Quasi-static response of sandwich steel beams with corrugated cores

Quasi-static response of sandwich steel beams with corrugated cores

Engineering Structures 97 (2015) 80–89 Contents lists available at ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/en...

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Engineering Structures 97 (2015) 80–89

Contents lists available at ScienceDirect

Engineering Structures journal homepage: www.elsevier.com/locate/engstruct

Quasi-static response of sandwich steel beams with corrugated cores Sushrut Vaidya a, Linhui Zhang a, Dharma Maddala b, Rainer Hebert b, Jefferson T. Wright c, Arun Shukla c, Jeong-Ho Kim a,⇑ a

Department of Civil and Environmental Engineering, University of Connecticut, 261 Glenbrook Rd., U-3037, Storrs, CT 06269, USA Department of Chemical, Materials and Biomolecular Engineering, University of Connecticut, 97 N. Eagleville Road, Storrs, CT 06269, USA c Dynamic Photomechanics Laboratory, Department of Mechanical Engineering and Applied Mechanics, University of Rhode Island, Kingston, RI 02881, USA b

a r t i c l e

i n f o

Article history: Received 23 June 2014 Revised 26 January 2015 Accepted 3 April 2015

Keywords: Sandwich beam Corrugated core Graded core Quasi-static loading Three-point bending test Finite element simulation

a b s t r a c t The response of sandwich steel beams with corrugated cores to quasi-static loading is investigated by employing experimental and computational approaches. The sandwich steel beam consists of top and bottom substrates made of AISI Steel 1018 and four corrugated core layers made of AISI Steel 1008. Various arrangements of the corrugated core layers with both uniform and graded layer thicknesses are considered. Three core arrangements with identical relative densities are used to study the effects of uniform versus graded core layer thicknesses onto the quasi-static behavior of corrugated steel beams. Finite element models are validated against quasi-static tests, and lender themselves suitable for a parametric study. A parametric study is also carried out on large-scale structural size beams of a few meters in length. The deformation modes observed in this study include core crushing, and plate bending and shear. It is found that core arrangement and beam span is key factors governing the quasi-static response of sandwich beams with corrugated cores. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Structural systems such as building frames, bridges, cranes, ship hulls, and airframes are used in infrastructure, industrial, and transportation applications. These systems are usually constructed in the form of assemblies of structural components or members, which are subjected to various types of loads such as dead, live, wind, impact, and blast loads. The behavior of a structure varies with the type of load acting on it, depending on the characteristics of the load as well as the nature of the structure. It is common practice to distinguish between phenomena involving static and dynamic loading. The response of a structure to dynamic loading is time-dependent, and is characterized by the existence of inertia forces. In addition, material properties are usually strain rate-dependent, which may add to the complexity of structural response to dynamic loading. However, if the structure is gradually loaded, so that no inertia forces appear during the response of the structure, the problem may be considered static in nature. In order to obtain a complete picture of structural behavior, it is important to investigate the response of a particular structural system to both static and dynamic loading. In this paper, we use the term ‘quasistatic loading’ to refer to loading that is applied at a rate low enough to prevent the appearance of inertia forces, yet sufficiently ⇑ Corresponding author. E-mail address: [email protected] (J.-H. Kim). http://dx.doi.org/10.1016/j.engstruct.2015.04.009 0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.

high to allow experimental testing to be performed in a reasonable amount of time. In recent times, it has become necessary to consider loads imposed by blasts and explosions in the design of civil infrastructure systems [1,2]. Blast loads pose multiple threats – such as blast waves and shrapnel – which can cause extreme damage to the structure. Structural design strategies for limiting the damage caused by a blast or explosion include the following: provision of structural continuity and redundancy in load paths, enhancement of the structure’s capacity for energy absorption, and provision of reserve structural strength [2]. Researchers have investigated concepts such as sandwich structures [3], metal foams [4], and polymers [5], which can act as shock absorption devices to protect the primary structure. Due to the potentially superior performance of sandwich structures compared to monolithic structures under blast loading [6], significant research efforts have been devoted to the investigation of the behavior of sandwich structures under dynamic loads. The effects of different types of cores, such as trapezoidal, corrugated, honeycomb, tetrahedral, and pyramidal cores, on the blast resistance of sandwich structures, has been extensively studied in the literature [7–25]. To name a few, Xue and Hutchinson [9] demonstrated that sandwich beams outperform monolithic beams of the same material and the same total mass when subjected to blast. Fleck and Deshpande [10] used a three-phase model for sandwich plates: the fluid structure interaction phase; core compression phase; and plate bending and longitudinal stretching phase. They

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showed that the last two phases can be coupled. Tilbrook et al. [13] and Liang et al. [14] found that the soft core with a low transverse strength reduces the transmitted impulse during the fluid–structure interaction stage for water blast and increases the coupling between core compression and plate bending, but the soft core can be fully crushed and gives a very high support reaction. Avila [24] studied the behavior of sandwich beams with a functionally graded core, and proposed a failure mode criterion for such beams. Wang et al. [25] studied the dynamic behavior of sandwich panels with E-glass vinyl ester composite face sheets and a stepwise graded foam core under shock loading. They found that monotonically increasing the impedance of the foam core from the front of the structure (i.e. the region facing the shock loading) to the back can greatly enhance the dynamic performance of sandwich composites. Both quasi-static and dynamic response of corrugated core sandwich beams have been also investigated [26–29]. Zhang et al. [30] have recently investigated the dynamic response of steel sandwich structures with uniform and graded corrugated cores subject to shock tube induced dynamic air pressure and demonstrated that the beam corrugated with smoothly graded core layers outperform those with uniform and nonsmoothly graded cores. The main motivation and contribution of this paper is to provide the effects of internal core arrangements into the quasi-static behavior, such as initial crushing load and load-deformation history, of corrugated steel beams. The present

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study also complements the work by Zhang et al. [30] by providing the quasi-static response of steel sandwich beams with corrugated cores. There are significant qualitative and quantitative differences between the response of a structure to blast loading and its response to quasi-static loading. Consideration of the response of a structural system to quasi-static loads is an equally essential part of the complete structural design process. A critical evaluation of the quasi-static performance of sandwich beams with corrugated cores is necessary to achieve a complete understanding of the structural response of such systems. To address this gap, the quasi-static response of sandwich beams with uniform and graded corrugated cores is investigated by performing quasi-static tests on laboratory-scale specimens, and extending finite element simulations to large-scale beam specimens. Several core arrangements with uniform and graded core layer thicknesses are studied. Three of these arrangements have identical relative core densities. Also, the influence of layer arrangement on the quasi-static response and energy absorption capacity of corrugated core sandwich beams is extensively investigated. Sandwich beams considered in this paper are envisioned as protective elements which are attached to the primary members of a structure, e.g. the beams and columns of a building frame. It is important to note that the specimens tested in this study are laboratory-scale models but finite element study is extended to

Fig. 1. (a) Microstructure of Steel 1008 as received; (b) microstructure of Steel 1008 after heating and cooling; (c) die used for corrugation; (d) assembled sandwich beam with four layers.

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large-scale structural beams. It is anticipated that the full-scale protective elements will be fabricated to a much higher standard of manufacturing than that employed in this study. This paper is organized as follows. Section 2 describes fabrication and testing of corrugated core sandwich beams with uniform and graded core layer arrangements. Section 3 presents the detail of finite element simulations. Section 4 discusses testing and modeling results. Section 5 provides parametric study on structural beam and Section 6 concludes the work.

2. Fabrication and testing of corrugated core sandwich steel beams The corrugated core sandwich beam specimen (see Fig. 1) consists of two substrates: the front (i.e. top) substrate, which faces the load, and the rear (i.e. bottom) substrate, which rests on the supports. Between the two substrates are arranged four corrugated layers, as shown in Fig. 1d. Both substrates have the same dimensions, 50.8 mm (width)  203.2 mm (length)  3 mm (thickness), and each has a mass of 250 g. The substrates are made of Steel 1018 (AISI 1018 mild and low carbon steel), while the corrugated core layers are made of Steel 1008 (AISI 1008 carbon steel). In the fabrication process, Steel 1018 is used as received. As-received Steel 1008 (see Fig. 1a) was first heated to 900 °C, and then held for 10 min and cooled after the furnace is shut off, in order to make it soft (with lower yield strength than Steel 1018) and ductile. Fig. 1a shows the microstructure of Steel 1008 as received, while Fig. 1b shows the pearlite microstructure of Steel 1008 with alpha phase after the heating and cooling operations. A semi-circular die is used to fabricate the corrugated layers (see Fig. 1c). The substrates and corrugated layers are spot welded to one another along both ends of the surfaces in contact, as shown in Fig. 1d. Each repeated unit of a corrugated layer approximates a half-sine curve in profile, after the removal of the die. This half-sine shape is clearly seen in Fig. 1d. The height of each corrugated layer is around 6 mm. Uniform and graded thicknesses of the four corrugated layers have been considered to investigate the effects of varying core layer thicknesses on the quasi-static structural behavior. Core layers are designated by the letters A, B, and C denoting different thicknesses of 0.762 mm, 0.508 mm, and 0.254 mm, respectively. A, B, and C layers have average masses of 60 g, 37 g, and 18 g, respectively. Sandwich steel beams with five different corrugated core layer arrangements are considered: AAAA, BBBB, CCCC, AACC, and ABBC. Three of these core arrangements have identical relative densities: BBBB, AACC, and ABBC. The relative density of the core is defined as the ratio of the volume of core material to the total volume of the core [31,32]. In each core arrangement, the first letter denotes the layer adjacent to the bottom substrate, while the last letter denotes the core layer adjacent to the top substrate (i.e. the substrate on which the loading is applied). It has been

suggested that in designing graded-core sandwich structures to resist blast loads, it may be desirable to monotonically increase the mechanical impedance of the core from the front of the structure to the rear [25]. It is thus hypothesized that in the case of the corrugated core, these objectives may be achieved by increasing the thickness of the layers from the front to the back, i.e. from the top to the bottom. In this study, this hypothesis is tested by evaluating the quasi-static behavior and energy absorption capacities of core arrangements with increasing and decreasing layer thicknesses from the front of the beam to the rear. Quasi-static load–deflection tests were performed on the sandwich beams with corrugated cores. The beam specimen was placed on two rigid supports, with a span of 152.4 mm, as shown in Fig. 2a. A circular punch with a diameter of 38.1 mm was used to apply a load to the top face of the beam. The load was applied at the center of the span so that the arrangement was symmetrical with respect to the axis of the punch. The load on the specimen was increased at a constant rate of 1 kN/min. The nominal strain rates attained in our study were on the order of 2  103 s1 to 3  103 s1, which are considered quasi-static [29]. During the tests, the central deflection of the front face and the corresponding load on the structure was recorded. Two specimens were tested for each core arrangement. It is noted that quasi-static tests were not performed on the BBBB core arrangement. The results of the tests and simulations for the CCCC and AAAA core arrangements are used to predict the quasi-static response of the sandwich beam with the BBBB core arrangement.

3. Finite element analysis Finite element (FE) simulations were conducted using the commercial FE solver, ABAQUS/Standard. A quarter model of the sandwich beam is created (see Fig. 2b). The model was meshed with C3D8I elements. The C3D8I element is the first-order fully integrated three-dimensional 8-node solid element, enhanced by incompatible modes to improve its bending behavior [33]. A convergence study was conducted for various levels of mesh refinement. In a representative mesh, the total number of elements is around 175,000, while the total number of nodes is around 480,000. The support and punch were modeled as rigid shells meshed with discrete rigid elements. The contact interactions between the bottom substrate and the rigid support, and between the top substrate and the punch, were modeled as frictionless contact in the tangential direction and hard contact in the normal direction. In ABAQUS, contact interactions are defined by specifying surface pairings and self-contact surfaces. General contact interactions typically are defined by specifying self-contact for the default surface. The contact interactions between the various parts of the sandwich beam – the substrates and the core layers – were modeled using a surface-to-surface self-contact definition. In the tangential direction, these interactions were modeled as

Fig. 2. (a) Quasi-static test setup (b) finite element model.

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frictionless contact, while in the normal direction they were modeled through a linear pressure-overclosure relationship. Contact problems are inherently nonlinear, and severe convergence issues are usually encountered in the numerical solution of such problems. In order to address these issues, the automatic stabilization option available in ABAQUS/Standard [33] was used in the contact definition. The punch had a single degree of freedom (DOF) in the direction normal to the plane of the top substrate, while all other DOFs of the punch were constrained. Finite strain formulations were used in all FE simulations to account for large-displacement effects.

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arrangement began at a relatively small load of 500 N, and the bending of the beam was not obvious in the early stages of the test. In agreement with our experimental observations, the deformed

3.1. Stress–strain relations for finite element modeling The quasi-static stress–strain curve for the core material (Steel 1008) was obtained experimentally. A bilinear model closely approximates the experimental curve. The hardening part of the bilinear model is given by the following equation:

r ¼ 200 þ 400ep MPa

ð4:1Þ

where ep denotes plastic strain. Eq. (4.1) may be regarded as a special case of the Johnson–Cook constitutive model [34], where the terms accounting for the effects of temperature and strain rate have been dropped by setting to zero the appropriate parameters of the Johnson–Cook model. The stress–strain curve for the as-received substrate material (Steel 1018) is given by the Ramberg–Osgood model [35] as:



r ¼ 370

e

18

0:00195

MPa

ð4:2Þ

where e denotes true strain. The power-law stress–strain relation of Steel 1018 is given in Fig. 3. In the figure, the assumed bilinear stress–strain relation of Steel 1008 matches well with the measured stress–strain data. 4. Results and discussion The experimental load–deflection behavior of each core arrangement is compared with finite element results. The deformed shape of the sandwich beam predicted by the FE analysis is shown at several critical points of each load–deflection curve. Fig. 4 shows the load–deflection curves for the CCCC and AAAA core arrangements. It is noted that the crushing of the CCCC core

Fig. 4. Load–deflection behavior of corrugated core sandwich beam under quasistatic compression: (a) CCCC and (b) AAAA core arrangements.

700

True Stress (MPa)

600 500 400 300 200 Steel 1008 Test Steel 1008 Bilinear Steel 1018

100 0 0

0.05

0.1

0.15

0.2

True Strain Fig. 3. Stress–strain curves for Steel 1018 (substrate) and Steel 1008 (core).

Fig. 5. FEM results for load–deflection behavior of corrugated core sandwich beam under quasi-static compression: BBBB core arrangement.

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shapes predicted in the early stages of our FE simulation also showed core crushing as the major deformation mode. During the quasi-static tests, it is also noted that when the crushing of the core was nearly complete, the entire beam began to bend. During the quasi-static tests on the AAAA core sandwich beam, it is observed that the onset of core crushing occurred at a much higher load of 5500 N, due to the greater thickness of the core layers. Also, the load magnitude did not increase very significantly with further deformation. These deformation characteristics were reproduced in our simulation, although the load at the onset of core crushing was predicted to be around 8775 N, and the stiffness of the initial load–deflection response of the beam was higher in the simulations than observed in the tests. It is noted that similar discrepancies in the other FE simulations as well, although the magnitude of the discrepancy was smaller in the other cases. These differences between the computational and experimental results can be attributed to the fact that our computational model does not account for the effects of initial material imperfections (such as potential material degradation and local damage at the sharp ends of circular regions (see die shapes in Fig. 1c) due to non-smooth die configuration and corrugation process) in the core layers and imperfect bonding between the sandwich beam components. In the quasi-static tests, it is observed that the welds joining the substrates to the core layers, as well as those joining the core

layers to one another, were in general weaker than the rest of the structure, and debonding did occur in a few cases. Similar over-prediction of the peak stress due to the exclusion of initial imperfection effects is also reported by Lee et al. [29] in their study of the behavior of pyramidal truss cores. At this point, it is worth reiterating that the purpose of our simulations is to provide insight into the salient characteristics of the quasi-static response of protective elements that will be fabricated to meet stringent manufacturing standards. It is also found that the bending and core compression phases of the response of the AAAA sandwich beam were coupled during the deformation of the beam. Our simulation succeeds in reproducing this feature of the deformation history of the AAAA core sandwich beam, as seen in Fig. 4b. Fig. 5 shows the predicted quasi-static load–deflection behavior of the BBBB core arrangement. Quasi-static testing was not performed for the BBBB core arrangement. The simulation results for the BBBB core are consistent with the behaviors of the CCCC and AAAA cores. The load at the onset of core crushing is predicted to be around 3860 N, which is between the loads predicted at the onset of core crushing for the CCCC and AAAA arrangements (1159 N for CCCC and 8775 N for AAAA). This result is exactly as anticipated, given that the B layers have an intermediate thickness of 0.508 mm, which is greater than the thickness of C layers (0.254 mm) but less than that of the A layers (0.762 mm).

Fig. 6. (a) Load–deflection behavior of sandwich beam with AACC core arrangement in compression; (b) deformed shape of the first AACC specimen at a load of 3349 N; (c) permanent deformation after removal of the load.

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Fig. 6 shows the load–deflection behavior of the beam with the AACC core arrangement, the deformed shape of the first AACC specimen at a load of 3349 N and permanent deformation after removal of the load. It is observed that the two C layers at the top of the core were crushed first, at a load of 700 N. The subsequent response of the sandwich beam exhibited coupling of the core compression and bending phases, up to a load of 8000 N. After this point, the deformation mode of the sandwich beam was dominated by bending effects, and was similar to the bending of a monolithic beam. Again, the major characteristics of the experimental load–deflection behavior are successfully reproduced in the FE simulation. Fig. 7 shows the behavior of the sandwich beam with the ABBC core arrangement under the load, the deformed shape of the first ABBC specimen at a load of 3912 N, and permanent deformation after removal of the load. The crushing behavior of the ABBC core was somewhat different than that of the AACC core. The crushing of this core progressed more gradually through the layers, due to the presence of the B layers (of intermediate thickness) between the top C layer and the bottom A layer. The load required to cause the crushing of the top C layer was around 3000 N, which is much higher than the load required for crushing the top C layers in the AACC arrangement. After the top C layer was crushed, the core compression and bending phases were coupled. The deformation modes predicted by the FE simulation also show coupling between

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core compression and beam bending, after the initial crushing of the top C layer is complete. It is note that the experimental load–deflection behaviors of the sandwich beams with different corrugated core layer arrangements share certain characteristics. The load–deflection plots contain several regions of high stiffness: the load undergoes a large increase, while the displacement does not. These regions of high stiffness are separated by other regions of low stiffness, where the load exhibits almost no change, while the displacement undergoes a large increase. Such areas of low stiffness are characteristic of structures exhibiting buckling or collapse behavior [29]. The progressive nature of the collapse mechanism is reflected in the multiple regions showing abrupt increase of stiffness [29]. These regions correspond to contact between the core layers, as the core undergoes sequential crushing. These features can be clearly seen in the experimental and computational load–deflection plots of the graded AACC and ABBC cores. In order to provide a more complete picture of the quasi-static failure of corrugated core sandwich beams, the variation of energy quantities in these beams is also discussed. It should be noted that among the various core arrangements considered, three have identical relative densities: BBBB, AACC, and ABBC. In the following discussion, our attention is focused on these three core arrangements with identical relative densities, in order to understand the effects of thickness gradation on the quasi-static performance of the core.

Fig. 7. (a) Load–deflection behavior of sandwich beam with ABBC core arrangement in compression; (b) deformed shape of the first ABBC specimen at a load of 3912 N; (c) permanent deformation after removal of the load.

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S. Vaidya et al. / Engineering Structures 97 (2015) 80–89 Table 1 Load at the initiation of core crushing, and relative density with respect to AAAA, for all core arrangements.

120 External work Internal energy Plastic energy

100

Enegry (J)

80

60

Core arrangement

Load at the initiation of core crushing (N)

Relative density with respect to AAAA

AAAA BBBB CCCC AACC ABBC

8775 3860 1159 1241 2821

1 0.9224 0.8339 0.9224 0.9224

40

20

0 0

1

2

3

4 5 Load (kN)

6

7

8

9

6

7

8

9

6

7

(a) BBBB 120 External work Internal energy Plastic energy

100

Enegry (J)

80

60

40

20

0

0

1

2

3

4 5 Load (kN)

(b) AACC 120 External work Internal energy Plastic energy

100

Enegry (J)

80

60

40

20

0

0

1

2

3

4 5 Load (kN)

8

9

(c) ABBC Fig. 8. FEM results for energy quantities: (a) BBBB, (b) AACC, and (c) ABBC cores.

Fig. 8 illustrates the variation of energy quantities obtained from FE simulations with the load for the core arrangements with identical relative densities. The quantities shown in Fig. 8 are

external work, internal energy, and plastic energy. The internal energy is related to elastic and plastic strains and includes the elastic and plastic energies. The difference between the external work and the internal energy is the energy dissipated by the ABAQUS/ Standard solver to stabilize the contact problem. Artificial dissipation of energy is necessary for stabilization of the problem, in order to minimize convergence difficulties. The plots of the energy quantities also clearly reflect the progressive collapse mechanism of the layered cores, with the jumps in these plots indicating the layer-by-layer progress of the core crushing process. For all three core arrangements, there are multiple jumps in the energy plots. The beam with the BBBB core has the largest initial crushing load among the three. For the AACC beam, crushing starts at a load smaller than the corresponding loads for the other two beams. After the two C layers are crushed, the load is resisted mainly by the two remaining A layers, and the energy increases smoothly as the load increases. Comparing the energy plots for the AACC and ABBC cores, it is seen that the jumps in the plots for the ABBC core occur closer to one another than those in the plots for the AACC core. This is due to the fact that there are two steps in the thickness gradation of the ABBC core (C to B, and B to A), compared to only one thickness gradation step in the AACC core (C to A). For the sandwich beams with cores of equal relative density (BBBB, AACC, and ABBC), the initial crushing loads predicted by our FE simulations are (see Table 1): 3860 N for BBBB, 2821 N for ABBC, and 1241 N for AACC, while the corresponding values measured from the quasi-static tests are about 3000 N for ABBC (see Fig. 7), and 700 N for AACC (see Fig. 6). Researchers have studied the effects of graded cores on the dynamic performance of various types of sandwich structures, e.g. sandwich panels with graded foam cores [25], and sandwich plates with corrugated cores [30]. These studies have found that the best dynamic performance is shown by graded cores in which the density [25] or the thickness [30] of the core layers increases monotonically, and relatively smoothly, from the front of the structure to the rear. To demonstrate such behavior in quasi-static loads, the numerical study on the quasi-static response of core arrangements with reversed layers are done: CBBA versus ABBC, and CCAA versus AACC. Fig. 9 shows the quasi-static load–deflection responses of the ABBC and CBBA cores predicted by FE simulations. It indicates that the ABBC core has higher loading and energy absorption capacity than the CBBA core. Fig. 10 shows the quasi-static load–deflection responses of the AACC and CCAA cores predicted by FE simulations. It is noted that by observing the progressive collapse of the CCAA core during the simulation, the presence of the two weaker C layers at the bottom of the core led to a very unstable collapse mechanism in this core. A combined stabilization methodology to address both contact issues and unstable collapse issues needs further investigation. Fig. 10 may indicate that the AACC core absorbs a greater amount of strain energy than the CCAA core.

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Fig. 9. Load–deflection behaviors of ABBC and CBBA cores.

Fig. 10. Load–deflection behaviors of AACC and CCAA cores.

Fig. 11. Structural size beam: 50.8 cm (width)  200 cm (length)  32 cm (height).

5. Parametric study on a structural beam The small-scale corrugated beam is scaled up to structural size scale to perform parametric study on structural beams (see Fig. 11) A structural size beam has the dimensions, 50.8 cm (width)  200 cm (length)  32 cm (height). The span-to-span dimension between rollers is 160 cm. The beam is subjected to the load either at the center or a quarter point (S/4 = 40 cm) off from the left rigid roller. Core layer thicknesses A, B, and C are 7.62 mm, 5.08 mm, and 2.54 mm, respectively. Fig. 12 shows the load–deflection behavior of structural beams with BBBB, AACC and ABBC cores under symmetric (thick lines) and asymmetric (thin lines) loads. The load and deflection are obtained from the center point of the rigid punch where the load

is applied through. The BBBB core sustains the highest initial crushing loads and moth preferred core arrangements among the three. Non-symmetric case also shows similar trend to the symmetric case. Fig. 13 shows deformed shapes of the BBBB core at initial stage and at load = 1000 kN and deflection = 0.25 m subject to the symmetric load, and at load = 280 kN and deflection = 0.09 subject to the asymmetric load. Von Mises stress contours represent 0 MPa to 510 MPa from dark blue to red. It is observed that core crushing prevails initially and bending deformation follows along with further core crushing. For the non-symmetric case, all three beams undergo non-symmetric bending deformation and then become unstable by sliding horizontally along the rigid supports which stopped the analysis.

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1100 1000 900 800

Load (kN)

BBBB 700 600 500 400 300 200

ABBC

100 0

AACC 0

0.05

0.1

0.15

0.2

0.25

Deflection (m) Fig. 12. Load–deflection behavior of structural beams under symmetric (thick lines) and asymmetric (thin lines) loads.

Another structural size beam 400 cm long is considered to elaborate bending effects in longer spanned beam. It is placed with the span of 240 cm. A symmetric model with respect to the length and depth is created. Fig. 14 shows deformation modes at two different pusedo-time steps. Core crushing initially happens at the loaded region and spread across the beam length along with bending deformations. The plate bending is more significant in this long structural beam than the small-scale short-span beam which showed shear dominant mode along with core crushing and bending modes. It can be noted that core crushing interacts with plate bending and its interaction depend on the beam span.

6. Concluding remarks This paper addresses the quasi-static response of corrugated core sandwich beams with various core layer arrangements of uniform and graded thicknesses, and performs parametric study of structural size beams. The major conclusions that can be drawn on the basis of our experimental and computational investigations are summarized below.

1. Among all the core arrangements studied, the beam with the AAAA core resists the largest load at initial crushing of the core, while the beam with the CCCC core resists the smallest. 2. Among the three arrangements with equal relative densities, the BBBB core resists the largest load at initial crushing of the core, while the AACC core resists the smallest. The load bearing capacity of the uniform core (BBBB) before unstable crushing behavior is the highest, so is recommendable for use in quasistatic regimes. This observation for the quasi-static case is different from the dynamic case. Note that the dynamic response of steel sandwich structures with uniform and graded corrugated cores subject to shock tube induced dynamic air pressure [30] demonstrated that the beam corrugated with smoothly graded core layers (ABBC) outperform those with uniform (BBBB) and non-smoothly graded (AACC) cores. 3. For the three core arrangements with equal relative densities, multiple jumps are observed in energy absorption as the load is increased. These jumps are associated with sequential crushing of the layers of the core. 4. Comparison of the load–deflection curves of cores with increasing and decreasing layer thicknesses through the depth of the core indicates that the cores with increasing layer thickness from top to bottom – ABBC and AACC – perform better than their respective reversed counterparts – CBBA and CCAA, by absorbing greater strain energy during deformation. 5. The load–deflection response of the sandwich beam with corrugated core depends on core layer arrangement and the span of beam. The deformation of corrugated beams starts with crashing of the top layer(s), and then core crushing, plate bending and shear interact to each other which are influenced by the span of beam. It is thus concluded that core arrangement and beam span is a key factor governing the quasi-static response of sandwich steel beams with corrugated cores. An experimental investigation of the quasi-static and dynamic behaviors of full-scale protective elements with corrugated cores would be an interesting extension of the present study. Acknowledgements We gratefully acknowledge the financial support from the U.S. Department of Homeland Security (Award 2008-ST-061-TS0002-02)

Fig. 13. Deformed shapes of the BBBB core: (a) initial core crushing and (b) core crushing and plate bending modes at load = 1000 kN and deflection = 0.25 m subject to symmetric load, and (c) at load = 280 kN and deflection = 0.09 subject to asymmetric load. Von Mises stress contours represent 0 MPa to 510 MPa from dark blue to red. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 14. Deformed shapes of the BBBB core: (a) the crushing of the first core layer with mild top plate bending deformation, and (b) severe core crushing with plate bending deformation.

to the University of Connecticut through the Center for Resilient Transportation Infrastructure (Director: Professor Michael L. Accorsi).

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