REPORT. No.: D3.3 part 4. Axially loaded sandwich panels

Transcription

1 REPORT Axially loaded sandich panels Publisher: Saskia Käpplein Thomas isiek Karlsruher Institut für Technologie (KIT) Versuchsanstalt für Stahl, Holz und Steine Task: 3.4 Object: Design of axially loaded sandich panels, global loadbearing behaviour This report includes 63 pages and annexes. Date of issue: Tel.: +49 () Abt. Stahl- und Leichtmetallbau, D-7618 Karlsruhe, Deutschland Fax: +49 ()

2 page Project co-funded under the European Commission Seventh Research and Technology Development Frameork Programme (7-13) Prepared by Theme 4 P-anotechnologies, aterials and ne Production Technologies Saskia Käpplein, Thomas isiek, Karlsruher Institut für Technologie (KIT), Versuchsanstalt für Stahl, Holz und Steine Drafting History Draft Version Draft Version Draft Version 1.3 Draft Version 1.4 Final Dissemination Level PU Public X PP RE Restricted to the other programme participants (including the Commission Services) Restricted to a group specified by the Consortium (including the Commission Services) CO Confidential, only for members of the Consortium (including the Commission Services) Verification and approval Coordinator Industrial Project Leader anagement Committee Industrial Committee Deliverable D3.3 part 4: Axially loaded sandich panels Due date: onth 35 Completed: onth 33

3 page 3 Table of contents 1 Introduction 6 Load-bearing behaviour of panels subjected to transverse loads 7 3 Load-bearing behaviour of axially loaded sandich panels Distribution of normal stresses 1 3. Global buckling 1 4 Buckling tests on axially loaded sandich panels 13 5 umerical investigations FE-odel Preliminary investigations 5.3 Calculations for determination of ultimate stress Calculations to investigate the influence of nd order theory 9 6 Design methods General Design according to nd order theory Design according to the equivalent member method 35 7 Time-dependent behaviour Basics Creeping of axially loaded sandich panels Long-term tests Evaluation of long-term tests Consideration of long-term effects in design formulae Typical loads on all panels of frameless structures 54 8 Partial safety factors 6 9 Summary 61 1 References 61 Annex 1: Annex : Design according to nd order theory Design according to equivalent member method

4 page 4 Symbols and notations A C A F B B S C my D E F E C GA G C L T cr ki P V cross section area of core cross section area of face idth of panel bending rigidity of a sandich panel equivalent uniform moment factor (equivalent member method) thickness of panel, for panels ith flat or lightly profiled faces also distance of centroids of faces elastic modulus of face elastic modulus of core shear rigidity of panel, elastic buckling load due to shear rigidity shear modulus of core length of panel, span of panel bending moment fixed end moment due to eccentric axial force moment due to temperature differences maximum bending moment of a sandich panel normal force, axial load elastic buckling load of a sandich panel elastic buckling load due to bending rigidity (faces) axial rinkling force, maximum compressive axial force concentrated transverse load (point load) transverse force e f Cv k k yy q q e t F b T v st initial deflection, global imperfection shear strength of core relationship beteen deflection due to shear and deflection due to bending interaction factor (equivalent member method) linear load, distributed load equivalent load (due to initial deflection and axial force) thickness of face sheet deflection deflection due to bending deflection due to temperature difference deflection due to transverse force deflection due to short-term load

5 page 5 lt deflection due to long-term load α α α F γ λ λ ki λ GA σ F σ τ C φ t φ St χ ΔT amplification factor to consider effects of nd Order Theory imperfection factor (equivalent member method) thermal expansion coefficient of the face shear deformation slenderness bending slenderness shear slenderness normal stress in the face rinkling stress shear stress in the core creep coefficient sandich creep coefficient reduction factor (equivalent member method) temperature difference beteen inner face and outer face I, V I, i II, V II, ii moment, transverse force, deflection calculated by 1 st Order Theory moment, transverse force, deflection calculated by nd Order Theory

6 page 6 1 Introduction Until no the common application of sandich panels is restricted to the function of space enclosure. The sandich panels are mounted on a substructure and they transfer transverse loads as ind and sno to the substructure. The sandich panels are subjected to bending moments and transverse forces only. A ne application is to apply sandich panels ith flat or lightly profiled faces in smaller buildings such as cooling chambers, climatic chambers and clean rooms ithout any load transferring substructure (Fig. 1.1). Fig. 1.1: Building made of sandich panels but ithout substructure In this ne type of application in addition to space enclosure, the sandich panels have to transfer loads and to stabilise the building. In addition to the moments and transverse forces resulting from transverse loads, the all panels transfer normal forces arising from the superimposed load from overlying roof or ceiling panels. That implies the question about the loadbearing behaviour and the load bearing capacity of sandich panels subjected to axial load or a combination from axial and transverse loads. Design procedures for sandich panels subjected to bending moments and transverse forces are given in E 1459 [1] and in different (national) approvals. But there are no general design methods for panels subjected to axial loads or a combination of axial and transverse loads available. There are some publications, hich mainly deal ith the elastic buckling load of axially loaded sandich panels, e.g. [1]. Also some mainly theoretical investigations on the interaction of elastic global buckling and elastic local buckling (rinkling) have been done, e.g. [16], [17]. But the available investigations do not result in a design procedure, hich can be used to design axially loaded sandich panels, because these investigations basically deal ith elastic buckling behaviour. Furthermore the influence of long-term loads (creep of core material) is not considered at all.

7 page 7 Within the frameork of WP3 of the EASIE project, a design method for axially loaded sandich panels has been developed. Buckling and long-term tests on axially loaded sandich panels ere performed. All tests are documented in the test reports D3. part 3 [7] and D3. part 4 [8]. In report D3.3 - part 4, the evaluation of the tests and additional numerical investigations are presented. A design procedure is developed The design model is based upon the existing design model for panels subjected to transverse loads according to E 1459 [1] and the ECCS recommendations for the design of sandich panels []. The existing model is extended in a ay that consideration of axial forces and influences of nd order theory is possible. Also the behaviour due to long-term loads (creeping of the core material) can be considered by the design method presented in the report at hand. The report at hand only deals ith the global load bearing behaviour of axially loaded sandich panels. Panels ith openings are not covered. In addition the local load-bearing capacity at the areas of load application, i.e. the cut edges of the panel, is to be considered. Load-bearing behaviour of panels subjected to transverse loads In sandich panels ith flat and lightly profiled faces, bending moments are transferred by a force couple in both face sheets. In the face sheets, only normal tensile and compression stresses act. D /D /D Fig..1: Load bearing behaviour of panels subjected to a bending moment The stress in the face sheets of a sandich panel loaded by a bending moment is σ F = (.1) D A ith A t F B D F t F F = B cross sectional area of face sheet thickness of face sheer idth of panel thickness of panel

8 page 8 The load bearing capacity is mostly restricted by reaching the ultimate stress in the compressed face sheet (Fig..). Failure by yielding of the face sheet subjected to tension occurs very rarely. The face sheet represents a plate hich is elastically supported by the core material. The stability failure of the compressed face sheet is termed as rinkling, the ultimate compression stress as rinkling stress σ. Fig..: Wrinkling of the compressed face sheet The elastic rinkling stress of a plane elastically supported plate can be calculated according to the folloing equation [14]. σ = (.),8 3 EF EC GC ith E F elastic modulus of face sheet (steel: E F = 1. /mm ) E C G C elastic modulus of core material shear modulus of core material According to [1] and [] for design purposes the elastic rinkling stress has to be reduced to take imperfections and quality of face, core and bonding beteen face and core into account. In the ECCS recommendations for sandich panels [] the folloing formulae is suggested. σ 3 = k EF EC GC (.3) ith k =,65 for continuously laminated polyurethane sandich panels k =,5...,65 for other core materials and methods of manufacture In E 1459 k =,5 is given for all types of panels. The experimentally determined rinkling stress, hoever, often strongly deviates from the rinkling stress calculated according to (.3). Usually the rinkling stress determined by testing is higher than the rinkling stress determined by calculation. E.g. even a lightly lining of

9 page 9 the faces increases the rinkling stress in comparison to the theoretical value. Therefore for design purposes the rinkling stress is usually determined by bending tests. The ithstanding of transverse forces V is done by shear stresses in the core of the panel. V τ C = (.4) A ith A C C = B D cross sectional area of the core V τ C Fig..3: Load bearing behaviour of panels subjected to a transverse force When reaching the shear strength f Cv of the core material shear failure occurs (Fig..4). Fig..4: Shear failure of the core Because of the relatively soft core layer deflections caused by transverse forces have to be considered. So the deflection of a sandich panel consists of a bending part b and a shear part v. VV = b + v = dx + dx (.5) B GA ith bending stiffness: B A A S F1 F S = EF D (.6) AF 1 + AF

10 shear stiffness: page 1 GA = G C A C (.7) For a single-span panel loaded by a uniformly distributed load q, the deflection at mid-span is q L q L = b + v = + (.8) 384 B 8 GA S 3 Load-bearing behaviour of axially loaded sandich panels 3.1 Distribution of normal stresses The stiffness of the faces is very much higher than the stiffness of the core material. This is hy also for panels loaded by axial loads or by a combination from axial load and bending moment normal forces act only in the face sheets. In addition to normal stresses from the bending moment (force couple), compressive stresses from the axial load act in the face sheets Fig /D / D /D / Fig. 3.1: Load bearing behaviour for bending moment and axial force Also for panels ith axial load failure can occur by rinkling of the face sheet or by shear failure of the core material. 3. Global buckling For slender building components loaded by axial compression forces global buckling can occur. Also for axially loaded sandich panels this stability failure mode has to be considered. If an ideal panel ithout any imperfections is assumed, the axial load increases up to the elastic buckling load cr. Then, global buckling induces. Because of the deflection moments and transverse forces arise. The deflection strongly increases ith a constant axial force. Also the stress resultants moment and transverse force increase. Finally, the rinkling stress in midspan or the shear strength at the support is reached and failure of the panels occurs [16], [17].

11 page 11 In Fig. 3. the load-deflection curve for an ideal panel as ell as for an imperfect panel ith initial deflection is shon. cr ideal sandich panel imperfect sandich panel rinkling of compressed face Fig. 3.: Load-deflection-diagram for axially loaded sandich panels The elastic buckling load cr of a sandich panel loaded by a centric axial load consists of the part ki considering the bending rigidity of the face sheets and the part GA considering the shear rigidity of the core. The rate ki corresponds to the elastic buckling load of both face sheets. The elastic buckling load of a sandich panel can be calculated as follos cr 1 cr ki = ki (3.1) 1+ GA 1 1 = + (3.) GA ki ith elastic buckling load due to bending rigidity ki BS = π (3.3) L elastic buckling load due to shear rigidity GA = G C A C (3.4) The rate from bending rigidity ki on the elastic buckling load depends on the length of the panel, hereas the shear rate GA is independent of the length. With increasing length of the panel, the elastic buckling load of the sandich panel approaches the rate ki from bending, i.e. the elastic buckling load of the face sheets. For very short panels the elastic buckling load

12 page 1 cr of the panel approaches the rate GA. Therefore GA represents an upper limit of the elastic buckling load (Fig. 3.3). GA cr (sandich panel) ki (faces, bending) buckling load length of panel Fig. 3.3: Elastic buckling load of sandich panels If imperfections and effects of nd order theory are neglected, the maximum compressive axial force of a centrically loaded panel corresponds to the axial load, hich leads to rinkling in both face sheets. The axial rinkling force can be calculated by the folloing equation. ( A A ) = σ + (3.5) F1 F The axial rinkling force is comparable to the plastic normal force of a steel crosssection. Thus, the slenderness of a sandich panel can be calculated as follos = λ (3.6) λ = λki + cr ith bending slenderness: GA λ ki = (3.7) ki shear slenderness: λ GA = (3.8) GA Corresponding to the rate from shear representing an upper limit of the elastic buckling load, the shear slenderness λ GA represents a loer limit of the slenderness of a sandich panel.

13 page 13 The first buckling mode of a sandich panel loaded by a centric axial force corresponds to global buckling for a slenderness of the panel λ > 1. For λ < 1, the first buckling mode corresponds to rinkling of the face sheets. In order that global buckling occurs, the panels must be thin and the faces must be relatively stiff, i.e. the face sheets must be thick. The global buckling load, hoever, is only reached for panels ith rather unusual slenderness regarding building practice. For panels ith usual dimension failure previously occurs by rinkling or shear failure. For (slender) components subjected to an axial compression force effects of nd order theory have to be considered in the design procedure. These result in an amplification of deflections and thus, also of the stress resultants moment and transverse force. 4 Buckling tests on axially loaded sandich panels For investigating the global load-bearing behaviour of axially loaded sandich panels, buckling tests ere performed. In order to be able to determine the global load-bearing capacity, local failure at load application area e.g. by crippling of the face sheet must be prevented. Therefore, the load as not introduced into the cut edges of the face sheets via contact such as in a real building structure. For load application, at first aluminium angles ere stuck together ith the face sheets and then fixed on a stiff load application plate. In order that the core of the panel can freely deform, the angles ere stuck in a ay that a gap of about 5 mm developed beteen the front side of the panel and the load application plate (Fig. 4.1). Fig. 4.1: Introduction of loads in the buckling tests In addition, panels ith relatively big unusual slenderness regarding building practice ere used for the tests. With increasing slenderness the influence of effects of nd order theory increases and global buckling and rinkling occur more likely. Therefore, the tests are mainly used for verifying the FE-model and the assumptions made to develop the design model.

14 page 14 In the tests, the panel had a hinged support at each end (Fig. 4.3). At the beginning of the test, the panel as subjected to a deflection via an additional support in mid-span. In this deflected position the support as fixed for the remaining test period (Fig. 4.4). In a second step, the axial load as applied. axial load ooden board hinged support round steel d=3mm sheet of steel d=mm L-shaped aluminium profiles 6x4x4 glued to the face of the panel transverse load P support in mid-span strain gauges length L measurement of displacement measurement of force hinged support Fig. 4.: Test set up for buckling tests

15 page 15 Fig. 4.3: Hinged support Fig. 4.4: Support at mid-span In addition to the axial force and the deflection, the strain of the face sheet subjected to tension as measured. Therefore in mid-span strain gauges ere applied (Fig. 4.5).

16 page 16 Fig. 4.5: easurement of strain in mid-span If no failure previously occurs at load application area, the reaction force at the deflected midsupport decreases ith increasing axial force. When roughly reaching the elastic buckling load the panel lifts from the fixed mid-support and the deflection increases ithout an essential change of the axial force (Fig. 4.6). 3 Stress resulting from force P and = rinkling failure stress [/mm ] stress in face subjected to tension axial force [k] -5 axial force displacement [mm] increase of axial force decrease of force P at mid-support increase of deflection increase of stress in face P=; Lifting of panel from mid-support Fig. 4.6: Test performance of buckling tests (test no. 9)

17 page 17 Because of the increasing deflection also the stress resultants moment and transverse force and thus the normal stress in the faces and the shear stress in the core increase. Finally failure occurs by rinkling of the compressed face sheet or by shear failure of the core material. Fig. 4.7: Increasing of deflection and rinkling at mid-span In the buckling tests panels ith different core materials and different faces ere tested. Also the length of the tested panels and the initial deflection at mid-support ere varied during the tests. A compilation of the performed buckling tests is given in Tab Tab. 4. shos a summary of the test results.

18 page 18 o. type of panel * ) face core length of panel initial deflection 1 F steel,,75 mm PU, 6 mm 5 mm 15 mm F steel,,75 mm PU, 6 mm 5 mm 15 mm 3 L GFRP, 1,8 mm EPS, 6 mm 3 mm mm 4 K steel,,6 mm EPS, 6 mm 3 mm mm 5 F steel,,75 mm PU, 6 mm 3 mm 15 mm 6 F steel,,75 mm PU, 6 mm 3 mm 17,5 mm 7 F steel,,75 mm PU, 6 mm 35 mm mm 8 F steel,,75 mm PU, 6 mm 35 mm 16,5 mm 9 F steel,,75 mm PU, 6 mm 35 mm 5 mm *) number refers to test report D3. [7], [8] Tab. 4.1: Performed buckling tests o. max load [k] failure mode final failure after global buckling analytical buckling load [k] 1 34,4 51, failure of glue at load introduction delamination of face subjected to compression at load introduction - 58,8-58,4 3 1,6 global buckling failure of core near load introduction 1,4 4 5,3 failure of core near load introduction - 63,5 5 46, global buckling failure of core at load introduction 49,5 6 47, 7 36,7 failure at load introduction failure at load introduction - 5,4-43, 8 43, global buckling failure of core at load introduction 4, Tab. 4.: 9 43,4 global buckling rinkling of face 43,4 Results of buckling tests For each type of panel the material properties of core and face layers ere determined (Tab. 4.3).

19 page 19 Tab. 4.3: type of panel thickness of face sheet [mm] shear modulus of core [/mm ] elastic modulus of core [/mm ] F,698 /,7 3,81 3,66 K,554 /,551 5,57 - L aterial properties GFRP (approx. 1,8 mm) 11,8 - For panel type F additional bending tests ere performed to determine the rinkling stress and the rigidity of the panel. The buckling tests on axially loaded sandich panels are documented in detail in D3.-part 3 [7]. 5 umerical investigations 5.1 FE-odel The numerical investigations ere performed using the finite element program ASYS. The face sheets of the panel ere modelled ith shell elements of type Shell 181. This element is defined by four nodes ith three displacement degrees of freedom and three rotational degrees of freedom. It has bending, membrane and shear stiffness. As material behaviour, bilinear material equations ere arranged (linear-elastic, ideal-plastic), i.e. after reaching the yield strength, yielding occurs ithout strain hardening. The core layer of the panel as represented by volume elements of type Solid 185. This element has eight nodes ith three displacement degrees of freedom. For the numerical investigations homogenous and isotropic core material as assumed. The numerical investigations ere performed on single-span panels ith hinged supports. In order to be able to investigate the global load-bearing behaviour of the panels, failure at the load application areas must be prevented. Therefore, the support of the panels as modelled as illustrated in Fig Compared to the face sheets, a stiff plate is directly connected to both face sheets on the front side of the panel. The core of the panel and the plate has no sheared nodes, so the core can freely deform. This corresponds to the construction of the support in the buckling tests, here free deformation of the core as facilitated by a gap beteen core material and load application plate. On the base point, displacements ere prevented at a line of nodes; this corresponds to a hinged support. At the opposite end of the panel, the nodes are only constrained horizontally, but they are vertically displaceable. On these nodes, the axial load can either be applied as load or as displacement.

20 page hinged support stiff plate for introduction of load face core Fig. 5.1: FE-model and detailed vie on the support 5. Preliminary investigations 5..1 Preliminary remark For checking the efficiency of the FE model preliminary investigations ere performed. At first, the necessary fineness of the mesh as defined. Especially if rinkling of the face sheet has to be represented by the model, the element size in longitudinal direction of the sandich panel has a decisive influence on the results. The mesh must be so fine that the aves corresponding to the eigenmode of a rinkled face sheet can correctly be represented. For the performed investigations a length of 4 mm as sufficient, the thickness direction resulted in a necessary mesh length of about 1 mm, the idth direction in about mm. The FE model as checked by comparing results of the numerical calculation ith analytical values. For this purpose, the stiffness for bending load and different eigenvalues, e.g. elastic rinkling stress and elastic buckling load ere considered. In addition, some of the buckling tests ere recalculated. 5.. Rigidity of the panel In order to examine, hether the FE-model correctly represents the rigidity of a sandich panel, the deflection determined in the FE-analyses as compared ith the analytically determined deflection according to (.8). In addition, the analytically and numerically determined stresses in the face sheets ere compared to each other. In the numerical investigations the panel as loaded by a uniform transverse load hich as selected in a ay that the stress in the face sheets as about 1 /mm.

21 page 1 With the parameters E F = 1 /mm, G C = /mm, t F =,6 mm, B = 5 mm and a thickness range of 4 mm to 1 mm the folloing analytical and numerical results of stress σ F and deflection ere calculated. Length Thickness load analytical results FE deviation b v stress stress stress [mm] [mm] [/mm ] [mm] [mm] [mm] [/mm ] [mm] [/mm ] [%] [%] 3 Tab. 5.1: 1,53 8,9 3, 38,9 1 39,4 99,8 1,1, 8,43 11, 3, 41, 1 41,4 99,9,58,1 6,3 14,9 3, 44,9 1 44,8 99,9,18,1 4,1,3 3, 5,3 1 5, 99,9,61,1 1,8 4, 3, 34, 1 35,3 99,9 3,9,1 8,64 5, 3, 35, 1 35,5 99,9 1,54,1 6,48 6,6 3, 36,6 1 36,6 99,9,4,1 4,3 9,9 3, 39,9 1 39,5 99,9 1,5,1 Comparison of analytical and numerical results Based on the good agreement beteen analytical and numerical analysis, it can be assumed that the FE-model correctly reproduces the bending behaviour of a sandich panel Buckling behaviour In addition, the FE-model as checked by comparison of numerically and analytically determined eigenvalues. For this purpose, the rinkling stress for panels subjects to bending and axial load and the global buckling load ere considered. Wrinkling stress due to bending For panels subjected to a constant transverse load the first eigenvalue as determined by numerical analysis, and the corresponding stress in the face sheets as calculated. These stresses are compared to the rinkling stress calculated according to (.). For the parameters G C =3 /mm, E C = 4 /mm, t F =,5 mm and a thickness range of 4 mm to 1 mm, the folloing results ensue. Thickness [mm] length [mm] rinkling FE 111,6 111,6 111,6 111,6 stress σ analytical 11,5 111,6 111,6 111,6 Tab. 5.: deviation [%],77,3,,1 Comparison of analytical and numerical results for rinkling stress

22 page Fig. 5.: Wrinkling of the compressed face (buckling mode) Wrinkling stress due to centric axial load For panels ith a slenderness λ < 1 the first eigenmode upon centric axial load corresponds to rinkling of both face sheets. The corresponding axial rinkling force can be calculated according to (3.5) and (.). In the FE analysis, the panel as loaded in longitudinal direction at the nodes forming the displaceable support. For a panel ith the parameters given in the previous section, the folloing values result from the analytical and numerical analysis. Thickness [mm] Length [mm] 6 Tab. 5.3: FE analytical deviation [%] 3,3,3 1,5,4 Comparison of analytical and numerical results for axial rinkling load Fig. 5.3: Wrinkling caused by axial load (eigenvalue)

23 page 3 Elastic buckling load for global buckling For panels ith slenderness s λ > 1, the first eigenmode is global buckling of the panel. The analytical eigenvalue can be calculated according to (3.1) and (3.). For the parameters G C = /mm, E C = 4 /mm, t F =,7 mm and a thickness range of 4 mm to 1 mm the folloing numerical and analytical results ensue for the global buckling load cr. Thickness [mm] Length [mm] Tab. 5.4: cr FE analytical deviation [%],6,,5 1,6 Comparison of analytical and numerical results for global buckling load Fig. 5.4: Global buckling caused by axial load (eigenvalue) For all results listed above, a very good agreement beteen analytically and numerically determined eigenvalues can be observed Comparison to test results In addition to the verification of the FE-model by comparison to analytical values, the model as also checked by comparison to test results. For the numerical calculations, the thickness of the face sheets and the material properties of the core determined for the tested panels ere used. The correct representation of the rigidity as checked by comparison ith the bending test performed ithin the frameork of the tests on axially loaded sandich panels [7]. In the fol-

24 page 4 loing figure, the force-deflection relationships determined in the test and in the numerical analysis are opposed to each other. 7 6 load [] test FE deflection [mm] Fig. 5.5: Load-deflection-diagram for test and FE-calculation In addition, some of the buckling tests documented in [7] ere recalculated in order to verify the FE-model. In doing so, tests ere selected for hich the axial force could be increased up to the buckling load ithout failure at the load application area, i.e. the tests in hich lifting of the panel from the mid-support and thus, an increase of deflection took place (tests no. 3, 5, 8, 9). At the mid-support only compression forces, but no tensile forces are transferred. In the FE model, the mid-support as represented by longitudinal springs (element type combin39) (Fig. 5.7). The load-displacement relationship of the springs as selected in a ay that they have a high stiffness, if compression forces act, but they cannot transfer any tensile forces (Fig. 5.6).

25 page 5 load displacement Fig. 5.6: Load-displacement-relationship for springs at mid support support at mid-span sandich panel P spring element combin39 Fig. 5.7: id-support in the FE-model In the folloing table, the maximum loads determined in the tests are opposed to the maximum loads from the numerical calculation. There is a good agreement beteen test and numerical calculation. test o. result of test result of FE deviation analytical buckling load 3 1,6 k 1,6 k,4 % 1,4 k 5 46, k 45,1 k,4 % 49,5 k 8 43, k 4,5 k 1, % 4, k 9 43,4 k 4,6 k 1,8 % 43,4 k Tab. 5.5: Comparison of test results and numerical calculation

26 page Calculations for determination of ultimate stress For the design of panels subjected to transverse loads the rinkling stress σ is the ultimate compression stress, hich can act in a face sheet. It is determined in bending test. In order to determine the ultimate stress for panels loaded by an axial force or by a combination of axial load and transverse load, FE-analyses ere performed. For this purpose, the panels ere loaded by a centric axial compression load or a combination of an axial load and a constant transverse load. The transverse load as selected in a ay that the compressive stress caused by bending corresponds to about half the rinkling stress. In addition, calculations ith panels ith a global initial deflection and a centric axial load ere done. As global imperfection an initial deflection e corresponding to the first eigenmode of an axially loaded panel as applied (Fig. 5.8). The range of the initial deflection e as selected according to the maximum alloable longitudinal boing folloing E 1459 [1]. 1 e = L (5.1) 5 Fig. 5.8: Global imperfection for numerical calculation For all calculations, a local geometric imperfection as applied on the face sheet subjected to compression. The local imperfection corresponds to the eigenmode hen reaching the rinkling stress (Fig. 5.9).

27 page 7 Fig. 5.9: Local imperfection for numerical calculation The calculations ere performed for the panel types given in Tab The range of the local imperfection as varied beteen,5 t F and,1 t F. Type of panel P1 P P3 E C [/mm ] G C [/mm ] 4 3 t F [mm],6,5,7 Tab. 5.6: D [mm] Panels for calculation of ultimate stress Calculations on the load-bearing capacity ere performed in order to determine the ultimate compressive stress. For comparison, calculations on the load-bearing capacity ere carried out on panels ith the same local imperfections under bending load only and the ultimate compression stress as determined as ell. This stress corresponds to the rinkling stress, hich is usually determined by bending tests. Fig. 5.1 shos a comparison of the numerically determined ultimate stresses for different kinds of loadings.

28 page 8 18 Ultimate stress for axial load [/mm ] axial force and global imperfection axial force and bending moment axial force Wrinkling stress for bending load [/mm ] Fig. 5.1: Comparison of ultimate stresses Obviously, the failure stress upon normal forces or combined load from normal force and bending moment and the failure stress for pure transverse load are identical. Therefore, the rinkling stress used to design panels subjected to transverse loads can also be used to design axially loaded sandich panels. This assumption as confirmed by the tests carried out on axially loaded sandich panels. In one of the buckling tests (test no. 9, [7]) failure occurred by rinkling of the face sheet subjected to compression (Fig. 5.1). The stresses in the face sheets ere calculated from the axial load and the deflection measured in the test. The stress in the compressed face sheet as 163 /mm upon failure of the panel (Fig. 5.11). For the corresponding face sheet, the rinkling stress as also determined in a bending test on a panel of the same type (Fig. 5.1). The rinkling stress for transverse load as 165 /mm. This good agreement confirms the applicability of the rinkling stress determined by bending tests as ultimate stress for axially loaded panels.

29 page 9 4 face in tension - -4 stress [/mm ] measured face in compression calculated rinkling failure axial force [k] Fig. 5.11: easured and calculated stresses in the buckling test Fig. 5.1: Wrinkling of the compressed face in buckling and bending test 5.4 Calculations to investigate the influence of nd order theory If components are loaded by compressive axial loads, effects of nd order theory have to be taken into account, i.e. deformations are considered in determination of stress resultants. If stress resultants are determined according to 1 st order theory deformations are neglected.

30 page 3 Because of effects of nd order theory stresses do not increase proportionally to the axial load. Due to the axial force an increase of deflection also results in an increase of moment and transverse force. The amplification of the stress resultants moment and transverse force V as ell as the amplification of the deflection can approximately be determined by the amplification factor α. II I = α (5.) II V V I = α (5.3) II I = α (5.4) II I = (5.5) ith I, V I, I II, V II, II amplification factor: 1 α = 1 cr stress resultants and deflection according to 1 st order theory stress resultants and deflection according to nd order theory (5.6) If effects of nd order theory are considered, also geometrical imperfections such as initial deflections must be considered for determination of stress resultants. So the moment according to 1 st Order Theory is I = + e (5.7) moment caused by transverse load and fixed-end moment e : moment caused by initial deflection in conjunction ith axial force In order to check hether the approximation by the amplification factor α can also be applied for sandich panels, FE-analyses ere carried out and the load-deflection relationships ere compared to the curves calculated according to the equations given above. In Fig and Fig. 5.14, the load-deflection curves from FE-analyses are opposed to the corresponding curves determined ith the amplification factor α.

31 page 31 4 Axial load and initial deflection 35 3 G C =4/mm E C =8/mm D=8mm t F =,6mm L=3mm FE nd Order Theory axial force [] 5 15 G C =3/mm E C =6/mm d=6mm t F =,75mm L=4mm 1 5 initial deflection deflection [mm] Fig. 5.13: Load-deflection-curves for FE and calculation by nd Order Theory 5 Axial and transverse load G C =4/mm E C =8/mm D=8mm t F =,75mm L=3mm FE nd Order Theory 3 force [] 5 15 G C =3/mm E C =6/mm D=6mm t F =,75mm L=4mm 1 5 deflection caused by transverse load deflection [mm] Fig. 5.14: Load-deflection- curves for FE and calculation by nd Order Theory The applicability of the amplification factor α on axially loaded sandich panels as also confirmed by the buckling tests. The stresses in the face sheets ere calculated in dependence

32 page 3 on the axial force and compared to the stresses determined from the measured strains. For selected tests, the calculated and the measured stresses are compared to each other (Fig to Fig. 5.17). Also in this case, a good agreement beteen test and calculation by nd Order Theory can be observed. 7 o 5 5 test calculation 3 stress [/mm ] axial force [k] Fig. 5.15: Comparison beteen test and calculation by nd Order Theory, test 5

33 page 33 o test calculation 3 stress [/mm ] axial force [k] Fig. 5.16: Comparison beteen test and calculation by nd Order Theory, test 8 o 9 1 test calculation stress [/mm ] axial force [k] Fig. 5.17: Comparison beteen test and calculation by nd Order Theory, test 9 In [1], the approximate calculation ith the amplification factor α is compared ith the exact analytical solution by means of an example. Also in this case, the results agree ell.

34 page 34 6 Design methods 6.1 General The influence of an axial load can be considered either by design according to nd order theory or by design according to equivalent member method. For design according to nd order theory the stress resultants moment and transverse force determined according to 1 st order theory are increased by the amplification factor α. In addition, geometrical imperfections must be considered for the determination of stress resultants, since they also result in moments and transverse forces if an axial force exists. Finally, the panels are designed on stress level. For design according to equivalent member method, the stress resultants according to the 1 st order theory are used. In dependence on the geometric imperfection and the slenderness of the component, hoever, the maximum compressive axial force (axial rinkling force ) has to be reduced. 6. Design according to nd order theory The stress resultants according to nd order theory are determined by approximation by amplification of the stress resultants according to 1 st order theory. In addition to moments from transverse load (e.g. ind loads) and fixed-end moments, also geometrical imperfections must be considered in the moment according to 1 st order theory. As geometrical imperfection an initial deflection e can be applied. The range of this imperfection can be selected according to the maximum alloable longitudinal boing folloing E 1459 [1]. 1 e = L (6.1) 5 Also deflections T from a temperature difference ΔT beteen internal face and external face result in moments and transverse forces if the panel is axially loaded. These moments can be determined just as the moments from geometrical imperfections. I T = (6.) T α F T α F L = T D 8 coefficient of thermal expansion (6.3) So the stress resultants according to nd order theory are calculated as follos. II I = α (6.4) II V V I = α (6.5) II I = (6.6)

35 page 35 amplification factor: 1 α = 1 I cr = + e + T moment caused by transverse load and fixed-end moment ( e) e : moment caused by initial deflection in conjunction ith axial force T : moment caused by temperature deflection (6.7) (6.8) After determination of moment and transverse force according to nd order theory, design is done on the stress level using the rinkling stress as ultimate stress. II σ F = + σ A + A A D F1 F F1 (6.9) For transverse load, the shear strength f Cv of the core material can also become decisive. II V τ C = f Cv (6.1) A C Since the rinkling stress σ and the shear strength f Cv are usually determined by testing, local imperfections of the panels such as surface irregularities of the face sheet and the material properties of the core are already considered in the ultimate stress. 6.3 Design according to the equivalent member method Alternative to the design according to nd order theory on the stress level also design by equivalent member method is possible. For determining the reduction factor of the axial rinkling force a buckling curve for the corresponding sandich panel is necessary. For a centrically loaded panel ith the initial deflection e e get the folloing equation. This equation considers the stress resultants according to nd order theory. The ultimate stress of the compressed face sheet is the rinkling stress σ. ( A + A ) F1 F σ e + D A σ F1 1 1 cr 1 (6.11) Insertion of the axial rinkling force and the slenderness λ as ell as the reduction factor χ = (6.1) results in the equation

36 page 36 e χ + D χ 1 χ λ = 1 (6.13) As solution of (6.13) the knon formula for determining the reduction factor is obtained. 1 χ = (6.14) φ + φ λ ith e φ =, λ D (6.15) By application of the length-dependent initial deflection e e = (6.16) L (6.15) can be transformed to ( 1+ α λ λ ) φ =,5 GA + λ (6.17) ith the imperfection factor e = π D α B S (6.18) The imperfection factor depends on the geometry of the panel (thickness D, bending rigidity B S ). In addition, the geometrical imperfection ē is considered in α. By consideration of the axial rinkling force and thus, the rinkling stress σ also local imperfections as ell as material properties of the core layer are indirectly considered in the imperfection factor. As an example, Fig. 6.1 shos a buckling curve for a sandich panel. For checking the method, additionally some reduction factors determined by FE-analyses are shon in the figure.

37 page 37 1, 1, G C =/mm E C =4/mm D=8mm t F =,75mm σ =9/mm e =1/5 L FE buckling curve,8 reduction factor χ,6,4,, λ,,5 GA 1, 1,5,,5 3, 3,5 slenderness λ Fig. 6.1: Example of a buckling curve for sandich panels In order to be able to consider also moments from transverse load or eccentric axial loads, besides a practically never existing pure centric axial load, the above equation can be extended by a part for the bending moment folloing E (section 6.3 and annex B). + k yy 1 (6.19) χ ith k yy A F = D σ (6.) = C my (1 +,8 χ ) C my : equivalent uniform moment factor according to E , table B.3 (Tab. 6.1) (6.1)

38 page 38 moment distribution C my uniformly distributed load point load ψ 1 ψ 1,6 +,4ψ, 4 α s 1 1 ψ 1, +,8α s, 4, +,8α s, 4 h s α s = h / s ψ h 1 α s < ψ 1,1,8α s, 4,8α s, 4 1 ψ <,1( 1 ),8α s, 4 ψ,( ψ ),8α, 4 s h s α h = h / s ψ h α h 1 1 ψ 1 1 α h < ψ 1 Tab. 6.1: C my according to table B.3 of E [3],95 +,95 +,5α h,5α h 1 ψ <,95 +,5α ( 1 + ψ ),9 +,9 +,1α h,1α h h,9,1α h ( 1 + ψ ) In Tab. 6. some calculations performed by means of the above equivalent member method are opposed to the results of a FE-analysis and a calculation according to nd order theory. For three different systems calculation is done for three different values of transverse load or end moment each. With the different transverse load or end moment the quotient / is determined. In a second step the axial load and the quotient / is determined. If the quotients / determined by the different methods are compared, a good agreement becomes apparent. / calculated by Tab. 6.: system / equivalent member method nd Order Theory,,46,46,45,39,31,3,31,79,1,1,1,19,47,46,46,39,33,3,33,78,1,1,11,18,44,46,46,37,3,33,33,74,1,1,1 Comparison of different possibilities for design FE G=3/mm E=6/mm D=8mm t F =,6mm L=35mm e =1/5 L

39 page 39 7 Time-dependent behaviour 7.1 Basics In addition to the deflection b due to bending, the deflection of a sandich panel consists of the deflection v due to shear deformation resulting from transverse force (formula (.5)). Both, organic core materials such as polyurethane or expanded polystyrene and mineral ool sho creep phenomena under long-term loads, e.g. dead-eight load and sno. If a constant load acts on a panel over a long period of time, the shear deformation of the core material and thus the shear deflection v increase. The shear strain γ increases ith constant shear stress. For sandich panels the time-dependent shear strain is usually described by a creep function φ(t). ( 1 ϕ( )) γ ( t) = γ () + t (7.1) Therefore the time-dependent deflection is ( 1 + ϕ( )) ( t) = + ( t) = + () t (7.) b v b v If the panels are loaded only by transverse forces creep effects can be considered by a reduction of the shear-modulus G C. So for design e have a notional time-dependent shearstiffness GA(t) ( t) = dx + VVdx ( 1+ ϕ ( t) ) = dx + VVdx (7.3) B GA B GA( t S S ) GC GA( t) = A 1 + ϕ( t) C (7.4) To design sandich panels, hich are subjected to long-term loads, not the complete creepfunction is necessary. Usually only to creep coefficients are used. The creep coefficient φ is used to consider sno loads; the creep coefficient φ 1. is used to consider dead eight loads. To determine the creep coefficients φ t, the creep function φ(t) is determined by (bending-) creep tests on panels subjected to a constant transverse (dead) load. In the test, the deflection of the panel is measured for a period of 1 hours and the creep function is recalculated [1], []. ( t) () ϕ ( t) = (7.5) () b ith (t) deflection measured at time t () initial deflection measured at time t=

40 page 4 b bending deflection, determined by calculation Based on the experimentally determined creep-function, both creep coefficients φ and φ 1. are determined by extrapolation. For design purposes the creep coefficient φ determined by testing is increased by %. So it is taken into account that a part of the creep deformation caused by sno loads during inter time is not recovered during summer time. 7. Creeping of axially loaded sandich panels Creeping of the core material results in an increase of the deflection of a panel. If, in addition to the transverse load, an axial load acts on the panel the increase of the deflection results also in an increase of the stress resultants moment and transverse force (Fig. 7.1). The normal stresses in the faces and the shear stress in the core also increase. Therefore, the longterm behaviour of the panels has not only to be considered in the design for serviceability limit state (deformation limit) but also in the design for ultimate limit state (load-bearing capacity), i.e. for the determination of the stress resultants. t o = t t = o o Fig. 7.1: Creeping of an axially loaded sandich panel 7.3 Long-term tests To investigate the influence of axial loads on the creep behaviour of sandich panels, longterm tests on axially loaded sandich panels ere performed. In creep tests, hich are usually performed on panels ith transverse load, the panels are loaded by a constant load and the time-dependent deflection is measured (Fig. 7.).

41 page 41 load measured displacement time t time t Fig. 7.: Creep test To test panels ith axial load this is an inapplicable procedure. So the long-term test ith axial load ere not performed as classical creep tests but as relaxation tests, i.e. a constant displacement is applied and a resulting time-dependent force or stress is measured (Fig. 7.3). displacement measured force time t time t Fig. 7.3: Relaxation test During the long-term tests, the panels ere loaded by an axial load corresponding to the orking load to be expected in reality. For determination of the axial load it as assumed that all panels are loaded by the loads applied from an overlying roof panel. For the roof panel a span of 6 m and a dead eight of about,1 k/m ere assumed. With a sno load s k =,68 k/m (sno load zone, m above normal zero [4], [5]) an axial load of,4 k/m ensues for the all panels. This also approximates the indications in [9]. Here, an axial design load of 3,67 k/m is assumed for the external all of a building, hich corresponds to a load of,6 k/m for an averaged load factor of γ F = (1,5+1,35)/ = 1,45. The axial load in the tests as,8 k/m (including test set-up). Since the creep behaviour of different core materials differs from each other, panels ith the common core materials (polyurethane, expanded polystyrene, mineral ool) ere tested.

42 page 4 ooden board hinged support round steel d=3mm sheet of steel t=mm L-shaped aluminium profiles 6x4x4 glued to the face of the panel transverse load P support in mid-span constant deflection strain gauges length L Axial loading x 48 kg measurement of displacement measurement of force hinged support Fig. 7.4: Test set up for long-term tests on axially loaded sandich panels The test set-up for the long-term test is given in Fig To apply the axial load the same test set-up as used as in the buckling tests (section 4). Per type of panel to tests ere performed. In the first test, the axial load as applied centrically, i.e. into both face sheets, in the second test the axial load as applied eccentrically only into one face sheet (Fig. 7.5).

43 page 43 / / Fig. 7.5: Centric and eccentric application of axial load After applying the axial load, a deflection as applied to the panel via an additional support in mid-span (Fig. 7.6). The support as fixed and the deflection as kept constant over the test period. Fig. 7.6: Support at mid-span The reaction force at the mid-support as continuously recorded during the test (Fig. 7.7). To measure the reaction force strain gauges ere used hich ere applied on the thread rods, used to install the support (Fig. 7.7).

44 page 44 Fig. 7.7: easurement of reaction forces at mid support (strain gauges) In addition, in mid-span the strains in the face sheet subjected to tension ere measured ith strain gauges and also continuously recorded. Fig. 7.8: Strain gauges at the face sheet subjected to tension A compilation of the performed long-term tests is given in Tab A detailed documentation of the test results can be found in D3.-part 4 [8].

45 page 45 Tab. 7.1: o. type of panel face core application of axial load initial deflection F-1 F steel,,75 mm PU, 6 mm centric 15,5 mm F- F steel,,75 mm PU, 6 mm eccentric 1,5 mm G-1 G steel,,6 mm EPS, 6 mm centric 15,5 mm G- G steel,,6 mm EPS, 6 mm eccentric 1, mm H-1 H GFRP, 1,8 mm EPS, 6 mm centric 15,5 mm H- H GFRP, 1,8 mm EPS, 6 mm eccentric 1,5 mm I-1 I steel,,6 mm W, 6 mm centric 15, mm I- I steel,,6 mm W, 6 mm eccentric 1, mm Performed long-term tests For the purpose of comparison additionally to the long-term tests on axially loaded panels creep tests according to E 1459 [1] ere performed for each tested type of panel. The tests ere performed on single-span panels hich ere subjected to a uniformly distributed dead load (Fig. 7.9, Tab. 7.). measurement of displacement dead load measurement of displacement L = 7mm Fig. 7.9: Test set up for creep tests according to E 1459 Fig. 7.1: Test set up for creep tests according to E 1459

46 page 46 o. F G H I Tab. 7.: dead load 1,44 k 1,44 k,96 k,64 k Dead load in creep tests In addition the material properties of core and face layers ere determined for each tested type of panel. Also the results of the creep tests and the material properties are documented in D3.-part 4 [8]. 7.4 Evaluation of long-term tests In the folloing it is verified that an axial force does not influence the creep behaviour of sandich panels. So the creep coefficients determined according to E 1459 by simple bending tests ith transverse loading can also be used to design panels, hich are axially loaded. Comparatively complex tests on axially loaded panels are not necessary. For this purpose the reaction force P at mid support and the stresses in the face sheet are determined by calculation using the creep functions determined according to E 1459 (bending creep tests). These calculated values are compared to the values determined in the tests on panels ith axial load. Based on the creep behaviour of the core material the reaction force P measured at midsupport decreases during the test (Fig. 7.11). For this reason, also the stresses and strains in the face sheets decrease.