small compared

ity.

a

elas-

3. SONLINE.ARITIES

bution

B,. = 0322Ef’t.

small

contains

seams.

Davies[4],

analytical

Et” C’ B, = I?(, - F*?)7 *

a finite

the

fasteners,

the inherent

and strength

connector

ac-

later.

The flexural

moment

determine

is its usual mode of failure1 181. Hence.

tic response

slip at the sheet fasteners.

Because

to this factor

inertia

shear

in most prac-

are the weakest

spaced

fastening

It is known

-

of shear

to be a function

eters (sheet deformation,

where

that

diaphragm

of closely

denser

the failure

in more

corrugation reduces

would

if a light-gauge

number

by

+ /.LL)I.

21

corded

corrugation

I41

D,,. = E,,.r = p ~

other

it was found

capacity.

sufficient

ments

in determining

at every

failure

of adja-

p:E,IE,

t.~~ = k and so. p,, =

that

The shear stiffness

negligible

properties

part

the connectors

and their

between

Thus.

Ha[ 171 states

diaphragms.

However.

(2)

at the connectors

an important

third

shear Niison

Also.

the seams

of a diaphragm.

than at every

diaphragm

= E,t.

D,

along

plays

components

D, = E,t =

at the fasteners[l4].

that the use of end connectors

tical

by [3. 7. 12. I-l]

are fevr in number.

is due to localized

state that deformation

actual

rather

are given

fasteners

of a shear diaphragm

cent panels

nominal-

in .V and .v for the spe-

corrugated

(based

failure

the response

rigidities

corrugation

and

sheet.)

case of standard

analytically

ratios

quantities

star. CJ,. .L’,. .W,. which

paper. within

Poisson’s

If the diaphragm

elastic

sheet.

the

is the shell effects

diaphragm

Thus.

shells has to be supplemented

and it

or buckling.

influenced

buckling

The main itself

flexibility

by the nonlinear

the use of the orthoanalysis

of corrugated

by a separate

analysis

is

Table I. Extensional rigidities of the corrugated sheet’ Finite elementsZ Rigidity rkips,in.) D, (rE,) D, (fE,) D; (rEv) D,,CrG)

.Analytical Old pr = &E, E.. *I = 0

439 -709

I794 550 7673

1800

25’0

i-1-10

1794 0 7673 2820

while. consistently.

Present 1421 434 7673 1885

* E = ill X iOh psi. I* = 0.3. I = 0.25 in. i Calculared p2 = 0.3 = p. ei = 0.056 = plE.JE,.

Equation (6) is a result of pure statics of the corrugated sheet and indicates that the extensional rigidity in .Y has to be modified accordingly. This is done in Appendix A with the use of Casti@iano‘s second theorem on an approximated sinusoidal corrugated protile. The modified rigidity E, is

tE, for local buckling. Liboce[ 191. ho\t.ever. expresses doubts centering on the question of whether the flexural and torsional elastic constants of the “equivalent” orthotropic shell properly account for the deformability of the cross sections in the “soft” planes perpendicular to the corrugations. According to Abdel-Sayed[ I]. the prebuckling behavior of curved corrugated shear panels is found to be the same as that of plane ones. if bile the buckling limit of curved corrugated panels is considerably higher compared to the strength of plane sheets.

The finite element mesh and the corrugation contiguration are shown in Fig. 7. The standard profile of the corrugated sheet analyzed here is that of the arc-and-tangent corrugation[lS. 161. The parameters used in the numerical analysis are: c = 2 in.. L = 3 in.. f = 0.21875 in., I= 2.046in., CL= 17.54”, R = 2.0208 in., and I = 0.25 in. The actual extensional rigidities obtained with finite element analysis by imposing states of constant strain on the corrugated sheet (Fig. 7). are compared with those of the analytical expressions given by eqn (2) (for both E, = 0 and f& = pE,f in Table 1. It can be seen that the expression for E, is about 235%higher than the quantity obtained by the finite elements. Also. E, is not negligible in this case and the assumption that F: = k is verified by the numerical analysis. The discrepancy that exists in E, is probably due to the fact that the strain energy due to the axial force Zy: has been neglected in the derivation of the old expression[ I]. even thou_ph this is not stated clearly in the literature reviewed. From the finite element analysis of the constant E: case, one can Fee that in addition to the constant X, force. there are also developed moments M, and M, which vary as shown in Fig. 3. (Note that all stress resultants developed in the finite element model are actual and not nominal.) M, increases tvith the distance from the center line of the corrugations and its value is found to alivays be

which for a plate (f = 0) yields E,r = E as it should. In Appendix A it is also shown that

and

The values of E, and E, given by eqns (8) and (9) are very close to the ones obtained by the finite elements (Table I. “present”). The shear stiffness was obtained numerically by

Table 2. Flexural rigidities of the corrugated sheet+

i

Rigidity

B; B,

Analytical Present Old

Finite elements fkips. in.) 41.5 11.5 216.5

4

B r\

42.0

0.00 187.6(2X.6)$

30.0

12.0 12.4 717 4 -__. 30.0

30.7

+ E = 30 x 106 psi, FL = 0.3. I = 0.25 in + Formula by Blodgett[lS]. i Mang er ni.[lO].

Fig. 1. .V, force developed in a state of conslant curvature II,.,?,.in the corrugated sheet with finite element analysis.

applying constant shear strain ET,.. Assuming some distance away from fasteners or assuming boundary conditions ensuring continuity (many fasteners) p is found to be I (Table 1). The ratio c/I in expression (3) comes from the expression of the shear strain energy for a panel of breadth 1. which is the developed width of a single corrugationf?]. However. the shear stress resultant developed in the direction parallel to the corrugations as a result of a constant nominal shear strain. should be identical to that developed on a plate sheet. Hence. eqn (3) is modified to

D,,. = E,,.t =

z?(I

+ jL)

which is valid away from connections or other boundaries or when continuity of the connection is satisfactory.

The bending stiffnesses were obtained numerically from the application of constant curvature conditions on the sheet. The rigidities obtained are compared in Table 1 where one can see that 8, = pB, as in the case of isotropic plates while 8, < p-B,. An error of 13% between the analytical and numerical B, values is found. which however decreases to 4% if one uses the more accurate expression for C,[lS. 161. The flexural stiffness B,. actually consists of two parts: flexural stiffness from the sheet itself and flexural stiffness from the geometry of the corrugation profile. This results in a uniform moment due to the flexural stiffness of the sheet (of uniform thickness) and an additional nonuniform moment due to the axial forces N, developed in the corrugated shell (Fig. 4). The development of the N, forces is a result of the statics of a “beam” which is formed by the corrugated sheet and can be visualized by the projection of the sheet on they: plane (Fig. 5). Thus the N, forces are required to keep lines normal to the middle surface before deformation to remain normal, while the uniform moment A4 is developed as a result of the constant curvature state of the sheet itself (irrespective of geometry). The statics of the constant curvature state w.,*, are utilized through Castigliano’s second theorem in Appendix A to obtain a simple analytical expression for B,.. which is B,. =

Et3 121

-

+ CL21

Etj? 2

*

(1 I)

This the

expression

finite

gives

element

“present”).

a stiffness

analysis

In addition.

&I and :V,. (Fig.

even

results

closer (Table

the relationship

5) is established

to

twisting

2.

the finite

A as

follows:

veloped. Since

t2

.\I -

corrugated

the value

a unit length about

moment

a unit

length

shell moment

Poisson’s

i 12)

i’

about M:.

of the axis .V (Fig.

another

the y axis

(Fig.

this

5). Then,

length

twi\ting

forces

.I/,,

.I- and J’ do-

in the J’ direction of a plate

de-

moment

is rhe same in both

to be the same as that moditied

that in the C;ISL’

sheet.

it has

eqn (5) i\

accordingly:

Et’

=

I 1-J)

1311 t

/.lL)

1). which

or.

gives

a result

identical

to that obtained

nu-

merically.

the

of the cor-

= cJIII.

moment

as a result

moment

.-ictuall)

sheet.

are no membrane

and in addition.

B,,

about

shell midsurface.

axis .i- is ;\I:

thsrs

at the ridge

numerically

per unit

twist

agrees exactI>

.\I in Fig. 4 is applied

of ,W applied

rugated

about

sin -

of the corrugated

of this oped

j

sheet which

of X, obtained

The uniform

component

Ti.i-

( 12)gives .L’, = 160 kipsiin.

Equation (: = f) ofthe with

p2)

I3 I -

corrugated

anal) si‘i indicates

but onI>, ;I conytant

rections.

s_

ot’the

element

of constant

between

in Appendix

stiffness

Because

,\I;

I-

is devel-

It should

of the material‘s

of

behavior

the

present for both

shell

thin

(Table

sheet

With

the

in

the

derived

3). the equivalency

is valid

shells.

equivalent

corrugated

of an

simulates

shell.

espressions

and thick

new

adequately

of a corrugated

modified

section

The

by now that the concept

orthotropic

the actual use

COSC~STR.\TIOSS

be clear

equivalent

ratio:

STRESS

orthotropic

have

been

rigidities

for

implemented

in

a II.

Yll-l:[20].

The

or

local

the

strain

cases were obtained

(13)

stresses

factors

tained The Table

(13) gives

by finite results 2 verify

the exact

elements. for that

the

values

as in Table twisting

eqn (5) adequately

E,i

-

effects

finite

B. This stresses

rigidities

Present: El

I - p.2)(l’ic’ - (li2nc) sin 2://c)]

Er 1 c

c

El

c

p2(1i

31 -r l-4

IZCI -

Et3

c p2) i

0 0.22Erf’

/ ~- Ef3 IZ(I + +) c

c p’) i

I?(1 -

d3.r Et’ I2(1 -

&-

Erf’

kZ)

2

Ef’ IZ(I

within

the use of

[eqn (BY) and

B].

and flexural

enables

for possible

through

elements

con-

maximum

lLZDr

Et’

BX,

for extensional

localized

maximum

or buckling

Appendix

[I + (flr)‘6(

0

El

B,

(B9).

the

in Appendix

orthotropic

constant

in Appendix

shell and so. to investigate

yielding

equivalent

are devel-

for these local forces,

the actual

I z

D,

B,

the

expressions

6(1-r*?)? I*:&

Bx

in

Old expressions’

D,

D,

local shown

predicts

Et D,

a corrugated

ob-

2.

rigidity

Table 3. Analytical Rigidity

for B,

which in certain

analytically

for

are obtained

one to obtain Equation

corrugations

the expressions

centration

= pB,.

or moments

within

A. Using

B, = p;B:’

forces

oped

T pL)

f Davies[3]: Easley[lJ]; El-Atrouzy er al.[lZ]; Marzouk er u/.[7]. assume G, = 0. [I/( I CLIP:) = I. Dp = ct,D, = p2D., = 01. Nilson et a/.[91 assume ~2 = I*. I*, = D,!D,. $ Assume away from connections or other boundaries which constitute discontinuities.

Equivalent

orthotropic

properties

of corrugated

\heety 1

20

I c t

0

b”

Fig. 6. Effect

of the ratio .fI! on localized maximum

strexseb

at the ridge of the corrugations

The concentration factors depend primarily on Ac~norc,ledgmenrs-The author wishes to express his thanks to Professor D. A. Pecknold (Department of Civil the geometry of the corrugations expressed by the ratio fir. The effect of j/t on the stresses CT,and CT, Engineering, University of Illinois. Urbana) for his encouraging and helpful discussions, and to Professor J. 0. at the extreme fibers at the ridges of the corrugaCurtis (Department of Agricultural Engineering. Univertions is shown in Fig. 6. It can be said that the consity of Illinois. Urbana) for reviewing this paper and for his useful suggestions. centration factor due to a constant moment MT is insignificant, the maximum being Us,,;,, = 1.134~~: only for relatively thick shells. For thin REFERESCES shells there is a reduction factor rather. which I. G. Abdel-Sayed. Critical shear loading of curved “stiffens” the shell right at the ridge of the corrupanels of corrugated sheets. J. Engng Mech. Div. Am. gations. On the other hand, the maximum tensile Sm. Civil Engng 96, 895-911 (1970). and compressive stresses crrmnxat the extreme fi2. R. E. Bryan and M. W. El-Dakhakhni. Shear flexibers of the ridges increase linearly with f/t and may bility and strength of corrugated decks. J. Srrrrcr. Dir. Am. Sot. Ci\,i/ E~$!rr~~91, 2549-2580 (1968). reach very high values under the presence of NF 3. J. M. Davies. Calculationofsteel diaphragm behavior. membrane forces in the case of thin shells. Then, J. Sfrrrcl. Dit,. Am. Sot. Cit,i/ Engng 102, I11 l-1430 local yield and buckling may become serious prob(1976). lems. 4. J. M. Davies. Simplified diaphragm analysis. J. 6.

StiIIXlARY

The simulation of a corrugated shell with an orthotropic equivalent one is based on a set of analytical expressions for the extensional and flexural rigidities of the orthotropic shell. These expressions are reviewed and modified in certain cases in the present paper to give a better simulation of the actual thin shell. In addition, analytical expressions are derived for the nonuniform forces or moments which are actually developed within the corrugated shell but are hidden in an analysis with orthotropic shell elements. Based on these nonuniform forces and moments, maximum stresses are obtained which make possible a secondary analysis for checking against local yielding or buckling. This could be very important in nonlinear analysis, especially if an appropriate model of material nonlinearities is established to account for the interaction between overall and localized behavior of a given structure.

5.

6.

7.

8.

9.

IO.

II.

12.

Slrtrcf. Dit,. Am. Sot. Ci1.d En,~rrg 103, 2093-1 IOY (19710. H. K. Ha. N. El-Hakim and P. P. Fazio. Simplified design of corrugated shear diaphragms. J. S/IX!. Dir. AI,,.-Sec. Ciri~En,~,,p 10-i. 1365 i377 ( 1978). H. A. Nilson. Discussion of “Bucklina of lieht-eaee corrugated metal shear diaphragms” b; J. T.-Eailei. 1. Stnrcr. Div. Am. Sot. Civil Engng 95, 3004-3006 (1969). A. 0. Marzouk and G. Abdel-Sayed. Linear theory of orthotropic cylindrical shells. J. S~rtrcr. Dir,. Am. Sot. Ci\,i/ Oy~r,~ 99. X87-2306 (1973). T. J. Easley and E. D. McFarland, Buckling of lightgage corrugated metal shear diaphragms. J. Strucf. Div. Am. Sot. Civil Engng 95, 1497-1516 (1969). H. A. Nilson and R. A. Ammar. Finite element analysis of metal deck shear diaphragms. J. Srrrrc,r. Dit,. Am. Sot. Cil,i/ Engrrg 100. 71 l-726 (1973). A. H. Mang. V. C. Girya-Vallabhan and H. J. Smith. Finite element analysis of doubly corrugated shells. J. S/rrrcr. Dit,. ;\tu. Sot,. Cil.i/ Eugug 102. 2033-X50 ( 1976). M. W. El-Dakhakhni. Shear of light-gage partitions in tall buildings. J. Slr~cr. Div. Am. Sot. Civil Engng 102, 1431-1445 (1976). N. hl. El-Atrouzy and G. Abdel-Sayed. Prebuckling

D.

I i6 analysis

of orthotropic

barrel

BRIASSOC'LIS

J. Srr//c,r. .!I;\,.

shells.

I 1978,. S. kt’oinowsky-Kricger. TIreor\ of P/tires ofrtl Slrc,//.\. 2nd Edn. .LlcGra\c-Hill. New York I 19591.

.Am. SOCK.Ciril Engtro 10-t. I775- I786 13. S. P. Timoshenkoand

14. T. J. Easley.

Buckling formulas for corrugated metal diaphragms. J. Srnrc,r. Di\,. Am. SW. Cit.;/ 101, 1-103-1-117 I IY71). Blodgett. Moment of inertia of corrugated Cid En.q,y 1. 492-493 (IY34t. Wolford. Sectional properties of corrugated determined b), formula. Cil,i/ O~,gtr,g 103. 5Y-

shear

&uruu Ii.

B. H. sheets. 16. S. D. sheets 60 t 1Y.i-l). 17. H. K. Ha.

Dii.. Am

Corrugated

shear

SCK. Civil Grcqc

Fig.

2.

Al.

rorios p,.

Poisso~7’s

the length

Constant

1. Then.

strain

state ET.

pL1.Assume

for e:’

E: = one has

= ,,“‘/2/

pc,”

along

J. Srrrrc~r.

diaphragms.

( IY7YJ.

105. 577-586

IX. T. J. Easley.

Strength and stiffness of corrugated metal shear diaphragms. J. S/r/tc,r. ht.. A/u. S~K. Ci~,i/ GI~VI,~ 103. 169-180 (lY77). plates in shear. I/r19 C. ‘Libove. Buckline of corrugated

loutr/iorrtrl Co//oq;ticrfrr OII Sttrhilily C’rulu Srrrtic md Lhutrnric~ Lorrdr.

of’ .Srr.r~c~lrrrcs

pp. 135-462. Engineers. New York

70

American Society of Civil (1977). L. A. Lopez. Finite: An approach chanics systems. /!I/. J. .Vlr/tfcr. 81I-866 ( 19771.

Hence.

to structural

(AZ)

and

me-

.Uc//r.r Gfp~rp

E,,

I I,

c LE,

c E,IE

piI

pL2 -lQ=qEE,=

(A3)

or

Mod~/iet/c~.rrw~iomr/ .sr~(fk~,s

I,

3. /+.rwtr/ E,.

Assume

sr$fiwss

B,.

Referred

to Fig. AZ. one has

the pro-

file of the corrugation is given by ; = .f sin ax/c,. At a distance .t-. the forces applied on the sheet under constant strain E: we (Fig. Al)

H = ,Vf cos t). M

= iv::

= NY f sin xv/c. or

Under

constant

strain

iv:

r:

= /(EXE:

+ E,r:I/lI

0 = [(E,e: Then,

F,

=

-

r:lr:

one has -

)I,ll:).

+ E,E?J.

so iv,

and

A -

f

M=l IV:

= +--&

IE,cF

+ k?E,(_

we:1

The displacement the strain

dLJ

I, = x

II over a projected

energy

llr

L

=

I311 -

p2) f sin 2,

/2

(’

L’

IA41

= f&E:. From

from

sin

length

Fig. 4 then.

for an average

M:

moment

over

%:

1~. is given

U by

= 2

or

Assuming

WT

II = E Then. tropic

$fJ c+ 7

i

,’ I 7 - 5

= rE,11/(2c) N: = rE,et shell of length 2. or

rE, =

(ASI

IV = (I “$)

cos H = (,I/.

E/

2d sm 7

1

for an equivalent

ortho-

)I IAl)

Fig. AI.

Axial forces and moments curvature

state

developed ,I,.:,

in ;I constant

Equivalent

orthotropic

properties

of corrugated 2

and

Since for an equivalent orthotropic stant IV.:, . .\r: is uniform. one has

sheers

Nrf

sheet under con-

Fig. Bt. Stress concentrtttions at ridees of corrugations under constant r: s&in.

vr

B,

=EI(I +$)

Using eqns (A5) and (A6) from Appendix

or

N, 0, = Z-“-TF

I

(-47)

,116

I-

A. one has tBJ)

or

SO

(5,. =

APPENDIX

B

Stress concentrations in corrugated sheets As it is shown in the finite element analysis the stress distributions in the corrugated shell under certain constant strain conditions are not uniform, as the equivalent orthotropic shell would indicate. It is possible however, through the use of the analytical expressions developed earlier to obtain the actual stress distribution from the solution of an orthotropic equivalent shell. Thus. in the case of constant strain CT. there IS a moment ‘M, developed in the corrugations which varies linearly wi!h the distance ; (Fig. I). This moment produces compressive and tensile stress in the extreme fibers of the shell in addition to the constant axial stress. Hence, one has

N: I

a,=---_T_

NT,6

I

IBS)

,,”

=s t c

I1

(Bf)

For positive :. oV is maximum at : = f at the outer fiber of the corrugations t Fig. 82). only if fir < l#‘X Then.

UV., =

Also. for negative :. o, is maximum at ; = .f at the outer fibers of the corrugations (Fig. 82) only if fir < Ii 3:

r

This stress becomes maximum at the inner fibers of the corrugations at : = f. for any f/l ratio (Fig. B I ). Then.

A maximum stress of opposite sign is developed on the outer fibers at : = f but only if j/f > 113. So.

A similar problem is found in the case of constant curvature I,’ .FV(Fig. 3). In this case. there is no constant moment .\I, developed but. instead. a much smaller uniform moment M is developed along the corrugated edge plus an axial force N, which varies linearly with :. As a result. the stress o, varies along the corrugated sheet while this can not be seen within an equivalent orthotropic shell.

Hence, in a state of constant curvature w.:, . when the outer fibers of the top ridges of the corrugations are in compression. those of the bottom ridges are in tension and vice versa. It is probable now that maximum stressconcentrations occur on the outer and inner fibers along the ridge lines of the corrugations. On the inner fibers. stress concentration can occur only as a result of an axial force XT of the equivalent orthotropic shell. On the outer fibers. on the other hand. there may be concentrated stresses because of either the presence of NT or MT in the equivalent orthotropic shell depending on whether fit is greater or smailer than 113. But whatever fir ratio is. there will always be stress concentrations on both inner and outer fibers of the corrugations not revealed by an analysis using orthotropic shell properties. In general. by knowing the

of the same sign.

Alio.

-(;/i,-ik3

0

-Id

0

I .z )c: 0

Fig.

--:/!:I

BZ. Stress concentrations at ridges of corrugation5 under constant II,.:, curvature. where

forces and moments at any point of the orthotropic shell. one can obtain the maximum slresses developed in the actual corrugated shell using eqn (B?). (83). (86). and (87). Then. eqn (88) and (BY) are derived which give the maximum stress a~ the ridges ofthe corrugations when the forces and moments of the equivalent orthotropic shell (plate) are known (the subscripts iand o indicate inner and outer fibers. respectively):

_f>l

61

3

1

I

C, =

f,! / -3 (fir) 31 - )LZ)+ I (fir)’

6(l

-

f,!

I

CL:) +

3

I

“2 =

i

f,!

I.

t-3

IL(fll) (fir)’

6(l

21 -

~‘1

f

CL21 +

I

1

-<3 I

(‘3 =

f,!

I.

I

1

3

The basic geometric parameter which determines the magnitude of the stress concentraGons is the ratio fir. Increase of this ratio results in increase of the c concentration factor but not necessarily the (‘2 concentration fattor. In the last case. it can be shown that the concentration factor C: reaches a maximum when

f

-= / Thus, positive

maximum : ridges

o, occurs

(1 -

and the maximum

+2)/l.5 = 0.147

21

-

1*2)

at the inner

of the corrugations

_fibers ofc the u hen ,V; and MT are

-

-II\1

is c2 =

I, 134.

(BIO)