A general condition for the existence of unconnected

equilibria for symmetric arches

Yang Zhou, Ilinca Stanciulescu

∗

Rice University, Department of Civil and Environmental Engineering, Houston, TX, 77005, U.S.A

Abstract

This paper presents a semi-analytical study of unconnected equilibrium states for symmetric

curved beams. Using the Fourier series approximation, a general condition for the existence

of unconnected equilibria for symmetric shallow arches is derived for the ﬁrst time. With

this derived condition, we can directly determine whether or not a shallow arch with speciﬁc

initial conﬁguration and external load has remote unconnected equilibria. These unconnected

equilibria can not be obtained in experiments or nonlinear ﬁnite element simulations without

performing a proper perturbation ﬁrst. The derived general condition is then applied to

curved beams with diﬀerent initial shapes and external loads. It is found that initially

symmetric parabolic arches under a uniformly distributed vertical force can have multiple

groups of unconnected equilibria, depending on the initial rise of the structure. However,

small symmetric geometric deviations are required for parabolic arches under a central point

load, and half-sine arches under a central point load or a uniformly distributed load to have

unconnected equilibria. The validity of the analytical derivations of the nonlinear equilibrium

solutions and the general condition for the existence of unconnected equilibria are veriﬁed by

nonlinear ﬁnite element methods.

Keywords: Unconnected equilibrium states, A general condition, Symmetric curved beams,

Post-buckling responses

1. Introduction

Curved beams have been studied extensively due to their rich nonlinear structural be-

havior and broad applications in aerospace, civil and mechanical engineering. When these

∗

Corresponding author. Tel.: +1713 348 4704; fax: +1713 348 5268.

Email address: ilinca.s@rice.edu (Ilinca Stanciulescu)

Preprint submitted to Elsevier November 18, 2017

arches are loaded transversely, they can jump between coexisting equilibrium conﬁgurations,

phenomenon accompanied by rapid curvature reversals. This phenomenon is often referred

to as snap-through buckling and it greatly exacerbates fatigue failure.

Motivated by the goal of avoiding snap-through, a great portion of studies in the early

literature focused on calculating the buckling loads of shallow arches with diﬀerent geome-

try, boundary and loading conditions. One of the earliest work was included in [1], where

Timoshenko used the ﬁrst symmetric mode to obtain the solutions for a half-sine arch under

a half-sine load. This study provided some guidance for analyzing snap-through but failed to

consider possible bifurcated branches that involve higher order modes. Following this work,

Fung and Kaplan [2] used few sine terms to obtain the stability boundaries of sine arches

with several diﬀerent types of transverse loads. Instead of using Fourier series, Schreyer and

Masur [3] directly solved the diﬀerential equations analytically and made a special eﬀort to

diﬀerentiate symmetric and asymmetric snap-through. These pioneering studies have been

extended by many researchers to examine the inﬂuence of elastic boundary constraints on

critical equilibrium states [4–9], to characterize the sensitivity of buckling loads to geometric

and load imperfections [10–17], to derive exact solutions for critical loads without truncat-

ing the Fourier series [18, 19], to incorporate thermal eﬀects in nonlinear buckling analyses

[20–23], and to study the eﬀects of pre-buckling deformations, initial shapes and loading on

the nonlinear stability responses [23–29]. In these investigations, the snap-through buckling

loads were obtained but few eﬀorts were made to characterize the post-buckling responses.

Despite inducing fatigue, snap-through has recently been utilized in developing vibratory

energy harvesters [30, 31], switches [32, 33], sensors [34] and non-volatile memories [35]. In

these applications, the curved beams are designed to undergo snap-through and to work in

the post-buckling region. Therefore, only determining snap-through buckling loads is not

suﬃcient and it is also necessary to gain insights into the post-snap responses. In fact,

recent studies have continued to reveal new and interesting post-snap responses. Pi and

Bradford [36–38] found that curved beams with unequal boundary conditions may have four

limit points and looping equilibrium states in the post-buckling responses. Moghaddasie and

Stanciluescu [39] identiﬁed that more pairs of critical points and more complex post-snap

behaviors can exist for shallow arches in thermal environments. Similar phenomena were also

observed by Hung and Chen [40] for buckled beams on elastic foundations. In addition to the

2

identiﬁcation of connected complex looping equilibria, Harvey and Virgin [41] demonstrated

that remote equilibria exist for curved beams under an oﬀ-center point load in physical

experiments by manually perturbing the structure to a remote conﬁguration. The current

authors Zhou et al. [42] showed that shallow arches with asymmetric geometric imperfections

can have remote unconnected “hidden” equilibria, which are diﬃcult to obtain numerically if

a prior knowledge of these “hidden” equilibria is not available. It was also identiﬁed in [42]

that asymmetric geometric imperfections make the bifurcated paths disappear and equilibria

split around these secondary paths.

In the current paper, we ﬁnd that initially symmetric arches may also have unconnected

“hidden” equilibria, which is not as initially expected. Unlike arches with asymmetric geo-

metric imperfections, it is in general diﬃcult to tell when symmetric arches have unconnected

equilibria. Here, a general condition for the existence of unconnected equilibria is derived

for shallow symmetric arches. We apply this general condition on perfect parabolic, half-sine

arches and these arches with small symmetric geometric deviations when they are subjected

to uniformly distributed load and point load in the middle and verify the validity of the gen-

eral condition by comparing the equilibrium states of the structures with parameters below

and above the general condition. It is found that the existence of unconnected equilibria is

very sensitive to the arches’ initial conﬁgurations and to the type of external load that they

are subjected to. This work can also help to guide the exploration of the “hidden” equilibria

in numerical and physical experiments.

Remote unconnected equilibria may not aﬀect the static buckling loads of the structures.

However, they can inﬂuence the dynamic responses of the structures. Under dynamic pertur-

bations, these structures may jump to remote unconnected conﬁgurations. Pi and Bradford

[38] identiﬁed that it is crucial to identify all coexisting equilibria of the structures to cor-

rectly determine the dynamic snap-through buckling loads. More recently, Wiebe and Virgin

[43] demonstrated that the presence and location of unstable equilibria can also have a great

impact on the long-term recurrent behavior of a dynamic system. We conjecture that the

identiﬁcation of remote unconnected equilibria may provide useful information for the design

of devices that take advantage of dynamic snap-through and work under post-snap regions

such as energy harvesters, switches, sensors and non-volatile memories [30–35].

3

2. Governing equations

Using the Fourier series and following [42], the dimensionless algebraic equilibrium equa-

tions of shallow arches under transverse loads (Fig. 1) can be written as

(α

n

− β

n

)n

4

− pn

2

α

n

+ q

n

= 0, n = 1, 2, ..., k, (1)

u(ξ)

u

ξ

π

0

q

(ξ)

Figure 1: A shallow arch under a transverse load.

Here, α

n

represent the mode coeﬃcients of the deformed conﬁgurations u(ξ) =

k

P

n=1

α

n

sin(nξ),

β

n

are the mode coeﬃcients of the initial conﬁguration u

0

(ξ) =

k

P

n=1

β

n

sin(nξ), q

n

denote

the mode coeﬃcients of the non-dimensional external load q and can be calculated from

q

n

= (2/π)

R

π

0

q sin (nξ) dξ, k is the number of Fourier Sine terms included, and p is the

dimensionless axial force that can be calculated from

p =

k

X

i=1

i

2

(β

2

i

− α

2

i

)

4

. (2)

A brief description of the procedure involved in deriving Eq. (1) is included in the Appendix

and more details can be found in [42].

2.1. Equilibrium states

Let the external load q(ξ) be expressed as q(ξ) = λq

0

(ξ), where q

0

(ξ) is a reference function

that can be chosen as needed and λ is the loading parameter. It can then be found that the

load coeﬃcients q

n

have a linear relationship with the loading parameter λ for an arbitrary

4

transverse load. The linear relationship is q

n

= γ

n

λ, and γ

n

will be referred as the reference

loading coeﬃcients that can be calculated from

γ

n

=

2

π

Z

π

0

q

0

(ξ) sin(nξ)dξ (3)

Substituting the load coeﬃcients q

n

= γ

n

λ into Eq. (1), the non-dimensional algebraic

equilibrium equation for shallow arches under an arbitrary vertical load can be obtained

(α

n

− β

n

)n

4

− pn

2

α

n

+ λγ

n

= 0, n = 1, 2, ..., k, (4)

The deformation mode coeﬃcients α

n

can then be calculated as

α

n

=

n

4

β

n

− λγ

n

n

2

(n

2

− p)

, n = 1, 2, ..., k (5)

With these deformation mode coeﬃcients α

n

and Eq. (2), we can ﬁnd that the loading

parameter λ satisﬁes the following quadratic equation

Aλ

2

+ Bλ + C + 4p = 0 (6)

For a given arch shape and loading case, A, B and C are functions of the axial load p and

are deﬁned as follows:

A(p) =

k

P

n=1

γ

2

n

n

2

(n

2

− p)

2

B(p) =

k

P

n=1

−2n

2

γ

n

β

n

(n

2

− p)

2

C(p) = 4p +

k

P

n=1

n

6

β

2

n

(n

2

− p)

2

− n

2

β

2

n

(7)

Substituting the mode coeﬃcients α

n

into the deformed conﬁguration u(ξ), the vertical

displacement ﬁeld ∆u(ξ) can be obtained as

∆u = − (u (ξ) − u

0

(ξ))

=

k

X

n=1

β

n

sin(nξ) −

k

X

n=1

n

2

β

n

(n

2

− p)

sin(nξ) −

k

X

n=1

γ

n

λ

n

2

(n

2

− p)

sin(nξ) (8)

Setting ξ = π/2, the displacement at the midspan can then be derived as

∆u

mid

= D(p)λ + F (p) (9)

5