post-buckling analysis of curved beamssourcedb.imech.cas.cn/zw/rck0/zgjzj/fxxlx/201504/w... ·...

Zhichao FanAML,

Department of Engineering Mechanics,

Tsinghua University,

Beijing 100084, China;

Center for Mechanics and Materials,


Beijing 100084, China

Jian Wu1

AML,







e-mail: [email protected]

Qiang MaAML,







Yuan LiuAML,







Yewang SuState Key Laboratory of Nonlinear Mechanics,

Institute of Mechanics,

Chinese Academy of Sciences,


Keh-Chih Hwang1

AML,







e-mail: [email protected]

Post-Buckling Analysis of CurvedBeamsStretchability of the stretchable and flexible electronics involves the post-buckling behav-iors of internal connectors that are designed into various shapes of curved beams. Thelinear displacement–curvature relation is often used in the existing post-buckling analy-ses. Koiter pointed out that the post-buckling analysis needs to account for curvature upto the fourth power of displacements. A systematic method is established for the accuratepost-buckling analysis of curved beams in this paper. It is shown that the nonlinear termsin curvature should be retained, which is consistent with Koiter’s post-buckling theory.The stretchability and strain of the curved beams under different loads can be accuratelyobtained with this method. [DOI: 10.1115/1.4035534]

Keywords: post-buckling, stretchability, curved beam, curvature, finite deformation

1 Introduction

Curved beams have simple geometry and facilitative fabricationand are widely used in the fields of electronic, aerospace, andarchitecture. Recently, curved beams are treated as internal con-nectors of stretchable electronics due to their high stretchabilitywith small strain [1–13]. There are many functional stretchableand flexible electronics based on different shapes of curvedbeams, such as epidermal health/wellness monitors [14–18], sensi-tive electronic skins [19–23], and spherical-shaped digital cameras

[24–26]. The post-buckling behaviors, especially lateral and out-of-plane buckling, provide the flexibility and stretchability of theelectronics [27], which are the essential properties of a robusttechnology of assembling various microstructures [28–30].

There are many researchers studying up on the stability of pla-nar curved beams. Timoshenko and Gere [31] presented initialbuckling behaviors of circular beams. The analytical models forpost-buckling behaviors of the inextensible ring under uniformradial pressure were developed by Carrier [32] and Budiansky[33]. A systematic variational approach of space curved beamswas developed by Liu and Lu and employed on buckling behaviorand critical load of the serpentine structure [34]. Many results ofthe post-buckling behaviors of curved beams are obtained by thesemi-analytical energy approach and finite-element method

1Corresponding authors.Manuscript received November 16, 2016; final manuscript received December

15, 2016; published online January 24, 2017. Assoc. Editor: Daining Fang.

Journal of Applied Mechanics MARCH 2017, Vol. 84 / 031007-1Copyright VC 2017 by ASME

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(FEM) [35,36]. Koiter’s approach of energy minimization forpost-buckling expanded the potential energy to the fourth powerof displacement because the third and fourth power terms actuallygovern post-buckling [37–39], which require the elongation andcurvature that at least express the third power of displacement.

This paper presents a systematic study on post-buckling ofcurved beam, the nonlinear relations between the deformationcomponents (elongation and curvatures) and displacements arederived, and the perturbation method is used to obtain the analyti-cal solution of the post-buckling behaviors of curved beams.

2 Deformation of Curved Beam

2.1 The Initial and Deformed Curved Beams. The materialpoints on the centroid line of the initial curved beam are denotedby ~PðSÞ (Fig. 1), where S is the arc-length of centroid line. Thelocal triad vectors at the centroid line of the curved beam areorthogonal unit vectors Eiði ¼ 1; 2; 3Þ. E1 and E2 are along twolines of symmetry of the cross section of beam (Fig. 1). E3 is theunit vector along the tangential direction of the centroid line andcan be given as

E3 ¼d~P

dS(2.1)

The curvature of the curved beam was defined by Love [40] as

dEi

dS¼ K � Ei i ¼ 1; 2; 3ð Þ (2.2)

where the curvature K ¼ K1E1 þ K2E2 þ K3E3, K1 and K2 are thecurvatures along the axes in cross section, and K3 denotes thetwist along the tangential direction of the centroid line.

The centroid line of curved beam is deformed from ~P to~p ¼ ~P þ U, where U is the displacement of the beam. The localtriad vectors of the deformed curved beam are eiði ¼ 1; 2; 3Þ,where e1 and e2 are the orthogonal unit vectors in the cross sectionof deformed curved beam, and e3 is the unit vector along the tan-gential direction of the centroid line of deformed beam, which isdefined as

e3 ¼d~p

ds(2.3)

where s is the arc-length coordinate of the deformed beam, whichis the function of the arc-length of initial curved beam, S. Theelongation of the deformed beam, k, can be obtained from

k ¼ ds

dS(2.4)

The curvature of the deformed curved beam also can be givenas [40]

dei

ds¼ j� ei i ¼ 1; 2; 3ð Þ (2.5)

where j ¼ j1e1 þ j2e2 þ j3e3, j1 and j2 are the components ofcurvature along sectional vectors e1 and e2, respectively, and j3

denotes the twist along the tangent direction, e3. Here, the twistangle, /, is defined with j3 as

/ ¼ð

j3ds (2.6)

where the axis of the twist angle changes with the location.

2.2 Equilibrium Equations and Constitutive Relation ofCurved Beam. The internal force, t ¼ tiei, and moment,m ¼ miei, of the curved beam satisfy the following equilibriumequations [41]:

dt

dsþ p ¼ 0

dm

dsþ e3 � t þ q ¼ 0

(2.7)

where p ¼ piei and q ¼ qiei are the distributed force and momentper unit length in the deformed beam, respectively. The derivativesymbols have the relation, ð1=kÞðdðÞ=dSÞ ¼ ðdðÞ=dsÞ.

The internal force, t3, is conjugate with the elongation, k, andhas the constitutive relation

t3 ¼ EAðk� 1Þ (2.8)

where E and A are the elastic modulus and the cross section areaof the beam, respectively.

The moment, m, is not conjugate with the curvature, j, but it isconjugate with the Lagrangian curvature, j, which is defined as

dei

dS¼ j � ei i ¼ 1; 2; 3ð Þ (2.9)

where the Lagrangian curvature, j, has the relation with curva-ture, j, as j ¼ kj.

The constitutive relation between Lagrangian curvatures(abbreviated hereafter simply to curvatures) and moments is

m1 ¼ EI1ðj1 � K1Þ

m2 ¼ EI2ðj2 � K2Þ

m3 ¼ GJðj3 � K3Þ

(2.10)

where G is the shear modulus of the beam, I1 and I2 are the sec-tion area moment of inertia about the local coordinates, and J isthe polar moment of inertia of the cross section of beam.

The components of the equilibrium equation expressed with thecomponents of internal forces, moments, and the curvatures canbe given as

dtidSþ �ijkjjtk þ kpi ¼ 0 i ¼ 1; 2; 3ð Þ

dmi

dSþ �ijkj jmk � �ij3ktj þ kqi ¼ 0 i ¼ 1; 2; 3ð Þ

(2.11)

Fig. 1 Schematic illustration of initial and deformed curvedbeam

031007-2 / Vol. 84, MARCH 2017 Transactions of the ASME


where �ijk are the components of Eddington tensor.

3 The Deformation Variables in Terms of

Displacement for Planar Curved Beam

The planar curved beam, which has only one nonzero compo-nent of initial curvature, K1, is widely used. We will focus onthe planar curved beam in Secs. 3–7. The displacement, U, candecompose in the local coordinates, Eiði ¼ 1; 2; 3Þ, asU ¼ U1E1 þ U2E2 þ U3E3. The elongation defined by Eq. (2.4)can be expressed through the components of the displacementas

k ¼ 1þ 2dU3

dSþ U2K1

� ��

þ dU1

dS

� �2

þ dU2

dS� U3K1

� �2

þ dU3

dSþ U2K1

� �2" #)1

2

(3.1)

The relations between the local triads on the initial anddeformed curved beams are given by the coefficients aij as

ei ¼ aijEj ði ¼ 1; 2; 3Þ (3.2)

E3 and e3 are along the tangent directions of the initial anddeformed curved beams, respectively. By substituting Eqs. (2.1)and (2.3) into Eq. (3.2), the coefficients, a3jðj ¼ 1; 2; 3Þ, can bederived as

a31 ¼1

kdU1

dS; a32 ¼

1

kdU2

dS� U3K1

� �;

a33 ¼1

k1þ dU3

dSþ U2K1

� � (3.3)

The local coordinates, ei ði ¼ 1; 2; 3Þ and Ei ði ¼ 1; 2; 3Þ, areorthonormal (i.e., ei � ej ¼ dij and Ei � Ej ¼ dij), which lead thecoefficients, aij, to satisfy the following equations:

a2i1 þ a2

i2 þ a2i3 ¼ 1 ði ¼ 1; 2; 3Þ

aikajk ¼ 0 ði 6¼ jÞ(3.4)

The relations between the coefficients, aijði ¼ 1; 2; j ¼ 1; 2; 3Þ,the displacements, Ui, and the twist angle, /, are complicated andcannot be explicitly expressed as Eq. (3.3). Substitution ofEqs. (3.1) and (3.3) into Eq. (3.4) and expansion of the coeffi-cients, aij, to the powers of generalized displacements lead to

aijf g ¼1 0 0

0 1 0

0 0 1

8><>:

9>=>;

þ

0 w 1½ � � dU1

dS

�w 1½ � 0 � dU2

dSþ U3K1

dU1

dS

dU2

dS� U3K1 0

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

þ ah2iij

n oþ a

h3iij

n oþ � � �f g i; j ¼ 1; 2; 3ð Þ (3.5)

where w½1� ¼ /�Ð

K1ðdU1=dSÞdS, and the superscripts h2i and

h3i in ah2iij and a

h3iij refer to the second and third power of

generalized displacements (i.e., Ui and /). They are given inAppendix A, and f� � �g are the terms of the fourth and higherpower of displacements and twist angle.

After substituting Eqs. (3.2) and (3.5) into Eq. (2.9), the curva-tures can be expressed in terms of generalized displacements as

j1

j2

j3

8><>:

9>=>; ¼

K1

0

0

8><>:

9>=>;þ

� d2U2

dS2þ d U3K1ð Þ

dS

d2U1

dS2� w 1½ �K1

d/dS

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;þ jh2ii

n o

þ jh3ii

n oþ � � �f g (3.6)

where jh2ii and jh3ii are the second and third power of displace-ments, Ui, and twist angle, /, which are given in Appendix A.

The nonlinear terms (i.e., jh2ii and jh3ii ) are important for the anal-ysis of the post-buckling behaviors of the curved beams based onthe Koiter’s theory. As far as the authors are aware, the powerexpansion of deformation for Euler–Bernoulli curved beams(without assuming inextensibility of the beams) is a new result.The results obtained by Su et al. [41] for straight beams follow asa special case.

4 The Perturbation Solution for Post-Buckling of the

Planar Curved Beam

The planar curved beam will be buckling in-plane or out-of-plane due to the planar loads (e.g., p1 ¼ 0). The method of pertur-bation is used to solve differential equations (2.11) substitutedwith variables of the deformation and displacement, where a smallratio, a, of the maximum deflection to the characteristic length(e.g., the beam length, LS, or the initial curvature radius of beam,R) is introduced. The generalized displacements from buckling,expanded to the powers of a, can be written as

U1 ¼ aU1ð1Þ þ a2U1ð2Þ þ a3U1ð3Þ þ Oða4ÞLS



/ ¼ a/ð1Þ þ a2/ð2Þ þ a3/ð3Þ þ Oða4Þ

(4.1)

By substituting Eq. (4.1) into Eqs. (3.1) and (3.6), the elonga-tion and curvatures can be also expanded to the powers of thesmall ratio, a, as

k ¼ 1þ akð1Þ þ a2kð2Þ þ a3kð3Þ þ Oða4Þ (4.2)

ji ¼ jið0Þ þ ajið1Þ þ a2jið2Þ þ a3jið3Þ þ Oða4ÞL�1S ði ¼ 1; 2; 3Þ

(4.3)

where j1ð0Þ ¼ K1, j2ð0Þ ¼ j3ð0Þ ¼ 0, kðkÞ and jiðkÞ are the func-tions of generalized displacements, and

k 1ð Þ ¼dU3 1ð Þ

dSþ U2 1ð ÞK1 (4.4)

j1 1ð Þ ¼ �d2U2 1ð Þ

dS2þ

d U3 1ð ÞK1

� �dS

j2 1ð Þ ¼d2U1 1ð Þ

dS2� w 1½ �

1ð ÞK1

j3 1ð Þ ¼d/ 1ð Þ

dS

(4.5)

where w½1�ð1Þ ¼ /ð1Þ �Ð

K1ðdU1ð1Þ=dSÞdS.The coefficient, aij, can be expanded to the powers of the small

ratio, a, as

Journal of Applied Mechanics MARCH 2017, Vol. 84 / 031007-3


aij ¼ 1þ a2aijð2Þ þ a3aijð3Þ þ Oða4Þ ði ¼ jÞaij ¼ aaijð1Þ þ a2aijð2Þ þ a3aijð3Þ þ Oða4Þ ði 6¼ jÞ

(4.6)

where aijðkÞ are the functions of generalized displacements

aij 1ð Þ ¼

0 w 1½ �1ð Þ �

dU1 1ð ÞdS

�w 1½ �1ð Þ 0 �

dU2 1ð ÞdSþ U3 1ð ÞK1

dU1 1ð ÞdS

dU2 1ð ÞdS� U3 1ð ÞK1 0

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;(4.7)

The expressions of kðkÞ, jiðkÞ, and aijðkÞ for k ¼ 2 are given inAppendix B.

Substituting Eqs. (4.2) and (4.3) into constitutive relations (2.8)and (2.10) and considering the equilibrium equations, the internalforces and moments can be expanded to powers of the small ratioa as

ti ¼ tið0Þ þ atið1Þ þ a2tið2Þ þ a3tið3Þ þ Oða4ÞEA (4.8)

mi ¼ mið0Þ þ amið1Þ þ a2mið2Þ þ a3mið3Þ þ Oða4ÞEI1L�1S (4.9)

where tið0Þ and mið0Þ are the forces and moments at the onsetof buckling, and the relation between t3ðkÞ, miðkÞ, and kðkÞ, jiðkÞðk � 1Þ can be written as

t3ðkÞ ¼ EAkðkÞ

m1ðkÞ ¼ EI1j1ðkÞ

m2ðkÞ ¼ EI2j2ðkÞ

m3ðkÞ ¼ GJj3ðkÞ

(4.10)

After substituting the forces (4.8) and moments (4.9), equilib-rium equations (2.11) can be decomposed as

dti 0ð ÞdSþ �ijkj j 0ð Þtk 0ð Þ þ pi 0ð Þ ¼ 0 i ¼ 1; 2; 3ð Þ

dmi 0ð ÞdSþ �ijkjj 0ð Þmk 0ð Þ � �ij3tj 0ð Þ þ qi 0ð Þ ¼ 0 i ¼ 1; 2; 3ð Þ

(4.11)

and

dti nð ÞdSþXl¼n

l¼0

�ijkj j lð Þtk n�lð Þ þ k lð Þpi n�lð Þ� �

¼ 0

i ¼ 1; 2; 3; n ¼ 1; 2; 3; :::ð Þdmi nð Þ

dSþXl¼n

l¼0

�ijkjj lð Þmk n�lð Þ � �ij3k lð Þtj n�lð Þ þ k lð Þqi n�lð Þ� �

¼ 0

i ¼ 1; 2; 3; n ¼ 1; 2; 3; :::ð Þ(4.12)

where kð0Þ ¼ 1, and the loads are expanded to the power series ofa as

pi ¼ pið0Þ þ apið1Þ þ a2pið2Þ þ a3pið3Þ þ Oða4ÞEAL�1S

qi ¼ qið0Þ þ aqið1Þ þ a2qið2Þ þ a3qið3Þ þ Oða4ÞEI1L�2S

(4.13)

where pið0Þ and qið0Þ are the critical loads at bifurcation point.The differential equations in this section, in which internal

forces and moments are substituted with deformation variablesand displacements, are solved in the following steps.

Step 1: Solve Eq. (4.12) for n ¼ 1 with the correspondingboundary conditions to determine the buckling mode for the lead-ing order, kð1Þ, jið1Þ, and Uið1Þ, of elongation, curvatures, and dis-

placements, and the critical loads at the onset of buckling, pið0Þand qið0Þ.

Step 2: Solve Eq. (4.12) for n ¼ 2 with the correspondingboundary conditions to determine the second order kð2Þ, jið2Þ, and

Uið2Þ of elongation, curvatures, and displacements, and the incre-

ment of loads, pið1Þ and qið1Þ.Step 3: Solve Eq. (4.12) for n ¼ 3 with the corresponding

boundary conditions to determine the buckling mode for the thirdorder kð3Þ and jið3Þ of elongation and curvatures, and the incre-ment of loads, pið2Þ and qið2Þ.

5 In-Plane Post-Buckling Behavior of Elastic Ring

As illustrated in Fig. 2, the uniform distributed radial load, p2,which remains normal to the centroid line during the deformation,is applied on an elastic thin ring. The arc-length, S, is clockwisecounted from A. The ring at A is simply supported, and the dis-placement along tangent direction at B is constrained, theseboundary conditions can be written as

UiðnÞjS¼0 ¼ 0 ði ¼ 1; 2; 3; n ¼ 1; 2; 3;…Þ (5.1)

U3ðnÞjS¼pR ¼ 0 ðn ¼ 1; 2; 3;…Þ (5.2)

where R ¼ 1=K1 is the initial curvature radius of ring. The dis-placements, deformations, and forces/moments are periodical dueto the periodical deformation of the ring, i.e.,

ðÞjS¼S0¼ ðÞjS¼S0þ2pR; ð0 � S0 � 2pRÞ (5.3)

where ðÞ denotes k, kðnÞ, j i, jiðnÞ, ti, tiðnÞ, mi, miðnÞ, Ui, UiðnÞ, /,/ðnÞ, and aij, aijðnÞ.

The width of the ring section, w, is much larger than its thick-ness, t, and the ring will be buckling in the initial plane of thecurved beam under the load, p2 ¼ ��p2. The out-of-plane compo-nents of force, moment, and displacement are zero, i.e., t1 ¼ 0,m2 ¼ 0, m3 ¼ 0, U1 ¼ 0, and / ¼ 0. The internal force andmoment at the onset of buckling, t2ð0Þ, t3ð0Þ, and m1ð0Þ, satisfyEq. (4.11) and have the relation with critical load, �p2ð0Þ, as

Fig. 2 Schematic illustration of elastic ring under uniformcompression



t3ð0Þ ¼ ��p2ð0ÞR; t2ð0Þ ¼ 0; m1ð0Þ ¼ 0 (5.4)

After substituting constitutive relation (4.10) into equilibriumequations (4.12) for n ¼ 1 and eliminating t2ð1Þ and j1ð1Þ, the dif-ferential equation for elongation, kð1Þ, is derived as

d3k 1ð ÞdS3

þ k21

dk 1ð ÞdS¼ 0 (5.5)

where k21 ¼ ½1þ �p2ð0ÞR=ðEAÞ þ �p2ð0ÞR

3=ðEI1Þ�=R2. Solution ofEq. (5.5) is

kð1Þ ¼ C12 cos ðk1SÞ þ C11 sin ðk1SÞ þ C10 (5.6)

where C10, C11, and C12 are the parameters to be determined. Thecritical load, �p2ð0Þ, can be determined by the periodical conditionof elongation, kð1Þ, as

�p2 0ð Þ ¼3EI1

R31þ c1ð Þ�1

(5.7)

where c1 ¼ EI1= EAR2ð Þ. j1ð1Þ is also obtained by substitutingEqs. (4.10) and (5.6) into Eq. (4.12) as

j1 1ð Þ ¼1

3Rc1

C10 þ c1

�p2 1ð ÞR3

EI1

1þ c1ð Þ þ 4C10

" #(

�3 C11 sin2S

Rþ C12 cos

2S

R

� �� (5.8)

The differential equations for U2ð1Þ and U3ð1Þ can be derivedfrom Eqs. (4.4) and (4.5) as

dU3 1ð ÞdSþ

U2 1ð ÞR¼ k 1ð Þ

d2U2 1ð ÞdS2

� 1

R

dU3 1ð ÞdS

¼ �j1 1ð Þ

(5.9)

The solutions of U2ð1Þ and U3ð1Þ can be obtained by the boundaryconditions (5.1) and (5.2) and the periodical condition (5.3) as

U2 1ð Þ ¼ �R cosS

Rþ R cos

2S

R� 1þ c1ð Þc1

1þ 4c1

�p2 1ð ÞR4

EI1

1� cos2S

R

� �

U3 1ð Þ ¼ R sinS

R� R

2

1þ 4c1

1þ c1

þ c1

�p2 1ð ÞR3

EI1

!sin

2S

R

(5.10)

and the parameters, C10, C11, and C12, are also determined as

C10 ¼ ��p2 1ð ÞR

3

EI1

c1 1þ c1ð Þ1þ 4c1

; C11 ¼ 0;

C12 ¼ �3c1

1þ c1ð Þ þ3c2

1

1þ 4c1ð Þ�p2 1ð ÞR

3

EI1

" #(5.11)

where U2ð1Þ is assumed to be symmetrical about AB, andmaxðU2ð1ÞÞ ¼ 2R.

Substitution of constitutive relation (4.10) into equilibriumequations (4.12) for n ¼ 2 and elimination of t2ð2Þ and j1ð2Þ givethe differential equation for elongation, kð2Þ, as

d3k 2ð ÞdS3

þ k21

dk 2ð ÞdS¼ F21 Sð Þ (5.12)

where

F21 Sð Þ ¼3 1þ 4c1 þ

c1 �p2 1ð ÞR3

EI1

1þ c1ð Þ

" #

R3 1þ c1ð Þ2 1þ 4c1ð Þ2

�2�p2 1ð ÞR

3

EI1

c1 1þ 3c21 þ 4c3

1

� �sin

2S

R

�45c1 1þ 4c1 þ�p2 1ð ÞR

3

EI1

c1 1þ c1ð Þ

" #sin

4S

R

8>>>><>>>>:

9>>>>=>>>>;

The solution of Eq. (5.12) is

k 2ð Þ ¼ C22 cos k1Sð Þ þ C21 sin k1Sð Þ þ C20

þðS

0

ðf

0

sin k1 S� nð Þ½ �F21 nð Þk1

dndf (5.13)

where kð2Þ is also periodical, which determined the increment of

load, �p2ð1Þ ¼ 0, and the parameters, C20, C21, and C22 will be

determined with the boundary conditions and orthogonality condi-

tion [37,38],Ð 2pR

0kð1ÞðnÞkð2ÞðnÞdn ¼ 0.

The curvature, j1ð2Þ, is also obtained by substitution of Eqs.(4.10) and (5.13) into Eq. (4.12) as

j1 2ð Þ ¼�3 23þ68c1ð Þ16R 1þ c1ð Þ2

þ1þ c1

3R

�p2 2ð ÞR3

EI1

þ1þ4c1

3Rc1

C20�C21

Rc1

sin2S

R

� 1

4R

45

1þc1ð Þ2þ4C22

c1

" #cos

2S

Rþ 9 1�4c1ð Þ

16R 1þ c1ð Þ2cos

4S

R

(5.14)

The differential equations for U2ð2Þ and U3ð2Þ are given by sub-stitution of Eqs. (5.13) and (5.14) into Eqs. (B1) and (B2) as

d2U2 2ð ÞdS2

� 1

R

dU3 2ð ÞdS

¼ F22 Sð Þ

dU3 2ð ÞdSþ

U2 2ð ÞR¼ F23 Sð Þ

(5.15)Fig. 3 The ratio of load to critical load, �p2=�p2ð0Þ, versus thenormalized displacement, U2max=ð2RÞ, during post-buckling,which is consistent with the results of Carrier’s model



where

F22 Sð Þ ¼ �j1 2ð Þ þd

dS

dU3 1ð ÞdS

dU2 1ð ÞdS

� �þ 1

2R

d2 U22 1ð Þ � U2

3 1ð Þ

� dS2

� 1

R2

d U2 1ð ÞU3 1ð Þ� �

dS

24

35

F23 Sð Þ ¼ k 2ð Þ �1

2

dU2 1ð ÞdS� 1

RU3 1ð Þ

� �2

and the underlined terms in F22ðSÞ come from the nonlinear part of curvature (B2).The solutions of Eq. (5.15) with the boundary conditions (5.1) and (5.2) are

U2 2ð Þ ¼R

80 1þ c1ð Þ2 1þ 4c1ð Þ

180c1 1þ 2c1ð Þ � 3c1 61þ 124c1ð ÞcosS

R

�5 9þ 16c1 1þ c1ð Þ3�p2 2ð ÞR

3

EI1

" #1� cos

S

R

� �þ 3c1 1þ 4c1ð Þcos

4S

R

8>>>><>>>>:

9>>>>=>>>>;

þ 3Rc1

5 1þ c1ð Þ2cos

S

R� cos

4S

R

� �

U3 2ð Þ ¼ �R

320 1þ c1ð Þ2 1þ 4c1ð Þ

4 45� 3c1 61þ 124c1ð Þ þ 80c1 1þ c1ð Þ3�p2 2ð ÞR

3

EI1

" #sin

S

R

þ3 1þ 4c1ð Þ �15þ 76c1ð Þsin4S

R

8>>>><>>>>:

9>>>>=>>>>;þ 3Rc1

20 1þ c1ð Þ2�4 sin

S

Rþ sin

4S

R

� �

(5.16)

where the underlined terms are derived from the nonlinear termsof the curvature, and U2ð2Þ is assumed to be symmetrical aboutAB. The parameters, C20; C21; and C22, are also determined as

C20 ¼�c1 1þ c1ð Þ

1þ 4c1ð Þ�p2 2ð ÞR

3

EI1

þ 9c1 23þ 68c1ð Þ16 1þ c1ð Þ2 1þ 4c1ð Þ

;

C21 ¼ 0; C22 ¼�45c1

4 1þ c1ð Þ2(5.17)

Substitution of constitutive relation (4.10) into equilibriumequation (4.12) for n ¼ 3 and elimination of t2ð3Þ and j1ð3Þ givethe differential equation for elongation, kð3Þ, as

d3k 3ð ÞdS3

þ k21

dk 3ð ÞdS¼ F3 Sð Þ (5.18)

where

F3 Sð Þ ¼ 6c1

R3 1þ c1ð Þ 1þ 4c1ð Þ

27 3� 24c1 � 16c21

� �32 1þ c1ð Þ2

� 1� c1 þ 4c21

� � �p2 2ð ÞR3

EI1

" #sin

2S

R

þ 1053c1 �3þ 4c1ð Þ16R3 1þ c1ð Þ3

sin6S

R


k 3ð Þ ¼ C32 cos k1Sð Þ þ C31 sin k1Sð Þ þ C30

þðS

0

ðf

0

sin k1 f� nð Þ½ �F3 nð Þk1

dndf (5.19)

where C30, C31, and C32 are the parameters, and the increment ofload, �p2ð2Þ, can be determined by the periodical condition of elon-gation, kð3Þ

�p2 2ð Þ ¼3� 24c1 � 16c2

1

� �1þ c1ð Þ2 1� c1 þ 4c2

1

� � 27EI1

32R3(5.20)

The thickness, t, of the cross section of beam is much smallerthan the radius, R, which indicates c1 � 1. The load, �p2, normal-ized by critical load, �p2ð0Þ, can be simplified as

�p2

�p2 0ð Þ¼ 1þ a2

�p2 2ð Þ�p2 0ð Þ

1þ 27a2

32¼ 1þ 27

32

U2max

2R

� �2

(5.21)

where a ¼ U2max=ð2RÞ has been used, and it is the same with theresult of Budiansky [33].

Figure 3 shows that the normalized load, �p2=�p2ð0Þ, increases withthe normalized maximum displacement, U2max=ð2RÞ, which is con-sistent with Carrier’s model [32]. It indicates that the elongation canbe neglected due to the inextensibility of elastic ring in Carrier’smodel.

6 Lateral Buckling of Circular Beam

The curved beam is widely used as interconnector, which isoften freestanding and connects the sensors in the stretchable and



flexible electronics [2]. The lateral buckling of the freestandinginterconnector will happen because the thickness of the beamcross section, t, is much larger than its width, w [1,2,16]. Thethickness direction is along the radial direction of the circularbeam, which is consistent with it in Sec. 5. The displacement, U1,and rotation, /, which are the odd powers of the small ratio, a, ofthe maximum deflect of U1 to beam length, are the primary dis-placements, while the secondary displacements, U2 and U3, arethe even powers of the small ratio, a [41]. The curvatures, j2 andj3, are the odd powers of the small ratio, a, and the elongation, k,and curvature, j1, are the even powers of the small ratio, a.

6.1 The Lateral Buckling of Circular Beam Under BendingMoment. As shown in Fig. 4, the bending moment, M, is appliedon the circular beam at the ends. This beam with length, LS, whichsubtends the angle, a, is simply supported in the plane and theout-of-plane, and beam ends cannot rotate around the centroid ofbeam, but the right beam end can freely slide in the plane of thebeam, i.e.,

U2jS¼0 ¼ U3jS¼0 ¼ 0

U2E2 þ U3E3ð ÞjS¼LS� cos

a2

E2 � sina2

E3

� �S¼LS

¼ 0

tjS¼LS� sin

a2

E2 þ cosa2

E3

� �S¼LS

¼ 0

m1jS¼0 ¼ m1jS¼LS¼ M

(6.1)

U1jS¼0 ¼ U1jS¼LS¼ 0

ðe2 � E1ÞjS¼0 ¼ ðe2 � E1ÞjS¼LS¼ 0

m2jS¼0 ¼ m2jS¼LS¼ 0

(6.2)

By substituting Eq. (3.2) into the above equations, the boundaryconditions can be expanded with respect to the perturbationparameter, a, as

U2 nð ÞjS¼0 ¼ U3 nð ÞjS¼0 ¼ 0

U2 nð Þ cosa2� U3 nð Þ sin

a2

� �S¼LS

¼ 0

Xl¼n

l¼0

ti lð Þai2 n�lð Þ sina2þ ti lð Þai3 n�lð Þ cos

a2

� �S¼LS

¼ 0

m1 nð ÞjS¼0 ¼ m1 nð ÞjS¼LS¼ M nð Þ

(6.3)

U1ðnÞjS¼0 ¼ U1ðnÞjS¼LS¼ 0

a21ðnÞjS¼0 ¼ a21ðnÞjS¼LS¼ 0

m2ðnÞjS¼0 ¼ m2ðnÞjS¼LS¼ 0

(6.4)

The internal forces and moments at the onset of buckling, tið0Þ,mið0Þði ¼ 1; 2; 3Þ, satisfy Eq. (4.11) and have the relation with crit-ical load, Mð0Þ, as

t1ð0Þ ¼ t2ð0Þ ¼ t3ð0Þ ¼ 0; m2ð0Þ ¼ m3ð0Þ ¼ 0; m1ð0Þ ¼ Mð0Þ (6.5)

Substitution of constitutive relation (4.10) into equilibriumequations (4.12) for n ¼ 1 and elimination of j3ð1Þ and t1ð1Þ givethe differential equation for j2ð1Þ as

d2j2 1ð ÞdS2

þ k22j2 1ð Þ ¼ 0 (6.6)

where k22 ¼ ½Mð0ÞR=ðEI2Þ � 1�½ðMð0ÞRc2=ðEI2Þ � c3�=ðR2c3Þ,

c2 ¼ EI2=ðEAR2Þ, and c3 ¼ GJ=ðEAR2Þ. The solution of Eq. (6.6)is

j2ð1Þ ¼ C41 cos ðk2SÞ þ C40 sin ðk2SÞ (6.7)

where the parameters, C40 and C41, will be determined bythe boundary conditions. The boundary conditions,j2ð1ÞjS¼0 ¼ j2ð1ÞjS¼LS

¼ 0, which are derived by substitution of

constitutive relation (4.10) into the boundary conditions (6.4),determine the critical load as

M 0ð Þ ¼EI2

2R1þ c3

c2

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� c3

c2

� �2

þ 4

�a2

c3

c2

s24

35 or

M 0ð Þ ¼EI2

2R1þ c3

c2

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� c3

c2

� �2

þ 4

�a2

c3

c2

s24

35

(6.8)

where the normalized angle, �a ¼ a=p, and k2 can be simplified to1=ðR�aÞ. When �a ¼ 1, one of the two values of the critical load inEq. (6.8) is zero, which corresponds to the freedom of a semicir-cular beam to rotate about the diameter connecting the two ends,the other value, Mð0Þ ¼ pEI2ð1þ c3=c2Þ=LS, is the critical load

for the semicircular beam. When the curved beam is shallow, i.e.,R LS, the critical load, Mð0Þ, will approach to the critical load

for the straight beam, ðpEI2=LSÞffiffiffiffiffiffiffiffiffiffiffic3=c2

p[31].

The j3ð1Þ can be obtained by the substitution of constitutiverelation (4.10) and Eq. (6.7) into equilibrium equations (4.12) forn ¼ 1 as

j3 1ð Þ ¼�ac2

c3

M 0ð ÞR

EI2

� 1

� �C40 1� cos

S

�aR

� �þ C41 sin

S

�aR

� þ C42

(6.9)

where the parameter, C42, will be determined by the boundarycondition.

The differential equations for U1ð1Þ and /ð1Þ can be derivedfrom Eq. (4.5) as

d/ 1ð ÞdS¼ j3 1ð Þ

d2U1 1ð ÞdS2

þ 1

R2U1 1ð Þ ¼ j2 1ð Þ þ

1

R/ 1ð Þ

(6.10)

The solutions of Eq. (6.10), which satisfy the boundary conditions(6.4), are

/ 1ð Þ ¼pc2�a �a2 � 1ð Þ M 0ð ÞR

EI2

� 1

� �

�a2c2

M 0ð ÞR

EI2

� 1

� �� c3

sinS

�aR

U1 1ð Þ ¼ p�aR sinS

�aR

(6.11)

where the maximum of U1ð1Þ is LS, and the parameters, C40, C41,and C42, are

C40 ¼p

�aR

c3 �a2 � 1ð Þ

c3 þ �a2c2 1�M 0ð ÞR

EI2

� � ; C41 ¼ 0;

C42 ¼pR

c2 �a2 � 1ð Þ 1�M 0ð ÞR

EI2

� �

c3 þ �a2c2 1�M 0ð ÞR

EI2

� �(6.12)

Substitution of constitutive relation (4.10) into equilibriumequations (4.12) for n ¼ 2 and elimination of kð2Þ and t2ð2Þ givethe differential equation for leading terms of j1 and j1ð2Þ, as



d3j1 2ð ÞdS3

þ k23

dj1 2ð ÞdS¼ F51 Sð Þ (6.13)

where k23 ¼ 1=R2, and F51ðSÞ ¼ ðc2 � c3Þ=c1½ðd2=dS2Þðj2ð1Þj3ð1ÞÞ þð1=R2Þj2ð1Þj3ð1Þ�. The solution of Eq. (6.13) is

j1 2ð Þ Sð Þ ¼ C52 cos k3Sð Þ þ C51 sin k3Sð Þ þ C50 þðS

0

ðf

0

sin k3 f� nð Þ½ �F51 nð Þk3

dndf (6.14)

The elongation, kð2Þ, can be obtained by substitution of constitutive relation (4.10) and the above equation into equilibrium equations(4.12) for n ¼ 2 as

k 2ð Þ ¼ �c1R C52 cosS

Rþ C51 sin

S

R

� �þ

p2c2c3 c2 � c3ð Þ �a2 � 1ð Þ2 M 0ð ÞR

EI2

� 1

� �

c3�a � c2�a3M 0ð ÞR

EI2

� 1

� �� 2cos

2S

�aR

þc1R3

ðS

0

cosS� n

RF51 nð Þdn

(6.15)

The differential equations for the leading terms, U2ð2Þ and U3ð2Þ, of displacements, U2 and U3, are derived by substitution ofEq. (6.14) into Eqs. (B1) and (B2) as

d2U2 2ð ÞdS2

þU2 2ð ÞR2¼ F52 Sð Þ

dU3 2ð ÞdS

¼ k 2ð Þ �U2 2ð Þ

R� 1

2

dU1 1ð ÞdS

� �2 (6.16)

where

F52ðSÞ ¼ �j1ð2Þ þ kð2Þ=R� ðdU1ð1Þ=dSÞ2=ð2RÞþw½1�ð1Þðd2U1ð1Þ=dS2Þ �

h�w½1�ð1Þ

2

þ ðdU1ð1Þ=dSÞ2i=ð2RÞ

and the underlined terms are derived from the nonlinear terms of curvature.The solutions of Eq. (6.16), which satisfy the boundary conditions (6.3), are

U2 2ð Þ Sð Þ ¼ C53 cosS

Rþ C54 sin

S

Rþ R

ðS

0

sinS� nð Þ

RF52 nð Þdn;

U3 2ð Þ Sð Þ ¼ðS

0

k 2ð Þ nð Þ � 1

RU2 2ð Þ nð Þ � 1

2

dU1 1ð Þ nð Þdn

� �2" #

dnþ C55

(6.17)

and the parameters, C50; C51; C52; C53; C54; and C55, are determined as

C55 ¼ C53 ¼ C51 ¼ 0

C52 ¼p2 c2

c1

c2

c3

� 1

� ��a2 � 1ð Þ2 M 0ð ÞR

EI2

� 1

� �

�a2R 1� �a2c2

c3

M 0ð ÞR

EI2

� 1

� �� 2

C54 ¼ �p�a3R2

8C52 þ

p�aR

2

c2

c1

M 2ð ÞR

EI2

þ p3 �a2 � 1ð ÞR8�a

1þ �a2 � 1

1� �a2c2

c3

M 0ð ÞR

EI2

� 1

� �� 2

8><>:

9>=>;

C50 ¼ �C52 þc2

Rc1

M 2ð ÞR

EI2

(6.18)

Substitution of constitutive relation (4.10) into equilibrium equations (4.12) for n ¼ 3 and elimination of j3ð3Þ and t1ð3Þ give the dif-ferential equation for j2ð3Þ as

d2j2 3ð ÞdS2

þ k22j2 3ð Þ ¼ F6 Sð Þ (6.19)

where

F6ðSÞ ¼ fð1� c1=c2Þ½Mð0ÞRc2=ðEI2c3Þ � 1�j1ð2Þj2ð1Þ � Rðc1=c2 � c3=c2Þ½dðj1ð2Þj3ð1ÞÞ=dS�g=R




j2 3ð Þ ¼ C61 cos k2Sð Þ þ C60 sin k2Sð Þ þðS

0

sin k2 S� nð Þ½ �F6 nð Þk2

dn (6.20)

where C60 and C61 are the parameters. j2ð3Þ satisfies the boundary conditions j2ð3ÞjS¼0 ¼ j2ð3ÞjS¼LS¼ 0, which determine the load

increment

M 2ð Þ ¼EI2

R

p2 �a2 � 1ð Þ2 c2

c3

� 1

� �1þ 3

c3

c2

� 4c3

c1

� 4� c3

c1

� 3c2

c1

� �M 0ð ÞR

EI2

" #1�

M 0ð ÞR

EI2

� �

8 1�M 0ð ÞR

EI2

� ��a2

c2

c3

þ 1

� 2

2c3

c1

� c3

c2

� 1þ 2� c2 þ c3

c1

� �M 0ð ÞR

EI2

� (6.21)

For the narrow rectangular section of beam, which thickness is much larger than its width, i.e., t w, the stiffness ratios, c2 and c3 aremuch smaller than c1, i.e., c1 c2 � c3. The load increment in Eq. (6.21) can be simplified to

M 2ð Þ EI2

R

p2 �a2 � 1ð Þ2 c2

c3

� 1

� �1þ 3

c3

c2

� 4M 0ð ÞR

EI2

� �1�

M 0ð ÞR

EI2

� �

8 1�M 0ð ÞR

EI2

� ��a2

c2

c3

þ 1

� 2

2M 0ð ÞR

EI2

� 1� c3

c2

� � (6.22)

The ratio of c3 to c2 is 2=ð1þ �Þ, which only depends on the Poisson’s ratio, Eq. (6.22) can be written as Mð2Þ ðEI2=RÞf1ða; vÞ. Theratio of bending moment load, M, to the critical load, Mð0Þ, is

M

M 0ð Þ¼ 1þ a2

M 2ð ÞM 0ð Þ

1þ U1max

LS

� �2 p2 � � 1ð Þ �a2 � 1ð Þ2 7þ �1þ � �

4M 0ð ÞR

EI2

� �1�

M 0ð ÞR

EI2

� �

4M 0ð ÞR

EI2

�a2 1þ �ð Þ 1�M 0ð ÞR

EI2

� �þ 2

� 2 2M 0ð ÞR

EI2

� 3þ �1þ �

� � (6.23)

where the small ratio was defined as a ¼ U1max=LS.The maximum principal strain in the beam can be obtained as

emax ¼ awð Þmax1� �

4

j2 1ð Þ

þ 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� �ð Þ2

4j2

2 1ð Þ þ j23 1ð Þ

s24

35

(6.24)

The shortening ratio of the distance, cd, between the beam ends due to the bending moment is defined as

cd ¼jUjS¼LS

j

2R sinp�a2

¼ a2

2R

U2 2ð Þ þ U3 2ð Þcotp�a2

� �S¼LS

(6.25)

Fig. 4 Schematic illustration of boundary conditions ofcircular beam under bending moment load

The ratio of the maximum principal strain in beam toffiffiffiffifficd

p ðw=RÞ is

ce ¼emaxffiffiffiffiffi

cd

pR

w¼ f2 �a; vð Þ (6.26)

where the function, f2ð�a; vÞ, depends only on the shape and thePoisson’s ratio of beam. As shown in Fig. 5(a), the effect of Pois-son’s ratio on the ratio, ce, can be neglected, especially for the cur-vature with the nonlinear terms. But the effect of the normalizedangle, �a, on the ratio, ce, is significant as shown in Fig. 5(b),where the Poisson’s ratio is 0.42 for gold, which is the primarymaterial of the interconnector in the stretchable and flexible elec-tronics. The value gap between the ratio, ce, with and without thenonlinear terms in curvature would be larger than 100% for�a > 5=6. Figure 6 shows the normalized maximum principalstrain, emaxR=w, versus the shortening ratio of the ends distance,cd, for � ¼ 0:42, which indicates that the nonlinear terms in curva-tures should be considered for the strain of beam. There is 73%increase in the normalized maximum strain, emaxR=w, without thenonlinear terms of curvature for �a ¼ 2=3 and cd ¼ 0:3 from the



normalized maximum principal strain with the nonlinear terms,which indicates that the stretchability of circular beam is underes-timated when the nonlinear terms are neglected. Figure 6 alsoshows that the relation between the normalized strain, emaxR=w,and the shortening ratio, cd, depends on the normalized angle, �a,and the longer circular beams have the higher stretchability withthe same critical normalized strain. This model can be used toanalyze the stretchability of the serpentine bridge, which can besimplified as two semicircular beams [2]. The maximum strain inthe bridge fabricated with gold can be obtained as emax ¼ 0:9365

ðw=RÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiepre=ð1þ epreÞ

pand emax ¼ 2:264ðw=RÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiepre=ð1þ epreÞ

pfor the curvatures with and without the nonlinear terms, where theprestrain, epre, which is applied on the soft substrate, is related tothe shortening ratio, cd, by epre ¼ cd=ð1� cdÞ. For the design ofthe stretchability of the serpentine bridge, the nonlinear terms ofthe curvature should be considered in the analysis process of thepost-buckling behavior of the curved beam.

The finite-element method (FEM) of the commercial softwareABAQUS, where the shell element S4R is used since the width ofthe cross section, w, is much smaller than its thickness, t, isadopted to simulate the post-buckling behaviors of the curvedbeams under the bending moments. The distributions of the twistangle / of the circular beams for the shortening ratios, cd ¼ 0.1,0.2, and 0.3, are shown in Fig. 7, where the elastic modulus andPoisson’s ratio are 79.5 GPa and 0.42 for gold, and the length, LS,thickness, t, width, w, and the normal angle, �a, of the circular

beam are 900 mm, 30 mm, 1 mm, and 2/3, respectively. Figure 7shows that the theoretical results are consistent with the FEMresults.

6.2 The Lateral Buckling of Circular Beams UnderUniform Pressure. As shown in Fig. 8, the uniform pressure,�p2 ¼ �p2, which is along the thickness direction of cross section(i.e., �e2) during the deformation, is applied on the circular beam,the lateral buckling will happen because the thickness of the beamsection, t, is much larger than its width, w. The power orders ofdisplacements are similar to those in Sec. 6.1. The beam withlength, LS, which subtends the angle, a, is simply supported in theplane and out-of-plane, and the beam ends cannot rotate aroundthe centroid of beam, but can freely slide toward the arch center inthe plane, i.e.,



ðt � E2ÞjS¼0 ¼ ðt � E2ÞjS¼LS¼ 0

(6.27)


ðe2 � E1ÞjS¼0 ¼ ðe2 � E1ÞjS¼LS¼ 0


(6.28)

The deformation of the model is symmetrical about the midline(i.e., the dotted–dashed line in Fig. 8) to avoid the rigid move-ment. By substituting Eq. (3.2) into the above boundary condi-tions, the expanded formulas with respect to the perturbationparameter, a, of the boundary conditions (6.27) and (6.28) are



Xl¼n

l¼0

ðtiðlÞai2ðn�lÞÞjS¼0 ¼Xl¼n

l¼0

ðtiðlÞai2ðn�lÞÞjS¼LS¼ 0

(6.29)


a21ðnÞjS¼0 ¼ a21ðnÞjS¼LS¼ 0


(6.30)

The formulas and solving of the governing equations are similarto those in Sec. 6.1. Replacing k2, k3, F51ðSÞ; F52ðSÞ; and F6ðSÞin Sec. 6.1 with the following �k2, �k3, �F51ðSÞ; �F52ðSÞ; and �F6ðSÞ,which are

Fig. 5 The ratio, ce, with and without the nonlinear terms in curvature: (a) the ratio, ce,versus the Poisson’s ratio m for the normalized angle �a 5 1=4, 1=2, and 2=3. (b) The ratio,ce, versus the normalized angle �a for gold (i.e., m 5 0:42).

Fig. 6 The normalized maximum principal strain emaxR=w withand without the nonlinear terms in curvature versus the shorteningratio cd for m 5 0:42 and �a 5 1=4, 1=2, and 2=3



�k2

2 ¼1

R21þ

�p2 0ð ÞR3

EI2

!�k

2

3 ¼1

R21þ c2

c1

1þ c1ð Þ�p2 0ð ÞR

3

EI2

" #(6.31)

�F51 Sð Þ ¼ 1

R2c1

�2Rc3j3 1ð Þdj3 1ð Þ

dSþ R2 3c2 � 2c3ð Þ

dj2 1ð ÞdS

dj3 1ð ÞdSþ R2 2c2 � c3ð Þ

d2j2 1ð ÞdS2

j3 1ð Þ

þc2j2 1ð Þ 1þ�p2 0ð ÞR

3

EI2

c2

!j3 1ð Þ � R

dj2 1ð ÞdS

� �þ R2 c2 � c3ð Þj2 1ð Þ

d2j3 1ð ÞdS2

2666664

3777775

�F52 Sð Þ ¼ �j1 2ð Þ þk 2ð ÞR� 1

2R

dU1 1ð ÞdS

� �2

þw 1½ �1ð Þ

d2U1 1ð ÞdS2

� 1

2Rw 1½ �

1ð Þ

� 2

þdU1 1ð Þ

dS

� �2" #

�F6 Sð Þ ¼ 1

R2c2

j2 1ð Þ R2 c2 � c3ð Þj23 1ð Þ þ k 2ð Þ

h iþ Rj1 2ð Þ c1 � c2ð Þj2 1ð Þ þ R c3 � c1ð Þ

dj3 1ð ÞdS

�

�c3Rd

dSk 2ð Þj3 1ð Þ� �

þ R2c2

d

dSk 2ð Þ

dj2 1ð ÞdS

� �þ R2 �2c1 þ c3ð Þj3 1ð Þ

dj1 2ð ÞdS

8>>>><>>>>:

9>>>>=>>>>;

(6.32)

The critical load, �p2ð0Þ ¼ EI2=R3ð�a�2 � 1Þ, is obtained by the boundary conditions (6.30), which is consistent with the result of previousstudy [42]. When 1 c1 c2 � c3, the load increment can be simplified to

�p2 2ð Þ ¼EI2

R3f31 �; �að Þ þ f32 �; �að Þh i

(6.33)

where

f31 �; �að Þ ¼p2 1� �a2ð Þ2 �a4 17� �ð Þ 1þ �ð Þ þ 16 3þ 2�ð Þ � 2�a2 22þ 33� þ �2ð Þ

� �8�a2 �a2 � 4ð Þ 2þ �a2 1þ �ð Þ

� �2

f32 �; �að Þ ¼p2 1� �a2ð Þ2 8þ 4� þ �a2 1þ �ð Þ2

h i8 1� �a2ð Þ 1þ �ð Þ þ p�a �a2 � 4ð Þ 3þ �ð Þcot

p�a2

� 16�a2 �a2 � 4ð Þ 2þ �a2 1þ �ð Þ

� �2(6.34)

The function f32ð�; �aÞ is derived from the nonlinear terms of curvature.The ratio of the uniform pressure to the critical load is

�p2

�p2 0ð Þ¼ 1þ a2

�p2 2ð Þ�p2 0ð Þ

1þ U1max

LS

� �2 f31 �; �að Þ þ f32 �; �að Þh i

�a�2 � 1(6.35)

Fig. 7 The distributions of the twist angle of the circular beamfor the different shortening ratios, cd 5 0.1, 0.2, and 0.3

Fig. 8 Schematic illustration of boundary conditions of circu-lar beam under uniform pressure



Figure 9 shows the normalized load, �p2=�p2ð0Þ, versus the nor-malized maximum displacement, U1max=LS, which indicates thatthe nonlinear terms of curvature have significant impact on theanalysis of the post-buckling behaviors of circular beam. The uni-form pressure increases in the buckling process, but it decreasessince the nonlinear terms of curvature is neglected.

7 Conclusions and Discussion

A systematic method is established for post-buckling analysisof curved beams in this paper. The deformation variables of thecurved beam are up to the third power of generalized displace-ments due to the necessity of the fourth power of generalized dis-placements in the potential energy for the post-buckling analysis[37–39]. The currently prevailing post-buckling analyses, how-ever, are accurate only to the second power of generalized dis-placements in the potential energy since some authors assume alinear displacement–curvature relation. Although the effect of thenonlinear terms of curvature on in-plane post-buckling behaviorof planar beam is not so critical, it has significant impact on the

lateral post-buckling. When the bending moment is applied on cir-cular beam, the strain in beam is overvalued (�73%) due to theneglect of the nonlinear terms in curvature expression, which willlead to underestimate of the stretchability of circular beam. Thedifference of post-buckling behavior also stems from the neglectof the nonlinear terms for the lateral buckling under the uniformpressure. In summary, the nonlinear terms in curvature can not beneglected in the lateral buckling behavior of the curved beam.

Acknowledgment

J.W. and K.C.H. acknowledge support from the National BasicResearch Program of China (973 Program) Grant No.2015CB351900 and NSFC Grant (Nos. 11172146, 11320101001,and 11672149).

Appendix A

The second and third powers of generalized displacements incoefficients and curvatures are

ah2i11 ¼ �

1

2w 1½ �� 2

þ U021

�

ah2i12 ¼ �

1

2U01U02 þ

1

2U01U3K1 þ w 2½ �

ah2i13 ¼ U03U01 � w 1½ �U02 þ U01U2 þ w 1½ �U3

� K1

8>>>>>>><>>>>>>>:

(A1)

ah2i21 ¼ �

1

2U01U02 þ

1

2U01U3K1 � w 2½ �

ah2i22 ¼ �

1

2w 1½ �� 2

þ U02 � U3K1

� �2

�

ah2i23 ¼ w 1½ �U01 þ U03 þ U2K1

� �U02 � U3K1

� �

8>>>>>><>>>>>>:

(A2)

ah2i31 ¼ � U03 þ U2K1

� �U01

ah2i32 ¼ � U03 þ U2K1

� �U02 � U3K1

� �ah2i33 ¼ �

1

2U0

21 þ U02 � U3K1

� �2h i

8>>>>><>>>>>:

(A3)

ah3i11 ¼ �

1

2w 1½ �U01U02 � w 1½ �w 2½ � þ U0

21U03 þ U0

21U2 þ

1

2w 1½ �U01U3

� �K1

ah3i12 ¼

1

4w 1½ � U0

21 � U0

22

� þ U01U02U03 þ w 3½ �

þ U01 U02U2 � U03U3

� �þ 1

2w 1½ �U02U3

� K1 þ � 1

4w 1½ �U2

3 � U01U2U3

� �K2

1

ah3i13 ¼

1

2w 1½ �� 2

U01 þ w 1½ �U02U03 � w 2½ �U02 � U01 U023 �

1

2U0

21 þ U0

22

� �

þ U3w2½ � � w 1½ � U03U3 � U02U2

� �� U01 U02U3 þ 2U2U03

� �h iK1

þ �w 1½ �U2U3 þ1

2U01U2

3 � U01U22

� �K2

1

8>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>:

(A4)

Fig. 9 The ratio of load to critical load, �p2=�p2ð0Þ, with and with-out the nonlinear terms in curvature versus the normalized out-of-plane displacement, U1max=LS , for m 5 0:42 and �a 5 1=4, 1=2,and 2=3



ah3i21 ¼

1

4w 1½ � U0

21 � U0

22

� þ U01U02U03 � w 3½ � þ U01 U02U2 � U03U3

� �þ 1

2w 1½ �U02U3

� K1 þ � 1

4w 1½ �U2

3 � U01U2U3

� �K2

1

ah3i22 ¼

1

2w 1½ �U01U02 � w 1½ �w 2½ � þ U0

22U03 þ U2U0

22 �

1

2w 1½ �U01U3 � 2U02U03U3

� �K1

þ U03U23 � 2U02U2U3

� �K2

1 þ U2U23K3

1

ah3i23 ¼

1

2w 1½ �� 2

U02 � w 1½ �U01U03 þ w 2½ �U01 � U02 U023 �

1

2U0

21 þ U0

22

� �

þ �w 1½ �U01U2 �1

2U3 w 1½ �

� 2

þ U021 þ U0

22

� þ U3 U0

23 � U0

22

� � 2U2U02U03

� �K1

þ 2U2U3U03 þ3

2U02U2

3 � U02U22

� �K2

1 þ U22U3 �

1

2U3

3

� �K3

1

8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:

(A5)

ah3i31 ¼ U01 U023 �

1

2U0

21 þ U0

22

� � þ U01U02U3 þ 2U01U2U03� �

K1 þ U01U22 �

1

2U01U2

3

� �K2

1

ah3i32 ¼ U02 U023 �

1

2U0

21 þ U0

22

� � þ 1

2U3 U0

21 þ U0

22

� þ U3 U0

22 � U0

23

� þ 2U2U02U03

� K1

þ U02U22 � 2U2U3U03 �

3

2U02U2

3

� �K2

1 þ1

2U3

3 � U22U3

� �K3

1

ah3i33 ¼ U03 U021 þ U022

� þ U2 U021 þ U022

� � 2U02U03U3

h iK1 þ U03U2

3 � 2U02U2U3

� �K2

1 þ U2U23K3

1

8>>>>>>>>>>>><>>>>>>>>>>>>:

(A6)

jh2i1 ¼* w 1½ �U001 þ U03U02

� �0 þ U2U02 � U3U03� �

K01

þU2U002 � U3U003� �

þ U022 � U03

2� �

� 1

2w 1½ �� 2

þ U012

� � 2U2U3K01

8><>:

9>=>;K1 � U2U3ð Þ0K2

1

+

jh2i2 ¼w 1½ �U002 � U03U01

� �0 � w 1½ �U3 þ U01U2

� K01

� w 1½ �U03 þ w 2½ � þ U001 U2 þ3

2U01U02

� �K1 þ

1

2U01U3K2

1

2664

3775

jh2i3 ¼ 0

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

(A7)

jh3i1 ¼ w 2½ �U001 þ1

2w 1½ �� 2

U002 � w 1½ � U001 U03 þ U01U003� �

þ U0021

2U0

21 þ U0

22 � U0

23

� �þ 1

2U001 U01U02 � 2U003 U02U03

þ U3 U023 � U022

� � 1

2U3 w 1½ �

� 2

þ U021

� � U2 w 1½ �U01 þ 2U02U03

� � �K01

þ�w 1½ �w 2½ � � 2U2 U02U03

� �0 þ 1

2U01 U01U03 � U001 U3

� �þ 4U2U3U03 þ 2U02U2

3 � 2U02U22

� �K01

þU033 þ 2U3U03U003 �3

2U01U02w

1½ � � U001 U2w1½ � � 2U02U002 U3 � 3U0

22U03 �

1

2U03 w 1½ �� 2

2664

3775K1

þU002 U2

3 � U22

� �þ 2U2 U023 � U022

� þ 3U2

2U3 � U33

� �K01

þ4U02U03U3 þ 2U2U3U003 þ U021U2 þ1

2U01U3w

1½ �

2664

3775K2

1

þ U03 U22 � U2

3

� �þ 2U2U02U3

� �K3

1

jh3i2 ¼ w 2½ �U002 �1

2w 1½ �� 2

U001 � w 1½ � U002 U03 þ U02U003� �

� U0011

2U0

22 þ U0

21 � U0

23

� �� 1

2U002 U01U02 þ 2U003 U01U03

þ �w 1½ � U2U02 � U3U03� �

� w 2½ �U3 þ 2U01U2U03 þ1

2U01U02U3

� K01

þ2U01U2

2 �1

2U01U2

3 þ 2w 1½ �U2U3

� �K01 � w 1½ � 5

4U0

22 �

1

4U01

2 þ U2U002 � U3U003 � U023

� �

�w 2½ �U03 � w 3½ � þ U001 U02U3 þ 2U2U03� �

þ U01 2U2U003 þ1

2U002 U3

� �þ 7

2U01U02U03

26664

37775K1

þ U001 U22 �

1

2U2

3

� �� 3

2U3 U01U03 � w 1½ �U02

� þ w 1½ �U2U03 þ 3U01U2U02

� K2

1 þ � 1

4w 1½ �U2

3 � U01U2U3

� �K3

1

jh3i3 ¼ 0

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

(A8)

where ðÞ0 ¼ dðÞ=dS, and



w 2½ � ¼ð

1

2U01U002 � U001 U02� �

� U01U3K01 þ1

2U01U3K1

� �0 þ U01U2K21

� dS (A9)

w 3½ � ¼ð

� 1

2w 1½ � 0 w 1½ �

� 2

þ 1

4�w 1½ � U0

21 þ U0

22

� h i0þ U001 U02 � U01U002� �

U03

� �

þ 1

2w 1½ �U02U3 þ U01U3U03

� �K01

þ

1

2w 1½ �U02U3

� 0þ 1

2U01 U01

2 þ U022

� �� U001 U3U03 � U2U02

� �� U01U2U002

þU3 U01U2 �1

2w 1½ �U3

� �K01

2664

3775K1

� U01U2U3

� �0 þ U3

1

2w 1½ �U03 þ

1

4w 1½ � 0U3

� �� K2

1 � U01 U22 �

1

2U2

3

� �K3

1

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>;

dS (A10)

Appendix B

The expressions of kð2Þ, jið2Þ, and aijð2Þ are

k 2ð Þ ¼ U03 2ð Þ þ U2 2ð ÞK1 þ1

2U0

21 1ð Þ þ U02 1ð Þ � U3 1ð ÞK1

� �2h i

(B1)

j1 2ð Þ ¼

�U002 2ð Þ þ U3 2ð ÞK1

� �0 þ w 1½ �1ð ÞU

001 1ð Þ þ U03 1ð ÞU

02 1ð Þ

� �0 þ U2 1ð ÞU02 1ð Þ � U3 1ð ÞU

03 1ð Þ

� �K01

þU2 1ð ÞU

002 1ð Þ � U3 1ð ÞU

003 1ð Þ

� �þ U022 1ð Þ � U023 1ð Þ

� � 1

2w 1½ �

1ð Þ

� 2

þ U021 1ð Þ

� � 2U2 1ð ÞU3 1ð ÞK

01

8>><>>:

9>>=>>;K1 � U2 1ð ÞU3 1ð Þ

� �0K21

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

j2 2ð Þ ¼U00

1 2ð Þ � w 1½ �2ð ÞK1 þ w 1½ �

1ð ÞU00

2 1ð Þ � U03 1ð ÞU01 1ð Þ

� 0þ �w 1½ �

1ð ÞU3 1ð Þ � U01 1ð ÞU2 1ð Þ

� K01

þ �w 1½ �1ð ÞU

03 1ð Þ � w 2½ �

2ð Þ � U001 1ð ÞU2 1ð Þ �

3

2U01 1ð ÞU

02 1ð Þ

� �K1 þ

1

2U01 1ð ÞU3 1ð ÞK

21

2664

3775

j3 2ð Þ ¼ /0 2ð Þ

8>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>:

(B2)

a11 2ð Þ ¼ �1

2w 1½ �

1ð Þ

� 2

þ U021 1ð Þ

�

a12 2ð Þ ¼ w 1½ �2ð Þ �

1

2U01 1ð ÞU

02 1ð Þ þ

1

2U01 1ð ÞU3 1ð ÞK1 þ w 2½ �

2ð Þ

a13 2ð Þ ¼ �U01 2ð Þ þ U03 1ð ÞU01 1ð Þ � w 1½ �

1ð ÞU02 1ð Þ þ U0

1 1ð ÞU2 1ð Þ þ w 1½ �1ð ÞU3 1ð Þ

� K1

8>>>>>>>><>>>>>>>>:

(B3)

a21 2ð Þ ¼ �w 1½ �2ð Þ �

1

2U0

1 1ð ÞU02 1ð Þ þ

1

2U0

1 1ð ÞU3 1ð ÞK1 � w 2½ �2ð Þ

a22 2ð Þ ¼ �1

2w 1½ �

1ð Þ

� 2

þ U02 1ð Þ � U3 1ð ÞK1

� �2

�

a23 2ð Þ ¼ �U02 2ð Þ þ U3 2ð ÞK1 þ w 1½ �1ð ÞU

01 1ð Þ þ U03 1ð Þ þ U2 1ð ÞK1

� �U02 1ð Þ � U3 1ð ÞK1

� �

8>>>>>><>>>>>>:

(B4)

a31 2ð Þ ¼ U01 2ð Þ � U03 1ð Þ þ U2 1ð ÞK1

� �U0

1 1ð Þ

a32 2ð Þ ¼ U02 2ð Þ � U3 2ð ÞK1 � U03 1ð Þ þ U2 1ð ÞK1

� �U02 1ð Þ � U3 1ð ÞK1

� �a33 2ð Þ ¼ �

1

2U0

21 1ð Þ þ U02 1ð Þ � U3 1ð ÞK1

� �2h i

8>>>>><>>>>>:

(B5)

where w½1�ð2Þ ¼ /ð2Þ �Ð

K1U01ð2ÞdS

w 2½ �2ð Þ ¼

ð1

2U0

1 1ð ÞU00

2 1ð Þ � U001 1ð ÞU

02 1ð Þ

� � U0

1 1ð ÞU3 1ð ÞK01 þ

1

2U0

1 1ð ÞU3 1ð ÞK1

� 0þ U0

1 1ð ÞU2 1ð ÞK21

� dS.



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