Appendices for:

Buckling of chiral, anti-chiral and hierarchical honeycombsBabak Haghpanah, Jim Papadopoulos, Davood Mousanezhad, Hamid Nayeb-Hashemi and Ashkan

Vaziri*

Department of Mechanical and Industrial Engineering

Northeastern University, Boston, MA

* Corresponding author; email: vaziri@coe.neu.edu

The appendices include:

A) Characteristic matrix method applied to classical column problemsB) Buckling of an axially loaded beam on a rotational spring foundationC) Buckling of first order hierarchical honeycombD) Buckling of tri-chiral honeycomb

Appendix A) Characteristic matrix method applied to classical column problem

In this section, the matrix representation of the beam-deflection relations [1] is

used to find well-known buckling loads of standard column problems. Figure 1A shows

the three different boundary conditions considered. The beam-deflection relations

(Equations (1) of manuscript) and column boundary conditions for each case can be

presented in the matrix form that is given for each case. The rigid body rotation β of the

beam (i.e. the line joining the beam ends) is considered as an additional degree of

freedom since in the general case it could be deferent from end slopes.

Case I – Fixed / Free

[−Ψ (q ) −Φ (q ) 1 0 0−Φ (q ) −Ψ (q ) 0 1 0

1 −1 0 0 −q2

0 1 0 0 00 0 1 0 −1

][M a LEIM b LEIθaθbβ

]=0

Here, the first two rows correspond to the beam-deflection relations for the beam-column. The third row expresses the balance of moments about the fixed end. The fourth row specifies that the moment at the free end is zero. The final row identifies θa with β. Setting |A|=0 yields 1−q2Ψ (q )=0, leading to the well-known critical buckling load of π2EI / (2L )2.

Case II – Fixed / Laterally Supported

[−Ψ (q ) −Φ (q ) 1 0 0−Φ (q ) −Ψ (q ) 0 1 0

0 1 0 0 00 0 0 0 10 0 1 0 −1

] [M aLEIM bLEIθa

θb

β]=0

Here, the first two rows correspond to beam-deflection relations for the beam-column. The third row specifies that M b=0, and the fourth that β=0. The final row sets θa=β. Setting |A|=0 yields Ψ (q )=0, leading to a critical buckling load of π2EI / (0.699∗L )2.

Case III – Fixed / Fixed Slope

[−Ψ (q ) Φ (q ) 1 0 0Φ (q ) −Ψ (q ) 0 1 0

1 1 0 0 −q2

0 0 1 −1 00 0 1 0 −1

] [M aLEIM bLEIθa

θb

β]=0

Here, the first two rows correspond to the beam-deflection relations, and the third row expresses moment equilibrium about point a. The fourth row identifies θa with θb, and the fifth equates θa to β. Setting |A|=0 yields 1−q2 (Ψ (q )−Φ (q ) )=0, leading to a critical buckling load of π2EI /L2.

Comparing the critical buckling loads found above for cases I, II, and III to the Euler’s buckling load F cr=π 2EI /Leff

2 of an axially loaded column of effective length Leff , the effective length is found as 2 L, 0.7 L and L for cases I, II and III, respectively.

Figure A1 – Cases of an axially loaded beam subjected to three different boundary conditions: fixed/free, fixed/laterally supported (no moment), and fixed/fixed slope.

Appendix B) Buckling of an axially loaded beam on a rotational spring foundation

In this section, the governing differential equation for an axially loaded beam on a foundation of rotational elastic springs is derived. Assume that springs of rotational stiffness ΔK [N.m/rad] are repeated over length spans of ∆ x [m]. Note that in this case it is assumed that each rotational spring only resists a change in the angle of the beam (from the horizontal) and does not resist against the vertical or horizontal deflection of the beam. When the springs are considered close enough, their effect on the beam can be estimated as a distributed rotational spring over the length of the beam with the coefficient K t=ΔK /∆ x [N]. Figure B1 shows the free body diagram of such a beam where vertical and horizontal reaction forces and reaction moments are applied to beam ends. Equilibrium of moments about the origin is expressed by the following equation:

M 0+R .x−P .v ( x )+∫0

x

k tdv (x )dx

dx=M=EI d2 v (x)d x2

The second derivative of the above equation leads to the following differential equation governing the beam deformation

EI d4 v (x )d x4 +(P−k t )

d2 v ( x )d x2 =0

The analogy of the above relation to the well-known relation governing the instability of

an axially loaded beam with compressive force P (i.e. EI d4 v

d x4 +P d2vd x2 =0) implies that the

effect of distributed rotational spring can be regarded as an axial tensional force of magnitude K t superposed to the loaded beam. As a result, the critical load for the instability of a beam on a distributed rotational spring foundation of intensity K t is equal to Pcr=K t as the length of beam approaches infinity.

Figure B1 – Free body diagram of an axially loaded beam on a rotational spring foundation.

Appendix C) Buckling of hierarchical honeycomb structure

Hierarchical, iterative refinement of hexagonal honeycombs was recently shown to

enhance stiffness and plastic collapse strength compared to regular honeycomb of the

same density [2]. A first order hierarchical honeycomb is shown in figure C1. This

structure is obtained by the first iteration of a hierarchical refinement scheme in which all

three-edge nodes are replaced with smaller, parallel hexagons at each refinement level.

The length ratio, γ, is defined as the side of the newly added hexagons to the side of the

original hexagonal network in a first order honeycomb, and is geometrically bound to the

range 0≤γ ≤0.5. In the buckling analysis presented here the smaller hexagons in the

hierarchical lattice are considered small enough to be regarded as rigid parts. The

coordinate system of choice is the abc coordinate.

Based on finite element computations, the hierarchical honeycomb lattice buckles

based on two modes which are similar to modes I (uniaxial) and II (biaxial) observed in a

regular hexagonal honeycomb lattice. Figure C1 shows the post-buckling free body

diagram of the RVE of the hierarchical lattice according to mode I, where the rigid small

hexagon at the center of the RVE rotates by the angle α during buckling. The set of

beam-column and equilibrium relations are expressed in the following matrix form

[−Ψ (qa )+Φ (qa ) 0 0 −1 1

0 −Ψ (qb )−Φ (qb ) 0 1 00 0 −Ψ (qc )−Φ (qc) 1 0

1 −1 −1 (qa2+qb2+qc

2) γ1−2γ

0

2 0 0 0 −qa2][M a L (1−2 γ )

EIM b L (1−2 γ )

EIM c L (1−2 γ )

EIαβ

]=[000

Ga L2 (1−2 γ ) γEI

−G aL2 (1−2 γ )2

EI]

where qa=2 i√3√3σaa and cyclically for qb and qc, and the bar above the stresses means

they are normalized according to σ=(σ /E )/ (t /L )3. The first row expresses the moment

equilibrium of central node (hexagon) O, the second row satisfies the moment

equilibrium of beam OA, and the three last rows correspond to beam-column relations for

beam OA, OB, and OC. Equating the determinant of the characteristics matrix equal to

zero leads to the following relation for the buckling of hierarchical lattice of length ratio γ

according to mode I of bucklingγ

1−2 γ (qa2+qb

2+qc2 )+qa tan (qa/2 )−qbcot (qb/2 )−qc cot (qc/2 )=0

or in the abc stress space, considering all three 2π /3 rotations corresponding to this mode

(σ aa+σbb+σ cc ) 2√3√3 γ1−2 γ

+√σ aa tanh (√3√3σ aa )+√σbb coth (√3√3σbb )+√σ cccoth (√3√3σcc )=0

(σ aa+σbb+σ cc ) 2√3√3 γ1−2 γ

+√σ aacoth (√3√3σaa )+√σbb tanh (√3√3σbb )+√σ cccoth (√3√3σcc )=0

(σ aa+σbb+σ cc ) 2√3√3 γ1−2 γ

+√σ aacoth (√3√3σaa )+√σbb coth (√3√3σ bb)+√σcc tanh (√3√3σcc )=0

Similar to regular honeycomb lattice, the set of beam-column and equilibrium relations for mode

II buckling can be written in the following matrix form

[−Φ (qa )−Ψ (qa ) 0 0 0 0 1 0 0

0 −Ψ (qb ) −Φ (qb ) 0 0 0 1 00 −Φ (qb ) −Ψ (qb ) 0 0 1 −1 00 0 0 −Ψ (qc ) −Φ (qc ) 0 0 10 0 0 −Φ (qc ) −Ψ (qc) 1 0 −1

1 1 0 1 0 −(qa2+qb

2+qc2 ) γ

1−2 γ0 0

0 1 −1 0 0 qb2 −qb

2 00 0 0 1 −1 qc

2 0 −qc2

][M aL (1−2 γ )

EIM ob L (1−2 γ )

EIM bo L (1−2 γ )

EIM oc L (1−2 γ )

EIM coL (1−2 γ )

EIαθbθc

]=[00000

−(Ga+Gb+Gc )L2 (1−2 γ ) γEI

Gb L2 (1−2 γ )2

EIGc L

2 (1−2 γ )2

EI

]Here, the first five rows are the beam-column relations on beams OA, OB, and OC, the

sixth line corresponds to equilibrium of node (hexagon) O, and the last two relations

satisfy the moment equilibrium in beams OB and OC. Using the symbolic toolbox in

MATLAB software to set |A|=0, the relation expressing the instability of hierarchical

honeycomb lattice according to biaxial mode under a general loading is

−γ1−2 γ (qa

2+qb2+qc

2 )+qa cot (qa/2 )+qb cot (qb )+qc cot (qc)=0

Similar to regular honeycomb structure, the corresponding macroscopic state to this mode

of buckling is equal to that of uniaxial mode under x-y bi-axial loading, and is not

dominant under any other loading condition.

Figure C1 – (A) representative volume element for a hierarchical lattice. (B) Notations for the beam

and nodal rotation in the RVE, and free body diagram of RVE beam-elements.

Appendix D) Buckling of tri-chiral honeycomb structure

Chiral honeycombs have attracted a great deal of attention in recent years due to their

auxetic properties [3-7], and are suggested for design of compliant structures featuring

large multi-axial deformations under targeted loads including micro electo-mechanical

systems (MEMS) [8, 9], aircraft morphing components [7, 10-12], etc. According to full-

scale finite element analysis on the tri-chiral honeycomb buckling patterns similar to the

uniaxial modes in regular honeycomb is developed under different states of in-plane

loading - see figure D1. Here, we consider the circular elements in the structure of tri-

chiral lattice as rigid parts. This assumption is rather reasonable when the radius r of

circular elements is small enough relative to the length L of straight beams. The

coordinate system of choice is the a ' b ' c ' coordinate system oriented along the three

beam directions in the chiral lattice, and is obtained by rotating the abc coordinate system

by θ=tan−1 (2 r /L ) in the counter-clock-wise direction. Figure D1 shows the free body

diagram of the RVE of the structure, where the rigid circular element O at the center of

the RVE rotates by the angle α during buckling. Compared to the regular hexagonal

honeycomb, the non-central forces in the tri-chiral lattice will affect the equilibrium

requirements on the RVE central node (here rigid circle) O, so that the set of beam-

column and equilibrium relations for instability of tri-chiral honeycomb by uniaxial mode

of buckling is expressed in the following matrix form

[ −1 1 1 −2 (qa'2 +qb'2 +qc '

2 )( rL )2

0

−2 0 0 0 qa'2

ψ (qa' )−ϕ (qa' ) 0 0 1 −10 ψ (qb' )+ϕ (qb' ) 0 −1 00 0 ψ (qc ')+ϕ (qc ' ) −1 0

][M a' L/ (EI )M b' L/ (EI )M c ' L/ (EI )

αβ

]=[(Ga+Gb+Gc )L2/ (2EI )G aL

2/ (EI )000

]where qa'=2i √3√3σ a'a' and cyclically for qb' and qc', and the bar above the stresses means

they are normalized according to σ=(σ /E )/ ( t /L )3. The first row expresses the moment

equilibrium of central node O, the second row satisfies the moment equilibrium of beam

OA, and the three last rows correspond to beam-column relations for beam OA, OB, and

OC. Setting the determinant of the characteristics matrix equal to zero leads to the

following relation for the instability of tri-chiral lattice

2( rL )2

(qa'2 +qb'

2 +qc '2 )+qa' tan (qa'/2 )−qb' cot (qb' /2)−qc 'co t (qc ' /2)=0

or in terms of normalized stress components in a ’ b’ c ’ stress space, considering all three

(2π ⁄ 3 ) rotations corresponding to this mode

√3√3 (σa' a'+σb' b'+σ c 'c ') tan2θ+√σa' a' tanh (√3√3σ a'a' )+√σb'b' coth (√3√3 σb' b')+√σc 'c ' coth (√3√3σ c' c ')=0

√3√3 (σa' a'+σb' b'+σ c 'c ') tan2θ+√σa' a'coth (√3√3σa' a')+√σb' b' tanh (√3√3 σb' b')+√σc 'c ' coth (√3√3σ c' c ')=0

√3√3 (σa' a'+σb' b'+σ c 'c ') tan2θ+√σa' a'coth (√3√3σa' a')+√σb' b' coth (√3√3σb' b' )+√σ c' c ' tanh (√3√3σ c' c ')=0

The results are in agreement with results from FE eigenvalue analysis.

Figure D1 – (A) representative volume element for a tri-chiral lattice. (B) Notations for the beam

and nodal rotation in the RVE, and free body diagram of RVE beam-elements.

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