biosystem engineering 2003

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Biosystems Engineering (2003) 84 (3), 303314doi:10.1016/S1537-5110(02)00241-6

Available online at www.sciencedirect.com

SE}Structures and Environment

Analysis and Design of Low-density Polyethylene Greenhouse Films

D. Briassoulis1; E. Schettini1,2

1Department of Agricultural Engineering, Agricultural University of Athens, Iera Odos 75, 11855 Athens, Greece; e-mail of correspondingauthor: [email protected]

2Department PROGESA, University of Bari, Via Amendola 165/a, 70126 Bari, Italy; e-mail: [email protected]

(Received 20 August 2001; accepted in revised form 9 October 2002)

In the present paper, the mechanical behaviour of low-density polyethylene (LDPE) films under various

combinations of pre-tension and uniform pressure schemes is investigated experimentally and numerically

using the finite element method of analysis. The behaviour of the film is simulated by means of numerical

models and with the material properties obtained in the laboratory by using standard testing methods. The

finite element models used include both a commercial finite element program and a recently developed

research non-linear finite shell element, capable of modelling membrane behaviour. The numerical analysis

results obtained under appropriate boundary conditions and different analysis options are compared with

experimental results obtained from a specifically designed experimental arrangement. For the cases tested

experimentally, the two numerical approaches gave results in a good agreement with the experiment results, in

the linear elastic region. Subsequently, using the research finite element model, design criteria are developed

for the reliable design of LDPE greenhouse films.# 2003 Silsoe Research Institute. All rights reserved

Published by Elsevier Science Ltd

1. Introduction

The use of low-density polyethylene (LDPE) films as

covering materials for greenhouses has increased sig-

nificantly in the southern European countries (Briassou-

lis et al., 1997a; Osswald & Menges, 1996). Research

concerning these plastic films is a significant issue in

order to formulate technical suggestions for a reliable

design of the greenhouse and the covering, to make

more efficient use of the film and to achieve a longer

useful lifetime of LDPE films. The present work aims at

investigating the mechanical behaviour of the LDPE

greenhouse covering materials because the durability of

these materials strongly depends on their mechanicalbehaviour under various loading conditions (Billmeyer,

1984; Briassoulis et al., 1997b; Courtney, 1990; Ogor-

kiewicz, 1977; Shah, 1984; Simonds, 1961) as well as on

various degradation factors (Dilara & Briassoulis,

2000).

The mechanical behaviour of LDPE films under

various combinations of pre-tension and uniform

pressure schemes is investigated experimentally and

numerically using the finite element method of analysis.

Material properties used in the numerical models (e.g.

modulus of elasticity, yield stress) are based onmeasurements of the corresponding mechanical proper-

ties of LDPE film samples obtained in the laboratory

through standard testing methods. The finite element

models used to simulate the membrane behaviour of the

film include both a commercial finite element program

and a recently developed research non-linear finite shell

element. Comparisons are carried out between the

numerical analysis and the experimental results where

the latter were obtained by implementing a specially

designed experimental arrangement. Technical sugges-

tions are offered to assist with the reliable design of

LDPE greenhouse films.

2. Mechanical properties of low-density polyethylenegreenhouse plastic film

2.1. Laboratory testing of the mechanical behaviour of

low-density polyethylene films

2.1.1. Tensile properties

The stressstrain curve of greenhouse LDPE

films obtained from a tensile test gives information

1537-5110/03/$30.00 303 # 2003 Silsoe Research Institute. All rights reserved

Published by Elsevier Science Ltd


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concerning the materials elastic properties, the char-

acter and extent of its plastic deformation and its yield

and tensile strength. The time dependence of the

mechanical viscoelastic behaviour of the LDPE film is

not taken into account in the present work (e.g. creep

behaviour). In general, the polyethylene film exhibitstensile stressstrain relationships typical of a ductile

material with low modulus, low yield stress but very

high elongation and high stress at break (Briassoulis

et al., 1997a).

The mechanical characteristics of the films under

investigation were measured by applying standardised

testing methods (Dilara & Briassoulis, 1998; Briassoulis

et al., 2000). In particular, the tensile properties were

obtained by employing the standard testing method of

ASTM D882 (1991) or its equivalent ISO 527-3 (1995),

which is suited for films less than 025 mm thick and

employs specimens in the form of strips of constant

width. Details of the procedure are given in Briassoulis

and Aristopoulou (2001). Figure 1 shows engineering

stressstrain curves obtained in the laboratory for the

LDPE greenhouse film in the parallel direction (i.e. the

machine direction of the film tube during extrusion)

[Fig. 1(a)] and in the transverse direction (i.e. the

direction along the circumference of the film tube)

[Fig. 1(b)]. In the parallel direction the behaviour of the

film may be idealised by a bilinear model exhibiting an

elasticplastic behaviour with strain hardening. In the

transverse direction, the curve after yielding may be

assumed to exhibit a bilinear behaviour (Briassoulis &

Aristopoulou, 2001): a linear strain hardening behaviourwith a very low tangent modulus up to a strain equal

half the strain at break, (at a strain of 300%) followed

by a steeper linear strain hardening behaviour, up to the

limit point of ultimate stress (Briassoulis & Aristopou-

lou, 2001). For practical engineering analysis purposes,

assuming that the strain remains relatively low (e.g. up

to 50%), the behaviour of the film in the transverse

direction may also be idealised by a bilinear model

exhibiting an elasticplastic behaviour with strain

hardening (using the low tangent modulus of the first

linear part of the strain hardening behaviour). In the

present work, the numerical analysis is performed

assuming linear elastic behaviour. It is important

though that the reader is aware of the fact that the

material behaviour is really non-linear even in the elastic

region (mainly for stress above half the yield stress;34 MPa).

2.1.2. Membrane behaviour

The mechanical behaviour of the film had to be

simulated with appropriate membrane models because,

in engineering terms, the LDPE film acts as a

membrane. A way to evaluate possible membrane

models, is by numerically simulating specific laboratory

tests where the membrane action is in effect and by

comparing the numerical results against the correspond-

ing results obtained in the laboratory. Since a membrane

has no resistance in compression, a laboratory test

for film specimens was sought which introduces a

complex state of stress, including compressive

stresses. In fact, such a test concerns the measure-

ment of the initial tear resistance, a property that

is of major importance for the greenhouse LDPE

film, and it is measured by the conventional tension

testing machines in terms of the total force for initia-

tion and propagation of tear according to the stan-

dard testing method ASTM D1004 (1990) or its

equivalent ISO 34-1 (1994). The specimen has a

unique shape that induces the development of bothtensile and compressive stresses. However, due to

membrane action and its unique shape, the state of

stress ends up to the development of a high stress

concentration at the limited pre-designed location

where tear initiates, whereas it is zero elsewhere. The

linear elastic behaviour of the initial tear resistance

specimen obtained in the laboratory at low level

loads was used in Briassoulis and Schettini (2000) to

check the membrane behaviour of the numerical

models described below.

Notation

a square plate side length, m

b/a aspect ratio

E modulus of elasticity, Pa

L side length, m

q pressure, Pa

t thickness, m

w0 deflection at the centre of the plate, m

n Poissons ratio

s tensile stress at yield, Pa

sc stress at the film centre, Pa

seff effective stress based on Von-Mises criterion,

Pa

s0 pre-tension stress, Pa

Subscripts

x parallel direction

y transverse direction

D. BRIASSOULIS, E. SCHETTINI304


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2.2. Numerical simulation of the mechanical behaviour of

low-density polyethylene films

ANSYS (2000), a commercial finite element program,

was initially employed to simulate the behaviour of the

LDPE plastic film during the tensile and initial tear tests.

Film specimens were modelled numerically with the

corresponding specific geometry and with the anisotro-

pic material properties of the film obtained in the

laboratory. The models were studied numerically under

appropriate boundary conditions and by applying

different analysis options such as linear elasticity,

membrane action and plasticity.

Two specific ANSYS finite elements, a membrane

shell and a finite large strain shell element, were

investigated with respect to the best possible simulation

of the films behaviour under various testing configura-

tions (i.e. complex states of stress, not just tensile)

(Briassoulis & Schettini, 2000). The membrane element

available in ANSYS has membrane (in-plane) stiffness

but no bending (out-of-plane) stiffness and it is capable

of modelling membranes with variable thickness, stress

stiffening and large deflection. This membrane element is

applicable only for linear elastic analysis and has a

membrane-on option that allows the element to

wrinkle when it goes into compression (i.e. allows for

the activation of membrane action). The finite largestrain shell element is suitable for analysing thin shell

structures for linear, large rotation and large strain non-

linear applications. This shell element is well suited to

model non-linear material behaviour including plasticity

but it does not allow for membrane action. So, there is a

need to introduce indirectly an artificial membrane

action (e.g. through the material model; a rather

complicated procedure).

The numerical results obtained by applying the finite

large strain shell element in order to simulate the

laboratory tensile test are shown in Fig. 1. The agree-

ment between the finite element analysis using theproposed bilinear model exhibiting an anisotropic

elasticplastic behaviour with strain hardening and the

laboratory test appears to be reasonably good in both

parallel and transverse direction (Briassoulis & Schetti-

ni, 2000).

The numerical results obtained with the membrane

element used to simulate the initial tear resistance test

are described in detail in Briassoulis and Schettini

(2000). The analysis was carried out by activating the

membrane option (i.e. the membrane stiffness acts only

in tension and collapses in compression) and by not

including material non-linearities. It should be noted

here that the membrane behaviour is a non-linear

procedure by itself for the finite element analysis even

for modelling linear elastic material behaviour. The non-

linear membrane behaviour, in this particular case, was

modelled at a low load due to the high stress

concentration developing locally at the region where

tear initiates and the subsequent convergence problems

encountered at higher loads. The results obtained in

Briassoulis and Schettini (2000) suggest a good model-

ling of the membrane behaviour of LDPE films by the

ANSYS membrane element in agreement with the

corresponding laboratory initial tear test results.

3. Experimental procedures

3.1. Experimental arrangement

The experimental arrangement to study the mechan-

ical behaviour of a square pre-tensioned LDPE film

under pressure was developed at the Department of

Agricultural Engineering, Agricultural University of

Athens, (Briassoulis & Schettini, 2001; Schettini &

0

5

10

15

20

25

0 1 2 3 4 5 6

Strain , mm/mm

Str

ess,

MPa

0

5

10

15

20

25

0 1 2 3 4 5

Strain , mm/mm

Stress,

MPa

(a) (b)

Fig. 1. Stressstrain curves in the parallel direction (a) and in the transverse direction (b) for a low density polyethylene greenhousefilm: , measured in laboratory following standard testing method; , measured in laboratory employing a slow strain rate

(10 mm/min); , predicted by ANSYS applying the finite large strain shell element and a bilinear material model

ANALYSIS AND DESIGN OF LOW-DENSITY GREENHOUSE FILMS 305


4/12

Briassoulis, 2001). An open wooden box is fixed within a

steel frame supporting construction (Fig. 2). A system of

two steel frames restrains the film and provides a test

area of 1 m2. Rubber with sand paper coating placed on

the inner face of the lower frame and on the top edge of

the wooden box hold the film and prevent slippage of

the film. The pre-tension of the film is obtained byapplying appropriate weights along all the sides of the

film before the frames are secured to the box using

screws.

The pressure inside the box, applied by means of

pressurised air, is monitored by a pressure transducer

HCXM100D6 V made by Sensor Technics (Aubinger

Weg 27, 82178 Puchheim, Germany). The deflection

along the central line of the film is evaluated by

employing an image capturing and processing technique

using a digital camera. A card PCI-1710 HG and itssoftware (made by Advantech Co. Ltd., Taiwan) is used

to read and store all the results obtained during the

execution of the tests.

0.17 m

0.9

0m

Weights

Weights

Strain gauge

Gauge length

0.15 m

Steel arch support

Fig. 3. The experimental arrangement used to evaluate the Poissons ratio of the low density polyethylene film tested with a sampleof an I cross-section

Fig. 2. (a) The experimental arrangement used for pre-tensioned film under pressure; (b) schematic layout of the apparatus used



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3.2. Poissons ratio

A second experimental arrangement was developed

to evaluate the Poissons ratio of the LDPE film

tested (Briassoulis & Schettini, 2002). The Poissons

ratio n, representing the negative ratio of lateral

strain to longitudinal strain under conditions of uniaxialstress within the proportional or elastic limit, is rarely

reported in literature for LDPE. Courtney (1990)

considers that a reasonable value for n for polyethylene

is 04.

Samples of an I-section shape were cut following

both the parallel and the transverse directions of the

plastic film. The central part (vertical section) of the

sample had a constant width of 017 m. The samples

were fixed at the top flange. The tensile load was applied

gradually along the direction of the longitudinal axis

(vertical section) of the sample at the lower part,

which was free to move downwards (Fig. 3). In thecentral part of the sample vertical and horizontal lines

corresponding to predefined distance (gauge lengths)

were clearly marked and the displacements of those lines

(of known initial length) were evaluated by using a

manual technique and also by an image capturing

and processing technique. It was possible to get an

estimation of the Poissons ratio of the LDPE film in

the range of 04 05.

The measurement of strains at selected points

of the film was also attempted by using specific

commercial strain gauges KLM-6-120-A9 for polymers

made by Kyowa Electronic Instruments Co. Ltd. (3-5-1

Chofugaoka, Chofu, Tokyo, 182-8520, Japan). Unfor-

tunately, many attempts to measure strain on the LDPE

film failed because of the failure of the available

sensors to work on LDPE film (Briassoulis & Schettini,

2002).

4. Analytical and numerical procedures

4.1. Analytical solution

As a first approach, an analytical solution for a

thin plate (i.e. a flat membrane) exhibiting largedisplacements was used to evaluate the performance of

the finite element models used in the present work.

In particular, Timoshenko and Woinowsky-Krieger

(1959) give an approximate solution for a uniformly

loaded square plate with sides of length 2a and

clamped edges. This solution does not account for

any pre-tension stress on the plate. This method

consists of a combination of the known solutions given

by the theory of small deflections and the membrane

theory.

Assuming an elastic-membrane behaviour, the pres-

sure q can be calculated by:

qw0Et

3

a4 137194

w20t2

1

where E is the modulus of elasticity in Pa, t is the

thickness of the plate-membrane in m, w0 is the

deflection at the centre of the plate in m; the numerical

constants depend on the geometry of the plate and on

Poissons ratio n (a value n of 025 is assumed in the

analytical derivations of Eqn (1)).

4.2. Numerical simulation of the mechanical behaviour of

low-density polyethylene film under various combinations

of pre-tension and uniform pressure schemes

The mechanical behaviour of the LDPE film tested

experimentally under various combinations of pre-tension and uniform pressure schemes was also simu-

lated numerically by using ANSYS and a research

program employing the Reformulated Four Node Shell

element (RFNS) (Briassoulis, 1996, 2002a, 2002b,

2002c). The material properties of the LDPE film used

were obtained in the laboratory by applying standard

testing methods (Briassoulis & Aristopoulou, 2001). The

modulus of elasticity in the parallel direction Ex was

10297 MPa while in the transverse direction Ey was

11035 MPa; the tensile stress at yield in the parallel

direction sx was 798MPa and in the transverse

direction sy was 699 MPa. Assuming average isotropic

properties, the material properties used in the numerical

models were: modulus of elasticity E of 10666 MPa,

thickness of the film t of 0195 mm and stress at yield

syof 798MPa. In some cases, the analyses were

performed assuming different values for the Poissons

ratio n in order to investigate the effect of the

Poissons ratio on the numerical results. In general, a

Poissons ratio n of 04 was assumed for the LPDE film

tested experimentally in accordance with literature

(Courtney, 1990) and with the relevant experimental

results (Briassoulis & Schettini, 2002). Material non-

linearities and time-dependent behaviour were not

considered in the present research work.The ANSYS membrane element was chosen to

simulate the films elastic behaviour. The models used

for a quarter of the square film were constructed with

416 elements. Non-linear analysis, including membrane

action and geometric non-linearities such as large strain,

large displacements and stress stiffening have been

considered. The material behaviour has been considered

to remain linear elastic.

As far as the RFNS element is concerned, the

formulation of this element is based on physical



6/12

concepts as it is presented in detail in Briassoulis (1996).

The reliability and efficiency of the RFNS element was

established by means of the classical benchmark tests

and many comparative numerical studies (Briassoulis,

1996). The performance of the RFNS element was also

re-confirmed in terms of its asymptotic behaviour by

means of classical and new limit tests (Briassoulis, 2002aand 2002b). The RFNS formulation was extended

currently to incorporate geometric non-linear behaviour

(Briassoulis, 2002c). The total Lagrangian formulation

is employed. The RFNS element allows for membrane

action, large displacements and large rotations. The

RFNS element was used in the present work by

employing a mesh with 100 elements (10 10 model)

and in few cases a finer mesh with 225 elements (15

15 model).

5. Results and discussion

At a first step, an elastic-flat membrane made of an

arbitrary material (i.e. not a LDPE film) with Poissons

ratio nof 025 was modelled by employing the two finite

element models. The choice of an arbitrary material with

n of 025 had to be made in accordance with the

analytical solution available through Eqn (1); so this

choice serves only comparison purposes for the finite

elements models used. The membrane is assumed to be

square, clamped along the four sites as assumed by the

analytical solution of Timoshenko and Woinowsky-

Krieger. The maximum deflections w0 at the centre ofthe membrane obtained numerically for different values

of the applied uniform pressure are shown in Table 1

against the corresponding analytical results. The agree-

ment obtained between the three solutions is good.

The interaction pressurepre-tension stress curves

(including a case of zero pre-tension) are shown in

Fig. 4 assuming an arbitrary material with Poissons

ratio v of 025 for comparison purposes against the

corresponding values obtained by Timoshenko and

Woinowsky-Krieger (1959) for the case of no pre-

tension. The true Cauchy stress s is employed with the

RFNS element. A very good agreement is verified in this

case for the no pre-tension case against the correspond-

ing analytical solution.

In a second step, comparisons were made between the

experimental results of the LDPE film tested in the

laboratory under the experimental arrangement ofFig. 2and those obtained from the numerical modelling of the

same LDPE film arrangement (in terms of material

properties, boundary conditions and loading; Poissons

ratio n is 04) for different levels of pressure and for a

pre-tension stress s0 of 03 MPa. For this purpose,

appropriate finite element models were used in order

to simulate the experimental arrangement (boundary

conditions, loading conditions and material properties)

of the film. As shown in Table 2, for low values of

pressure (i.e. in the linear elastic region) the agreement

between the experimental results and the numerical

Table 1Deflection at the centre of an elastic-membrane material with Poissons ratio mof 025 applying Eqn (1) and numerical analyses

Pressure (q), Pa Deflection at the centre (w0), m

Analytic solution Numerical solution Numerical solutionEqn (1) ANSYS membrane element RFNS element

250 00729 00661 00675800 01074 00960 01002

1500 01324 01234 012432400 01549 01470 01465

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 0.05 0.1 0.15 0.2 0.25

Normalised pressure qL(1- 2)/Et

Normalisedstress

c

(1-

2)/E

Fig. 4. Normalised pressure q and maximum stress sc at centreof pre-tensioned film curves applying Reformulated Four NodeShell element assuming Poissons ratio n of 025 for differentvalues of pre-tension stress s0: , s0 of 0 MPa; , s0 of

282 MPa; , s0 of 5 MPa; , analytical by Timoshenkoand WoinowskyKrieger; E, modulus of elasticity; L, side

length; t, thickness



7/12

analysis results is good. For higher values of the

pressure the numerical analysis is only approximate

(linear elastic). This is because for values of effective

stress (e.g. Von Mises stress) above 3 MPa, the

behaviour of the material becomes rather non-linear as

shown in Fig. 1 (i.e. material non-linearity). Thus, it

should be expected that the experimental values for the

maximum deflection become increasingly higher than

those calculated numerically based on the assumption of

a linear elastic behaviour.

The experimental results for the maximum deflection

w0 at the centre are compared with the corresponding

numerical analysis results inFig. 5 for two cases of pre-

tension stress s0 applied on the LDPE plastic film of 0and 117 MPa. Usually farmers stretch the greenhouse

plastic film by hand and so the pre-tension applied to the

film is never very high in reality. The agreement between

the experimental results and the two numerical solutions

is very good for low values of pressure (i.e. in the linear

elastic region). For high values of the pressure the

numerical analyses results are only approximate because

of the non-linear material behaviour of the film that is

not yet considered in the numerical simulations.

Introducing material non-linearities in the ANSYS

membrane element was not possible, while the use of

the plastic strain shell does not allow for membrane

action. As a result, a possible extension of the numerical

analysis to combine membrane action and material non-

linearities would require further research development

of the finite element models.

Figure 6 shows, for three different values of pre-

tension stress s0, the relationship between the maximum

deflection w0 at the centre of the film and the value of

pressureq, respectively, for the experimental results and

for the ANSYS numerical solution. In both cases the

increase of the pre-tension stress s0results in a decrease

of the maximum deflection w0 for the same value of

pressure, as expected. It should be noted however, thatthe pre-tension cases in the experiment yielded only

slightly different results.

The experimentally obtained deflections for different

values of pressure are compared against the correspond-

ing ANSYS (Fig. 7(a)) and RFNS (Fig. 7(b)) numerical

results for deflections, in the case of a fixed pre-tension

stress s0of 046 MPa. Both numerical models simulated

adequately the deflection pattern. For low values of

pressure the numerical analyses results are in good

agreement with the experimental results while for higher

Table 2Numerical and experimental deflection at the centre of the low density polyethylene film for a pre-tension stress r0 of 0.3 MPa and

for Poissons ratio mof 04

Pressure (q), Pa Deflection at the centre (w0), m

Experimental results Numerical solution Numerical solutionANSYS membrane element RFNS element

130 0047 0042 0046250 0050 0055 0059600 0079 0080 0082

1500 0120 0113 01132500 0159 0139 0134

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.5 1 1.5 2 2.5

Pressure q , kPa

Deflectionwo,m

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.5 1 1.5 2 2.5

Pressure q , kPa

Deflection

wo,m

(a) (b)

Fig. 5. Deflection w0 at the centre and pressure q for a pre-tension stress s0 of 0 MPa (a) and of 117 MPa (b): , experimentalresults; , applying ANSYS finite large strain shell element; , applying Reformulated Four Node Shell element



8/12

values of pressure the experimental deflections are

consistently higher. These differences have been ex-

plained already in terms of the non-linear behaviour of

the plastic film and the assumed linear behaviour by the

numerical analysis models.

The interaction curves between the pressure q and the

stress sc at the centre of the LDPE film for different

values of pre-tension stress so are shown in Fig. 8. The

stress scwas calculated at the film centre for the ANSYSelement. For the RFNS model using 100 elements, the

stress/strain is calculated at the Gaussian integration

point located closest to the film centre. It should be

noted that the stress measure used with the ANSYS

membrane element is the rotated Cauchy stress as

compared to the true Cauchy stress used with the RFNS

element (in this case, the true Cauchy stress scat centre

is equal to the corresponding effective stress seff based

on Von-Mises criterion; note that the true Cauchy stress

also coincides with the 2nd PiolaKirchhoff stress

measure at the centre of the film). For the no pre-

tension case the two models yield results in a very good

agreement. However, there is a deviation in the values of

the stress scat the centre of the film obtained by the two

different numerical models for the two pre-tension cases.

This is attributed to the different way the pre-tension is

applied in each model. In the ANSYS model the value of

the pre-tension was applied along the boundaries, which

were assumed to be free to move horizontally. Thisvalue consists of a combination of the pre-tension stress

value applied along the edges in the experimental

arrangement plus an additional stress equivalent to that

of the reaction forces calculated by assuming fixed

boundaries (since in the experimental arrangement the

film is actually fixed along the sides). With the RFNS

model the pre-tension is directly applied to the 2nd

PiolaKirchoff stress tensor (i.e. a stress measure which

is referred to the un-deformed configuration of the film)

by means of the specific in each case pre-tension stress

0

0.02

0.04

0.060

.08

0.1

0.12

0.14

0.16

0.18

0 0.5 1 1.5 2 2.5

Pressure q , kPa

Deflectionwo,m

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.5 1 1.5 2 2.5

Pressure q , kPa

Deflectionwo,m

(a) (b)

Fig. 6. Experimental (a) and ANSYS (b) interaction curves between the deflection w0at the centre and the pressure q for differentvalues of pre-tension stress s0: , 0 MPa; , 046 MPa; , 117MPa

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Abscissax , m

Deflectionw,m

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Abscissax, m

Deflectionw,m

(a) (b)

Fig. 7. Deflection patterns obtained experimentally and numerically for a pre-tension stress s0of 046 MPa: (a) applying ANSYS;

(b) applying Reformulated Four Node Shell element. Deflections are calculated for different values of pressure q. In the experimentalanalysis: , q of 650 Pa; , q of 1500Pa; , q of 1950Pa. In both the numerical analyses: , q of 650 Pa; , q of 1500Pa;

, q of 1950 Pa



9/12

s0. In both cases an increase of the pre-tension stress s0results in an increase of the stress scat the centre of the

film for the same value of pressure.

Figure 9 verifies the good agreement between the two

numerical models for the axial membrane stress sof the

LDPE film in the case of a fixed pre-tension stress s0of

046 MPa for different values of pressure (taking into

account the effect of the different ways the pre-tension

stress is applied in the two models; the 2nd Piola

Kirchoff stress is plotted in this particular figure for the

RFNS element for comparative purposes, since 2nd

PiolaKirchoff stress measure is closer to the rotated

Cauchy stress calculated by the ANSYS membrane

element). Based on these results, the RFNS element is

considered to model adequately the membrane beha-

viour of the LDPE greenhouse film.

6. Design criteria for the pre-tensioned LDPE greenhousefilms

The design of plastic greenhouses may follow the

recently developed European standard for greenhouses

(prEN-13031-1, 2001). This standard covers all aspects

of load calculations and structural design criteria, by

providing complementary information to the relevant

structural Eurocodes (e.g. ENV 1993-1-1, 1994; ENV

1991-2-4, 1994; ENV 1993-1-1, 2000). However, no

information is provided concerning the design criteria/requirements of the plastic film coverings of plastic

greenhouses. This missing information is very important

as it concerns the load carrying capacity of the film and

the load transfer or redistribution mechanisms from the

film to the main frame.

Consequently, based on the previous analysis, the

RFNS finite element model, already verified to model

adequately the membrane behaviour of the film tested

experimentally, was used to develop some preliminary

design criteria for LDPE greenhouse films. In particular,

the non-linear formulation of the RFNS element was

used in modelling thin rectangular a b elastic LDPE

films under uniform pressure in combination with

various pre-tension stresses. The model used had the

following geometric and mechanical characteristics: the

side length (L equal a) of the membrane was of 10 m,

the thickness t was of 01954 mm, the elastic modulusE

was of 106 MPa, the Poissons ratiov was of 04 and the

side length ratiosb/a (aspect ratios) considered were 05,

1, 2 and infinite. The results obtained were normalised

to be of general use for design purposes.

Starting from the case of a square panel supporting

the film and assuming for example that the allowable

effective stress seff (e.g. Von Mises stress) of the film is

equal to 4 MPa, that is approximately half of the yieldstress sy of the material (Fig. 1), one may use Fig. 10 to

determine for a given pre-tension stress directly the

maximum allowable pressure, or for a given pressure

(e.g. wind and/or snow design loads) the maximum

allowable dimension of the square panel of the film

(considering an appropriate partial safety factor for the

material;e.g. 2). The maximum effective stress, as shown

inFig. 10, corresponds to the stress at the film centre in

the range of the lower pressures (i.e. for normalised

stress: seff(1-v2)/Ebelow 0006, 0024 and 0041 for the

0

1

2

3

4

5

6

7

8

9

0 0.5 1 1.5 2 2.5 3

Pressureq, kPa

Stress

c,MPa

Fig. 8. ANSYS and Reformulated Four Node Shell elementinteraction curves between pressure q and stress sc at the centreof the low density polyethylene film for different value of pre-tension stress s0. In ANSYS analyses: , s0of 0 MPa; ,s

0 of 046 MPa; ....., s

0 of 117 MPa. In Reformulated FourNode Shell element analyses: , s0 of 0 MPa; , s0 of046MPa; , s0 of 115MPa

0

1

2

3

4

5

6

7

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Abscissax , m

Stress,

MPa

Fig. 9. ANSYS and Reformulated Four Node Shell elementaxial membrane stress sof the low density polyethylene film fora pre-tension stress s0 of 046 MPa and for different values of

pressure q. In ANSYS analyses: , q of 150 Pa; , q of650 Pa; , q of 1500Pa; , q of 1950Pa. In ReformulatedFour Node Shell element analyse: , q of 150 Pa; , q of 650 Pa;

, q of 1500MPa; , q of 1950Pa



10/12

cases of pre-tension stress s0of 000, 046 and 115 MPa,

respectively) whereas it corresponds to the effective

stress value calculated at the mid-side of the film edge

for higher pressures. Nevertheless, it is apparent in

Fig. 10 that the effective stress does not vary consider-

ably along the centre line of the film panel in contrast

with the single axial stress components perpendicular to

the film edge, at the mid-side, which may be significantly

higher than the stress at the film centre (Fig. 9). Mesh

refinement is shown to result in the calculation of almost

identical results for the stress. Thus the agreement

observed in Fig. 10 between the stress calculated by

using the model with 100 elements as compared to the

stress calculated by using a 225 elements model,

confirms that a satisfactory convergence is already

achieved by using the 100 elements model. Similar

observations may be made with the maximum mem-

brane strain calculations (i.e. the membrane strain

components directed along the film centre line), shownin Fig. 11. Notice, however, the significant difference

between the single membrane strain components calcu-

lated at the film centre and those calculated at the film

mid-side. These differences reflect the variation of the

corresponding axial membrane stress components,

shown in Fig. 9.

Taking into account the fact that the film supporting

systems may represent panels of different geometries, the

maximum displacement and the maximum effective

stress were calculated as a function of the panel aspect

ratio b/a, as shown in Figs. 1215 (no pre-tension is

considered in these cases). It is apparent that for an

aspect ratio greater than 2, the behaviour of the film

remains practically constant in terms of the normalised

quantities shown in Figs. 1215. Taking into account

more demanding situations (e.g. such as curved roofs or

tunnels covered by LDPE film) would require specific

finite element analysis.

The loads transferred to the main-frame structure (i.e.

to the supporting greenhouse structural elements)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.00 0.04 0.08 0.12

Normalised pressureqL (1- 2)/Et

Normalisedstress

eff(1-

2)/E

Fig. 10. Normalised effective stress seff at centre and middleedge of a square film under normalised pressure q for different

pre-tension stress s0. Applying Reformulated Four Node Shellelement model with 100 elements in the case of normalised stress

at centre: , s0of 0 MPa; , s0of 046 MPa; , s0of115 MPa; and for normalised stress at middle edge: , s0of0 MPa; , s0of 046 MPa; , s0of 115 MPa. Using 225elements in the case s0 of 0 MPa: , normalised stress atcentre; , normalised stress at middle edge; E, modulus of

elasticity; L, side length; t,thickness; v, Poissons ratio

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.00 0.04 0.08 0.12

Normalised pressureqL(1- 2)/Et

Normalisedstrain

c

(1-

2)

Fig. 11. Normalised strain ec at centre and middle edge of asquare film under normalised pressure q for different pre-tensionstress s0. Applying Reformulated Four Node Shell elementmodel with 100 elements in the case of normalised stress at

centre: , s0 of 0 MPa; , s0 of 046 MPa; , s0 of115 MPa; and for normalised stress at middle edge: , s0of0 MPa; , s0 of 046 MPa; , s0 of 115 MPa. Applying

225 elements in the case ofs0of 0 MPa: , normalised stressat centre; , normalised stress at middle edge; E, modulus of

elasticity; L, side length; t, thickness; v, Poissons ratio

0.00

0.05

0.10

0.15

0.20

0.00 0.04 0.08 0.12

Normalised pressureqL (1- 2)/Et

Normalisedmaximum

disp

lacement

w

max

/L

Fig. 12. Normalised maximum displacements wmaxat centre ofa film panel and normalised pressure applying ReformulatedFour Node Shell element model for different aspect ratio b/a:

, b/a of 05; , b/a of 1; , b/a of 2; , b/a infinite;E, modulus of elasticity; L, side length; t, thickness; v, Poissons

ratio



11/12

correspond to the reaction forces developing at the

supporting edges of the film, as they are calculated by

the finite element model (using the RFNS or any otherappropriate finite element model). The reaction forces

developing along the supporting sides of the panel do

not follow the uniform pressure distribution but they

exhibit a rather non-uniform distribution.

7. Conclusions and prospective

The mechanical behaviour of low density polyethy-

lene (LDPE) films under various combinations of pre-

tension and uniform pressure schemes is investigated

experimentally by using a specifically designed experi-

mental arrangement and numerically by using the finite

element method of analysis. The experimental results

validated the performance of the selected ANSYS

membrane element as well as the Reformulated Four

Node Shell (RFNS) element performance in the linear

elastic region. Both finite element models may be used to

simulate adequately the mechanical behaviour of the

films investigated within the linear elastic range.

The RFNS finite element model is used subsequently

to study the load carrying and load redistributionmechanisms developed in greenhouse covering LDPE

films in transferring external pressure loads (such as

wind and snow loads) to the main structure. Some

preliminary design criteria are developed in the present

work that may be used directly for the design of

greenhouse LDPE films. The design criteria developed

are expected to help to formulate more detailed

technical suggestions, of a pre-normative nature, for

the reliable design of LDPE film covered greenhouses,

making more efficient use of LDPE greenhouse plastic

films.

Acknowledgements

The present work has been carried out under the

Marie Curie SMT Research Training Grant funded by

the European Commission: Analysis of the state of

stress of greenhouse plastic films under various loading

conditions.

The two authors shared programming and editorial

work equivalently.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 1 2 3 4 5 6

Aspect ratiob/a

Normal

isedmaximum

disp

lacement

w

max

/L

Fig. 13. Normalised maximum displacements wmaxat centre ofa film panel and aspect ratio b/a for different normalised

pressures qL(1-n2)/Et: , qL(1-n2)/Et of 0006; , qL(1-n2)/Et of 0024; , qL(1-n2)/Et of 0061; E, modulus of

elasticity; L, side length; t, thickness; v, Poissons ratio

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.00 0.04 0.08 0.12

Normalised pressureqL(1- 2)/Et

Normalisedstress

eff

(1-

2)/E

Fig. 14. Normalised maximum effective stress seffat centre ormidside of a film panel and normalised pressure q applyingReformulated Four Node Shell element model for differentaspect ratio b/a: , b/a of 05; , b/a of 1; , b/a of 2;

, b/a infinite; E, modulus of elasticity; L, side length; t,thickness; v, Poissons ratio

0

0.01

0.02

0.03

0.04

0.05

0.06

0 1 2 3 4 5 6

Aspect ratiob/a

Norm

alisedstress

eff

(1-

2)/E

Fig. 15. Normalised maximum effective stress seff at centre ormidside of a film panel and aspect ratio b/a for differentnormalised pressures qL(1-n2)/Et: , qL(1-n2)/Et of 0006;

, qL(1-n2)/Et of 0024; , qL(1-n2)/Et of 0061; E,modulus of elasticity; L, side length; t, thickness; v, Poissons

ratio



12/12

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