Dynamic Snap-Through of a Laterally Loaded Arch Under Prescribed End Motion

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Dynamic snap-through of a laterally loaded arch

under prescribed end motion

Jian-San Lin, Jen-San Chen

*

Department of Mechanical Engineering, National Taiwan University, Taipei 10617, Taiwan Received 1 November 2002; received in revised form 5 March 2003

Abstract

Emphasis of this paper is placed on ﬁnding out whether dynamic snap-through will occur when the laterally loaded arch is under prescribed end motion with constant speed. The ﬁrst harmonic component q1sin n of the lateral load is

assumed to be the dominant one, while the eﬀects of higher components such as q2sin 2n are also discussed in detail. It is

found that dynamic snap-through may occur in either stretching or compressing process when the end speed of the loaded arch is in certain range. The dangerous zones for dynamic snap-through can be determined by comparing the energy barrier and the total energy gained by the arch when the end is moved with inﬁnitely large speed. Generally speaking, for a speciﬁed q1 it is easier for the arch under prescribed end motion to snap dynamically if q26¼ 0.

Keywords: Dynamic snap-through; Shallow arch; Prescribed end motion

1. Introduction

When a lateral load tending to flatten a sinusoidal arch is applied quasi-statically, the axial thrust in the arch will develop and grow due to the immovability of the ends. For a specified arch shape the arch may undergo snap-through buckling at certain critical load. Early classical investigation can be found in Timoshenko (1935), Fung and Kaplan (1952), Gjelsvik and Bonder (1962), Onat and Shu (1962), Franciosi et al. (1964), Schreyer and Masur (1966), Lee and Murphy (1968), and Simitses (1973). Experimental results have been reported by Roorda (1965). In the case when the lateral load is applied dynamically instead of in the quasi-static manner, the critical load will be different from the one predicted statically, see Hoff and Bruce, 1954; Humphreys, 1966; Lock, 1966; Hsu, 1967, 1968; Hsu et al., 1968; Patricio et al., 1998; Xu et al., 2002. A comprehensive review on the dynamic instability of shallow arches can be found in the book by Simitses (1990), which also includes other structures such as shallow spherical cap and thin cylindrical shell. Generally speaking, the methodologies used in estimating dynamic critical loads of elastic structures can be classified in two groups (Simitses, 1990). The first approach is to solve the equations of motion

International Journal of Solids and Structures 40 (2003) 4769–4787

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*

Corresponding author. Tel.: +886-2-2366-1734; fax: +886-2-2363-1755. E-mail address:jschen@ccms.ntu.edu.tw(J.-S. Chen).

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numerically to obtain the system response. The load parameter at which there exists a large change in the response is called critical, for example, see Budiansky and Roth (1962) on spherical cap, and Kistler and Waas (1998, 1999) on cylindrical panel. In Kistler and WaasÕ works, the effects of different in-plane boundary conditions on the limit points of the response are also considered. This direct approach requires large amount of calculation in a wide parameter range. The second approach is to study the total energy and the phase plane of the system. By this method sufficient conditions for dynamic stability and instability may be established, for example, see Hsu (1967, 1968). A combination of both approaches would prove useful in practical engineering applications.

In this paper we investigate a new elastic stability problem which involves a loaded arch under prescribed end motion. At time t¼ 0 we assume that one end of the loaded arch starts to move to a new position with constant speed, while the other end remains ﬁxed in space. We wish to ﬁnd out whether it is possible for the arch to be snapped to the other side dynamically and stay there if damping is present. This problem not only is new from the academic point of view, it may have practical application as well. Consider an arch structure designed and constructed to be in a stable equilibrium position. In the case when certain unex-pected disturbance (for instance, in an earthquake or landslide) occurs such that the distance between the two ends of the arch changes, it is important to predict whether the structure still holds or not.

We assume that the laterally loaded arch is in a stable equilibrium configuration before one of the two ends is moved as prescribed. The magnitude and direction of the lateral load remain unchanged when the end motion is in progress. We first investigate the quasi-static case when the end speed is negligible. For a specified arch rise parameter and a moving distance of the end, we will determine all possible equilibrium configurations and their stability properties. If for a specified moving distance there are more than one stable equilibrium configurations, it is possible for the arch to jump from one stable configuration to the other and stay there if some damping mechanism is provided. In general it is very difficult to determine the necessary and sufficient condition for dynamic snap-through to occur. However, it is possible to propose a sufficient condition against dynamic snap-through. Effects of various disturbances on the response of the arch under prescribed end motion are also studied in detail in this paper, which include (1) the non-ideal initial conditions at the instance when the end starts to move, and (2) the imperfect distribution of the lateral loads.

2. Equations of motion

Fig. 1 shows an elastic shallow arch with the two pinned ends being separated originally by a distance L. The initial shape of the unloaded arch is y0ðxÞ. The arch is subjected to lateral loading QðxÞ and stays in a

stable equilibrium conﬁguration. At time t¼ 0, the end at x ¼ L starts to move a distance d with constant speed c_{. d < 0 means that the arch is compressed. The equation of motion of the arch can be written as}

qAy;tt¼ EIðy y0Þ;xxxxþ p _y

;xx Q: ð1Þ

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The parameters E, q, A, and I are YoungÕs modulus, mass density, area, and area moment of inertia of the cross-section of the arch. p _{is the axial force,}

pðtÞ ¼AE L dðtÞ þ1 2 Z L 0 ðy2 ;x y 2 0;xÞ dx : ð2Þ

In writing Eqs. (1) and (2) we neglect the inertial eﬀect in the axial direction. The boundary conditions for y at x¼ 0 and L are

yð0Þ y0ð0Þ ¼ y;xxð0Þ y0;xxð0Þ ¼ yðLÞ y0ðLÞ ¼ y;xxðLÞ y0;xxðLÞ ¼ 0: ð3Þ

Eqs. (1) and (2) can be non-dimensionalized to the forms

u;ss¼ ðu u0Þ;nnnnþ pu;nn Q; ð4Þ p¼ e þ 1 2p Z p 0 ðu2 ;n u 2 0;nÞ dn; ð5Þ where u¼y r; u0¼ y0 r; n¼ px L ; s¼ p2_t L2 ffiffiffiffiffiffi EI Aq s ; p¼p _L2 p2_EI; e¼ Ld p2_r2; Q¼ Q_L4 p4_EIr: ð6Þ

r is the radius of gyration of the cross-section. p¼ 1 corresponds to the Euler buckling load for a per-fectly straight simply supported beam. The transverse loading QðnÞ is assumed to be distributed in the form

QðnÞ ¼X

1

j¼1

qjsin jn: ð7Þ

It is noted that q1is positive when this harmonic component points downward in Fig. 1. The initial shape of

the arch before the lateral load is applied is assumed to be in the form

u0¼ h sin n: ð8Þ

his the rise parameter of the arch.

It is assumed that the shape of the loaded arch after being stretched or compressed can be expanded as uðs; nÞ ¼ u0þ

X1 n¼1

anðsÞ sin nn: ð9Þ

After substituting Eqs. (7)–(9) into (4) and (5) we obtain the equations governing an,

€ a a1¼ a1 ðG þ eÞðh þ a1Þ q1; ð10Þ € a an¼ n4an n2ðG þ eÞan qn; n¼ 2; 3; . . . ð11Þ where G¼1 4 X1 k¼1 k2a2_kþh 2a1; ð12Þ eðsÞ ¼ cs: ð13Þ

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The dimensionless speed c is related to c _by c¼ L 3 p4_r3 c cl ; ð14Þ

where cl is the longitudinal wave speed of the arch. The overhead dots in Eqs. (10) and (11) represent

diﬀerentiation with respect to s. The axial force p in Eq. (5) can be calculated as

p¼ G þ e: ð15Þ

For the case without any initial disturbance, the initial conditions for Eqs. (10) and (11) are

anð0Þ ¼ a0n; n¼ 1; 2; 3; . . . ð16Þ

_a

anð0Þ ¼ 0; n¼ 1; 2; 3; . . . ð17Þ

a0

ncorresponds to the equilibrium conﬁguration of the arch under lateral load Q before any end motion.

3. Equilibrium conﬁgurations for Q = q1sin n

We ﬁrst consider the case when the lateral load is distributed in a manner such that q16¼ 0 and qi¼ 0 for

i6¼ 1. In the case when the speed c is small the acceleration terms in Eqs. (10) and (11) can be neglected. From this condition we can examine the equilibrium conﬁgurations for various values of e. Two types of solutions are possible, i.e., one-mode and two-mode solutions.

One-mode solution u¼ u0þ a1sin n:

a1satisﬁes the following cubic equation:

eða1þ hÞ ¼

a1

4 ða

2

1þ 3ha1þ 2h2þ 4Þ þ q1: ð18Þ

We ﬁrst deﬁne the parameter e1as

e1 eT ¼ 3₄ð2jh q1jÞ 2=3 ; ð19Þ where eT¼ h2 4 1: ð20Þ

After deﬁning e1we can make the following observations:

Case (1): q1< h

If e > e1, then there is only one equilibrium conﬁguration P0.

If e < e1, then there are three equilibrium conﬁgurations P0, P1þ, and P1,

where

a1ðP0Þ > h > a1ðP1þÞ > h ½2ðh q1Þ 1=3

>a1ðP1Þ:

Case (2): q1> h

If e > e1, then there is only one equilibrium conﬁguration P1.

If e < e1, then there are three equilibrium conﬁgurations P0, P1þ, and P1,

where

a1ðP0Þ > h ½2ðh q1Þ 1=3

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In the special case when q1¼ h and e > e1, then the P0solution in Case (1) approaches the P1solution in

Case (2) and the only equilibrium conﬁguration of the deformed arch is in the shape of a straight line. This can be readily veriﬁed as the lateral load q1¼ h always admits a root a1¼ h in Eq. (18).

Two-mode solution u¼ u0þ a1sin nþ ajsin jn :

For this case the solutions can be written explicitly, a1¼ q1 j2h j2₁ ; ð21Þ aj¼ 2 j ffiffiffiffiffiffiffiffiffiffiffiffi ej e p ; j6¼ 1; ð22Þ where ej eT¼ 1 j2 ðh q1Þ2 4ðj2_1Þ2: ð23Þ

These conﬁgurations are denoted by P1jþ and P1j, which exist only when

e < ej: ð24Þ

It is noted that a1ðP1jÞ is independent of e.

4. Stability properties

The method of determining the stability of the above equilibrium conﬁgurations has been laid out explicitly in Lin (2002), and the conclusions are summarized in the following.

One-mode solutions:

Case (1) q1< h: P0is always stable. P1þis always unstable. Ifjq1 hj 6 3

ffiffiffi 6 p

, then P

1 is stable if and only if

e < e1. Ifjq1 hj > 3

ffiffiffi 6 p

then P

1 is stable if and only if e < e2.

Case (2) q1> h: P1 is always stable. P þ

1 is always unstable. Ifjq1 hj 6 3

ffiffiffi 6 p

, then P0is stable if and only if

e < e1. Ifjq1 hj > 3

ffiffiffi 6 p

then P0is stable if and only if e < e2.

Two-mode solutions: P1jþ and P1j are always unstable.

Fig. 2shows the bifurcation set using q1 h and e eT as control parameters. The tip T of the e1 eT

curve is always at the origin. Fig. 2is divided into nine regions by the solid ej eT curves. The equilibrium

conﬁgurations in each region are listed in Table 1, in which the stable conﬁgurations are labeled with bold-type symbols.

It is noted that the e2 eTcurve touches e1 eT curve when q1 h ¼ 3

ffiffiffi 6 p

and e eT¼ ð3=2Þð1 j2Þ.

It is also noted that only in regions 3, 6, and 9 there exist two stable equilibrium positions. The hybrid curves corresponding to the boundary of these bi-stability regions are signiﬁed by the thick lines in Fig. 2. This boundary comprises parts of e2 eTand e1 eTcurves, and will be called e01 eTin the later sections.

For a speciﬁed rise parameter h we can draw a horizontal line for e¼ 0 and a vertical line for q1¼ 0,

whose intersection is denoted by a black dot in Fig. 2. These points signify the unloaded and unstretched arch for a particular h. Several diﬀerent hÕs are chosen in Fig. 2to demonstrate the quasi-static behavior of the arch. Point (a) atðq1 h; e eTÞ ¼ ð0; 1Þ is for h ¼ 0, an initially straight rod. For the conventional case J.-S. Lin, J.-S. Chen / International Journal of Solids and Structures 40 (2003) 4769–4787 4773

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when e¼ 0 (along the horizontal line passing through this point) there exists only P0 or P1 solution for

q1<0 or q1>0, respectively. On the other hand when q1¼ 0 (along the vertical line) and e is decreased to

4 < e < 1, there exist three solutions P0, P1þ, and P1. It is noted that when e¼ 1 the axial thrust p ¼ 1

corresponds to the ﬁrst buckling load. When e is decreased to the range 9 < e < 4, there exist ﬁve solutions, i.e., P0, P1þ, P1, P12þ, and P12. e¼ 4 corresponds to the second buckling load p ¼ 4.

Point (b) atð2; 0Þ is for h ¼ 2. The scenarios along the e ¼ 0 line and the q1¼ 0 line are similar to the

case at Point (a), except that the tip T is right on the e¼ 0 line. As a consequence for an arch with h < 2, there exists only one solution along the e¼ 0 line, and no snap-through is possible (Hoﬀ and Bruce, 1954). It is also noted that as e decreases along the q1¼ 0 line across the e1 eT curve, the number of possible

solutions increases from 1 to 3. On the e1 eT curve the solutions P1þ and P1 coincide and the degenerate

root can be found as a1¼ h ½2ðh q1Þ

1=3

: ð25Þ

Fig. 2. Bifurcation set with q1 h and e eTas control parameters. The small dots represent the parameters q1¼ 0 and e ¼ 0 for various h.

Table 1

Equilibrium conﬁgurations and stability of various regions in Fig. 2

Region Equilibrium conﬁgurations

1 P0 2 P1 3 P0, P1þ, P 1 4 P0, P1þ, P1 5 P0, P1þ, P 1 6 P0, P1þ, P 1, P12þ, P12 7 P0, P1þ, P1, P13þ, P13 8 P0; P1þ, P 1, P13þ, P13 9 P0, P1þ, P 1, P þ 12, P12, P þ 13, P13

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Point (c) is for h¼ 3, in which the tip T is above the e ¼ 0 line and to the right of the q1 ¼ 0 line. In this

case there exist q11and q12on the e¼ 0 line such that there are three solutions P0, P1þ, and P1 when q1is in

the range q11< q1< q12, where

q11¼ h ðh2_4Þ3=2 6pffiffiffi3 ; ð26Þ q12¼ h þ ðh2_4Þ3=2 6pffiffiffi3 : ð27Þ

When q1< q11the only equilibrium conﬁguration is P0. When q1> q12the only equilibrium conﬁguration is

P₁. If q1is increased from 0 quasi-statically, the arch will be snapped from P0to P1 when q1¼ q12. On the

other hand when q1is decreased from beyond q12down to q11, then the arch will be snapped from P1 to P0.

Point (d) for h¼ 4 is right on the left branch of the e1 eT curve. Also the e2 eT curve touches the

e¼ 0 line. Therefore, for the unloaded ðq1¼ 0Þ and unstretched ðe ¼ 0Þ arch multiple equilibrium positions

are possible only when h > 4. In addition, Pþ

12, and P12 exist along the e¼ 0 line only when h > 4.

Points (e), (f), and (g) are for h¼ 4:2,pffiffiffiffiffi22, and 6, and are above, right on, and below the horizontal line passing through the touching points of e1 eTand e2 eT curves, respectively. The e2 eTcurves intersect

the e¼ 0 line in each of these three cases at q21 and q22, where

q21¼ h 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2₁₆ p ; ð28Þ q22¼ h þ 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2₁₆ p : ð29Þ

For h¼ 4:2when the load increases along the e ¼ 0 line static snap-through will occur when q1reaches

q12. On the other hand when h¼ 6, static snap-through will occur when q1reaches q22. In the case when

h¼pffiffiffiffiffi22, q11¼ q21and q12¼ q22. Point (h) is for h¼ 3

ffiffiffi 6 p

, which is on the vertical line passing through the left touching point of the e1 eTand e2 eTcurves. All the black dots in Fig. 2are on a parabola, indicated

by the dashed line

q1 h ¼ 1 1₄ðe eTÞ 2

:

In Fig. 3 we plot the relation between a1and e for h¼ 6 (point (g) in Fig. 2). The loads q1Õs in Fig. 3(a),

(b), (c), and (d) are 1 3pffiffiffi6, 11 3pffiffiffi6, 11þ 3pffiffiffi6, and 1þ 3pffiffiffi6, respectively. These q1Õs satisfy the

con-ditions (a) q1< h, jq1 hj > 3 ffiffiffi 6 p , (b) q1< h, jq1 hj < 3 ffiffiffi 6 p , (c) q1> h, jq1 hj > 3 ffiffiffi 6 p , (d) q1> h, jq1 hj < 3 ffiffiffi 6 p

. Both e1and e2are positive in Fig. 3. There are ﬁve equilibrium conﬁgurations when e¼ 0,

among them either P0or P1 can be the initial position before any end motion takes place, as denoted by

small black dots. The solid and dashed lines represent the stable and unstable conﬁgurations, respectively. For the cases in Fig. 3(a) and (b), if the arch is in P0 position when e¼ 0 and is stretched or compressed

quasi-statically, the arch will always stay in P0position. On the other hand, if the arch is in P1position and

stretched quasi-statically, then the arch will snap to P0position either via a sub-critical pitchfork bifurcation

when e reaches e2as in Fig. 3(a), or via a saddle-node bifurcation at e1 as shown in Fig. 3(b). In Fig. 3(c)

and (d), if the arch is in P

1 position when e¼ 0 and is stretched or compressed quasi-statically, the arch will

always stay in P

1 position. On the other hand, if the arch is in P0 position and stretched quasi-statically,

then the arch will snap to P

1 position either via a sub-critical pitchfork bifurcation when e reaches e2 as

shown in Fig. 3(c), or via a saddle-node bifurcation at e1 as shown in Fig. 3(d).

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5. Dynamic snap-through criteria

In the case when the end speed is not negligible dynamic through may occur. For dynamic snap-through to occur there must exist at least two stable equilibrium configurations when the motion of the arch end stops. While it is in general difficult to determine the necessary and sufficient condition for dynamic snap-through to occur, we can establish the sufficient conditions against dynamic snap-through in terms of dimensionless strain energy U of the equilibrium configurations and the total energy H gained by the arch at the instant when the arch end stops, where

H¼ 2ðG þ eÞ2þX 1 n¼1 ½ _aa2_nþ n4 a2_n þ 2q1ða1þ hÞ; ð30Þ U¼ 2ðG þ eÞ2þX 1 n¼1 n4a2_nþ 2q1ða1þ hÞ: ð31Þ

The physical total energy H _{and strain energy U} _{are related to H and U by}

H¼p

4_EI2_H

4AL3 ; U

_¼p4EI2U

4AL3 : ð32Þ

The basic idea is that if the total energy gained by the arch during the prescribed end motion is smaller than the minimum energy barrier lying between the nearest stable equilibrium position and the distant stable one, then the arch has no chance to snap dynamically. The energy barrier can be proved to be the strain energy of either the unstable conﬁguration Pþ

1 or P12, depending on the parameters q1, h and e (Lin,

2002). If we assume that the arch is in position P0 before any end motion, then the suﬃcient conditions

against dynamic snap-through from P0 to P1 can be stated in the following: Fig. 3. Equilibrium conﬁgurations for h¼ 6 with various q1.

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Case (1) jq1 hj < 3

ffiffiffi 6 p

: If e2< e < e1, then the suﬃcient condition against snap-through is a1ðssÞ > a1ðP1þÞ

and HðssÞ < U ðP1þÞ, where ssis the time when the arch end stops. If e < e2, then the suﬃcient

con-dition against snap-through is a1ðssÞ > a1ðP12Þ and H ðssÞ < U ðP12Þ.

Case (2) jq1 hj > 3

ffiffiffi 6 p

: If e < e2then the suﬃcient condition against snap-through is a1ðssÞ > a1ðP12Þ and

HðssÞ < U ðP12Þ.

In the case when the arch is in position P1before any end motion, then the above statements are slightly

modiﬁed by changing a1ðssÞ > a1ðP12Þ and a1ðssÞ > a1ðP1þÞ to a1ðssÞ < a1ðP12Þ and a1ðssÞ < a1ðP1þÞ,

respec-tively. These two statements give the suﬃcient conditions against dynamic snap-through if the deﬂection a1ðssÞ and total energy H ðssÞ are known, which require direct integration of the equations of motion.

In-stead of calculating HðssÞ, we can estimate the upper bound of total energy gained by the arch during the

end motion. It can be proved that the total energy gained by the arch when the end is moved with inﬁnitely large speed is an upper bound of the total energy gained by the arch stretched with ﬁnite speed (Lin, 2002). The total energy corresponding to c! 1 is denoted by H1, and can be calculated as

H1¼ ða01Þ 2 þ 2 ða 0 1Þ 2 4 " þha 0 1 2 þ e #2 þ 2q1ðh þ a01Þ; ð33Þ where a0

1is the deﬂection of the arch under lateral load q1before any end motion. In the case when q1¼ 0,

then a0

1¼ 0 and H1 is reduced to 2e2. Finally we can establish a simpler, although more conservative,

suﬃcient condition against dynamic snap-through by replacing HðssÞ in the snap-through criteria

men-tioned above by H1. It can be shown that there exists a critical distance ecr, such that

H1ðecrÞ ¼ U ðP1þÞ; ð34Þ

whenjq1 hj < 3

ffiffiffi 6 p

and e26e < e1. In all other cases,

H₁ðecrÞ ¼ U ðP12Þ: ð35Þ

H₁ðecrÞ in Eqs. (34) and (35) represents the total energy gained by the arch when its end is moved a distance

ecr with inﬁnitely large speed. It is noted that we allow ecr to be either positive or negative.

In Fig. 4 we plot the ecrcurves with dashed lines on the q1–e plane. The q1¼ 0 line, e ¼ 0 line, and the e1

and e2curves are also presented for reference. For a speciﬁed q1if we move the arch end from e¼ 0 into the

cross-hatched area then there is a possibility of dynamic snap-through. The cross-hatched area may be called the ‘‘dangerous zone.’’ If the arch is in P0position when e¼ 0, then the lines in the cross-hatched area

are of positive slope. On the other hand if the arch is in P

1 position when e¼ 0, then the lines in the

cross-hatched area are of negative slope. In Fig. 4(a) for h¼ 2no dynamic snap-through is possible during stretching process because there is only one stable equilibrium position when e > 0 for h 6 2. However, dynamic snap-through is possible during the compressing process. In Fig. 4(b) for h¼ 4 the sharp corners of the two dangerous zones during stretching process are located atðq1; eÞ ¼ ð0; 0Þ and ð8; 0Þ. LetÕs take

q1¼ 6 as an example. If the arch is in P0position when e¼ 0, then there is a risk of dynamic snap-through

if the arch is stretched a distance in the range from 0.75 to 1.11, or compressed a distance over 5.31. On the other hand, if the arch is in P

1 position when e¼ 0, then the arch is moved to the dangerous zone when it is

compressed a distance over 8.96. There is no risk of snap-through from stretching in this case. In Fig. 4(c) for h¼pffiffiffiffiffi22the sharp corners of the dangerous zones during both stretching and compressing processes coincide with the two touching points of the e1and e2curves. In Fig. 4(d) for h¼ 3

ffiffiffi 6 p

the sharp corners of the dangerous zones during both stretching and compressing processes coincide and are on the e2-curves. In

addition, the other corners of the dangerous zones during stretching process coincide with the two touching points of the e1and e2curves. Fig. 4(d) indicates that for an unloaded arch in the P0position dynamic

snap-through during stretching is possible only when h > 3pffiffiffi6.

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6. Dynamic snap-through during compression

The above analysis predicts that the loaded arch can snap dynamically when the loaded arch is either stretched or compressed. While it is easier to visualize the snapping of a stretched arch, it is not obvious how a loaded arch snaps when it is compressed dynamically. In Fig. 4 we demonstrate that the arch may indeed undergo dynamic snap-through when it is compressed. The reason for this seemingly unreasonable result is that we consider the minimum energy barrier in establishing the suﬃcient condition against dynamic snap-through, which turns out to be the strain energy of the positions P

12if these positions exist

when the arch end stops. However, in the case when the arch is compressed with ideal initial conditions (16) and (17) with a0

n¼ 0 n 6¼ 1, then the only possible non-zero coordinate is a1. Therefore, the arch has no

chance to pass the saddle point P

12, which involves both a1 and a2. In order to reach the other stable

equilibrium position P

1, H1has to surpass the energy barrier at P1þ. It can be shown that H1< UðP1þÞ in

the compressing process. Therefore, an arch with ideal initial conditions will never undergo snap-through when it is compressed dynamically.

The suﬃcient conditions presented in Fig. 4 are conservative in the sense that it implicitly takes into account various minor imperfections, such as the non-ideal initial conditions or the imperfect initial shape. Due to these imperfections the coordinates other than a1, such as a2, may be aroused. It is then possible for

the arch to pass through the saddle point P

12. Therefore, it will be safer to stick to the true minimum energy

barrier UðP

12Þ in asserting the suﬃcient conditions.

To demonstrate how non-ideal initial conditions may cause dynamic snap-through when the arch is compressed, we consider the case when q1¼ 7, h ¼ 5, e ¼ 10, and c ¼ 40. The arch is in P0 position

before the prescribed end motion. Initial conditions are the same as (16) and (17), except that we change a2ð0Þ from 0 to 0.001. In the numerical simulation we add damping terms l _aa1and l _aa2in Eqs. (10) and (11),

where we choose l¼ 0:01. Fig. 5 shows the deformations a1and a2as functions of s. We observe that in the Fig. 4. The dangerous zones for (a) h¼ 2, (b) h ¼ 4, (c) h ¼pffiffiffiffiffi22, and (d) h¼ 3pffiffiffi6.

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early stage s < 2, a1 is mostly positive as we may expect and a2 remains small. However, after s > 2the

amplitude of a2-oscillation grows signiﬁcantly and a1shoots to the negative territory at s¼ 7 and remains

negative thereafter. The coordinates of equilibrium positions P0, P1 and P12 corresponding to e¼ 10 are

shown as dashed lines for reference. In Fig. 6 we use the thick line to trace the trajectory of the deformation history in the a1–a2space. Also shown is the strain energy contour and the locations of various equilibrium

positions corresponding to e¼ 10. The trajectory starts near the valley P0 and slides back and forth

several times along an almost horizontal line (recall the small a2 in the early stage). Since the system is

unable to climb to the hilltop at Pþ

1, it naturally swirls up the wall surrounding P0and ﬁnds an easier route Fig. 5. Response history during compression for q1¼ 7, h ¼ 5, e ¼ 10, c ¼ 40, and l ¼ 0:01. Initial conditions are the same as (16) and (17), except that a2ð0Þ is changed from 0 to 0.001.

Fig. 6. Trajectory of the response in Fig. 5 in the a1–a2space. The equilibrium positions and strain energy contour for e¼ 10 are also shown.

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passing the saddle point P

12and reaches the other valley P1. Consequently dynamic snap-through occurs

when the arch is under compression with non-ideal initial conditions.

7. Eﬀects of q2on equilibrium positions

7.1. Two-mode solution

In this section we consider the eﬀects of the second harmonic component q2 of the distributed load.

Apparently there is no more one-mode solution when both q1and q2are present. Instead, the a1and a2of a

two-mode solution satisfy the following equations:

a1þ ðG þ eÞðh þ a1Þ þ q1¼ 0; ð36Þ 16a2þ 4ðG þ eÞa2þ q2¼ 0; ð37Þ where G¼1 4ða 2 1þ 4a 2 2Þ þ h 2a1: ð38Þ

If the notation a1ðP12Þ is used to represent the two-mode solution when q2¼ 0, which is given in Eq. (21),

then after eliminating ðG þ eÞ in Eqs. (36) and (37) a2 and a1can be related by

a2¼

ða1þ hÞq2

12½a1 a1ðP12Þ

: ð39Þ

After substituting Eq. (39) into Eq. (36), one can derive the equation for a1

36½a1 a1ðP12Þ 2 ½a3 1þ 3a 2 1hþ 2a1ð2 þ 2e þ h2Þ þ 4ðhe þ q1Þ þ ða1þ hÞ 3 q2 2¼ 0: ð40Þ

The ﬁrst bracket represents the original two-mode solution and the second bracket represents the original one-mode solution when q2¼ 0. Eqs. (39) and (40) can be used to study the eﬀects of q2on the equilibrium

configurations. As expected the sign of q2has no effect on the root locus of a1. There are a total of five poles

and ﬁve zeros in Eq. (40), among them a1¼ h is a triple zero.

Case (1) q1< h:

For e > e1there are two possible scenarios for the root loci of a1and a2, as shown in Fig. 7(a) and (b). In

each scenario there is only one real equilibrium P0when q2¼ 0. In Fig. 7(a) as q2 increases from zero to

inﬁnity, a1(P0) approaches the zero a1¼ h while the corresponding a2approaches1 along the real axis.

The complex conjugate poles a1ðP1Þ approach inﬁnity while the corresponding a2break away from the real

axis at the origin and approach inﬁnity. When q2 ¼ 0 a1ðP12Þ are on the real axis but the corresponding a2

are complex conjugate pairs. As q2increases from zero a1ðP12Þ break away from the real axis and approach

the zero a1¼ h, while the corresponding a2remain complex and approach inﬁnity. The situation in Fig.

7(b) is similar to that in Fig. 7(a) except that in Fig. 7(b) the poles a1ðP12Þ approach inﬁnity and a1ðP1Þ

approach the zero a1¼ h. Therefore, in either Fig. 7(a) or (b) the number of real equilibrium position

remains to be one. In addition, q2tends to suppress the deformed conﬁguration ðh þ a1Þ sin n contributed

from q1. In the limiting case when q2! 1 the arch is deformed to the form a2sin 2n.

There are also two possible scenarios for the root loci of a1 and a2for e1> e > e2, in which there are

three equilibrium positions P0, P1þ, and P1 when q2¼ 0 as shown in Fig. 7(c) and (d). Careful observation

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case when e < e2, in which there are ﬁve equilibrium positions when q2¼ 0. As q2 increases from 0, the

number of equilibrium positions reduces from ﬁve to three and ﬁnally to one as q2 increases.

Case (2) q1> h:

The root loci in this case are similar to those in Fig. 7. The main diﬀerence between Case (2) and Case (1) is the exchange of the roles played by P0and P1. The conclusion remains the same that q2tends to reduce

the number of equilibrium positions to one.

The eﬀects of q2on the root loci discussed above can also be demonstrated in the ai–e curves as shown in

Fig. 8, in which we choose h¼ 6, q1¼ 1 3

ffiffiffi 6 p

, and q2¼ 10. The a1-curves for q2¼ 0 have been presented

in Fig. 3(a) and are replotted in Fig. 8 as thin lines for reference. It is seen that the original sub-critical pitchfork bifurcation at e¼ e2 is destroyed and two separate branches are created. The two new

saddle-node bifurcation points are at e¼ e0

1 and e02. It is noted that e01 is deﬁned in such a way that two stable

equilibrium positions coexist when e < e0

1. It is reminded that the e01 for q2¼ 0 corresponds to the thick

hybrid curve in Fig. 2. The new a1-curve for P0solution is slightly lower than and is indistinguishable from

the original one with the chosen parameters. As in Fig. 3 the solid and dashed lines represent stable and unstable solutions, respectively.

Case (3) q1¼ h:

This is a limiting case of both Cases (1) and (2), but there are some interesting phenomena worth mentioning separately. First of all Eqs. (39) and (40) for the root loci fail in this case. Instead, we rewrite Eq. (36) to the form

ða1þ hÞðG þ e þ 1Þ ¼ 0: ð41Þ

Fig. 7. Root loci for a1and a2as q2increases from 0 to1 when q1< h. (a, b) For e > e1, (c, d) for e1> e > e2and (e) for e < e2. J.-S. Lin, J.-S. Chen / International Journal of Solids and Structures 40 (2003) 4769–4787 4781

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For solution

a1¼ h; ð42Þ

the corresponding a2 satisﬁes the equation

4a2ða22þ e e2Þ þ q2¼ 0: ð43Þ

On the other hand, for solution Gþ e ¼ 1 the corresponding a2 satisﬁes the equation

a2¼

q2

12: ð44Þ

After substituting Eq. (44) back to Gþ e ¼ 1 the corresponding a1 are

a1¼ h h2 4 q 2 2 144 þ e þ 1 1 2 : ð45Þ

Eqs. (42)–(45) can be used to plot the root loci for a1and a2in this case. Alternatively, one can also use

Fig. 7(a), (c), and (e) and let q1approach h to explain the root behaviors in this limiting case.

Fig. 9(a) shows the case for e > e1. When q2 ¼ 0 the three a1Õs for P0 and P12 coalesce with the triple

zero a1¼ h in Fig. 7(a) and the three a2Õs for P0and P1 are at the origin. Therefore there exists only one

real equilibrium at ða1;a2Þ ¼ (h; 0Þ. As q2increases from zero, the three poles at a1¼ h do not move,

while the three poles at a2¼ 0 approach 1 along the real axis together. Consequently, the number of Fig. 8. Equilibrium conﬁgurations for h¼ 6 and q1¼ 1 3

ffiffiffi 6 p

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equilibrium position remains to be one. Fig. 9(b) and (c) are for the cases e1> e > e2, and e < e2,

res-pectively.

Fig. 10 shows the ai–e curves when q1¼ h ¼ 6. The thin lines and the thick lines represent the solutions

for q2¼ 0 and q2¼ 10, respectively. For q2¼ 0 the arch undergoes super-critical pitchfork bifurcation at Fig. 9. Root loci for a1and a2as q2increases from 0 to1 when q1¼ h. (a) For e > e1, (b) for e1> e > e2and (c) for e < e2.

Fig. 10. Equilibrium conﬁgurations for q1¼ h ¼ 6. The thin lines are for q2¼ 0 and the thick lines are for q2¼ 10. J.-S. Lin, J.-S. Chen / International Journal of Solids and Structures 40 (2003) 4769–4787 4783

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both e¼ e1 and e2. The numbers of equilibrium positions are 1, 3, and 5 when e > e1, e1> e > e2, and

e < e2, respectively. When q26¼ 0, the bifurcation points are shifted leftward to e01and e02. It is noted that the

arch undergoes a super-critical pitchfork bifurcation at e¼ e0

1. This is diﬀerent from the saddle-node

bifurcation in Fig. 8 when q16¼ h.

7.2. Three-mode solution

When both q1and q2are non-zero three-mode solutions are possible, with a1being given by Eq. (21) and

a2and ajbeing a2¼ q2 4ðj2_4Þ; ð46Þ aj¼ 2 j ffiffiffiffiffiffiffiffiffiffiffiffi e0 j e q ; j >2; ð47Þ where e0_j¼ ej q2 2 16ðj2_4Þ2: ð48Þ

In the case when q2 approaches zero, the three-mode solutions are reduced to the original two-mode

solutions with ajin Eq. (47) approaching the one in Eq. (22). All these three-mode solutions are unstable.

In addition, the existence of these three-mode solutions will not aﬀect the stability of the two-mode solutions as shown in Figs. 8 and 10. The eﬀects of other qj with j > 2can be studied in a similar

manner.

8. Eﬀects of q2on dynamic stability

In order to study the eﬀects of q2on the dangerous zone in Fig. 4, we have to consider three factors. First

of all for all other parameters being unchanged q2 tends to reduce the number of stable equilibrium

po-sitions from 2to 1, as explained in the last section. Secondly, when q2 is non-zero the strain energy of

positions Pþ

12and P12 are no longer the same,

UðP 12Þ ¼ 2 ha1 2 þa 2 1 4 þ a 2 2þ e 2 þ a2 1þ 16a 2 2þ 2q1ða1þ hÞ þ 2q2a2; ð49Þ

a1and a2in Eq. (49) are coordinates of the equilibrium positions P12 of the laterally loaded arch under end

displacement e, and are dependent upon q2. Further calculation shows that

oUðP 12Þ

oq2

¼ 2a2ðP12Þ: ð50Þ

It is noted that a2ðP12þÞ > 0 and a2ðP12Þ < 0 for q2>0. Therefore, UðP12þÞ increases and U ðP12Þ decreases

for q2>0. Without loss of generality we assume that q2>0 in the following discussion. Consequently,

the energy barrier becomes UðP

12Þ, and it decreases when q2 increases. Thirdly, the total energy H1

be-comes H1¼ ða01Þ 2 þ 16ða0 2Þ 2 þ 2 ha 0 1 2 " þða 0 1Þ 2 4 þ ða 0 2Þ 2 þ e #2 þ 2q1ðh þ a01Þ þ 2q2a02: ð51Þ

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It is noted that a0 1and a

0

2represent the equilibrium positions before any end motion and are dependent upon

q1 and q2. Further calculation shows that

oH1 oq2 ¼ 2a0 2þ 4e q1 h ða0 1þ hÞ 2 oa0 1 oq2 : ð52Þ

Eqs. (50) and (52) can be used to estimate the eﬀect of q2on ecr. For simplicity, we consider the variation of

H1 U ðP12Þ near q2¼ 0. We can show that a20¼ 0 and oa01=oq2¼ 0 when q2¼ 0. Therefore,

oðH1 U ðP12ÞÞ oq2 q2¼0 ¼ 2a2ðP12Þ > 0: ð53Þ

In other words, q2 tends to increase the diﬀerence between H1 over the energy barrier. Therefore, the

existence of non-zero q2 makes it easier for the arch to snap dynamically and the absolute value of the

critical distance ecr would be smaller.

In Fig. 11 we plot the dangerous zone for h¼ 3pffiffiffi6. The thin lines are for q2¼ 0 and have been plotted in

Fig. 4(d). The thick lines represent the boundary of the dangerous zone when q2¼ 10. To demonstrate the

movement of the dangerous zones due to q2 we label the corners of the dangerous zones for q2¼ 0 and

q2¼ 10 with unprimed and primed letters, respectively. It is seen that the absolute value of ecris smaller for

both stretching and compressing processes when q2¼ 10. In Fig. 12we show the history of a1for an arch

under stretching with h¼ 3pffiffiffi6, q1¼ 10, c ¼ 40, and e ¼ 4. The arch is in P0position before the prescribed

end motion. The damping constant l is chosen to be 0.005. The initial conditions are the same as Eqs. (16) and (17). This loaded arch is represented in Fig. 11 by a black dot, which is in the dangerous zone for q2¼ 10 and should be safe for q2¼ 0. These predictions are veriﬁed in Fig. 12, in which the solid lines and

the dashed lines are for q2¼ 10 and q2¼ 0, respectively. The a1for equilibrium positions P0, P1, and P12for

q2¼ 10 and q2¼ 0 are also plotted as dashed lines for reference.

Fig. 11. Dangerous zone for h¼ 3pffiffiffi6. The thin lines and the thick lines are for q2¼ 0 and q2¼ 10, respectively. The small dot is in safe zone when q2¼ 0, but is in dangerous zone when q2¼ 10.

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9. Conclusions

In this paper we investigate the eﬀects of lateral loading on the dynamic behaviors of a shallow arch under prescribed end motion. The ﬁrst harmonic component q1sin n of the lateral load is assumed to be the

dominant one, while the eﬀects of higher components such as q2sin 2n are also discussed in detail. First of

all when q2¼ 0 some conclusions can be summarized in the following:

(1) The laterally loaded arch may undergo snap-through buckling when the end is stretched quasi-stati-cally. On the other hand, no snap-through is possible when the arch is compressed quasi-statiquasi-stati-cally. (2) When the end speed of the loaded arch is not negligible, dynamic snap-through may occur in either

stretching or compressing process. The dangerous zones for dynamic snap-through can be determined by comparing the energy barrier and the total energy gained by the arch when the end is moved with inﬁnitely large speed.

(3) An arch with ideal initial condition will never undergo dynamic snap-through when it is compressed. On the other hand dynamic snap-through can occur during compressing process when the arch is dis-turbed at the instant when the end starts to move.

In the case when both q1and q2are non-zero, then some more conclusions can be made in the following:

(4) The equilibrium solutions involve either two or three harmonic modes.

(5) In the case when q16¼ h the original sub-critical pitchfork bifurcation for solutions P12in the bifurcation

diagram is destroyed, and new saddle-node bifurcation points are created. The new bifurcation points e0 j

are smaller than the original ejwith q2¼ 0.

Fig. 12. Response history during stretching for h¼ 3pffiffiffi6, q1¼ 10, c ¼ 40, e ¼ 4, and l ¼ 0:005. The solid and dashed lines are for q2¼ 10 and 0, respectively.

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(6) In the special case when q1¼ h the original super-critical pitchfork bifurcation in the bifurcation

dia-gram is destroyed, and new saddle-node and super-critical pitchfork bifurcation points are created. (7) The absolute value of the critical distance ecr is smaller. That means in order to snap the arch to the

other side dynamically, a smaller stretching or compressing distance will do the job if q26¼ 0.

Acknowledgement

The results presented here were obtained in the course of research supported by a grant from the National Science Council of the Republic of China.

References

Budiansky, B., Roth, R.S., 1962. Axisymmetric dynamic buckling of clamped shallow spherical shells. Collected Papers on Instability of Shell Structures. NASA TN D-1510.

Franciosi, V., Augusti, G., Sparacio, R., 1964. Collapse of arches under repeated loading. ASCE Journal of Structure Division 90, 165–201.

Fung, Y.C., Kaplan, A., 1952. Buckling of low arches or curved beams of small curvature. NACA Technical Note 2840.

Gjelsvik, A., Bonder, S.R., 1962. The energy criterion and snap buckling of arches. ASCE Journal of Engineering Mechanics Division 88, 87–134.

Hoﬀ, N.J., Bruce, V.G., 1954. Dynamic analysis of the buckling of laterally loaded ﬂat arches. Journal of Mathematics and Physics 32, 276–288.

Hsu, C.S., 1967. The eﬀects of various parameters on the dynamic stability of a shallow arch. ASME Journal of Applied Mechanics 34, 349–358.

Hsu, C.S., 1968. Stability of shallow arches against snap-through under timewise step loads. ASME Journal of Applied Mechanics 35, 31–39.

Hsu, C.S., Kuo, C.T., Lee, S.S., 1968. On the ﬁnal states of shallow arches on elastic foundations subjected to dynamical loads. ASME Journal of Applied Mechanics 35, 713–723.

Humphreys, J.S., 1966. On dynamic snap buckling of shallow arches. AIAA Journal 4, 878–886.

Kistler, L.S., Waas, A.M., 1998. Experiment and analysis on the response of curved laminated composite panels subjected to low velocity impact. International Journal of Impact Engineering 21, 711–736.

Kistler, L.S., Waas, A.M., 1999. On the response of curved laminated panels subjected to transverse impact loads. International Journal of Solids and Structures 36, 1311–1327.

Lee, H.N., Murphy, L.M., 1968. Inelastic buckling of shallow arches. ASCE Journal of Engineering Mechanics Division 94, 225–239. Lin, J.-S., 2002, Dynamic stability of a shallow arch under prescribed end motion. Master Thesis, Department of Mechanical

Engineering, National Taiwan University, Taipei, Taiwan.

Lock, M.H., 1966. The snapping of a shallow sinusoidal arch under a step pressure load. AIAA Journal 4, 1249–1256.

Onat, E.T., Shu, L.S., 1962. Finite deformation of a rigid perfectly plastic arch. ASME Journal of Applied Mechanics 29, 549–553. Patricio, P., Adda-Bedia, M., Amar, M.B., 1998. An elastica problem: instabilities of an elastic arch. Physica D 124, 285–295. Roorda, J., 1965. Stability of structures with small imperfections. ASCE Journal of Engineering Mechanics Division 91, 87–106. Schreyer, H.L., Masur, E.F., 1966. Buckling of shallow arches. ASCE Journal of Engineering Mechanics Division 92, 1–19. Simitses, G.J., 1973. Snapping of low pinned arches on an elastic foundation. ASME Journal of Applied Mechanics 40, 741–744. Simitses, G.J., 1990. Dynamic Stability of Suddenly Loaded Structures. Springer-Verlag, New York.

Timoshenko, S.P., 1935. Buckling of ﬂat curved bars and slightly curved plates. ASME Journal of Applied Mechanics 2, 17–20. Xu, J.-X., Huang, H., Zhang, P.-Z., Zhou, J.-Q., 2002. Dynamic stability of shallow arch with elastic supports––application in the

dynamic stability analysis of inner winding of transformer during short circuit. International Journal of Non-Linear Mechanics 37, 909–920.