Response of a sandwich panel subject to fire or elevated temperature on one of the surfaces

Transcription

1 Comosites: Part A 37 (26) Resonse of a sandwich anel subject to fire or elevated temerature on one of the surfaces V. Birman a, *, G.A. Kardomateas b, G.J. Simitses b,r.li b a Engineering Education Center, University of Missouri-Rolla, 81 Natural Bridge Road, St Louis, MO 63121, USA b School of Aerosace Engineering, Georgia Institute of Technology, Atlanta, GA , USA Abstract The aer resents the analysis of deformations and stresses in a large asect ratio sandwich anel subject to fire or another source of elevated temerature. The anel, assumed to bend into a cylindrical surface, is simly suorted at the edges. The edges are also revented from in-lane dislacements roviding an elastic restraint as the anel stretches due to bending. The solution is obtained in a closed form when the deformations are small and when geometrically nonlinear effects are incororated into the analysis. The solution is oen to modification with arbitrary temerature and roerty distributions through the thickness of the anel, enabling a designer to incororate the results from a multitude of heat transfer scenarios so long as the structural roblem can be treated as quasi-static. q 25 Elsevier Ltd. All rights reserved. Keywords: Sandwich anel; B. Thermomechanical; B. Strength; C. Analytical modeling 1. Introduction The roblems of durability, real-time and residual strength and stiffness of sandwich structures subject to fire or other sources of elevated temerature reresent a major interest for designers. The comlexity of the roblem is related to a number of couled henomena, including the dynamic roblem of heat transfer, roerty degradation due to an elevated temerature, resin decomosition in PMC facings, real-time strength and stiffness and residual strength of the structure after fire. In some situations, the issue is a redicted life of the structure subject to fire, as the stresses and deformation gradually build u to the instant when the structure collases. Although the temerature and roerties of engineering materials are affected by the stress [1], it is usually ossible to ignore this interaction. In this case, the roblems listed above can be uncouled, i.e. the dynamic roblem of temerature distribution and material decomosition is analyzed first and subsequently, real-time mas of distribution of temerature and roerties are alied to the stress and deformation analysis. In the case of a sandwich anel * Corresonding author X/$ - see front matter q 25 Elsevier Ltd. All rights reserved. doi:1.6/j.comositesa subject to fire, the former roblem has been addressed in a number of investigations [2,3]. While the heat transfer roblem is dynamic, the structural roblem can be formulated as a static case since the changes associated with fire are relatively slow. The resent aer is concerned with the resonse of a large asect ratio sandwich anel subject to fire or another source of elevated temerature on one of the surfaces. The anel, bending into a cylindrical surface, is simly suorted along the edges that are also revented from inlane dislacements by adjacent structures. The solution is obtained in a closed form for small deformations and in a geometrically nonlinear formulation. Numerical results are obtained using a simlified quasi-static aroach to a distribution of temerature through the thickness. In site of this simlification, the results obtained from the solution are in a qualitative agreement with available exerimental data. In the case where the thermal roblem is solved by a more accurate aroach, the corresonding adjustments to the roerties and temerature can be incororated in the resent analysis without altering its validity. 2. Analysis Consider a large asect ratio sandwich anel with crossly facings and a foam core that reresents a art of

2 982 V. Birman et al. / Comosites: Part A 37 (26) Fig. 1. Structure consisting of a number of identical sandwich anels suorted by frames or bulkheads. Each anel reresents a large asect ratio late. Long edges of such late are constrained against dislacements in the x-direction by adjacent lates. the structure consisting of a number of identical anels suorted by frames or bulkheads (Fig. 1). The adjacent anels severely limit in-lane extension and contraction of the anel in the x-direction. If the symmetry of load or geometry is violated, or if only one anel in the comartment is affected by an elevated temerature, the conservative aroach is to assume simle suort along the long edges. In the roblem considered in this aer, temerature is nonuniform through the thickness as may be the case if fire occurs on one side of the anel. The distribution of temerature through the thickness deends on whether thermal conductivities of the constituent materials are affected by temerature. The thickness of the facings is usually relatively small, so that temerature can be assumed constant in each facing [4]. The rincial temerature gradient is the core. For examle, if the conductivity of the core is a linear function of temerature, i.e. k Z k Ck 1 T (1) where k, k 1 are constants and T is a change of temerature from the reference value, the variation of temerature in the core (without taking account of a decomosition of the material) is given by [4] T ZK k C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S k 1 z CS (2) 1 Constants of integration S, S 1 are determined from the thermal boundary conditions at the facing core interfaces. If the effect of temerature on the thermal conductivity of the core material is negligible, temerature varies linearly from the heated facing to the cooler facing. In the subsequent discussion, it is assumed that the distribution of temerature and material roerties through the thickness are known. The analysis is conducted modeling the sandwich anel by a first-order shear deformable theory. As a result of a uniform over the surface temerature varying in the thickness direction the central art of the anel (at a sufficient distance from the short edges) deforms into a cylindrical surface. Accordingly, all derivatives with resect to the y-coordinate as well as the rotation in the yz-lane and the dislacement in the y-direction are equal to zero. Accordingly, the equations of equilibrium for the anel are obtained simlifying the general three-dimensional equations of a first-order theory for shear-deformable comosite laminates. For convenience, a geometrically nonlinear formulation [5] is recalled here and resented in the form simlified for the resent alication. The strain in the facings that are assumed to be in the state of lane stress is comosed of the contributions of the strain of the middle lane of the anel and the change of its curvature (both of them in the xz-lane) 3 x Z 3 x Czj (3) where 3 x Z u ; x C 1 2 w2 ; x k x Z j ; x (4) The core works in transverse shear and the corresonding strain is g xz Z j Cw ; x (5) In these equations, u is a dislacement of the middle lane in the x-direction, w is a deflection of the anel, j is the rotation in the xz-lane, and ð.þ; x Z dð.þ dx. The stresses in the ith layer of the cross-ly facings are given by " s x Z Q # ( ) T i Q 12 T i 3x Ka x T i (6) s y i Q 12 T i Q 22 T i Ka y T i where Q mn (T i ) and a (T i ) are transformed reduced stiffnesses and the coefficients of thermal exansion, resectively, evaluated at the temerature of the layer. The stress in the isotroic core is given by t xz Z G xz ðtðzþþg xz (7) where the shear modulus G xz (T(z)) is affected by the local temerature. The equations of equilibrium of a anel bent into the cylindrical surface are [5,6] N x; x Z ; Q x; x CN x w ; xx Z ; Q x Z M x; x (8) where stress resultants and stress coule are given by N x Z A 3x CB k x KNx T ; M x Z B 3x CD k x KMx T ; Q x Z g xz (9) In these equations, A,, B, D are the extensional, couling, and bending stiffnesses introduced according to the standard definition (though the engineering constants emloyed to evaluate these stiffnesses are affected by temerature). The thermally induced stress resultant and stress coule acting in the sandwich anel with cross-ly

3 V. Birman et al. / Comosites: Part A 37 (26) facings can be evaluated from ð h=2 fnx T ; Mx T g Z ½Q ðtðzþþa x ðtðzþþ Kh=2 CQ 12 ðtðzþþa y ðtðzþþštðzþf1; zgdz (1) where h is the total thickness of the anel. The contribution of the core to the in-lane stress resultants and to the bending stress coules is usually neglected, i.e. the integration can be erformed over the thickness of the facings only, excluding the core. The first art of the subsequent analysis (Section 2.1) resents the solution for the geometrically seudo-linear roblem where bending deformations of the anel are small. Nevertheless, nonlinearity is resent since the axial stress resultant is affected by stretching of the middle lane. The second art (Section 2.2) shows the stress analysis of the anel (this can be alied to both seudo-linear and nonlinear roblems). Finally, Section 2.3 illustrates the aroach to the nonlinear analysis, accounting for moderately large deflections. The solution of the linear roblem is exact. The nonlinear roblem is reduced to a system of seven algebraic nonlinear equations for six constants of integration and the axial stress resultant. Exact solution of this system may be imossible, but the accuracy is limited only by the method of solution Geometrically seudo-linear bending roblem In this formulation, the nonlinear term is neglected in the first relation in Eq. (4). The equations of equilibrium become A u ; xx CB j ; xx KNx; T x Z ; ðj ; x Cw ; xx Þ CN x w ; xx Z ; D j ; xx K ðj Cw ; x Þ CB u ; xx KMx; T x Z ðþ Note that the thermally induced axial stress resultant and bending moment do not vary in the axial direction (temerature is uniform over the surface of the anel). Accordingly, the last terms in the first and last relations in Eqs. () disaear. Although Eqs. () seems linear, the nonlinearity is introduced through the axial restraint N x that deends on the magnitude of deflections. The solution of Eqs. () must satisfy the boundary conditions. If the anel is simly suorted, these conditions are: At xz, xza: u Z w Z ; M x Z B u ; x CD j ; x KM T x Z (12) Omitting the last term in the first relation in () yields u ; xx ZK B A j ; xx (13) Substituting Eq. (13) into the last two Eqs. () gives the following result w ; x ZKj C 1 D A K B2 j 55 A ; xx (14) and j ; xxx Clj ; x Z (15) where KN l Z x (16) 1 C N x D K B2 A The roblem can be reduced to the classical result for a thin symmetrically laminated late if jzkw x, Z, B Z. Then if the late is simly suorted, it is easy to show that N x,cr ZKD (/a) 2. If thermally induced comressive stresses are large, N x is negative. Consider the case where 1 C N x! (17) so that lo. Accordingly, the solution of Eq. (15) is j Z C 1 CC 2 sin ffiffi l x CC3 cos ffiffi l x (18) where C i are constants of integration. The solution of Eq. (14) becomes w Z C 4 KC 1 x Cf ðxþc 2 cos ffiffi l x Kf ðxþc3 sin ffiffi l x (19) where f ðxþ Z 1 ffiffi C D KðB 2 =A Þ l ffiffi l (2) Note that the integration of Eq. (14) added an additional constant of integration. Finally, integrating Eq. (13) yields u Z C 5 CC 6 x K B ðc A 1 CC 2 sin ffiffi l x CC3 cos ffiffi l xþ (21) Six constants of integration in Eqs. (18), (19) and (21) can be determined from six boundary conditions (12). Therefore, given the rescribed value of the thermally induced stress coule, one can determine u, w, j as functions of the stress resultant N x. However, this solution is not sufficient to redict thermal bending in terms of alied temerature. This is because N x accounts for two effects: (i) the thermally induced stress resultant N T x based on the solution of the heattransfer roblem and Eq. (1) and (ii) deflections of the anel that result in stretching of its middle lane. Integrating the first Eq. (9) between xz and xza, one obtains the relationshi between the axial stress resultant

4 984 V. Birman et al. / Comosites: Part A 37 (26) and deformations u and j: N x Z Nx T K 1 ð a ð a a A u ; x dx CB j ; x dx o (22) Now, given a distribution of temerature and stiffness through the thickness of the anel, it is ossible to determine six constants of integration and N x from seven linear Eqs. (12) and (22). In the case where comression is small, the inequality (17) is not satisfied, l!, and the solution of Eq. (15) becomes ffiffiffiffiffi ffiffiffiffiffi j Z C 1 CC 2 sinh jlj x CC3 cosh jlj x (23) The aroach to the analysis, similar to that described above for the case where lo, is omitted for brevity. Of course, the sign of l is unknown in advance, and the solution may have to be reeated if this sign was not guessed correctly. Note that Thornton considered the roblem of bending of an isotroic beam subject to a nonuniform temerature using a similar aroach [7] Stress analysis The thermally induced strains in the facings assumed in the state of lane stress are given by (inequality (17) is satisfied) 3 x Z u ; x Cz j ; x Z C 6 K B ffiffi l C A 2 cos ffiffi l x KC3 sin ffiffi l x ffiffi Cz l C2 cos ffiffi l x KC3 sin ffiffi l x ð24þ where z is a coordinate of the oint where the strain is evaluated. Now the stresses can be calculated in each layer of the facings by Eq. (6). The transverse shear strains in the core are ffiffi g xz Z C 2 1 Kf ðxþ l sin ffiffi ffiffi l x CC3 1 Kf ðxþ l cos ffiffi l x (25) The core carries only shear stresses available from Eq. (7). Note that contrary to thin facing layers, where it is ossible to use the average-through-the thickness value of temerature, the variations of temerature through the thickness of the core are significant. Accordingly, the shear modulus is a function of the z-coordinate Geometrically nonlinear roblem The first equilibrium relation in Eq. (8), written to account for the nonlinear strain dislacement Eq. (4), yields the axial dislacement at the middle lane. It is a nonlinear function of deflections: u ; xx ZK B A j ; xx Kw ; x w ; xx (26) The second Eq. (8) is not exlicitly affected by nonlinear effects. It follows that j ; x ZK 1 C N x w A ; xx (27) 55 Substituting Eq. (27) into Eq. (26) results in the exression for u as a function of w: u ; xx Z B 1 C N x A w ; xxx Kw ; x w ; xx (28) Eq. (27) can be integrated yielding the exression for the rotation j ZK 1 C N x w A ; x CC4 (29) 55 where C 4 is a constant of integration. The nonlinear version of the third equilibrium Eq. (8) is D j ; xx CB ðu ; xx Cw ; x w ; xx Þ K ðj Cw ; x Þ KMx; T x Z (3) After the substitution of Eqs. (28) and (29) and transformations, this equation assumes the form w ; xxx Klw ; x KhC 4 Z (31) where l is defined by Eq. (16) and h Z (32) ðd KðB 2 =ÞÞ 1 C N x Nonlinear terms cancel out in Eq. (31), so that the solution can be obtained in the closed form. The integration of Eq. (31) yields (for the case where lo) w Z C1 CC2 sinh ffiffi l x CC 3 cosh ffiffi hc l x K 4 l x (33) in which we have three additional constants of integration. Note that a deviation of the nonlinear solution from its linear counterart is reflected in a difference between hyerbolic and trigonometric functions that becomes essential only at large values of the argument. Finally, integrating Eq. (28) twice results in the solution for the axial dislacement u Z C5 CC6x C B 1 C N x CC 3 A ffiffi ffiffi h l sinh l x K l C 4 C 2 ffiffi ffiffi l cosh l x KFðC2; C3; C4Þ ð34þ where F(C 2,C 3,C 4) is a nonlinear function that is easily evaluated. Two additional constants of integration in Eq. (34) bring the total number of constants that have to be secified to six (as in the linear case). Six constant of integration have to be determined from the boundary conditions that do not differ from those for

5 V. Birman et al. / Comosites: Part A 37 (26) the linear case, excet for the exression for the bending moment: At xz, xza: u Z w Z ; M x Z B ðu ; x C 1 2 w2 ; xþ CD j ; x KM T x Z ð35þ The solution rocedure is similar to the linear case, though it may be tedious due to nonlinearities. The axial stress resultant is related to the deformations by the nonlinear version of Eq. (22): N x ZNx T K 1 ð a a A ðu ;x C 1 ð a 2 w2 ;xþdxcb j ;x dx (36) o 3. Numerical results The solution of the nonlinear roblem requires us to determine seven unknowns (C 1, C 2, C 3, C 4, C 5, C 6, l) from six boundary conditions and Eq. (36). This nonlinear roblem was solved by the Newton Rahson iteration method. The room temerature roerties of the layers of crossly facings considered in the examles corresonded to a tyical grahite/eoxy: E 1 Z 12:87 GPa; E 2 Z 18:58 GPa; m 12 Z :276; a 1 Z 1:977 mm=m er 8 C; a 2 Z 3:2881 mm=m er 8 C The variations of the stiffness of the facings with temerature are reflected in Table 1, obtained by assuming that the effect of temerature is similar to that for glass/ eoxy [8]. The room-temerature roerties of two grades of Divinycell core considered in the examles are resented in Table 2. The variation of the shear modulus of the core with temerature is assumed in the form G c (T)ZG c [1KG 1 (T/T rf )], where G c is the value at the room temerature T rf and G 1 is a nonlinear function of T/T rf given by G 1 T T rf Z:68 T 3 T K:7 T rf T rf 2 C:27 T T rf K:29 (37) Eq. (37) is alicable if the ratio T/T rf O3 Note that Eq. (37) yields variations of the shear modulus of the core with temerature that are in a qualitative agreement with those reorted for a tyical thermolastic foam [9]. Table 1 Variations of the stiffness of the facing material with temerature T (8C) E1 (GPa) E2 (GPa) Table 2 Stiffness of the core materials at room temerature Material H45 H6 E (MPa) G (MPa) The length of the short edges of the sandwich anel considered in the examles was az.6 mm, the core was 2 mm thick and two thickness of the facings was either h f Z2.5 or 5. mm. As was shown in Ref. [4] for the case of quasi-isotroic glass vinyl ester facings and a olymeric 2 mm thick core, temerature remains ractically uniform in the facings and its variation is mostly limited to the core. An examle of such analysis resented in Table 3 confirms the validity of this statement. In this table, T, T 1, T 2 and T 3 are temeratures of the surface of the heated facing, heated facing core interface, colder facing core interface and the surface temerature of the colder facing, resectively. Accordingly, in the following examles, the temerature of the heated facing was assumed constant and equal to T, while the temerature of the oosite facing was found from the static heat transfer roblem, assuming that the air outside this facing is at 2 8C. The axial stress resultant is shown as a function of the temerature of the heated facing in Fig. 2 and Table 4. The axial force increases from zero to a maximum value but once temerature reaches a certain level, the tendency is reversed, i.e. the axial force begins to decrease (the increasing force at temeratures below 5 8C is not shown). Physically, such decrease is due to tensile reactive stresses at the immovable in the x-direction edges attributed to stretching roduced by bending (similar to the reaction of immovable suorts of a beam subject to large bending deformations). The thickness of the facings has little effect on the magnitude of the force. The shae and magnitude of deflections along the san of the anel are shown in Fig. 3 (a similar result obtained for a anel with thicker facings is omitted for brevity). As follows from this figure, deflections increase until temerature reaches the value of T Z1 8C and begin to decrease at higher temeratures. Deflections of the anels with various facing thickness and core materials are also shown as a function of temerature in Fig. 4. As is clearly observed in this figure, deflections exerience a raid buildu as temerature varies between the room value and about 6 8C. At higher temeratures, the increase of deflections Table 3 Temerature distribution along the facing core interfaces and on the surfaces of a sandwich anel (h f Z5. mm) T T T T

6 986 V. Birman et al. / Comosites: Part A 37 (26) N x, MN/m.4.3 Core H6 Core H T, C Fig. 2. Absolute value of the axial edge restraint stress resultant (MN!m) as a function of the exosed surface temerature T. The facings are 2.5 mm thick. Table 4 Absolute value of the axial edge restraint stress resultant (MN!m) as a function of the exosed surface temerature T T (8C) H45 (h f Z2.5 mm) H6 (h f Z2.5 mm) H45 (h f Z5. mm) H6 (h f Z5. mm) W(x), mm (a) T = 5 C T = 75 C T = 1 C T = 125 C Fig. 4. Maximum deflections of the anel as a function of the temerature of the heated surface T, the thickness of the facings and the material of the core. slows down and eventually reverses itself. Predictably, a thicker core resulted in a decrease in deflections. Note that the henomena of the reversal of deflections with a higher temerature have been reorted in literature. In articular, Meyers and Hyer observed such a reversal of deflections of a comosite anel subject to a linearly distributed through thickness temerature [1]. A recent aer of Lattimer et al. [] on deformations of sandwich anels subjected to fire also suorts the observations in the resent aer. A distribution of the maximum transverse shear stresses in the core throughout the san of the anel is deicted in Fig. 5. The variations of these stresses at the suorts where they reach the extreme values are shown in Fig. 6 as a function of temerature. The observed X, mm W(x), mm 5 (b) σ c Max, Ma T = 5 C T = 75 C T = 1 C T = 125 C 2 1 T = 5 C T = 75 C T = 1 C T = 125 C X, mm X,mm Fig. 3. Maximum deflections of the anel as a function of the temerature of the heated surface T. The thickness of the facings is h f Z2.5 mm. Case (a): core H45; Case (b): core H6. Fig. 5. A distribution of the maximum shear stresses along the san of the anel as a function of the temerature of the heated surface T. The core is H45, the thickness of the facings is h f Z2.5 mm.

7 V. Birman et al. / Comosites: Part A 37 (26) σ c Max, Ma hfu = 2.5 mm hfu = 5. mm Core H T, C tendency in variations of deflections with an increasing temerature is mirrored by the similar trend for the transverse shear stress. The magnitude of shear stress was reduced in the anels with a thicker core, reflecting a decrease in deflections in such anels. Note that the magnitudes of the stresses at the suorts shown in Fig. 6 indicate that the loss of shear strength may become a ossible mode of failure in the roblem considered in the aer. 4. Conclusions Core H45 Fig. 6. Maximum transverse shear stress as a function of the temerature of the heated surface T, the thickness of the facings and the material of the core. The roblem of deformations and stresses in a large asect ratio sandwich anel subject to an elevated temerature on one of the surfaces that results in cylindrical bending is considered in the aer. The anel is simly suorted and the long edges are revented from in-lane dislacements. The formulation enables us to account for a deterioration of the roerties of the constituent hases, i.e. the matrix of the facings and the core material. Both geometrically nonlinear as well as transverse shear deformations are taken into account. While a simultaneous effect of geometrically nonlinear and transverse shear deformations is seldom encountered in ractical roblems, in the resence of an elevated temerature it may become essential due to a degradation of the material roerties combined with resin decomosition. As follows from the analysis of deformations and stresses of reresentative anels, they exerience a significant buildu as temerature of the heated surface increases from the room value to about 6 8C. At higher temeratures, the rate of this buildu slows and eventually, at the value of the surface temerature exceeding about 1 8C, deflections and stresses begin to decline. Such behavior is in agreement with theoretical and exerimental results reorted in literature. An increase in the thickness of the facings does not substantially change the distribution of temerature through the thickness as the temerature of each facing ractically does not vary between the surface of the facing and its interface with the core. Such relatively little effect of the thickness of the facings on a distribution of temerature was exlained by a mismatch between thermal conductivities of tyical facing and core materials [4]. Thicker facings resulted in smaller deflections of the anel and reduced stresses. As follows from numerical examles dealing with the maximum transverse shear stress in the core that occurs along the edges of the anel, this stress may reach dangerous levels, even if temerature changes are relatively modest. In the anels with thin facings exeriencing significant deformations, the failure of the core may become a ossibility. It is emhasized that the stress analysis of sandwich anels oerating in high temerature environments should account for variations in the strengths of facings and core associated with the instantaneous temerature values at the oint of interest. In addition, it is necessary to account for the rocess of resin decomosition that may significantly affect the strength. Acknowledgements The financial suort of the Office of Naval Research, Grant N , and the interest and encouragement of the Grant monitor, Dr Luise Couchman, is gratefully acknowledged. References [1] Dunn SA. Using nonlinearities for imroved stress analysis by thermoelastic techniques. Al Mech Rev 1997;5(9): [2] Krysl P, Ramroth W, Asaro RJ. FE modeling of FRP sandwich anels exosed to heat: uncertainty analysis. Proceedings of the SAMPE meeting, Long Beach, CA; [3] Gibson AG, Wright PNH, Wu Y-S, Evans JT. Laminate theory analysis of comosites under load in fire. Proceedings of the SAMPE meeting, Long Beach, CA; [4] Birman V. Effect of elevated temerature on wrinkling in comosite sandwich anels. Proceedings of the SAMPE meeting, Long Beach, CA; [5] Tauchert TR. Temerature and absorbed moisture. In: Turvey GJ, Marshall IH, editors. Buckling and ostbuckling of comosite lates. London: Chaman & Hall; [6] Bert CW. Shear deformation and sandwich configurations. In: Turvey GH, Marshall IH, editors. Shear deformation and sandwich configurations. Buckling and ostbuckling of comosite lates. London: Chaman & Hall; [7] Thornton EA. Thermal stresses for aerosace alications. Reston, Virginia: AIAA Press; [chater 1].

8 988 V. Birman et al. / Comosites: Part A 37 (26) [8] Kulkarni AP, Gibson RF. Nondestructive characterization of effects of temerature and moisture on elastic moduli of vinyl ester resin and E- glass/vinylester comosite. In: Sankar BV, Ifju PG, Gates TS, editors. Proceedings of the American society for comosites 18th annual technical conference, CD-ROM, Paer #122, Gainesville, Florida; 23. [9] Elkin R. Personal communication; 24. [1] Meyers CA, Hyer MW. Thermally-induced geometrically nonlinear resonse of symmetrically laminated comosite lates. AIAA Paer AIAA CP; [] Lattimer BY, Ouellette J, Sorathia U. Large-scale fire resistance tests on sandwich comosites. Proceedings of the SAMPE meeting, Long Beach, CA;