G Maier

F Hofmann

Performance Enhancements of Polymer-Matrix Composites by Changing of Residual Stresses

Keywords: A. Polymer-matrix composites, B. Hygrothermal effect, Mechanical properties, C. Residual stress, Thermomechanical treatment

Manufacturing induced residual stresses of polymer-matrix composites (PMC) reduce the tensile load at which first ply failure occurs. Thermomechanical treatments offer the potential to change these residual strains, but their application is hindered because the shape stability of PMC components is limited at treatment temperatures, which must be above the glass transition temperature of matrix.

This study describes successful localized treatments performed on a stress concentration induced by a circular hole in a laminate. Localization of the treatment allows significant property improvements at much lower treatments loads. Investigation of the influence of moisture pick up on treatment effectiveness revealed it reduces the benefits of treatment to a large extent. Calculations of local stresses between single fibres and predictions of laminate properties based on laminate plate theory using a quadratic failure criterion confirm the experimental results and provide confidence in the physical explanations offered for the measured effects.

Introduction

Most of applied polymer-matrix composites (PMC) designs consist of many plies, which are dictated by the stiffness and strength requirements of the component or product. The single ply is characterized by anisotropic behaviour and very large property differences such as stiffness and thermal expansion between the fibre and the matrix. In high performance PMCs, these characteristics give rise to large internal residual stresses even immediately after manufacture, in the unloaded case at room temperature [START_REF] Pagano | Evaluation of Composite Curing Stresses[END_REF][START_REF] Hashin | Analysis of Properties of Fiber Composites With Anisotropic Constituents[END_REF][START_REF] Collings | Hygrothermal Effects in CFRP Laminates: Strains Induced by Temperature and moisture[END_REF][START_REF] Bailey | On transverse cracking and longitudinal splitting behaviour of glass and carbon fibre reinforced epoxy cross ply laminates and the effect of Poisson and thermally generated strain[END_REF][START_REF] Tsai | Introduction to Composite Materials[END_REF]. These stresses are fundamentally due to curing at elevated temperature. When injection technology is used instead of Prepreg technology, there is an additional source of residual stresses: matrix shrinkage [START_REF] Rudd | Liquid Moulding Technologies[END_REF].

During usage of a laminate, these internal stresses are changed by moisture pickup and the resulting volumetric expansion of the matrix. These effects depend on environmental conditions, time of exposure, diffusion coefficient and geometric parameters [START_REF] Tsai | Introduction to Composite Materials[END_REF][START_REF] Springer | Environmental effects on composite materials[END_REF]. Finally, the loading itself can induce additional inhomogeneities in the stress/strain distribution, particularly in those plies with reinforcementoriented transverse to the primary loading direction [START_REF] Tsai | Introduction to Composite Materials[END_REF]. The superposition of all of these spatially varying stresses results in a very inhomogeneous stress distribution and creates local stress concentrations in the matrix throughout the whole laminate. These stress concentrations in turn result in a much lower tensile load sustained at first ply failure stress/strain than the load sustained at final failure stress/strain [START_REF] Kies | Maximum Strains in the Resins of Fibre Glass Composites[END_REF][START_REF] Masters | Basic Failure Modes of Continuous Fiber Composites[END_REF][START_REF] Van Daele | Matrix Cracking in Cross Plied Thermosetting and Thermoplastic Composites During Monotonic Tensile Loading[END_REF]. This fact prevents a designer from fully exploiting the potential weight savings of PMCs, because in many applications, the components should be designed so as to preclude ply failure at any location. First ply failure makes the components more vulnerable to subsequent degradation, for example accelerating environmental effects (e.g. moisture pick up) as well as fatigue crack growth [START_REF] Brien | Characterization of Delamination Onset and Growth in a Composite Laminate[END_REF][START_REF] Jamison | Characterization and Analysis of Damage Mechanisms in Tension-Tension Fatigue of Graphite/Epoxy Laminates[END_REF][START_REF] Maier | Damage Development in Carbon Fibre-Reinforced Polyimides in Fatigue Loading as a Function of Stress Ratio[END_REF][START_REF] Maier | Notch Sensitivity of Carbon Fibre-Reinforced Polyimides in Fatigue Loading as a Function of Stress Ratio[END_REF][START_REF] Ch | Fatigue Damage Response of CFRP With Toughened Matrices and Improved fibres[END_REF][START_REF] Konur | Effect of the Properties of Constituents on the Fatigue Performance of Composites[END_REF].

In the past this problem has been accepted, due to the lack of a simple solution.

Approaches to change the internal stresses of PMC by applying thermomechanical treatment have been very successfully applied to laboratory scale specimens or simple components [START_REF] Schürmann | Gezielt eingebrachte Eigenspannungen erhöhen die Belastbarkeit von Bauteilen aus Faser-Kunststoff-Verbunden[END_REF][START_REF] Schürmann | Belastbarkeit von Bauteilen aus Faser-Kunststoff-Verbunden durch gezielt eingebrachte Eigenspannungen[END_REF][START_REF] Maier | Verfahren zur Herstellung eines festigkeitsgesteigerten Produkts aus Duroplast[END_REF][START_REF] Maier | Performance Increase of Polymer Composites by Thermal Treatment Under Load[END_REF][START_REF] Maier | Property Improvements After Loading at High Temperature[END_REF][START_REF] Schulte | Influence of artifical prestressing during curing of CFRP laminates on interfibre transverse cracking[END_REF][START_REF] Cuntze | Increasing the limit of usability of CFRP tubes by built-in-stresses[END_REF][START_REF] Maier | Festigkeitssteigerung von ebenen Hochleistungsfaserverbund-Laminaten mit polymerer Matrix durch thermomechanische Behandlung[END_REF][START_REF] Maier | Mechanisms of Property Improvements of Polymer Composites by Thermomechanical Treatment[END_REF][START_REF] Bartosch | Verbesserung des Zwischenfaserbruchverhaltens von CFK-Laminaten bei praxisrelevanten Belastungen durch thermomechanische Behandlung[END_REF]. However, applications to large or complex components have been limited, primarily due to problems concerning components` shape stability during relatively high stress loading at temperatures above glass transition. This is a fundamental step in the application of thermomechanical treatments.

In this study, thermomechanical treatments were performed so as to localize the treatment to failure critical areas, thereby reducing the overall load on a component during treatment. Additionally, the influence of residual stress change by moisture pick up on untreated and differently treated specimens was investigated.

The intent was to get a real view of potential property improvements under various service conditions by alteration of residual stresses. The results of this study should also provide insight into the potential property improvements which may result from future modification of matrix materials to contain nanoparticles [START_REF] Schadler | Polymer-based and polymer-filled nanocomposites[END_REF][START_REF] Adebahr | Reinforcing Nanoparticles in Reactive Resins[END_REF][START_REF] Preghenella | Thermo-mechanical characterization of fumed silicaepoxy nanocomposites[END_REF][START_REF] Zhang | Property improvements of in situ epoxy nanocomposites with reduced interparticle distance at high nanosilica content[END_REF].

Such matrix materials are thought to have less thermal expansion and/or less moisture pick up than conventional ones.

Experimental

Materials and specimens preparations

All investigations is this study were performed with a composite consisting of continuous carbon fibres T800 in a thermosetting matrix Cycom 5245C, former trade

Thermomechanical treatment of specimens

A load frame designed for uniaxial creep testing at elevated temperatures up to 250 °C and tensile loads up to 100 kN was used for thermomechanical treatments. In general, these treatments were all based on the following steps, namely quick heating up to treatment temperature, uniaxial loading at this temperature, cooling down under load and finally releasing from the treatment load. Different treatments were applied to completely cured specimens (Treatment A) and to partially cured specimens (Treatment B). In the first case, the treatment loading was performed above the glass transition temperature for only a short period of time (details in Table 1). In the second case, the specimens were held under reduced load at curing or post-curing temperatures until complete cure occurred. The load was then increased to a level higher than that applied in Treatment A (details in Table 1). The treatments were applied to both notched and unnotched specimens.

Unnotched specimens were notched following treatment. The determination of proper treatment conditions was described in detail in [START_REF] Maier | Property Improvements After Loading at High Temperature[END_REF][START_REF] Maier | Festigkeitssteigerung von ebenen Hochleistungsfaserverbund-Laminaten mit polymerer Matrix durch thermomechanische Behandlung[END_REF][START_REF] Maier | Mechanisms of Property Improvements of Polymer Composites by Thermomechanical Treatment[END_REF].

Determination of properties before and after treatment

Uniaxial tensile tests were performed according to DIN 65559, in order to determine the number of transverse cracks as increasing load was applied to the specimens. During these tests, specimens were loaded stepwise in increments (500 N for untreated and 1000 N for treated specimens) and monitored via acoustic emission (AE) measurements until rupture occurred. After receiving the first AE signals, the specimens were removed from the load frame and prepared for radiographic examination. The specimens were contrasted by storage in a contrast fluid (zinc iodide dilution containing 1200 g zinc iodide, 160 ml H 2 0, 200 ml wetting agent Agepon, 200 ml isopropanol) in a vacuum chamber for 4 hours. The contrasted specimens were than radiographed and the resulting images examined for indication of transverse cracks. The specimens were installed back into the load frame and loaded to the next load step, removed and radiographed again. This procedure was repeated until delamination occurred. After the onset of delamination the specimens were loaded up to final rupture and notch strength recorded.

Results

Moisture pick up of untreated and treated specimens

Fig. 1 illustrates the moisture pick up of untreated and different treated specimens versus storage time. It can be seen that treated specimens had less moisture pick up than the untreated ones, but there is essentially no influence between various types of treatment. The continuing moisture pick up over time is typical for the type of matrix used.

Tensile properties

Fig. 2 compares average laminate strains at first ply failure measured by AE on untreated and differently treated specimens in both dry and wet conditions. It can be seen that results depend on the measuring method applied. In dry and untreated wet specimens, approximately the same results were obtained by AE and radiography. However, a discrepancy is noted for wet treated specimens, particularly after Treatment B. It can be assumed this effect reflects the limited sensitivity of the X-ray method, because cracks initiated at the notch do not extend through the whole specimen width. The rapid decay of stress with distance from the notch [START_REF] Young | Roark's Formulas for Stress and Strain, Seventh Edition[END_REF] implies that cracks should arrest. Especially at first ply failure, crack length and opening displacements should be small, thereby limiting the amount of contrast fluid to infiltrate the specimen. Because this assumption is supported by calculations discussed later in this paper, AE results were chosen for evaluation of influence of different treatment and moisture pick up on first ply failure strain. As Fig. 2 illustrates, the first ply failure of specimens in both wet and dry conditions is shifted to higher strains by all types of thermomechanical treatments. The relative increases vary from 100 to 179% in the dry condition and from 30 until 49% in the wet condition, as compared to untreated specimens. This shows that the improvement by thermomechanical treatment is fundamentally diminished by subsequent moisture pick up. Treatment B provides a larger shift of these strains, namely 169 to 179% in the dry condition and 44 to 49% after moisture pick up compared to Treatment A with 100 to 121% dry and 30 to 41% wet. In spite of the approximately 50 % lower treatment load applied, specimens notched before treatment tend on average to show more substantial strain shifts (121% to 179% dry and 41 to 49% wet) than specimens notched after treatment (100 to 169% dry and 30 to 44% wet). These results confirm that localization of treatment to areas of stress concentrations allows a fundamental reduction in treatment load. Fig. 3 displays the transverse crack formation with increasing strain applied to dry specimens. As expected from the nature of such kinds of damage, there is a continuous increase in the number of transverse cracks with increasing strain. In the case of untreated specimens, there is a progressive increase of crack rate meaning that the rate of crack formation is low at low tensile strains and increases with increasing strain. A similar trend in crack rate can be seen for all specimens notched before treatment. Specimens notched after treatment reveal an approximately constant crack rate. The average crack rate (slope of the fitted crack number/tensile strain curve) of specimens notched before treatment is visibly larger than that of specimen notched after treatment. It is also significantly smaller in the case of Treatment B than in the case of Treatment A, despite much higher test strain level for Treatment B specimens.

The transverse crack formation of wet specimens is illustrated in Fig. 4. It can be seen that moisture pick up causes a fundamentally different trend in the crack number vs tensile strain curves. Untreated and Treatment A specimens show a decreasing crack rate with tensile strain, meaning that rate of crack formation is high at low tensile strains and decreases with increasing strains. In contrast, Treatment B specimens show still a progressively increasing crack rate, but the average crack rate is clearly higher than that of comparable dry specimens. As a consequence of this different crack formation trend, wet Treatment B specimens notched after treatment reach higher numbers of cracks at high tensile strains than all Treatment A specimens.

In contrast to transverse crack formation however, no correlation was observed between final failure stress and the changes of residual stresses by thermomechanical treatment and moisture pick up.

Calculation of internal stresses/strains

The fundamentals of thermomechanical treatment, the procedures to calculate internal stresses and strains before, during and after this treatment and the characteristics of the laminates used have all been previously reported in detail [START_REF] Maier | Property Improvements After Loading at High Temperature[END_REF][START_REF] Maier | Festigkeitssteigerung von ebenen Hochleistungsfaserverbund-Laminaten mit polymerer Matrix durch thermomechanische Behandlung[END_REF][START_REF] Maier | Mechanisms of Property Improvements of Polymer Composites by Thermomechanical Treatment[END_REF]. The following overview should provide a short introduction, to these procedures and serve to address the specifics of the new results published in this paper. Because of the well-defined relationship between stresses and strains and for the sake of clarity, mainly only strains are considered in the following discussion.

Laminate ply strains

Laminate properties like modulus and ply strains caused by manufacture or external loading of a laminate were calculated according to Laminate Plate Theory (LPT) [START_REF] Tsai | Introduction to Composite Materials[END_REF] using the properties of single plies. It has to be kept in mind from previous papers [START_REF] Maier | Property Improvements After Loading at High Temperature[END_REF][START_REF] Maier | Festigkeitssteigerung von ebenen Hochleistungsfaserverbund-Laminaten mit polymerer Matrix durch thermomechanische Behandlung[END_REF][START_REF] Maier | Mechanisms of Property Improvements of Polymer Composites by Thermomechanical Treatment[END_REF] that the permanent deformations which occur transverse to the reinforcement direction of single plies, govern the property improvement after treatment. These deformations can be realized by inducing strains at a temperature above the glass transition temperature of the matrix, and locking those strains in during cool down to a temperature below the glass transition temperature. In addition to the mechanical loading itself, temperature changes are a fundamental source of induced strains during treatment. They result also in strains mainly perpendicular to the ply reinforcement due to anisotropy of the thermal expansion and stiffness of the single ply.

The simplest condition from which to calculate the internal strains during and after thermomechanical treatment is the completely cured laminate at curing temperature, because it can be assumed to be stress free. In this case the first change of internal strains is caused during heating up to treatment temperature. A second change of ply strains is given by the mechanical loading while at treatment temperature. During the subsequent cooling down back to curing temperature, the first changes of internal strains are reversed. This means that heating up from curing temperature to treatment temperature and the reverse process have no influence on internal strains. The subsequent cooling down to test temperature causes further changes of ply strains. To enhance the accuracy of the calculation for this treatment step, the temperature range should be divided into two parts, from cure temperature to glass transition temperature and from glass transition temperature to test temperature, because permanent deformations can only be produced above the glass transition temperature. The dependence of glass transition temperature on loading condition [START_REF] Maier | Bestimmen des Glasübergangs von Faserverbundwerkstoffen im statischen Zugversuch[END_REF][START_REF] Maier | Zur Lastabhängigkeit der Glasübergangstemperatur von Faserverbundwerkstoffen[END_REF] as well as the above mentioned reduced stiffness contribution of plies and its effect on single ply strain during loading above the glass transition temperature must also be considered. Additionally, moisture pick up of a PMC in a humid environment causes swelling of the matrix and contributes to its internal strains. The cumulative resulting residual strain components of a ply "i" can be calculated by

{ε} (i)r = { (ε (i)Th-Mech + ε (i)Tg-Shift ) * F r + ε (i)Thermal + ε (i)Moisture } (1)
where {ε (i)Th-Mech } are the strain components by loading at treatment temperature, {ε (i)Tg-Shift } are thermal induced strain components during cooling down from curing temperature to shifted glass transition temperature, {ε (i) Thermal } are residual thermal induced strain components during cooling down from curing temperature to property test temperature, {ε (i) Moisture } are residual strain components caused by moisture pick up and F r is the quotient of applied strain and permanent deformation of a ply.

All of these strain components can be calculated on the basis of LPT [START_REF] Tsai | Introduction to Composite Materials[END_REF]. For calculation of stresses/strains at the notch during treatment and property testing, a stress concentration factor of 2.57 was applied, which was calculated as described for isotropic materials in [START_REF] Young | Roark's Formulas for Stress and Strain, Seventh Edition[END_REF] using the dimensions of the specimens researched in this study.

The accuracy of the ply strain/stress predictions depends on the selecting the proper ply characteristics, which in turn depend on load/strain level, temperature and moisture content. In Table 2 the best possible choice of ply characteristics used for prediction of laminate properties and the alteration of residual strains/stresses due to thermomechanical treatment and moisture pick up are listed.

The predicted transverse and shear strain components in untreated and different treated laminates in dry and wet conditions are compared in Fig. 5 These residual stresses can be added to the mechanically applied strains previously described [START_REF] Tsai | Introduction to Composite Materials[END_REF] using the following equation:

{ε} (i)res = {ε (i)r + ε (i)m } ( 2
)
where {ε} (i)res are the resulting ply strain components and ε (i)m the ply strain components caused by mechanical loading both given during property testing.

For the prediction of failure strain in this study a quadratic failure criterion [START_REF] Tsai | Introduction to Composite Materials[END_REF] was applied using a normalized interaction term of -0.5 (bounded by the generalized von Mises criterion). Fig. 8 to 10 show examples of failure envelopes for an artificial laminate in the absence of residual strains and for untreated and treated laminates in dry and wet conditions tested. The comparison of the envelope of a lami-nate without residual strains to that of a dry untreated laminate (Fig. 8) illustrates that production induced residual strains cause a reduction of the strain space/area without first ply failure, which takes place mainly in the tensile strain space. Thermomechanical treatment reduces or inverts this effect (Fig. 9). It cause an enlargement of first ply failure free strain space to higher tensile strains in laminate axis 1 compared to untreated laminates (Fig. 8), due to the reduction or inversion of residual strains in this direction mentioned above. Treatment B effects a greater enlargement of this failure free strain space than Treatment A because of higher permanent transverse ply deformation (Fig 5 to 7) due to the higher final treatment load (Table 1) and the higher degree of permanent retention of treatment strain following this treatment (ratio "permanent deformation"/"treatment strain", Table 2) (Fig. 9). In laminates without residual strains and in all dry laminates (untreated and treated), first ply failure is expected to occur in 90°-plies under uniaxial tensile load in laminate axis 1 (Fig. 8 and9) as applied during property testing in this study. Considering only the swelling effect, moisture pick up should also lead to an enlargement of first ply failure free strain space, due to the reduction or inversion of residual strains. But the large decrease of tensile strength in ply axis 2 limits this effect in the case of untreated laminates, and negates this effect in the case of treated laminates in the tensile strain space (Fig. 10). In the first case, the line of strain path for uniaxial tensile loading intersects the failure envelope of 90°-and 45°-ply at very similar strain values, which predicts a simultaneous ply failure initiation and development in both type of plies. This effect could be one reason for the much higher rate of transverse crack formation compared to untreated dry laminates (Fig. 3 and4). The predicted first ply failure of all treated laminates shifted from 90°-plies to 45°-plies by moisture pick up. The reason is that in the case of a poor transverse tensile loading, a reduction or inversion of residual tensile strains should contribute directly to the loading capacity to first ply failure, because the given internal material change and loading have approximately the same direction of effectiveness. This is not the case however in 45°plies, because an inversion of residual strain transverse to fibre direction does not improve the shear loading capacity of 45°-plies. The reason for lower crack formation rates for specimen notched before treatment compared with specimens notched after treatment cannot be derived from the calculations done in this study, because the influences of stress gradients around the notch were not investi-gated. But it can be assumed this effect is based in similar loading conditions during treatment and property testing, which cause a reduction of internal stresses similar to cases considered above.

In Fig. 11

Local micromechanical strains between fibres

A square fibre arrangement model was used to calculate local stresses/strains of matrix between single fibres. The smallest single element of volume/area of this arrangement, containing quarter segments of 2 fibres in cross section and the matrix filling the space between, was divided in subelements (slices) with very small thickness. This model allows an easy illustration and understanding of the cause of variation of local strain between two fibres. In principal, the dominant effects depend on the length fraction of fibre and matrix within the various subelements.

For instance, the slices closest to a line joining to fibres' axes will include most of each fibres' radii, but only a relatively small length of matrix. Those slices further from this line would have a correspondingly higher matrix/fibre ratio. By considering the differences in matrix/fibre ratio between subelements, the magnitude of the strains are given in [START_REF] Maier | Festigkeitssteigerung von ebenen Hochleistungsfaserverbund-Laminaten mit polymerer Matrix durch thermomechanische Behandlung[END_REF][START_REF] Maier | Mechanisms of Property Improvements of Polymer Composites by Thermomechanical Treatment[END_REF].

The following examples were selected to illustrate the variation of influences on residual stresses. Fig. 12 and 13 show local stresses transverse to fibre and parallel to the loading direction in unidirectional 90°-laminates (UDL) and 90°-plies of quasi-isotropic laminates (QIL), in dry and wet condition calculated with properties given in [START_REF] Maier | Mechanisms of Property Improvements of Polymer Composites by Thermomechanical Treatment[END_REF] or back calculated from UDL values (e.g. moisture expansion coefficient = 0.5/%). As already stated in [START_REF] Maier | Property Improvements After Loading at High Temperature[END_REF][START_REF] Maier | Festigkeitssteigerung von ebenen Hochleistungsfaserverbund-Laminaten mit polymerer Matrix durch thermomechanische Behandlung[END_REF][START_REF] Maier | Mechanisms of Property Improvements of Polymer Composites by Thermomechanical Treatment[END_REF], production of UDL forms fundamentally compressive stresses in areas where the fibre surfaces are relatively close together (small matrix/fibre ratio), balanced by fundamentally tensile stresses areas where the fibre surfaces are further apart (large matrix/fibre ratio) (Fig. 12). Loading transverse to the fibre direction itself creates an inverted stress distribution compared to the production induced one. The superposition of local stresses from production and loading up to failure results in approximately uniform local stresses between fibres, so that the transverse strength of dry UDLs is very close to that of the neat resin.

In a QIL, production induced stresses are positive and raise the average value of local residual stresses, thereby diminishing the load at which first ply failure of the laminate occurs (Fig. 12). Thermomechanical treatment shifts the local stress curves to lower values due to permanent deformation of 90°-plies transverse to fibre direction. The magnitude of this effect depends on the intensity of treatment.

In the most intensive treatments applied (Type B on specimens notched before treatment) predominantly compressive stresses occur, which fundamentally increase the loading capacity of the laminate. It should be noted that the contact of single fibres does not influence the maximum local stress. In [START_REF] Maier | Festigkeitssteigerung von ebenen Hochleistungsfaserverbund-Laminaten mit polymerer Matrix durch thermomechanische Behandlung[END_REF][START_REF] Maier | Mechanisms of Property Improvements of Polymer Composites by Thermomechanical Treatment[END_REF], it was already shown that no significant change of local stress distribution is caused by thermomechanical treatment beyond that of the previously described permanent deformation.

Moisture pick up reduces or inverts the production induced local stresses (Fig 13).

The extent of moisture pick up measured in this study is sufficient to induce tensile stresses in areas where the fibre surface are closer together (fibre centre).

The contact of single fibres amplifies this effect. Loading transverse to fibre direction creates a similar change of stress distribution as moisture pick up, which further increases the local stress magnification near the fibre centres. These combined effects lower the loading capacity of an UDL and are reflected in a low transverse strength of this type of laminate in the wet condition. In an unloaded QIL, the average swelling transverse to fibre direction reduces or inverts average production-induced residual stresses. It reduces the stress magnification in areas where the fibre surfaces are closer together (above the fibre centre). Subsequent loading would cause an increase of stress magnification as before, but compared to the situation in a wet UDL, the stresses are reduced somewhat by the previously mentioned swelling effect. The effect of thermomechanical treatment was already shown for dry QIL. In the case of subsequently moisturized specimens it produces a relatively high level of compressive stresses transverse to fibre direction. Assuming no compressive failure occurs, it should elevate the load capacity up to first ply failure of the laminate partly compensated by moisture reduced transverse ply strength. This prediction was confirmed experimentally.

Summary and Conclusions

In this paper, different thermomechanical treatments and/or storage in humid environment were applied to illustrate the potential for property improvements in polymer-matrix composites by alteration of residual stresses. Notched specimens were researched, because such stress concentrations are often responsible for damage initiation in system components. Additionally, they offer local limited changes of residual stresses by thermomechanical treatment.

The results demonstrate that application of thermomechanical treatments increases the tensile strain at first ply failure at a circular notch in a quasi-isotropic laminate by 30% to 179%, depending on treatment type and moisture content.

Treatments on notched specimens lead on average to noticeably higher increase of strain at first ply failure than treatment on unnotched specimens. That means a localized treatment is possible and may be more effective than a treatment of a whole structure. This knowledge facilitates application of thermomechanical treatments to real structures, because the level of load required for treatment is a critical parameter and is often limited by the reduced shape stability of composite structures at treatment temperature.

In dry laminates (untreated and treated), first ply failure at a notch in quasiisotropic laminates occurs exclusively in 90°-plies. Moisture pick up and its related matrix swelling reduce or invert residual stresses, but also reduce the transverse ply strength by magnifying the stresses between fibres as well as by weakening of fibre/matrix interface strength. The elongation effect transverse to fibre di-rection produces, especially in treated laminates, compressive stresses in 45°plies, which reduce the loading capacity of these plies. This in turn results in a shift of first ply failure from 90°-plies to 45°-plies. Moisture pick up therefore reduces the potential for property improvement by thermomechanical treatment.

It can be concluded that moisture pick up in realistic service environment neutral-

  name NARMCO 5245C, (Resin type: Modified cynate ester), because most characteristics of this PMC needed for predicting first ply failure and micromechanical calculation of local stresses and for achieving thermomechanical treatment were available from previous studies [21,24,25]. The fabricated laminates had a quasiisotropic lay-up [(+45°/0°/-45°/90°) 2 ]s. A fraction of them were completely cured (curing cycle to manufacturer's specification + post-curing), the remainder were produced using a shortened curing cycle (reduction of curing time = 30 %) without post-curing. From these laminates, specimens of 36 mm by 280 mm were cut. A certain number of specimens were notched before thermomechanical treatment by drilling a central hole of a diameter of 6 mm (notched specimen) while being water-cooled. The remaining specimens were similarly notched after thermomechanical treatment. Before thermomechanical treatment and testing in dry condition or storage in defined climate, specimens were stored in a dry environment at 70°C for three days and than at 90°C until constant mass was reached. A certain number of untreated and treated specimens were moisturized by storage in a constant climate of 70°C/95% relative humidity for 200 days. The moisture pickup was measured by weighing.

  and 6. Because the first ply failure is expected to occur between fibres in 90° -and 45°plies, only the transverse strains (in ply axis 2) and shear strains (in ply axis 6) are considered. Due to the laminate lay-up used (quasi-isotropic), shear strains are induced by treatment loading only. It can be seen that production initiated transverse strains are positive, while those induced by moisture and treatment are negative. That means moisture pick up or thermomechanical treatment tend to offset the production induced residual strains, depending on the amount of water picked up and the effectiveness of treatment (Fig. 5 to 7). A thermomechanical treatment and subsequent moisture pick up of the magnitude studied here result in an inversion of residual strains induced during production. One consequence of this inversion, especially in unloaded wet specimens, is that the compressive ply strains/stresses limit the infiltration of contrast fluid, thereby hindering the detection of cracks by radiography. These compressive stresses explain the discrepancy in first ply failure measurements between AE and radiography, and provide a rationale for using only the AE measurements for evaluation of property changes.

  , predicted and experimental first ply failures are compared. For all considered cases a nearly perfect correlation of predicted and measured first ply failure was observed. That means in the dry condition the change of strain at first ply failure by thermomechanical treatment goes along with the amount of introduced permanent deformation depending on condition of the treatment applied (Treatment A or B, notched before or after treatment). The influence of moisture pick up has been already discussed. Essential additional effects on the improvements obtained by thermomechanical treatment cannot be expected from these results.

  various factors affecting residual local stress can be determined. These factors include the different expansion of matrix and fibre by temperature change, permanent deformation of matrix caused by thermomechanical treatment, expansion by swelling due to moisture pick up and strains induced by mechanical loading. Calculations were performed for two cases while respecting the statistical distribution of fibres in the matrix. In the first case, all fibres were assumed to have an average separation. In the second case, single fibres with approximately contact (fibre distance/fibre diameter=1.019) were assumed to be surrounded by fibres with average separation. A detailed description of this model including these extreme cases, as well as a simplification used to calculate micromechanical stresses and

Fig. 1 :Fig. 2 :Fig. 3 :Fig. 4 :Fig. 5 :Fig. 6 :Fig. 7 Fig. 8 :Fig. 9 :Fig. 10 :Fig. 11 :Fig. 12 :Fig. 13 :

 12345678910111213 Fig.1: Influence of thermal treatment on water pick up

Table 1 :

 1 Conditions of thermomechanical treatments Average stress at the smallest cross section.

	Type of	Unnotched specimen		Notched specimen
	Treatment/Step Tempera-	Stress	Duration	Temperatur	Stress 1)	Duration
			ture [°C]	[MPa]	[min]	e [°C]	[MPa]	[min]
	A	Heating up	23 to	0	15	23 to 210	0	15
			210
		Loading	210	250	5	210	133	5
		Cooling	210 to 23 250	15	210 to 23	133	15
		down
		Unloading	23	250 to 0	1	23	133 to 0	1
	B	Heating up	23 to180	0	15	23 to180	0	15
		Loading	180	222	115	180	117	115
		Heating up	23 to 190 222	5	23 to 190 117	5
		Loading	190	222	115	190	117	120
		Heating up 190 to 200 222	5	190 to 200 117	5
		Loading	200	222	120	200	117	120
		Heating up 200 to 210 222	5	200 to 210 117	5
		Loading	210	222	235	210	117	235
		Loading	210	306	5	210	167	5
		Cooling	210 to 23 306	15	210 to 23	167	15
		down
		Unloading	23	306 to 0	1	23	167 to 0	1
		1)


A H Sheikh

O T Thomsen

An efficient beam element for the analysis of laminated composite beams of thin-walled open and closed cross sections

Keywords: Thin-walled laminated beams, Shear deformation, Warping deformations, Crosssectional rigidities, Finite element approximation, Bending-torsion-extension-shear coupling

A condensed fully coupled beam element for thin-walled laminated composite beams having open or closed cross sections is presented. An analytical technique is used to derive the cross-sectional stiffness of the beam in a systematic manner considering all the deformation effects and their mutual couplings. An efficient finite element approximation is adopted for the transverse shear deformation, which has helped to conveniently implement the C 1 continuous formulation required by the torsional deformation due to incorporation of warping deformation. The performance of the element is tested through the solution of numerical examples involving open section I and channel (C) beams and closed section box beams under different loading conditions, and the obtained results are compared with model as well as experimental results available in literature.

Introduction

The problem of modelling thin-walled laminated composite beam/beam like slender structures having open or closed cross section as one dimensional condensed beam elements has drawn a significant amount of attention of researchers over the last decades, and a number of investigations have been carried out to study the different aspects of this problem. A few representative studies relevant to the present context are given in [1][START_REF] Chandra | Thin-walled composite beams under bending, torsion, and extensional load[END_REF][START_REF] Cesnik | VABS: A new concept for composite rotor blade cross-section modelling[END_REF][START_REF] Lee | Center of gravity and shear center of thin-walled open-section composite beam[END_REF][START_REF] Jung | Refined structural model for thin and thick walled composite rotor blades[END_REF][START_REF] Kollar | Mechanics of composite structures[END_REF][START_REF] Salim | Torsion of open and closed thin-walled laminated composite sections[END_REF][START_REF] Lee | Flexural analysis of thin-walled composite beams using shear-deformable beam theory[END_REF][START_REF] Shan | Flexural-torsional buckling of fiber-reinforced plastic composite open channel beam[END_REF][START_REF] Librescu | Thin-walled composite beams[END_REF]. One of the initial applications of composite beam theory was found in the analysis of helicopter rotor blades. It has subsequently been applied to the analysis of pultruded composite profiles and other applications, including the analysis of long and slender wind turbine blades made of composite materials.

The studies carried out so far may broadly be divided into two groups, based on the approach adopted for the evaluation of the constitutive matrix of the beam, defined as the cross-section stiffness coefficients. The first and most common approach is based on an analytical technique, while the other approach requires a two-dimensional finite element analysis to obtain the cross-section stiffness matrix. Hodges and co-workers [START_REF] Cesnik | VABS: A new concept for composite rotor blade cross-section modelling[END_REF] pioneered the second approach, which is referred to as the so-called variational asymptotic beam section analysis (VABS). It is based on a method known as the variational asymptotic method (VAM) [START_REF] Berdichevsky | Variational-asymptotic method of constructing a theory of shell[END_REF], where the three dimensional elasticity problem is systematically divided into a two dimensional cross-sectional problem, and a one dimensional beam problem (length direction). VABS has the advantage that beams having solid or thick-walled cross sections may be analyzed, where the three dimensional stress state can be extracted in the post processing stage. Opposed to this, the fully analytical approach may be preferred specifically for the analysis of beams with thin walled cross sections in order to avoid the additional twodimensional finite element analysis required in VABS.

In this context it should be mentioned that Hodges and co-workers (e.g., Volovoi et al. [START_REF] Volovoi | Asymptotic theory for static behavior of elastic anisotropic I-beams[END_REF], Volovoi and Hodges [START_REF] Volovoi | Theory of anisotropic thin-walled beams[END_REF], Volovoi et al. [START_REF] Volovoi | Assessment of beam modeling methods for rotor blade applications[END_REF], Volovoi and Hodges [START_REF] Volovoi | Hodges DH Single-and multi-celled composite thin-walled beams[END_REF], Ye et al. [START_REF] Yu | A generalized Vlasov theory for composite beams[END_REF]) further applied the concept introduced by the variational asymptotic method to two dimensional cross-sectional problem and derived closed-form expressions for the crosssectional stiffness coefficients of thin-walled beams. As the resulting theory [START_REF] Volovoi | Asymptotic theory for static behavior of elastic anisotropic I-beams[END_REF][START_REF] Volovoi | Theory of anisotropic thin-walled beams[END_REF][START_REF] Volovoi | Assessment of beam modeling methods for rotor blade applications[END_REF][START_REF] Volovoi | Hodges DH Single-and multi-celled composite thin-walled beams[END_REF][START_REF] Yu | A generalized Vlasov theory for composite beams[END_REF] evolved through rigorous mathematical treatments, it involves some additional terms including those for in-plane warping of the section, which is found to be important in some specific situations. Thus, the generality of the modelling has been enhanced in the asymptotic approach, but at the same time the mathematical formulation is very complex. However, the approach being a fully analytical one produced final results in the form of relatively simple closed-form expressions. The present approach proposes a fully coupled beam model which is significantly less mathematically complex, and which includes all effects except in-plane cross sectional warping.

In the present investigation, an analytical (closed-form) approach is adopted for the derivation of the cross-sectional stiffness matrix considering different effects and their coupling to yield a general formulation, which includes a torsional warping moment in addition to the classical (and well known) de St. Venant torsion contribution, an axial force, bi-axial bending moments and bi-axial transverse shear forces. In total this yields a 7x7 crosssectional stiffness matrix. All the elements of this matrix are explicitly derived for open I and channel (or C) sections, and closed box section profiles. For the constitutive equation of any beam wall, defined locally, provision is kept to enable the specification of either plane stress conditions (zero normal stress along the wall profile, i.e. in the circumferential direction) or plane strain conditions (zero normal strain along the wall profile, i.e. in the circumferential direction).

For the finite element approximation of the beam element, the torsional deformation requires C 1 continuity of the twisting rotation due to incorporation of the out of plane warping deformation. This requirement is satisfied with the use of a Hermetian interpolation function considering the twisting rotation and its derivative with respect to the length coordinate as the nodal unknowns. The association of the derivative of the twisting rotation helps to impose warping restraints or warping free conditions by constraining or releasing this nodal unknown. At the same time the bending deformation requires C 0 continuity of the transverse displacements due to the incorporation of the transverse shear deformation of the beam walls. This requires a reduced integration technique for the evaluation of the stiffness matrix in order to avoid shear locking. As the bending deformation is not uncoupled from the other modes of deformation, including torsion, it is difficult to implement a C 0 formulation with a C 1 formulation having different orders of integration schemes. Lee [START_REF] Lee | Flexural analysis of thin-walled composite beams using shear-deformable beam theory[END_REF] tried to solve the problem by an amended representation of the torsional deformation, so as to model it with a C 0 formulation like for bending deformation, but this involved a non-physical parameter in the formulation. Moreover, the C 0 formulation with reduced integration technique is susceptible to display inherent numerical disturbances like the occurrence of spurious modes.

Considering these aspects, the finite element implementation of the bending deformation in the present formulation is carried out with a different approach based on a concept proposed by the first author [START_REF] Sheikh | A new concept to include shear deformation in a curved beam element[END_REF]. It does not require a reduced integration technique, and this effectively eliminates the problem described above. Based on this methodology a three node beam element, as shown in Fig. 1, has been developed, where the nodes at the two ends contain seven degrees of freedom (three translations, three rotations and the derivative of the twisting rotation), while the internal node contains five degrees of freedom (three translations and two bending rotations). The results show a very good performance of the proposed element in terms of convergence and solution accuracy. The developed element is also utilized to derive some new results, which are presented for future references.

Formulation

Figure 2 shows a portion of the composite beam shell wall, with its local coordinate system x-s-n, the local displacement components, the global coordinate system x-y-z and finally the global beam displacement components.

In Fig. 2, O is the centroid, and P is the shear centre/pole of the beam section. The displacement components at the mid-plane of the shell wall in the local coordinate system (xs-n) may be expressed in terms of the global displacement components of the beam [1] as

x z y z y U u θ ϕ θ θ ′ + + + = (1) x r W V v θ α α - + = sin cos (2) x q W V w θ α α + + - = cos sin ( 3
)
where ϕ is the warping function, =

y θ y V Ψ + ′ - (V ′ is the derivative of V with respect to x,
and y Ψ is the rotation of beam section about z for the transverse shear deformation) is the bending rotation of the beam section with respect to z, and

z z W Ψ + ′ - = θ
is the bending rotation of the beam section with respect to y. For beams with closed cross sections, the effect of the warping function is not as significant as for open cross section beams, as has been nicely demonstrated by Ye et al. [START_REF] Yu | A generalized Vlasov theory for composite beams[END_REF]. However, cross section warping has been included for both both closed and open cross section profiles in order develop a general formulation for both type of beam profiles. Again, Volovoi and Hodges [START_REF] Volovoi | Hodges DH Single-and multi-celled composite thin-walled beams[END_REF] have shown that the effect of in-plane warping is significant in some specific cases of closed section beams. However, this effect has not been incorporated in the present approach as mentioned earlier.

Considering the effect of shell bending and transverse shear deformation, the corresponding displacement components at a point away from the shell mid-plane may be expressed as With the above equations, the displacement components at any point within the shell wall may be expressed in terms of the global beam displacement components as

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ∂ ∂ - + = xn x w n u u ψ (4) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ∂ ∂ - + = sn s w n v v ψ (5) w w = (6
Ψ + Ψ - = (7)
( ) ( ) ( ) x z y nq n z n y U u θ ϕ θ α θ α ′ - + + + - + = cos sin (8) ( ) x n r W V v θ α α + - + = sin cos (9) x q W V w θ α α + + - = cos sin ( 10
)
The strain components at the corresponding point in the local axis system (x-s-n) may be expressed as

{ } ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ = ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ = sn xn sn xn xs s x x v s u s v x u ψ ψ ε ε ε ε ε εˆ (11)
Assuming the normal strain component s ε to be zero (plane strain condition), or the normal stress component s σ to be zero (plane stress condition), the reduced strain vector may be expressed in terms of the global displacement parameters with the help of the above Eqs. (7-10) as

{ } ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ = ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ = xn xn xs x x v s u x u ψ ε ε ε ε ( ) ( ) ( ) ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ Ψ + Ψ - ′ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ - + - Ψ + Ψ ′ ′ - + ′ + + ′ - + ′ = α α θ ϕ α α θ ϕ θ α θ α
= ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ + ′ + ′ ′ ′ ′ ′ ′ ′ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ - - + - - + - = cos sin 0 sin cos 0 0 2 0 0 0 0 0 cos 0 0 sin 0 0 1 , ( 13
)
Considering the transverse shear deformation of the laminated shell wall [START_REF] Reddy | Mechanics of laminated composite plates and shells: theory and analysis[END_REF], the stress strain relationship at any point within a lamina of the laminated shell wall in the local coordinate system parallel to x-s-n may be expressed as Using Eqs. ( 13) and ( 15), the strain energy of the system can be written as

⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ sn xn xs s x sn xn xs s x Q Q Q Q Q Q Q Q Q Q Q Q Q ε ε ε ε ε σ σ σ σ σ
/ ~Q Q Q Q Q - = , / ~26 12 16 16 Q Q Q Q - = 22 Q , 22 16 16 66 66 / ~Q Q Q Q Q - =
{ } { } { } [ ]{ } ∫ ∫ = = dx F dv U T T ε ε σ ε 2 1 2 1 ~ ( 16
)
where [ ] For the one dimensional finite element implementation, a quadratic approximation has been adopted for the axial deformation, which follows a Lagrangian type formulation. The torsional deformation is based on a Hermetian formulation as mentioned earlier, where a cubic approximation has been adopted. The approximation of the bending deformation coupled with transverse shear deformation is based on the concept proposed by Sheikh [START_REF] Sheikh | A new concept to include shear deformation in a curved beam element[END_REF]. Based on this the field variables are approximated as follows at the two external nodes as

[ ] [ ][ ] [ ] [ ][ ] ( ) [ ] ∫ ∫ ∫ ∫ ∫ = = = ds C ds dn H Q H dsdn H Q H F
{ } [ ]{ } a R = δ or { } [ ] { } δ 1 - = R a (26) where{ } [ ] T a a a a a a 19 18 3 2 1 " = , { } [ 1 1 1 1 1 1 1 x z y x W V U θ θ θ θ δ ′ = ] T x z y x z y W V U W V U 3 3 3 3 3 3 3 2 2 θ θ θ
′ , and the matrix [R] having an order of 19x19 contains the element nodal coordinates.

The generalized strain vector { } ε in Eq. ( 13) may be expressed in terms of { } a using Eqs. [START_REF] Reddy | Mechanics of laminated composite plates and shells: theory and analysis[END_REF][START_REF] Lee | Flexural-torsional behavior of thin-walled composite beams[END_REF][START_REF]ABAQUS/Standard User's Manual[END_REF][START_REF] Vo | Flexural-torsional behavior of thin-walled closed-section composite box beams[END_REF][START_REF] Chandra | Experimental and theoretical analysis of composite I beams with elastic couplings[END_REF][START_REF] Smith | Formulation and evaluation of an analytical model for composite box-beams[END_REF] as

{ } [ ]{ } a S W V U W V U z y x x z y z y x x z y = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ Ψ′ Ψ′ ′ ′ ′ Ψ′ + ′ ′ - Ψ′ + ′ ′ - ′ = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ + ′ + ′ ′ ′ ′ θ θ θ ε ( 27
)
where the matrix [S] having an order of 7x19 is a function of x. The strain vector { } ε can be finally expressed in terms of the nodal displacement vector { } δ using Eq. ( 26) as

{ } [ ][ ] { } [ ]{ } δ δ ε B R S = = -1 (28)
With the above equation, the strain energy (16) of the system may be expressed as

{ } [ ] [ ][ ] { } { } [ ]{ } δ δ δ δ K dx B F B U T T T 2 1 2 1 ~= = ∫ (29)
where [K] is the element stiffness matrix. Using Eq. ( 20) and Eq. ( 26), the element load vector due to a distributed transverse load of intensity q acting in the direction of z may be expressed as

{ } [ ] [ ] ∫ - = dx S q R P T q T (30)
where the row matrix [ ] q S having an order of 19 is function of x. The integrations involved in the evaluation of the element stiffness matrix [K] and the load vector {P} are carried out numerically following the Gauss quadrature technique.

Results and discussions

In the following a number of numerical examples involving composite beams with I, channel (C) or box cross sections are analysed using the proposed element, and the results obtained are compared with analytical, experimental and/or numerical results available in literature for most of the cases. The analyses are usually based on plane stress conditions, unless specified otherwise. In all examples the beam walls are assumed to be constituted by identical layers of the same thickness, but the layers may have different orientations. The geometry of the beam sections are defined in terms of their centre line dimensions. A 2.5m long open section I beam simply supported at its two ends, and subjected to a uniformly distributed transverse load of 1kN/m along the web of the beam, is analysed using the proposed element. The beam has a depth of 50mm, a flange width of 50mm and same thickness (2.08mm) for the flanges and the web. The study is made with different symmetrical stacking sequences, where the flanges and the web are having identical lay-ups for all cases. The material properties of the layers are: E 1 = 53.78GPa, E 2 = 17.93GPa, G 12 = 8.96GPa, G 13 = 8.96GPa, G 23 = 3.45GPa, ν 12 = 0.25. The analysis is carried out assuming both plane stress and plane strain conditions. The values of deflection at the centre of the beam and the bending slope at its supports obtained in the present analysis are presented in Table 1 and Table 2, respectively. The results for the deflections are compared with those of Lee [START_REF] Lee | Flexural analysis of thin-walled composite beams using shear-deformable beam theory[END_REF] and Lee and Lee [START_REF] Lee | Flexural-torsional behavior of thin-walled composite beams[END_REF] in Table 1. Lee [START_REF] Lee | Flexural analysis of thin-walled composite beams using shear-deformable beam theory[END_REF] considered the effect of transverse shear deformation, whereas the study of Lee and Lee [START_REF] Lee | Flexural-torsional behavior of thin-walled composite beams[END_REF] is based on classical lamination theory. In both the studies [START_REF] Lee | Flexural analysis of thin-walled composite beams using shear-deformable beam theory[END_REF][START_REF] Lee | Flexural-torsional behavior of thin-walled composite beams[END_REF], one dimensional finite element analysis has been applied after obtaining the cross-sectional stiffness matrix analytically. For the validation of the results in [START_REF] Lee | Flexural analysis of thin-walled composite beams using shear-deformable beam theory[END_REF][START_REF] Lee | Flexural-torsional behavior of thin-walled composite beams[END_REF], the beam structure was analysed using ABAQUS [START_REF]ABAQUS/Standard User's Manual[END_REF], where the S9R5 shell element was used to model the composite beam. The results produced by ABAQUS [START_REF]ABAQUS/Standard User's Manual[END_REF] are also included in Table 1. The table shows an excellent agreement between the predictions of the present model and the results of the other models mentioned, especially the results of Lee [START_REF] Lee | Flexural analysis of thin-walled composite beams using shear-deformable beam theory[END_REF]. Moreover, the present results based on plane stress conditions are found to be closer to the results produced by ABAQUS [START_REF]ABAQUS/Standard User's Manual[END_REF] than the results in [START_REF] Lee | Flexural analysis of thin-walled composite beams using shear-deformable beam theory[END_REF], [START_REF] Lee | Flexural-torsional behavior of thin-walled composite beams[END_REF]. It is further observed that in all cases convergence of the results was achieved using only two elements with the approach proposed herein.

Clamped box beam subjected to uniformly distributed transverse load

A box beam clamped at both ends, and subjected to uniformly distributed transverse load of 6.5kN/m along the mid-plane of one of the webs, is analysed using the proposed element assuming both plane stress and plane strain conditions. The beam is assumed to be free from axial and warping restraints at the supports. The beam is 1.0 m long, 70 mm deep and 50 mm wide, and all the beam walls are 2 mm thick having a stacking sequence of (45/-45) 2 /(0/0) 6 /(45/-45) 2 . The material properties assumed for all layers are: E 1 = 148.0 GPa, E 2 = 9.65 GPa, G 12 = G 13 = G 23 = 4.55 GPa, ν 12 = 0.3. The values of the deflection and angle of twist at the centre of the beam obtained in the present analysis are presented in Table 3, along with results obtained by Kollar and Springer [START_REF] Kollar | Mechanics of composite structures[END_REF] assuming plane strain conditions, and Vo and Lee [START_REF] Vo | Flexural-torsional behavior of thin-walled closed-section composite box beams[END_REF] assuming both plane stress and plane strain conditions. Kollar and Springer [START_REF] Kollar | Mechanics of composite structures[END_REF] solved the problem analytically (closed form), while Vo and Lee [START_REF] Vo | Flexural-torsional behavior of thin-walled closed-section composite box beams[END_REF] applied one dimensional finite element analysis after obtaining the cross-sectional stiffness matrix analytically. As both of the studies [START_REF] Kollar | Mechanics of composite structures[END_REF][START_REF] Vo | Flexural-torsional behavior of thin-walled closed-section composite box beams[END_REF] did not consider the effect of transverse shear deformation, the present analysis was also carried out with very high values of transverse shear moduli (G 13 = G 23 = G 12 x 10 6 ), and the results obtained are included in Table 3 (marked ‡). The results show a significant effect of the transverse shear deformation and a wide variation of the values of torsional rotation obtained on the basis of plane stress and plane strain assumptions. The table shows that the convergence of the present finite element formulation is very good. It also shows a good agreement between the results obtained by the different techniques having similar basis.

Cantilever I beam subjected to tip transverse load with restrained warping

A 0.762 m (30 inch) long cantilever I beam subjected to a transverse unit load of 4.4482 N (1.0 lb) at the free end is analysed using the proposed element assuming restrained warping at both ends. The beam has a depth of 12.7 mm (0.5 inch), a flange width of 25.4 mm (1.0 inch) and the same thickness of 1.016 mm (0.04 inch) is assumed for the flanges and the web. The stacking sequence of the flange is 0/90/0/90/90/0/15/15, while that of the web is 0/90/0/90/90/0/90/0. The assumed material properties are: E 1 = 141.96 GPa (20.59 x10 6 Psi), E 2 = 9.7906 GPa (1.42 x10 6 Psi), G 12 = G 13 = G 23 = 6.1363 GPa (0.89 x10 6 Psi), ν 12 = 0.42. The variation of deflection, bending slope and twisting rotation along the length of the beam are plotted in Fig. 3, Fig. 4 and Fig. 5, respectively. The results for the bending slope and twisting rotation are compared with the numerical results of Jung et al. [START_REF] Jung | Refined structural model for thin and thick walled composite rotor blades[END_REF] and the experimental results of Chandra and Chopra [START_REF] Chandra | Experimental and theoretical analysis of composite I beams with elastic couplings[END_REF] in Fig. 4 and Fig. 5. Jung et al. [START_REF] Jung | Refined structural model for thin and thick walled composite rotor blades[END_REF] has produced results based on a mixed formulation, as well as on the displacement formulation of Smith and Chopra [START_REF] Smith | Formulation and evaluation of an analytical model for composite box-beams[END_REF], where one dimensional finite element has been applied after obtaining the cross-sectional stiffness matrix analytically. The figures show an excellent agreement between the results of the approach proposed herein and the results given in [START_REF] Jung | Refined structural model for thin and thick walled composite rotor blades[END_REF][START_REF] Chandra | Experimental and theoretical analysis of composite I beams with elastic couplings[END_REF][START_REF] Smith | Formulation and evaluation of an analytical model for composite box-beams[END_REF].

Cantilever box beam subjected to tip transverse load/twisting moment

A 0.762 m (30 inch) long cantilever box beam having a depth of 12.7 mm (0.5 inch), a width of 23.444 mm (0.923 inch) and assuming the same thickness of 0.762 mm (0.03 inch) for all the walls consisting of 6 layers is analysed using the proposed element. The assumed loading is either a unit transverse load of 4.4482 N (1.0 lb) or a unit twisting moment of 0.11299 Nm (1.0 lb-inch) applied at the free end. The stacking sequence of the top and bottom walls is (45/45) 3 , while that of the left and right walls is (45/-45) 3 . The material properties of all layers are assumed to be identical and the same as those adopted in the previous example (I beam). The variation of the bending slope and twisting rotation along the length of the beam are plotted in Fig. 6 to Fig. 9 along with the analytical and experimental results of Chandra et al. [START_REF] Chandra | Thin-walled composite beams under bending, torsion, and extensional load[END_REF], and the finite element results of Stemple and Lee [START_REF] Stemple | A finite element model for composite beams with arbitrary crosssectional warping[END_REF]. The results of the different methods are found to be in excellent agreement.

A channel beam clamped at one end and simply supported at the other end

A 2.5 m long beam having a channel (C) section clamped at one end and simply supported at the other end is analysed using the proposed element. The channel section has a depth of 50 mm, a width of 25 mm and the same thickness of 2.08 mm for all the walls consisting of 16 layers. The layers are having identical orientation, which is varied from 0 0 to 90 0 with an interval of 15 0 . The material properties of the layers are: E 1 = 53.78GPa, E 2 = 17.93GPa, G 12 = 8.96GPa, G 13 = 8.96GPa, G 23 = 3.45GPa, ν 12 = 0.25. The beam is subjected to a unit twisting moment 1.0 Nm at the simply supported end. The results for the angle of twist at the simply supported end obtained by the proposed element are presented with those obtained by Kim et al. [START_REF] Kim | Exact solution for thin-walled open-section composite beams with arbitrary lamination subjected to torsional moment[END_REF] in Table 4. Kim et al. [START_REF] Kim | Exact solution for thin-walled open-section composite beams with arbitrary lamination subjected to torsional moment[END_REF] applied an analytical technique (closed form) and two different finite element models to analyse the problem. They have also analysed the structure using ABAQUS [START_REF]ABAQUS/Standard User's Manual[END_REF] where the structure is modelled with shell elements. Table 4 shows that the results obtained from the different sources are showing an excellent agreement with the predictions of the coupled beam element proposed herein. Moreover, Table 4 also shows a very good convergence of the results obtained by the proposed element.

Conclusions

A fully coupled beam element has been developed for the analysis of thin-walled laminated composite beams of open and closed cross sections including axial displacement, torsion, out of plane warping, bi-axial bending and transverse shear deformations. The constitutive equations of the beam element are derived analytically considering the coupling of all the modes of deformation, i.e. the beam model is fully coupled. The resulting composite beam theory is applied to laminated composite beams of open I and channel cross sections, as well as to laminated composite beams of closed box cross sections. The incorporation of transverse shear deformations demands a C 0 formulation for the one dimensional finite element approximation of the bending deformations, while the torsional deformation demands a C 1 formulation for the incorporation of out-of-plane warping. The difficulty in implementing both formulations in the present coupled problem is successfully overcome by adopting an efficient approach for the finite element approximation of the bending deformations. Numerical examples of composite open and closed cross section beams having different load and boundary conditions are analysed using the proposed element. The results obtained are compared with analytical, experimental and/or other finite element results available in literature, and the comparisons show a very good performance of the proposed fully coupled beam element. In addition to the comparative studies, some new results are also presented for future references.

Fig. 1 .

 1 Fig. 1. Three node beam element

  the bending rotations of the shell wall about x and s, respectively. Similarly, xn ψ and sn ψ are the shear rotations of the shell wall sections about s and x, respectively, for the transverse shear deformation. It is assumed that sn ψ = 0, while xn ψ may be expressed in terms of the corresponding global beam parameter as

Fig. 2 .

 2 Fig. 2. Cross-section of a portion of a shell wall of the thin-walled composite beam with local and global coordinate systems and displacement components

  s σ = 0 along with ns ψ = 0, the above equation reduces to

  the plane stress ( s σ = 0) condition.

  of the cross-sectional stiffness matrix [F] are explicitly derived for open I and channel (C) sections as well as for closed box section profiles. The explicit expressions for the different elements of the matrix [F] are presented in Appendix B, C and D for I section, channel section (C) and box section, respectively. The matrix [C] is common for all the considered cross-sections, and its elements are presented in Appendix A. For this purpose the warping function ϕ used in the above equations are taken as is the cross-sectional area enclosed by the mid-plane contour in the case of beams having closed section profiles. For an open section profile, the warping function may be simply obtained by dropping the second term associated with secondary warping, thus giving as ∫ = rds ϕ .

  should be noted that y Ψ and z Ψ are adopted as the field variables instead of y θ and z θ , which are usually used in a typical C 0 formulation. However, the corresponding nodal unknowns are interestingly y θ and z θ , and these are not simply the field variables y Ψ and z Ψ . Now, with the help of Eqs. (19-22), the bending rotations y the above Eqs. (18-23) are expressed in terms of the nodal displacement vector { } δ after substitution of U (18), V (19), W (20), y θ (24) and z θ (25) at all the three nodes of the beam element (see Fig. 1); and x θ (23) and its derivative x θ ′

  † G 13 = G 12 , ‡ G 13 = G 23 = G 12 , x 10 6 , ♠ Number of elements

Fig. 3 .Fig. 4 .Fig. 5 .

 345 Fig. 3. Variation of deflection of the I beam

Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .

 6789 Fig. 6. Variation of bending slope of the box beam under a transverse load at the tip

11

 11

Table 1

 1 Deflection w (cm) at the centre of the simply supported I beam

	Stacking sequence	[0/0] 4S	[15/-15] 4S	[30/-30] 4S	[45/-45] 4S	[60/-60] 4S	[75/-75] 4S	[0/ 90] 4s
	Present ( s σ = 0) -2 †	6.264	6.929	9.320	13.45	17.00	18.46 9.387
	Present ( s σ = 0) -4 †	6.264	6.929	9.320	13.45	17.00	18.46 9.387
	Present ( s ε = 0) -2 †	6.134	6.640	8.309	11.37	15.15	17.68 9.192
	Present ( s ε = 0) -4 †	6.134	6.640	8.309	11.37	15.15	17.69 9.192
	Lee [8] ( s σ = 0)	6.259	6.923	9.314	13.45	16.99	18.45 9.381
	Lee [8] ( s ε = 0)	6.129	6.637	8.307	11.36	15.15	17.68 9.189
	Lee and Lee [19] ( s σ = 0)	6.233	6.899	9.290	13.42	16.96	18.41 9.299
	Lee and Lee [19] ( s ε = 0)	6.103	6.610	8.281	11.34	15.12	17.64 9.153
	ABAQUS [20]	6.340	6.989	9.360	13.48	17.02	18.49 9.400
	† Number of elements

Table 2

 2 Bending rotation θ z x 100 (rad) at the support of the simply supported I beam

	Stacking sequence	[0/0] 4S	[15/-15] 4S	[30/-30] 4S	[45/-45] 4S	[60/-60] 4S	[75/-75] 4S	[0/ 90] 4s
	Present ( s σ = 0) -2 † 7.978 8.830 11.89 17.18	21.71	23.57	11.97
	Present ( s σ = 0) -4 † 7.978 8.830 11.89 17.18	21.71	23.57	11.97
	Present ( s ε = 0) -2 †	7.812	8.461	10.60 14.52	19.35	22.58	11.72
	Present ( s ε = 0) -4 †	7.812	8.461	10.60 14.52	19.35	22.58	11.72
	† Number of elements

Table 3

 3 Deflection and angle of twist at the centre of the clamped box beam w x 10 4 (m) θ x x 10 3 (rad)

	Number of elements	2	4	6	2	4	6
	Present † ( s σ = 0) -2♠	7.811	7.811	7.811	6.705	6.703	6.703
	Present † ( s ε = 0) -2♠	5.779	5.779	5.779	2.755	2.754	2.754
	Present ‡ ( s σ = 0) -2♠	4.940	4.940	4.940	6.705	6.703	6.703
	Present ‡ ( s ε = 0) -2♠	4.378	4.378	4.378	2.755	2.754	2.754

Table 4

 4 Maximum angle of twist of a beam having a channel section

	Stacking sequence	[0] 16	[15] 16 [30] 16 [45] 16 [60] 16 [75] 16 [90] 16
	Present Element -2 †	0.2441 0.2195 0.1934 0.2029 0.2394 0.2793 0.2967
	Present Element -4 †	0.2444 0.2198 0.1939 0.2038 0.2405 0.2804 0.2978
	Present Element -6 †	0.2444 0.2199 0.1940 0.2038 0.2405 0.2805 0.2979
	Present Element -8 †	0.2444 0.2199 0.1940 0.2038 0.2406 0.2805 0.2979
	Closed form solution [25]	0.2444 0.2199 0.1940 0.2038 0.2406 0.2805 0.2979
	Finite Element -I [25] -1 † 0.2444 0.2199 0.1940 0.2038 0.2406 0.2805 0.2979
	Finite Element -II [25] -4 † 0.2444 0.2198 0.1939 0.2038 0.2404 0.2804 0.2978
	ABAQUS [20]	0.2459 0.2250 0.2011 0.2090 0.2427 0.2795 0.2952
	† Number of elements

Acknowledgement

The work presented was carried out as part the Innovation Consortium "Integrated Design and Processing of Lightweight Composite and Sandwich Structures" (abbreviated "KOMPOSAND") funded by the Danish Ministry of Science, Technology and Innovation and the industrial partners Composhield A/S, DIAB ApS (DIAB Group), Fiberline Composites A/S, LM Glasfiber A/S and Vestas Wind Systems A/S. The support received is gratefully acknowledged.

Appendix A

The explicit expressions for the elements of the matrix [C] found in Eq. ( 16) are presented in this Appendix. The matrix [C] is valid for all the three sections considered in this study. As the matrix is symmetric, the elements in the upper triangle of the matrix are listed below.

Appendix B

For the open section I profile as shown in Fig. 10, the warping function may be expressed as yz = ϕ . Based on this the expressions for the elements in the upper triangle of the symmetric matrix [F] found in Eq. ( 16) are as follows.

Appendix C

For the open section channel profile as shown in Fig. 11, the warping function may be expressed as ( )

, where the location of the centroid [START_REF] Lee | Center of gravity and shear center of thin-walled open-section composite beam[END_REF] and shear centre [START_REF] Shan | Flexural-torsional buckling of fiber-reinforced plastic composite open channel beam[END_REF] are:

Appendix D

For the closed section box profile as shown in Fig. 12, the warping function may be simply expressed as yz

, where ( ) ( )

β if all the walls are identical and placed symmetrically with respect to the centroid [START_REF] Chandra | Thin-walled composite beams under bending, torsion, and extensional load[END_REF]. With this the elements of the matrix [F] may be expressed as follows.

(

) ( )

4 11

2 11