Buckling Analysis of Sandwich Plate Systems with Stiffening Ribs : Theoretical , Numerical , and Experimental Approaches

-is paper discusses a global buckling analysis approach for sandwich plates with stiffening ribs.-e approach is based on theoretical study and is implemented by the finite element method (FEM).-e equilibrium equation corresponding to critical global buckling of the sandwich plate with stiffening ribs under simple supported boundary condition is established by the energy method. -e critical buckling solutions for a typical rectangular sandwich plate system (SPS) with a single stiffening rib in the longitudinal direction are then investigated while varying the potential influencing factors. -e shear rigidity within the inner core exerts little effect on global buckling and can be neglected. An FEM study on elastic buckling was then conducted via ANSYS software.-e advantages of the SPS were highlighted via its elastic eigenvalue buckling numerical analysis with multiple stiffeners. -e ultimate buckling loads were computed similarly for different influential factors. Finally, an SPS specimenwas tested in a compression test.-e results showed that when the rib spacing is large, the local buckling of the plates in the grillage is controllable and the SPS is more resistant to both local and global buckling. -e results based on our theoretical method agreed well with those of the FEM and experimental results.


Introduction
Steel sandwich panels comprise two solid faceplates and one low-density core.Initially installed in aerospace equipment, they are now increasingly used in shipbuilding.A new type of sandwich plate, called the sandwich plate system (SPS), comprises two metal faceplates sandwiching a continuous polyurethane (PU) elastomer core.SPSs are important for weight reduction, rapid reparation, and impact resistance in bridge engineering [1] and the shipbuilding fields [2]. Figure 1 shows the application of SPS in different fields and differentiates the shapes of the SPS from those of conventional structures.
PU elastomer is significantly viscoelastic with high damping characteristics.erefore, a PU elastomer core can potentially raise the noise isolation and damping behaviors of the whole SPS structure [3].In shipbuilding, stiffeners are usually bonded with steel plates or sandwich plates to increase the stiffness of the whole structure under variable loads.SPSs with stiffening ribs have several advantages over conventional steel plates; in particular, they increase the corrosion resistance and reduce the processing cost by diminishing the space requirements of the discrete stiffeners [4].
In recent years, applications and simulations of SPS have been extensively researched.Martin [5] conducted static and fatigue tests of an SPS bridge deck.Chunlei et al. [6] calculated the stress distribution in an orthotropic steel bridge deck that was stiffened by SPSs.e lateral load-distribution characteristics of SPS bridges were delineated via numerical simulations by Harris et al. [7].Based on the finite element method (FEM), Feng et al. [8] developed a new numerical simulation with shell elements for studying SPS structures.Shang [9] studied the typical failure mode and ultimate strength of an SPS in a model test under combined biaxial compression and lateral pressure.Liu et al. [10] analyzed the crashworthiness of a ship with an SPS hull using FEM software.ey demonstrated the superior collision resistance and energy-absorbing capacity of the SPS-incorporated structure over the conventional structure.Zou et al. [11] analyzed the effect of adhesive in an equivalent modelling of SPS.
Rib-stiffened plates are the most common engineering applications of SPSs.As analysis approaches and manufacturing techniques of sandwich materials continue to develop, different kinds of sandwich plates are being considered for engineering practice.Two popular examples in the literature are stiffened sandwich plates and plates with viscoelastic properties.Liu and Hollaway [12] presented an optimization procedure for composite panel structures with stiffening ribs under multiple loading cases.Wang and Zhao [13] presented an FEM for vibrating stiffened sandwich plates with moderately thick viscoelastic cores.Gara et al. [14] developed a series of experimental tests and a numerical model for wall sandwich panels.John and Li [15] proposed a new sandwich design with an orthogrid-stiffened syntactic foam core based on shape-memory polymer.Xin and Lu [16] analytically modeled the wave propagation in orthogonally rib-stiffened sandwich structures.Briscoe et al. [17] presented a model of shear buckling and local bearing failure in web-core sandwich panels and validated it in experiments.Goel et al. [18] presented a model and numerical simulation of foam sandwich panels subjected to impulsive loading.Adopting a virtual testing approach, Wadee et al. [19] characterized the mechanical behavior of folded core structures for advanced sandwich composites under flatwise compression loads.Yuan and Dawe [20] numerically analyzed the free vibrations and buckling of sandwich plates using a B-spline finite strip method.
Buckling failure is an important failure mode of plates with stiffening ribs under various conditions.Global buckling of plates with stiffeners was first studied by Timoshenko and Gere [21].Heder [22] approximated the buckling load of a simply supported stiffened sandwich panel using an energybased method.Zhao et al. [23] introduced an equivalent laminate modelling method that analyzes the global buckling of stiffened panels with different sectional shapes and stringer distribution forms.Al-Qablan [24] developed a semianalytical buck analysis of stiffened sandwich plates based on first-order shear deformation plate theory (SDPT).
Amadio and Bedon [25][26][27][28] studied the effect of variable factors, including multiple mechanical and geometrical aspects, on the buckling behavior of laminated glass elements by combining theoretical methods, numerical simulations, and experiments.In their paper, the normalized resistant domain was proposed for the assessment of the proposed stability check.It was proved that the method was very effective for evaluating buckling behavior of laminated structures through comparing the critical buckling load derived by different analytical interaction formulations.e study was also beneficial to provide us a new theoretical approach to the research of buckling of sandwich panel with ribs.
e present paper theoretically analyzes global buckling in SPSs with stiffening ribs and compares the buckling properties of SPS and conventional rib-stiffened structures.Global buckling is then analyzed via FEM.Finally, an SPS panel stiffened by inner ribs is fabricated and tested in a compression testing facility.Parts of this paper have been published in conference proceedings by the corresponding author [29,30].

Theoretical Buckling Analysis of Sandwich
Plates with Stiffening Ribs  e global theoretical buckling analysis for sandwich plates with sti ener ribs can be combined with the bending theory of sandwich plates and conventional sti ened plates.
e analysis was simpli ed by the following assumptions: (1) e sandwich plate is modeled by traditional SDPT sandwich plate theory (Ho 's theory), in which only the core bears shear e ects under transverse loads.e bending rigidity of the faceplates is also considered.(2) e SPS sti eners are perfectly connected with the sandwich plate, and they deform along with the plate at all times.
e torsional e ect of the sti eners during buckling is ignored when calculating the critical buckling loads.
As shown in Figure 3, the sandwich plate with rectangular sti ening ribs experiences in-plane pressure under the simply supported boundary conditions.e buckling shape of the global sti ened sandwich plate was expressed as the following bitrigonometric series: where w is the out-of-plane de ection function and a and b are the plate length and width, respectively.a mn are the polynomial parameters of the bitrigonometric series, where m is the number of semiwaves and n is the number of ribs.
Ignoring the bending sti ness of the core (justi ed under the above assumptions), the total deformation energy of the global sandwich structure was divided into two parts: the bending deformation energy of the faceplates and the shear deformation energy of the PU elastomer core.e total energy is thus calculated as where ΔU bf and ΔU tc are the bending and shearing deformation energy of the sandwich plate, respectively.
Based on traditional sandwich plate theory, the deformation of the sandwich plate was considered as the deformation of a homogeneous plate with the same bending sti ness under external loads plus the shear deformation of the core: where C G c (h + t) and G c is the torsional modulus of the core; D h is the equivalent sti ness of the whole Sandwich plate; and w 0 denotes the deformation corresponding to the bending e ect, which is similar (in form) to the whole deformation of the sandwich plate: Substituting equations ( 1) and (4) into equation (3), the relation between a mn and b mn is given as Setting the parameter B mn (1 − (D h /C)((m 2 /a 2 ) + (n 2 / b 2 ))π 2 ), the bending deformation energy becomes where D h de nes the bending sti ness of a single face of the panel.Similarly, the bending deformation energy of core was determined as where D c is the bending sti ness of the core.ese quantities are, respectively, calculated as follows: where D 0 is the bending sti ness of the whole sandwich plate (ignoring the sti ness of the faceplates).ΔU tc is expressed as

Advances in Civil Engineering
Under the assumptions of Ho 's plate theory, shear stress is uniformly distributed along the core section.Following the arguments in [31], the shear stress is calculated as e external shear loads Q cx and Q cy are assumed to be uniformly distributed along the section.Using the equilibrium equation of thin-plate bending theory, they are, respectively, expressed as follows: where E f and υ f are Young's modulus and Poisson's ratio of the faceplates, respectively.e subscript c, f, and h represent the core, faceplates, and the whole sandwich plate, respectively.Substituting equations ( 10) and ( 11) into equation (9) gives As the sti eners are bonded to the structure, the ribs deform along with the plate; thus, the bending deformation energy ΔU i is given by where EI i is the bending sti ness of the rib at distance c i from the side of the panel (y 0).When the sandwich panel is buckling, the work ΔT done by the in-plane pressure N x is given by and the work done by the force P i acting on each rib is Neglecting the torsional deformation energy of the ribs, the total potential energy is computed as adopting the following symbols: where A i is the section area of the rib.Substituting equations ( 12)-( 15) and equation ( 17) into equation ( 16), the total potential energy becomes  Advances in Civil Engineering Equating Π to 0 in equation ( 18), we obtain Equation ( 18) computes the critical buckling load for different numbers of ribs.

Global Buckling Calculation for a Sandwich Plate with
One Rib.To verify the effectiveness of previous analyses and investigate the differences between sandwich plates and conventional stiffening plates, we first explored the influences of various parameters on a single-ribbed sandwich plate.As shown in Figure 4, the sandwich plate is arranged with only one rib in the center of the plate, and the width of the plate is divided into equal intervals, meaning that c i equaled to b/2.Assuming one buckling wave along the y-axis (i.e., n � 1) and half a wave along the x-axis (i.e., m � 1) as a typical example, equation ( 18) can be written as follows: In thin-plate bending theory with stiffeners, the firstorder critical buckling loads are approximated by equation (20).All terms a mn are equaled to 0 except a 1 , the first term of the double triangular series.Introducing the parameter φ, σ cr can be written as e parameter φ relates to the shear deformation work done by the core.Introducing the coefficient of critical buckling k, σ cr finally becomes When n is even, the parameter a mn in equation ( 20) takes a single value, and equation ( 20) expresses the deflection of the internode line.When the sandwich panel is uniformly buckled to generate an even number of semiwaves along the y direction, the internode line remains straight, meaning that the ribs arranged along the center of the plate are nonfunctional.
As β increases, multiple semiwaves may appear in the compression loading direction (m ≠ 1).erefore, substituting m in equation (20), we obtained the total potential energy.e potential energy of a sandwich plate with n ribs is calculated by substituting c i � b/(n + 1) in equation (19).
e first-order critical buckling loads are approximately given by

Examples of Solutions to Buckling eory.
Stiffeners influence sandwich plates with single and multiple ribs in a similar manner.erefore, as an example, we analyzed a simple supported rectangular sandwich panel with a single rib (Figure 4).e length b is 2030 mm, and the width a is calculated from β. e faceplate and core thicknesses are t � 2 mm and h � 13.6 mm, respectively.For steel faceplates,

Advances in Civil Engineering
Young's modulus is E 2.1 × 10 5 MPa and Poisson's ratio is 0.3.e shear modulus G c of the core is taken as 308 MPa.
According to equation ( 22), the critical stress is governed by the bending sti ness of the components in the sandwich plates.e critical stresses for di erent sti ness (equation ( 8)) in the above example are listed in Table 1.
As shown in Table 1, the total bending sti ness of the sandwich plate is 0.55% higher when considering the bending sti ness of the faceplate than when not considering the sti ness factor.Moreover, the bending sti ness of the core is approximately 0.32% of the overall bending of the bending rigidity of the sandwich plate.erefore, we concluded that the bending sti ness of the faceplates can be ignored when calculating the bending sti ness.
In engineering practice, the cross-sectional area of the ribs is 2%-10% of the section area of the plates.In the present example, the sti ener is considered to cover 5% of the cross-sectional area of the plate.e e ect of the inertia moment of the sti ening rib is then investigated under critical buckling loads.
When the faceplate thickness is xed, increasing the core thickness will increase the bending sti ness of the whole structure.Figure 5 plots the e ect of core thickness on the critical buckling load for di erent β. e faceplate thickness and inertia moment are xed at 2 mm and (418 × 10 4 ) mm 4 , respectively.Increasing the core thickness increased the bending sti ness of the SPS.Here, we considered only the rst buckling semiwave.Lowering the β value reduced the in uence of the core thickness: at low β, the corresponding curve changed gently; at higher β, it increased more sharply.
Figure 6 shows the critical buckling loads in the case of multiple semiwave buckling, corresponding to di erent core thicknesses.
e inertia moment of the rib is xed at 418 × 10 4 mm 4 .
As shown in Figure 6, the critical buckling load changed slowly with the thickness of the core in sandwich plates.Increasing the core thickness from 5 to 50 mm signi cantly increased the critical buckling load from approximately 5000 N/mm to 11000 N/mm. is result can be explained by the sharp increase in bending sti ness of the SPS as the core thickness increased.
Figure 7 relates the critical buckling coe cient to the inertia moment of the rib for di erent numbers of semiwaves, where the section area of the ribs is 2% of the total section area.e critical buckling coe cient is quite sensitive to the inertia moment of the sti ening ribs.When the bending sti ness of the ribs is ignored, the critical coe cient is minimized when the SPS panel buckled into two semiwaves (n 1, m 2); as the bending sti ness of the ribs increased above 81.7 cm 4 , the critical coe cient appeared at

6
Advances in Civil Engineering lower values in the curve with the hollow circles. is result indicates a change in the buckling of the SPS structure from two semiwaves to one semiwave (m 1) along the x direction.
As the three curves in Figure 7(b) do not intersect, changing the inertia moment of the sti ening rib did not a ect the buckling wave number, and the corresponding buckling waveform remained as one semiwave.As seen in Figure 7(c), when the sti ening rib is not installed on the sandwich panel, the buckling waveform is three half-waves.For small inertia moments of the sti ening rib, the buckling form exhibited two semiwaves.As the inertia moment of the sti ening rib increased, the sandwich panel buckled into one semiwave, indicating a major role of the sti ening rib in the buckling of the overall structure.e critical buckling stress in a conventional plate with ribs is given by e β values at the critical buckling load for di erent inertia moments of a single rib on the SPS and steel plates are displayed in Figure 8.According to these curve, the inertia moment signi cantly a ects the critical buckling load.A low critical buckling load always accompanies a small inertia moment but cannot fall below the critical buckling load of the structure without the rib.
e critical buckling load tended to increase with increasing inertia moment of the  Advances in Civil Engineering ribs, but the growth trend of critical buckling stress gradually slows down.Meanwhile, the inertia moment is positively correlated with the buckling wavelength.
Clearly, when the inertia moment of the rib is small, the critical buckling load is also small but is lower-limited by the load without the ribs.After decreasing by a certain degree, the load grew slowly with increasing inertia moment.Increasing the inertia moment also increased the buckling wavelength.
Figure 8(b) in a zoom-in of the area delineated by the rectangular frame in Figure 8(a).e upper and lower curves are the results of SPS considering the shearing deformation energy and the equivalent conventional steel structure, respectively.e small gap between the two curves implies that the shearing deformation energy can be neglected.
Figure 9 plots the critical buckling stresses as functions of β for di erent inertia moments of a single rib on the SPS and conventional steel plate.
e critical buckling stress signi cantly di ered between the two curves.Moreover, the absolute value of this di erence increased roughly proportionally to the critical buckling stress.e reasons are explained as follows: as the critical buckling loads are similar for the SPS and conventional plates, the larger cross-sectional area of the sandwich plates in comparison with the conventional plates reduces the critical stress in the former relative to the latter.e ratio of the critical buckling stresses in the sandwich and homogeneous plates is computed from equations ( 24) and ( 26) as With their large Young's modulus, the faceplates in the sandwich structure are the main load-bearing components under in-plane compression stress.For the same compression load and cross-sectional area, the compressive stress under a critical buckling load will be higher in the faceplates of SPS than in a conventional steel plate because the faceplates take most of compression load in SPS.
e multiplication factor is (1 + (h/2t)).At this time, the ratio of the critical buckling stresses in the sandwich and homogeneous plates is given by   Advances in Civil Engineering Figure 10 relates the critical buckling stress to β for di erent inertia moments of the SPS and steel plates.e critical buckling load under in-plane pressure, at which buckling rst appears, is larger in the SPS than in the conventional plate, implying higher internal stresses in the SPS than in the conventional plate.
In typical applications, the ratio δ of the faceplate thickness to the core thickness lied between 1 : 5 and 1 : 10.Substituting δ in equation ( 28), the ratio of the critical buckling loads of the faceplate and core ranges from 3 to 4.5.For the common beam-plate composite structures in engineering practice, the stress in the plate component is far from the yield strength when overall buckling occurs.
erefore, when the steel plate is made into a sandwich structure, buckling occurs under large inner stress, which improves the utilization of the material.

FEM Analyses of SPS and Conventional
Plates with Stiffening Ribs

FEM Eigenvalue Buckling Analysis.
e calculations in Section 2 cannot determine the local buckling in SPSs.e critical buckling stress in ribbed sandwich plates is sometimes controlled by local buckling in a single grid.FEM analysis can estimate both the global and local buckling in SPSs and can assess the in uences of di erent factors on the critical buckling loads of conventional plates and SPSs with sti ening ribs.
In this analysis, the material properties are unchanged from those in Section 2, and the sti ening ribs are simulated as beam elements.e conventional steel plates and SPSs are constructed from shell elements.Figure 11 shows the FEM model with β 2 and β 5.

Analysis of SPS with a Single Sti ened Rib.
To verify the accuracy of the above analysis, the above theoretical calculations are compared with the FEM results of an eigenvalue analysis.e theoretical and FEM results of an SPS with a single sti ening rib for di erent β values and inertia moments are shown in Figure 12.Here, the inertia moment of the single rib is varied from 5 × 10 4 to 1000 × 10 4 mm 4 , and β is set to 1.0 (critical buckling load ∼450-3000 N/mm) and 2.0 (critical buckling load ∼450-1500 N/mm).
e results of both methods are highly consistent and exhibited the same general trends.At some points, the di erence between the results is increased by local buckling in certain grids.e error is larger for β < 2, re ecting the higher inertia of the sti ener rib in this case.When global buckling occurred, the calculation and numerical results are relatively close.

Result Comparisons for Steel Grillage and SPS with Sti ening Ribs.
is subsection rst investigates the in uence of β on the critical buckling load.In the FEM, the conventional plate is 2000 mm wide and 4 mm thick.Meanwhile, the core and faceplate thicknesses of the SPS are set to 16 mm and 2 mm, respectively.e sandwich panel is sti ened by four ribs with rectangular cross sections.e inertia moment of each rib is 22500 mm 4 .
e critical buckling loads for the di erent β values are shown in Figure 13.
e critical buckling loads are approximately three times higher in the ribbed SPS than in the conventional plates, con rming that the core in SPS enhances the bending sti ness of the whole structure.
e small bump in both curves indicated a transformation of the buckling shape from one half-wave to two half-waves.
Figure 14 plots the critical buckling load versus inertia moment of the ribs.
e critical buckling load is an approximately linear function of number of ribs.However, in the conventional plates, the critical buckling loads tended to remain steady after the inertia moment of the ribs reached a certain threshold.At this time, rst-order buckling is not global buckling but in fact local buckling in a single cell of the grillage.erefore, when the number of ribs in the plates is xed, enlarging the sti eners can signi cantly improve the stability until the appearance of local buckling in one cell of the grillage.
Figure 15 plots the critical buckling loads in the x and y directions in conventional and SPS plates with di erent numbers of ribs.For a given number of sti eners, the critical buckling loads are larger in the x direction than in the y direction because the compressive loads are applied in the x direction.Moreover, the critical buckling load along the y direction (Figure 15(b)) increased slightly more rapidly with rib number than the critical buckling load along the x direction (Figure 15 Advances in Civil Engineering the SPS structure can withstand higher critical buckling loads than conventional plates with the same number of sti eners.

Buckling Experiment for SPS with
Stiffening Ribs   e specimen is prepared as shown in Figure 17.First, the mixed liquid of polymer and curing agent is poured into a closed container.Under air pressure applied by an air compressor, the compound liquid owed to the opening hole beneath the specimen.Twenty minutes later, the polyurethane elastomer had solidi ed. e specimen is laid aside for two weeks to fully solidify the elastomer; then, it is tested for its buckling behavior.
Although the sti eners in the sandwich structure mentioned in the previous calculations have di erent shapes, the buckling analysis therein is unchanged.To ensure the same constraint conditions in the experiment and the previous analysis, a steel constraining device with V-shaped grooves is designed and installed.
is device Advances in Civil Engineering constrained the de ections in the out-of-plane direction only.

Experimental Results and Discussion
. Buckling experiments in a compression test machine are conducted on an SPS model with and without the injected PU elastomer (Figure 18).e out-of-plane displacements of the SPS experimental model with the sti eners are constrained by the V-shape grooves installed at all sides, but the boundary de ections in the vertical direction are unrestrained.e pressure is applied by the compression testing machine.e loading speed is 2 mm/min, and the experiments are terminated when the total vertical displacement reached 20 mm. e critical buckling loads obtained in di erent ways are given in Table 2.
Figure 19 shows the load-displacement curve obtained in the compression experiment.e theoretical and FEM results di ered by 5.2% and 3.9%, respectively, from those of  12 Advances in Civil Engineering the sandwich plate with a PU core (Table 2 and Figure 19).e di erence can be explained by the nonideal boundary constraints in the experiment.In the real case, the boundary constraints are a ected by initial bending during welding of the SPS product, which creates a slight gap between the boundary and the xture device.e results of the sandwich plate without the PU core specimen di ered from the FEM and calculation results by almost the same amount (4.0%).However, in the specimen with the PU core, the theoretical and experimental results di ered by 29.6%, and the curve change at the position of buckling is less obvious in the experimental specimen.
is discrepancy is attributed to local buckling in the corner grids and the assumption of the rst-order buckling model, meaning that local buckling appears before the whole structure buckling.In the real case, the PU core can prevent local buckling and raise the buckling strength.

Conclusions
A global buckling analysis approach for sandwich plate with sti ening ribs and PU core is discussed here based on theoretical, FEM, and experiments, the following conclusions can be obtained:

Figure 1 :
Figure 1: Comparison SPS versus conventional structure: (a) comparison between conventional structure and SPS structure; (b) SPS application as bridge deck; (c) SPS applied as partition board in superstructure of ship; (d) SPS stairs in stadium.

Figure 3 :
Figure 3: Section of sandwich plate and SPS with sti eners.(a) Structure sketch of SPS reinforced by cross wall.(b) Simpli ed loading schematic diagram.

Figure 4 :
Figure 4: Straight rib when the plate is buckling.

Figure 5 :Figure 6 :
Figure 5: Relation between critical buckling load and core thickness for di erent β.

Figure 8 :Figure 9 :
Figure 8: Relation between β and critical buckling load for di erent inertia moments of a single rib on the SPS and steel plates.(a) Original relations between β and critical buckling loads.(b) Local enlargement of the curves in (a).

Figure 10 :
Figure 10: Relation between β and critical buckling stresses (actual) for di erent inertia moments of the single rib in the SPS and steel plates.

Figure 11 :Figure 12 :
Figure 11: Finite element analysis of plates with sti eners and the rst-order buckling model: (a) ribbed SPS structure in the FEM model, (b) FEM model with β 2, and (c) FEM model with β 5.

Figure 13 :
Figure 13: Critical buckling loads in ribbed SPS and conventional plates with di erent length-to-width ratios.

Figure 14 :Figure 15 :
Figure 14: Critical buckling loads in ribbed SPSs and conventional plates with di erent inertia moments of sti eners.

Figure 16 :Figure 17 :
Figure 16: Shape and sizes of the test specimens.(a) Side view of specimen.(b) Top view of specimen.(c) Front view of specimen.

Figure 19 :
Figure19: Load-displacement curve obtained in the experiment.

Table 1 :
Di erent rigidities of the sandwich plate.
4.1.Design and Preparation of the Test Specimen.To verify the calculation result, a typical SPS model test is conducted in a compression testing machine.e shape and sizes of the

Table 2 :
Critical buckling loads of the test specimens.