Materials Science and Engineering A 397 (2005) 132–137
Pyramidal lattice truss structures with hollow trusses
Douglas T. Queheillalt∗, Haydn N.G. WadleyDepartment of Materials Science and Engineering, 116 Engineers Way, P.O. Box 400745, University of Virginia, Charlottesville, VA 22904-4745, USA
Received 22 November 2004; accepted 8 February 2005
Lightweight sandwich panel structures consisting of lowensity cores and their solid face sheets are widely used inerospace and other applications[1]. Cellular core structures
ncluding honeycombs[2,3] and prismatic forms[4–7] areften used because of their very low density, good crush re-istance and high in-plane shear resistance. Recently, latticeruss cellular structures have begun to be explored as a can-idate sandwich panel core material[8–11]. The lattice truss
opology can be designed to efficiently support panel bendingoads by aligning them so they are subject to only axial defor-
ations[12–14]. For trusses with lattice relative densities of–10%, inelastic buckling then determines the strength of the
russes and the cellular materials out-of-plane compressionnd in-plane shear strength[11].
Diamond “textile” lattice truss structures have recentlyeen constructed with hollow trusses and found to be signif-
cantly stronger than their solid truss counterparts[15]. Theollow trusses in these structures had higher second area mo-ents than their solid truss counterparts and this was found to
increase their inelastic buckling strength. The use of hotruss structures also provided a means for varying the cestructures relative density without changing the cell siztruss slenderness ratio[15].
Here, we modify the process devised for diamond lastructures and explore a method for making a pyramhollow truss lattice structure like that shown inFig. 1. Wedescribe the fabrication of a representative structurecompare its out-of-plane compressive response to thasolid pyramidal truss of similar relative density.
2. Hollow lattice truss fabrication
Solid truss pyramidal lattice structures are fabricateda folding operation that bends a diamond perforated sto create a single layer of trusses arranged with a pyramtopology[16]. Fig. 2schematically illustrates this processsolid truss structure with a relative density of 2.5% was mfrom 304L stainless steel by a similar process using thetailed procedure initially described by Sypeck and Wa
[16]. Briefly, the process consisted of punching a metal sheetto form a periodic diamond perforation pattern, folding noderow by node row using a paired punch and die tool set and
D.T. Queheillalt, H.N.G. Wadley / Materials Science and Engineering A 397 (2005) 132–137 133
Fig. 1. A pyramidal lattice truss core sandwich structure.
then brazing this core to solid face sheets to form a sand-wich structure. The sheet thickness,t= 0.81 mm, truss width,w = 0.81 mm and the truss length,l = 12.7 mm. The inclina-tion angle,ω = 45◦. This resulted in a pyramidal lattice withsquare cross section trusses and a measured relative densityρ̄ = 0.025± 0.003.
A similar process was used to create a layer of circularcross section hollow trusses with the trusses arranged in asimilar diamond arrangement. The process began by takingtwo collinear 304L stainless steel tube arrays and overlayingthem at 60◦. They were then vacuum brazed to form the di-amond pattern shown inFig. 3(a). The brazed structure wasthen folded node row by node row using a paired punch anddie with the sheets folded to create a square base pentahedron.Fig. 4shows a cartoon of the folding process. The punch anddie were again designed so that the trusses were inclined at anangleω = 45◦. An example of the hollow lattice truss diamondsheet after folding to create a hollow truss pyramidal lattice isshown inFig. 3(b). The tubes had an o.d.,do = 1.47 mm andan i.d.,di = 1.07 mm. The truss lengthl = 14.3 mm and theinclination angleω = 45◦. This structure was then brazed to304L stainless steel face sheets. The measured relative den-sity of the hollow truss pyramidal lattice was 0.028± 0.003.
The hollow and solid truss systems used identical brazingprocesses for both brazing steps. The samples were placed ina vacuum furnace at a base pressure of∼10−4 Torr. They wereh e
F layerp hapedd
Fig. 3. Photographs of (a) the diamond hollow tube sheet prior to the foldingoperation and (b) the hollow lattice truss pyramidal structure after folding.
binder), then heated to the brazing temperature of 1050◦C.They were held for 60 min at this temperature before fur-nace cooling at∼25◦C/min to ambient. A braze alloy witha nominal composition of Ni–25.0Cr–10.0P–0.03C (wt.%)was used for the bonding process. It was applied by sprayingthe samples to be bonded with a mixture of the braze powderand a polymer binder.Fig. 5 shows photographs of the asfabricated solid and hollow truss pyramidal lattice sandwichstructures.
Fig. 4. An illustration of the folding operation used to create the single layerpyramidal truss sandwich structures. Also shown is the hollow truss shapedd
eated at 10◦C/min to 550◦C, held for 1 h (to volatilize th
ig. 2. An illustration of the folding operation used to create the singleyramidal truss sandwich structures. Also shown is the solid truss siamond cells prior to the forming.
iamond cells prior to the forming.
134 D.T. Queheillalt, H.N.G. Wadley / Materials Science and Engineering A 397 (2005) 132–137
Fig. 5. Photographs of the brazed pyramidal truss sandwich structures fab-ricated from (a) solid truss and (b) hollow truss elements.
3. Unit cell densities
Fig. 6 shows the unit cells of a pyramidal lattice trussstructure with solid and hollow trusses. Ignoring the addedweight of the braze alloy; only the truss cross sectional areaand length determine the relative density of the two structures.For any regular pyramidal structure of truss lengthl, the unitcell volume:
Vc = 2 cos2 ω sinωl3 (1)
whereω is the angle between the truss members and the baseof the unit cell.
For the solid trusses with a rectangular cross section,Fig. 6(a), the volume occupied by metal:
Vs = 4twl. (2)
F oresf
For hollow trusses,Fig. 6(b), with an o.d.,do, and an i.d.,di ,the volume occupied by solid metal is:
Vs = π(d2o − d2
i )l. (3)
The relative density of the cell,̄ρ, is the ratio of the trussvolume to that of the unit cell. For the solid pyramidal latticethe relative density is:
ρ̄ = 2wt
l2 cos2 ω sinω. (4)
For the hollow pyramidal lattice:
ρ̄ = π(d2o − d2
i )
2 l2 cos2ω sinω. (5)
The predicted relative densities were slightly lower than thatmeasured because of the added weight of the braze alloy andnon-ideal node conditions.
4. Compressive responses
The solid and hollow truss structures were tested at am-bient temperature in compression at a nominal strain rate of4× 10−2 s−1. The measured load cell force was used to calcu-late the nominal stress applied to the structure. The nominalt xten-s s.
trainr s ares lars nse,y wedb apidh
F ses fort
ig. 6. Schematic illustrations of the pyramidal lattice unit cells for cabricated from (a) solid and (b) hollow trusses.
hrough thickness strain was obtained using a laser eometer to monitor the displacements of the face sheet
The through thickness compressive nominal stress–sesponses for both the solid and hollow lattice structurehown inFig. 7. Both exhibit characteristics typical of cellutructures including a region of nominally elastic respoielding, plastic strain hardening to a peak strength, folloy a drop in flow stress to a plateau region and finally rardening associated with core densification[17]. The solid
ig. 7. Through thickness compressive nominal stress–strain responhe solid and hollow lattice truss pyramidal sandwich structures.
D.T. Queheillalt, H.N.G. Wadley / Materials Science and Engineering A 397 (2005) 132–137 135
Fig. 8. Photographs showing the deformation characteristics of the hollowtruss pyramidal lattice at plastic strains of 2, 5, 10, 20, 30 and 40%.
truss structures plastically buckled shortly after yielding (ata strain of∼0.03). The hollow trusses exhibited significantpost yield hardening to a strain of∼0.1 before they under-went buckling. After the onset of plastic buckling, significantcore softening was observed in both structures, followed byplateau region. The plateau stress of the solid truss structurewas∼25% of its peak value, whereas the hollow truss struc-ture was reduced to only∼75% of its peak value.
Photographs of the hollow lattice truss during plastic de-formation are shown inFig. 8. Lateral deflections of the trussmembers began to occur at loads prior to the attainment ofthe peak strength. Softening coincided with the formation ofa plastic hinge near the middle of the truss members. Neithertruss member nor node fracture was observed in any of thecompression experiments performed.
Tensile tests were performed on 304L stainless steel sam-ples that had followed the same thermal cycle used for fabri-cation of the brazed sandwich structures. Three tensile tests
were performed according to ASTM E8-01. The uniaxial ten-sile response is shown inFig. 9. The average Young’s mod-ulus,Es, and 0.2% offset yield strength,σys, were 203 GPaand 176 MPa, respectively.
5. Compressive response predictions
The effective properties of a solid truss pyramidal corehave been discussed in detail by Deshpande and Fleck[13]. Athigh relative densities (low aspect ratio trusses), the strengthof a truss structure made from a rigid ideally plastic mate-rial is controlled by plastic yielding. The peak compressivestrength,σpk, of a pyramidal lattice failing by plastic yieldingis then given by[13]:
σpk = σy sin2 ωρ̄. (6)
A non-dimensional peak strength coefficient,Σ =σpk/ρ̄σy (whereσpk is the peak compressive strength,σy thesolid materials yield stress) was determined from the dataand is plotted against̄ρ in Fig. 10. The average value of thepeak strength coefficient was 0.43 for the solid truss struc-ture. The hollow truss structures had a value of 0.75. If asolid truss pyramidal structure (withω = 45◦) is made froma rigid—ideally plastic material,Σ = 0.5. Values ofΣ > 0.5,t ancec
dert g ax-i engthi ns s,
F 304Ls ollowt
herefore exploit the strain-hardening and buckling resistharacteristics of the alloy from which it is made.
If a pyramidal lattice is constructed from very slenrusses, the trusses collapse by elastic buckling durinal compression. In this case, the peak compressive strs found by replacingσy in Eq.(6)with the elastic bifurcatiotress,σc, of the trusses[18]. The elastic bifurcation stres
ig. 9. Average uniaxial tensile response (true stress–true strain) oftainless steel and the critical inelastic buckling stress for the solid and hrusses.
136 D.T. Queheillalt, H.N.G. Wadley / Materials Science and Engineering A 397 (2005) 132–137
Fig. 10. Predicted and measured non-dimensional peak strength coefficientsΣ = σpk/ρ̄σy plotted against̄ρ for both the solid and hollow lattice trusspyramidal sandwich structures. Built in (k= 2) assumed for all predictions.
σc, for an axially loaded column is given by:
σc = π2k2IEs
Al2(7)
whereEs is the elastic modulus,I the second area moment ofinertia of the truss andA is the truss cross sectional area[18].
Expressions for the second area moment of inertia andcross sectional area for the solid and hollow trusses are sum-marized inTable 1together with the calculated values forthe trusses studied here. The factork depends on the rota-tional stiffness of the nodes; for a pin-joint that can freelyrotatek= 1. The casek= 2 corresponds to a built in-joint,which does not rotate. In the experiments conducted here noevidence of node rotation was observed and we takek= 2.
If the truss material has a non zero post yield strainhardening rate, inelastic buckling strength defined byShanley–Engesser tangent modulus theory determines thelattice strength[18,19]. The peak compressive strength isthen obtained by replacingσy in Eq. (6) with the inelastic
Table 1Cross sectional area and area moment of inertia for solid and hollow trussgeometries (w = 0.81 mm,t= 0.81 mm,do = .47 mm,di = 1.07 mm)
Truss geometry Cross sectional area,A (mm2)
Area moment of inertia,I (mm4)
bifurcation stress of a compressively loaded column. An ex-pression for this stress,σc, is obtained by replacingEs byEt, (the tangent modulus of the material used for making thetrusses) in Eq.(7) [18]:
σc = π2k2IEt
Al2. (8)
The tangent modulus is defined as the slope dσ/dε of theuniaxial stress versus strain curve of the solid material atthe inelastic bifurcation stress levelσc [18]. The inelasticbifurcation stress for the solid and hollow trusses can bededuced from a Ramberg–Osgood fit of the solid materialstress–strain response as shown inFig. 9. We also show thestress predicted by Eq.(8)(the dashed curve) calculated usingthe local tangent modulus obtained by differentiation of theRamberg–Osgood response. The inelastic bifurcation stress,σc, is obtained at the intersection of these curves and for thesolid truss was 170 MPa and the hollow truss was 240 MPa.
Predictions for the normalized peak compressive strengthsare shown inFig. 10 for three characteristic deformationmodes: plastic yielding, elastic buckling and inelastic (plas-tic) buckling of the trusses as a function of the core’s relativedensity. Some ambiguity in the precise length of a truss (l)exists in practice because of the accumulation of braze alloyat the nodes,Fig. 5. We show predictions for the hollow pyra-midal lattice truss using truss lengths (l) of 14.7 and 11.7 mmt truel razea ussa inF asticb heetsa
oeffi-c russs d be-c lign-m (i.e.a -d s. Itc lizeds solidt f gy-r ibest uteda a-d ling.T turea
6
hasb withh roach
wt = 0.656 wt3
12 = 0.0359
π4 (d2
o − d2i ) = 0.798 π
64(d4o − d4
i ) = 0.1649
o account for braze material build up at the nodes. Theength appears to lie between these bounds. Minimal blloy build up was observed for the solid pyramidal trnd predictions for truss length (l) of 12.7 mm are shownig. 10. We also assume that during both elastic and ineluckling, the trusses have built-in nodes at the face snd takek= 2 in Eqs.(7) and(8).
Recall that the average value of the peak strength cient for solid trusses was 0.43 and 0.75 for the hollow ttructure. Some scatter of the test results was observeause they are sensitive to the accuracy of fabrication, aent of the applied load and the end support conditionsmount of braze material at the nodes).Fig. 10shows the preicted normalized strength coefficient for both truss typean be seen that the hollow pyramidal lattice has a normatrength that is nearly twice that of a comparable densityruss lattices. This increase is controlled by the radius oation,
√I/A, of the trusses. The radius of gyration descr
he way in which the area of a cross section is distribround its centroidal axis[18]. Increasing the value of the rius of gyration increases a columns resistance to buckhe radius of gyration was 0.23 for the solid truss strucnd 0.46 for the hollow truss structure.
. Summary
A simple hollow tube lay up and node folding processeen devised to create pyramidal lattice truss structuresollow metal trusses. We have demonstrated the app
D.T. Queheillalt, H.N.G. Wadley / Materials Science and Engineering A 397 (2005) 132–137 137
using 1.47 mm diameter 304L stainless steel tubes with awall thickness of 200�m. This process results in a lattice,which can be bonded to face sheets to form a metallic sand-wich structure with a pyramidal topology. Similar relativedensity pyramidal truss structures have been fabricated withsolid trusses and the compressive mechanical properties ofthe two systems have been computed. The use of the hollowtrusses as ligaments of pyramidal lattice truss structures in-creases the peak compressive collapse strength by increasingthe inelastic buckling resistance of the hollow tubes. The out-of-plane compressive strength of a hollow pyramidal latticewith a relative density of 2.8% is shown to be approximatelytwice that of similar relative density solid truss structures.
Acknowledgements
We are very grateful to Vikram Deshpande (CambridgeUniversity, UK) for helpful discussions. This work has beenperformed as part of theTopologically Structured Materi-als:Blast andMultifunctional Implementationsprogram con-ducted by a consortium of Universities consisting of HarvardUniversity, Cambridge University, the University of Califor-nia at Santa Barbara and the University of Virginia. We aregrateful for the many helpful discussions with our colleaguesin these institutions. The Office of Naval Research (ONR),m orku
References
[1] T. Bitzer, Honeycomb Technology: Materials, Design, Manufactur-ing, Applications and Testing, Chapman & Hall, 1997.
[2] R.M. Christensen, J. Mech. Phys. Solids 34 (1986) 563.[3] B. Kim, R.M. Christensen, Int. J. Mech. Sci. 42 (2000) 657.[4] W. Ko, NASA Technical Paper 1562, 1980.[5] T.S. Lok, Q.H. Cheng, J. Sound Vib. 229 (2000) 311.[6] T.S. Lok, Q.H. Cheng, Comp. Struct. 79 (2001) 301.[7] T.S. Lok, Q.H. Cheng, J. Sound Vib. 245 (2001) 63.[8] A.G. Evans, J.W. Hutchinson, M.F. Ashby, Curr. Opin. Solid State
