Appendices for:
Buckling of chiral, anti-chiral and hierarchical honeycombs
Babak Haghpanah, Jim Papadopoulos, Davood Mousanezhad, Hamid Nayeb-Hashemi and Ashkan
Vaziri*
Department of Mechanical and Industrial Engineering
Northeastern University, Boston, MA
* Corresponding author; email: [email protected]
The appendices include:
A)
B)
C)
D)
Characteristic matrix method applied to classical column problems
Buckling of an axially loaded beam on a rotational spring foundation
Buckling of first order hierarchical honeycomb
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)
−1
1
0
1
0
0
0
1
0
1
0
0
0
𝑀𝑎 𝐿
0
𝐸𝐼
𝑀
0
𝑏𝐿
−𝑞2 𝐸𝐼 = 0
𝜃𝑎
0
−1 ] 𝜃𝑏
[ 𝛽 ]
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 − 𝑞2 Ψ(𝑞) = 0, leading to the well-known critical buckling load
of 𝜋 2 𝐸𝐼/(2𝐿)2 .
Case II – Fixed / Laterally Supported
−Ψ(𝑞)
−Φ(𝑞)
0
0
[ 0
−Φ(𝑞)
−Ψ(𝑞)
1
0
0
1
0
0
0
1
0
1
0
0
0
𝑀𝑎 𝐿
0
𝐸𝐼
𝑀
𝑏𝐿
0
𝐸𝐼 = 0
0
𝜃𝑎
1
−1] 𝜃𝑏
[ 𝛽 ]
Here, the first two rows correspond to beam-deflection relations for the beam-column.
The third row specifies that 𝑀𝑏 = 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 𝜋 2 𝐸𝐼/
(0.699 ∗ 𝐿)2 .
Case III – Fixed / Fixed Slope
−Ψ(𝑞)
Φ (𝑞 )
1
0
[ 0
Φ (𝑞 )
−Ψ(𝑞)
1
0
0
1
0
0
1
1
0
1
0
−1
0
𝑀𝑎 𝐿
0
𝐸𝐼
0 𝑀𝑏 𝐿
−𝑞2 𝐸𝐼 = 0
𝜃𝑎
0
−1 ] 𝜃𝑏
[ 𝛽 ]
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 𝜋 2 𝐸𝐼/𝐿2 .
Comparing the critical buckling loads found above for cases I, II, and III to the Euler’s
buckling load 𝐹𝑐𝑟 = 𝜋 2 𝐸𝐼 ⁄𝐿2𝑒𝑓𝑓 of an axially loaded column of effective length 𝐿𝑒𝑓𝑓 , the
effective length is found as 2𝐿, 0.7𝐿 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 𝐾𝑡 = 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:
𝑥
𝑀0 + 𝑅. 𝑥 − 𝑃. 𝑣(𝑥) + ∫ 𝑘𝑡
0
𝑑𝑣(𝑥)
𝑑2 𝑣(𝑥)
𝑑𝑥 = 𝑀 = 𝐸𝐼
𝑑𝑥
𝑑𝑥 2
The second derivative of the above equation leads to the following differential equation
governing the beam deformation
𝐸𝐼
𝑑4 𝑣(𝑥)
𝑑2 𝑣(𝑥)
(𝑃
)
+
−
𝑘
=0
𝑡
𝑑𝑥 4
𝑑𝑥 2
The analogy of the above relation to the well-known relation governing the instability of
𝑑4 𝑣
𝑑2 𝑣
an axially loaded beam with compressive force 𝑃 (i.e. 𝐸𝐼 𝑑𝑥4 + P 𝑑𝑥2 = 0) implies that the
effect of distributed rotational spring can be regarded as an axial tensional force of
magnitude 𝐾𝑡 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 𝐾𝑡 is equal
to 𝑃𝑐𝑟 = 𝐾𝑡 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
0
−Ψ(𝑞𝑏 ) − Φ(𝑞𝑏 )
0
0
0
−Ψ(𝑞𝑐 ) − Φ(𝑞𝑐 )
[
1
−1
−1
2
0
0
−1
1
1
(𝑞𝑎2 + 𝑞𝑏2 + 𝑞𝑐2 )
0
𝛾
1 − 2𝛾
𝑀𝑎 𝐿(1 − 2𝛾)
0
𝐸𝐼
0
𝑀𝑏 𝐿(1 − 2𝛾)
0
2
𝐸𝐼
= 𝐺𝑎 𝐿 (1 − 2𝛾)𝛾
𝑀𝑐 𝐿(1 − 2𝛾)
0
𝐸𝐼
𝐸𝐼
𝐺𝑎 𝐿2 (1 − 2𝛾)2
2
𝛼
−𝑞𝑎 ]
[−
]
𝐸𝐼
[
𝛽
]
1
0
0
where 𝑞𝑎 = 2𝑖 √3√3σ
̅aa and cyclically for 𝑞𝑏 and 𝑞𝑐 , and the bar above the stresses
means they are normalized according to 𝜎̅ = (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 buckling
𝛾
(𝑞𝑎2 + 𝑞𝑏2 + 𝑞𝑐2 ) + 𝑞𝑎 𝑡𝑎𝑛(𝑞𝑎 ⁄2) − 𝑞𝑏 𝑐𝑜𝑡(𝑞𝑏 ⁄2) − 𝑞𝑐 𝑐𝑜𝑡(𝑞𝑐 ⁄2) = 0
1 − 2𝛾
or in the 𝑎𝑏𝑐 stress space, considering all three 2𝜋⁄3 rotations corresponding to this
mode
(σ
̅𝑎𝑎 + σ
̅𝑏𝑏 + σ
̅𝑐𝑐 )
2√3√3𝛾
+ √σ
̅𝑎𝑎 𝑡𝑎𝑛ℎ (√3√3σ
̅𝑎𝑎 ) + √σ
̅𝑏𝑏 𝑐𝑜𝑡ℎ (√3√3σ
̅𝑏𝑏 ) + √σ
̅𝑐𝑐 𝑐𝑜𝑡ℎ (√3√3σ
̅𝑐𝑐 ) = 0
1 − 2𝛾
(σ
̅𝑎𝑎 + σ
̅𝑏𝑏 + σ
̅𝑐𝑐 )
2√3√3𝛾
+ √σ
̅𝑎𝑎 𝑐𝑜𝑡ℎ (√3√3σ
̅𝑎𝑎 ) + √σ
̅𝑏𝑏 𝑡𝑎𝑛ℎ (√3√3σ
̅𝑏𝑏 ) + √σ
̅𝑐𝑐 𝑐𝑜𝑡ℎ (√3√3σ
̅𝑐𝑐 ) = 0
1 − 2𝛾
(σ
̅𝑎𝑎 + σ
̅𝑏𝑏 + σ
̅𝑐𝑐 )
2√3√3𝛾
+ √σ
̅𝑎𝑎 𝑐𝑜𝑡ℎ (√3√3σ
̅𝑎𝑎 ) + √σ
̅𝑏𝑏 𝑐𝑜𝑡ℎ (√3√3σ
̅𝑏𝑏 ) + √σ
̅𝑐𝑐 𝑡𝑎𝑛ℎ (√3√3σ
̅𝑐𝑐 ) = 0
1 − 2𝛾
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
−Ψ(𝑞𝑏 ) −Φ(𝑞𝑏 )
0
−Φ(𝑞𝑏 ) −Ψ(𝑞𝑏 )
0
0
0
0
0
0
[
0
0
0
0
0
0
−Ψ(𝑞𝑐 ) −Φ(𝑞𝑐 )
−Φ(𝑞𝑐 ) −Ψ(𝑞𝑐 )
1
1
0
1
0
0
0
1
0
−1
0
0
1
0
−1
1
0
1
0
1
−(𝑞𝑎2 + 𝑞𝑏2 + 𝑞𝑐2 )
𝑞𝑏2
𝑞𝑐2
𝛾
1 − 2𝛾
0
1
−1
0
0
0
−𝑞𝑏2
0
𝑀𝑎 𝐿(1 − 2𝛾)
0
𝐸𝐼
0 𝑀 𝐿(1 − 2𝛾)
0
𝑜𝑏
0
0
𝐸𝐼
0
0 𝑀𝑏𝑜 𝐿(1 − 2𝛾)
0
1
𝐸𝐼
2
−1 𝑀𝑜𝑐 𝐿(1 − 2𝛾) = −(𝐺𝑎 + 𝐺𝑏 + 𝐺𝑐 )𝐿 (1 − 2𝛾)𝛾
𝐸𝐼
𝐸𝐼
0
𝐺𝑏 𝐿2 (1 − 2𝛾)2
𝑀𝑐𝑜 𝐿(1 − 2𝛾)
𝐸𝐼
0
𝐸𝐼
𝐺𝑐 𝐿2 (1 − 2𝛾)2
𝛼
−𝑞𝑐2 ]
[
]
𝜃𝑏
𝐸𝐼
[
𝜃𝑐
]
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
−𝛾
(𝑞 2 + 𝑞𝑏2 + 𝑞𝑐2 ) + 𝑞𝑎 𝑐𝑜𝑡(𝑞𝑎 ⁄2) + 𝑞𝑏 𝑐𝑜𝑡(𝑞𝑏 ) + 𝑞𝑐 𝑐𝑜𝑡(𝑞𝑐 ) = 0
1 − 2𝛾 𝑎
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 fullscale 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 trichiral 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 (2𝑟⁄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 beamcolumn and equilibrium relations for instability of tri-chiral honeycomb by uniaxial mode
of buckling is expressed in the following matrix form
𝑟 2
−2(𝑞𝑎2′ + 𝑞𝑏2′ + 𝑞𝑐2′ ) ( )
𝐿
−2
0
0
0
𝜓(𝑞𝑎′ ) − 𝜙(𝑞𝑎′ )
0
0
1
0
𝜓(𝑞𝑏′ ) + 𝜙(𝑞𝑏′ )
0
−1
0
0
𝜓(𝑞𝑐′ ) + 𝜙(𝑞𝑐′ )
−1
[
−1
1
1
0
𝑀𝑎′ 𝐿⁄(𝐸𝐼)
(𝐺𝑎 + 𝐺𝑏 + 𝐺𝑐 )𝐿2 /(2𝐸𝐼)
𝑀𝑏′ 𝐿⁄(𝐸𝐼)
𝐺𝑎 𝐿2 /(𝐸𝐼)
𝑀𝑐′ 𝐿⁄(𝐸𝐼) =
0
−1
𝛼
0
0
]
𝛽
[
] [
0
0]
𝑞𝑎2′
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 𝜎̅ = (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
𝑟 2
2
2 ( ) (𝑞𝑎2′ + 𝑞𝑏2′ + 𝑞𝑐′
) + 𝑞𝑎′ 𝑡𝑎𝑛(𝑞𝑎′ ⁄2) − 𝑞𝑏′ 𝑐𝑜𝑡(𝑞𝑏′ ⁄2) − 𝑞𝑐 ′ 𝑐𝑜𝑡(𝑞𝑐 ′ ⁄2) = 0
𝐿
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(σ
̅𝑎 ′ 𝑎 ′ + σ
̅𝑏 ′ 𝑏′ + σ
̅𝑐 ′ 𝑐 ′ )𝑡𝑎𝑛2 𝜃 + √σ
̅𝑎′ 𝑎′ 𝑡𝑎𝑛ℎ (√3√3σ
̅𝑎 ′ 𝑎 ′ ) + √ σ
̅𝑏′ 𝑏′ 𝑐𝑜𝑡ℎ (√3√3σ
̅𝑏 ′ 𝑏 ′ )
+ √σ
̅𝑐 ′ 𝑐 ′ 𝑐𝑜𝑡ℎ (√3√3σ
̅𝑐 ′ 𝑐 ′ ) = 0
√3√3(σ
̅𝑎 ′ 𝑎 ′ + σ
̅𝑏 ′ 𝑏′ + σ
̅𝑐 ′ 𝑐 ′ )𝑡𝑎𝑛2 𝜃 + √σ
̅𝑎′ 𝑎′ 𝑐𝑜𝑡ℎ (√3√3σ
̅𝑎′ 𝑎′ ) + √σ
̅𝑏′ 𝑏′ 𝑡𝑎𝑛ℎ (√3√3σ
̅𝑏 ′ 𝑏 ′ )
+ √σ
̅𝑐 ′ 𝑐 ′ 𝑐𝑜𝑡ℎ (√3√3σ
̅𝑐 ′ 𝑐 ′ ) = 0
√3√3(σ
̅𝑎 ′ 𝑎 ′ + σ
̅𝑏 ′ 𝑏 ′ + σ
̅𝑐 ′ 𝑐 ′ )𝑡𝑎𝑛2 𝜃 + √σ
̅𝑎′ 𝑎′ 𝑐𝑜𝑡ℎ (√3√3σ
̅𝑎′ 𝑎′ ) + √σ
̅𝑏′ 𝑏′ 𝑐𝑜𝑡ℎ (√3√3σ
̅𝑏 ′ 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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