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ENOC-2005, Eindhoven, Netherlands, 7-12 August 2005

DROP-SHOCK STABILITY OF A THIN SHALLOW ARCH;CONSIDERING THE EFFECTS OF VARIATIONS IN SHAPE

Niels J. Mallon, Rob H.B. Fey, Kouchi Zhang and Henk NijmeijerDepartment of Mechanical Engineering

Eindhoven University of Technology

PO Box 513, 5600 MB Eindhoven, The Netherlands

[email protected], [email protected], [email protected], [email protected]

Abstract

Thin-walled structures can be sensitive to buckling:that is loss of stability under loading. In order to derive

design guides for such structures under dynamic load-

ing, a group of thin shallow arches subjected to a half-

sine drop-shock pulse is examined. All arches have

the same principle dimensions such as width, height

and cross-sectional area but differ in shape. Using

numerical means and a multi-degree-of-freedom semi-

analytical model, both a quasi-static and a non-linear

transient dynamical analysis are performed. The in-

fluence of various parameters, such as pulse duration,

damping and, especially, the arch shape is illustrated.

Moreover, results are validated with finite element re-

sults. The main results are firstly that the critical drop-shock level can be significantly increased by optimiz-

ing the arch shape and secondly, a small geometric im-

perfection decreases the critical drop shock level in a

much less fatal manner under dynamic loading as pre-

dicted from the quasi-static analysis.

Key words

Dynamic Stability, Buckling, Thin-Walled Structures.

1 IntroductionThin-walled structures possess a favorable stiffness-

to-mass ratio and are encountered in a wide variety

of applications, such as aircraft fuselages, civil engi-

neering structures and in micro-electro-mechanical

systems (MEMS). If such a thin-walled structure is

initially curved and is loaded above some critical mag-

nitude, the structure may buckle so that its curvature

suddenly reverses. This behaviour, also known as

snap-through buckling, is often undesirable and may

lead to failure of the structure. Therefore, stability

of thin-walled structures can not be neglected during

the design of the structure. The analysis of structures

liable to buckling under static loading is a wellestablished topic in engineering science. However,

often thin-walled structures are subjected not only

to a static load but also to a distinct dynamic load,

such as drop/impact loading, step loading or periodic

loading. The resistance of structures liable to buckling,

to withstand time-dependent loading is often addressed

as the dynamic stability of these structures. The corre-

sponding failure mode is often addressed as dynamic

buckling. Generally, dynamic buckling is related

to a large increase in the response resulting from a

small increase in some load parameter (Budiansky and

Hutchinson, 1964). However, still no rigorous criterion

exist for identifying dynamic buckling. In the past,

many studies have been already performed concerning

the dynamic stability of thin-walled structures. Design

strategies for such structures under loading with adistinct dynamic nature are, however, still lacking. The

research described in this paper is intended as a first

step in deriving such design strategies and deals with

thin shallow arches under drop-shock loading.

An early paper on dynamic buckling of shallow

arches subjected to shock loading is (Hoff and

Bruce, 1954). In this paper, the dynamic stability

problem of the arch is investigated following an energy

based approach (Hsu, 1966; Simitses, 1990). With

one of the first numerical studies on dynamic bucklingof shallow arches (Lock, 1966), two mechanisms for

the initiation of dynamic snap-through buckling are

revealed. These mechanisms are referred to as direct

dynamic snap-through buckling, i.e. snapping occur-

ring on the initially induced oscillations, and finite

time dynamic snap-through buckling, i.e. snapping

occurring after many oscillations. Both mechanisms

have also been found experimentally for a spherical

shell under step-pressure load (Lock, 1968). Direct

snap-through buckling of a clamped shallow arch

under impulsive loading following from detonating

sheet explosives is reported in (Humphreys, 1966).

The effect of damping on dynamic buckling loads forarches is addressed in (Lock, 1966; Hegemier and



Tzung, 1969; Johnson, 1980). All these studies reveal

that the presence of a little damping, as present in all

real-life structures, increases the dynamic buckling

loads significantly compared to the undamped case,

especially when the critical loads are not determined

by direct dynamic snap-through buckling. The effect

of various parameters on the dynamic stability of

the arch under impulsive loading is examined in

(Hsu, 1967). Some of the considered parameters are

the amplitudes of higher harmonics in the arch shape.

However, the amplitudes of the higher harmonics are

considered here as imperfections and thus only small

amplitudes are considered and positive side-effects

are not addressed. The effects of shape variation on

the static buckling of arches subjected to sinusoidal

distributed and constant distributed load are examined

in (Fung and Kaplan, 1952) and for arches subjected

to a concentrated point load at the center in (Fung and

Kaplan, 1952; Aubert and Rousselet, 1998; Bruns et

al., 2002). Moreover, the dynamic stability problem

of the arch shows a correspondence with the dynamicstability problem of buckled beams, see for example

(Noijen et al., 2005).

In this paper, a group of (very) flexible thin shallow

arches subjected to a short pulse-load in the form

of a drop-shock, is examined. All arches have the

same principle dimensions like width, height and

cross-sectional area. The main goal of the research

is to study the drop-shock stability of the arch for a

wide range of shapes, such that the influence of theseshape-variations on critical drop-shock loads can be

determined. Hereto, the arch-shape is parameterized

with a single shape-factor, such that the arch-shape

can be varied continuously. The considered shapes

are, therefore, not restricted to specific shapes like a

sinusoidal or circular curve. In addition, a geometric

imperfection in the form of an asymmetry in the

arch shape, is taken into account. The drop-shock

loading of the arch is modelled by a half-sine pulse in

transversal acceleration.

The outline for this paper is as follows. The next sec-

tion deals with the derivation of the equations of mo-

tion. In section 3, the quasi-static response to accelera-

tion loading is discussed, and the influence of the arch

shape on these responses is illustrated. Furthermore,

results are compared with FEM results. Dynamic buck-

ling of the arch is discussed in section 4. The influence

of the arch-shape, small geometric imperfections, the

level of damping and the shock-pulse duration on the

critical drop-shock magnitudes are examined and the

results are compared with the results from the quasi-

static analysis. Finally, in section 5 conclusions andrecommendations are presented.

y

y, w

d

y(t) y(t)

w

w0h

L

x

x, v

(a)

(b)

Figure 1. Arch geometry (a), pinned-pinned arch with transversal

end-point motion (b).

2 Equations of Motion

In this section the equations of motion for the arch are

derived. The arch has a thickness d, an initial height h

at the center and a width L. The initial (undeformed)shape of the arch is indicated with w0(x), the shape

after (elastic) deformation with w(x) and the axial dis-

placement with v(x), see figure 1-a. All geometrical

and material properties are considered to be constant

over the arch length and are fixed to the values as shown

in table 1. The internal normal force N and moment M in the arch are defined by N = EAε and M = EI κ,

where the kinematics are described by

ε = v′ +1

2

(w′)2 − (w′

0)2

κ =−

(w′′

−w′′0

) ,(1)

in which the prime denotes differentiation with respect

to x. The adopted kinematic model (1), often ad-

dressed as the curved beam model, is valid for slender

beams with small initial curvature and moderate dis-

placements and rotations and is beneficial for shape-

variation studies, since shape-variations can easily be

incorporated in the initial shape function w0. In or-

der to model the drop-shock loading via a prescribed

transversal motion, the arch is considered to be pinned-

pinned to a movable frame (see figure 1-b). The kine-

matic and natural boundary conditions for this load-

case read as

v(0, t) = v(L, t) = 0

w(0, t) = w(L, t) = w0(0, t)

M (0, t) = M (L, t) = 0

. (2)

The equations of motion are derived following the

variation principle of Hamilton, i.e. minimizing the in-

tegral

t2t1

(T − V + W nc) dt, (3)



Property Value Unit

A 2.056 · 10−5 [m2]

EI 0.232 [Nm2]

h 38.4 · 10−3 [m]

L 0.8315 [m]

d 0.803 · 10−3 [m]

ρ 7850 [kg/m2]

Table 1. Parameter values.

with respect to the fields v and w. First the expres-

sions for the the potential energy V , the work done by

non-conservative forces W nc, and kinetic energy T are

discussed. The potential energy of the arch takes the

form

V (v, w) = L0

1

2N ε +

1

2M κ

dx. (4)

Structural viscous damping of the arch in the form of

F d = −bw, leads to the work expression

W nc(w) =

L0

F dwdx. (5)

Neglecting rotary and in-plane (longitudinal) inertia

terms, the kinetic energy of the arch equals

T (w) =1

2ρA

L0

w2dx. (6)

Following Hamilton’s principle, the equations of mo-

tion for the arch can be derived to be

N ′ = 0, (7a)

ρAw + bw + M ′′ − (Nw′)′ = 0. (7b)

Using the fact N ′ = 0 (see (7a)) and v(0) = v(l), the

potential energy V may be expressed in terms of w and

its spatial derivatives solely

V (w) =N 2L

2EA+

L0

1

2Mκ + ρAgw

dx, (8)

where

N = EA2L

L0

(w′)2 − (w′0)2 dx. (9)

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 1−1

−0.5

0

0.5

1

x/Lx/L

w 0 / h

e

[ m ]

a = 0

a > 0

a < 0

shape variation imperfection

Figure 2. Shape variation and imperfection shape

2.1 Initial shape

In order to be able to study the effect of shape-

variations, a family of initial shapes are examined in

the analyses. All shapes of the family are symmetric

with respect to x = L/2, have a undeformed height of

w0(L/2) = h and are identified with a single shape-

variation parameter value a. Moreover, an imperfec-

tion in the form of a first harmonic asymmetry is in-

corporated, scaled by the imperfection parameter e. In

order to trigger not only the first harmonic asymme-

try sin(2πx/L), but a wide range of harmonic asym-

metries, a polynomial function is chosen to describe

the first harmonic asymmetrical shape. The complete

parametrization of the arch shape, including the (pre-

scribed) transversal movement y(t) of the end-points

of the arch, reads as

w0(x, t) = (h + a)sinπx

L

+ a sin

3πx

L

initial shape

+ e 36L3√

3x(x− L/2)(x− L)

imperfection shape

+y(t). (10)

The initial shape and imperfection shape are illustrated

in figure 2. The imperfection is considered to be very

small, i.e. e/d ≪ 1 with d the thickness of the arch.

2.2 Discretization

With (8), all energy expressions only depend on the

shape of the deformed arch w and its derivatives. In or-

der to be able to approximate the static and dynamic be-haviour of the arch according to the PDE (7), the field

w is discretized as

w(x, t) = w0(x, t) + f (x, t), (11)

where f (x, t) is a finite series of sine functions with

time varying amplitudes

f (x, t) =ni=1

Qi(t)sin

iπx

L

, (12)

satisfying the boundary conditions (2) a priori. Thecorresponding discrete set of equations of motion are



determined by following a Rayleigh-Ritz procedure,

i.e. substituting (11) with (12) for f , into the energy

and work expressions (4)-(6) and, subsequently, min-

imize (3) with respect to the generalized coordinates

Q = [Q1, Q2,..,Qn]T . In the sequel, a n-dof model

arch incorporates the first n harmonic modes, unless

stated otherwise. For illustration, the equations corre-

sponding to the 6-dof approximation are given

M Q+ C Q+ K (Q) = −By(t), (13)

where M = ρAL2

I , C = b2L

I with I the identity ma-

trix, B = 2ρALπ

1 0 1

30 1

50T

and

K (Q) =

π2

2LN (Q)(a + h + Q1) + EIπ4

2L3 Q1

2

LπN (Q)(18e

√3 + π3Q2) + 8EIπ4

L3 Q2

9π2

2LN (Q)(Q3 + a) + 81EIπ4

2L3 Q3

2

LπN (Q)(9e

√3 + 4π3Q4) + 128EIπ4

L3 Q4

25π

2

2L N (Q)Q5 +625

EIπ

4

2L3 Q5

6

πLN (Q)

2e√

3 + 3π3Q6

+ 648EIπ4

L3 Q6

,

where

N (Q) =EAπ2

4L2[Q1(2a + 2h + Q1)...

+18aQ3 +144

√3

π3e(Q2 + 1

2Q4 + 1

3Q6)...

+4Q2

2+ 9Q2

3+ 16Q4

4+ 25Q2

5+ 36Q2

6].

With the adopted discretization (12), Q = 0 alwaysrepresents the (undeformed) initial shape. As can be

noted, coupling of the individual modes is only attained

via the non-linear stiffness terms. Moreover, the asym-

metrical modes Q2, Q4,... are not excited directly by

the loading and are only triggered if e/d = 0 (assum-

ing the initial condition equals Q = 0¯

).

3 Static Buckling

For comparison with the dynamic analysis, buckling

of the arch under a time-invariant acceleration y(t) =A is investigated. For this analysis, the evolution of

static equilibrium points of (13), described by

K (Q) = −BA, (14)

are studied for a quasi-statically varying load A.

The equilibrium path or ’load-path’ is computed

using a pseudo-arc-length continuation scheme

(Kuznetsov, 1995). Stability of the equilibrium states

is assessed by evaluating the eigenvalues of the local

linear stiffness matrix dK (Q)/dQ.

Static buckling corresponds to loss of stability of

an equilibrium state at some critical point. Gener-ally, there are two types of buckling (Thompson and

−1 −0.75 −0.5 −0.25 0 0.25−40

−20

0

20

40

60

80

100

120

−0.02 0 0.02 0.04 0.060

10

20

30

40

50

60

stableunstable

W mid

(enlargement)

LP A [ m / s 2 ]

snap-through

Figure 3. Initial load-path for 3-sym-dof model of arch (symmetri-

cal modes only) with e/d = 0 and a/h = 0.

−0.1 − 0.05 0 0.050

10

20

30

40

50

60

−0.1 − 0.05 0 0.050

10

20

30

40

50

60

A [ m / s 2 ]

A [ m / s 2 ]

W midW mid

Perfect Imperfect

LP

B

e/d = 0.1

e/d = 0.5

e/d = 1

Figure 4. Initial load-path for 6-dof arch model with a/h = 0 for

e/d = 0 (left) and various e/d = 0 (right).

Hunt, 1973). Firstly, at a critical state two or more

load-paths may coincide, i.e. the initial load-path bi-

furcates. This type of buckling is often addressed as bi-

furcation buckling and the corresponding buckling load

as bifurcation load. Secondly, if the slope of the initial

load-path varies, the load-path can lose stability at a

limit-point, i.e. the load-path reaches a maximum. This

type of buckling is addressed as limit-point buckling.

At a limit-point there is an absence of local equilib-

rium states for load values greater than the limit-point

load. Consequently, under an incrementally increasing

load, the structure must jump to another (far) point on

the load-path, a phenomenon known as snap-through

buckling.

In order to plot a load-path for a multi-dof model such

as (13), some scalar measure for the deformation must

be chosen. Here, the following measure is adopted

W mid =w(L/2)−w0(L/2)

δh, (15)



where δh is the vertical distance between the unloaded

upward equilibrium position and the unloaded down-

ward equilibrium position measured at the mid-point

(δh = 2h for a/h = 0).

First, the quasi-static behaviour of a perfect arch

(e/d = 0) is examined. For this case it is known

the asymmetrical modes Q2

, Q4

,... are not triggered(see section 2.2). Consequently, for the analysis of a

perfect arch, the asymmetrical modes are redundant

and may be removed from (13). The initial load-path

for the perfect sinusoidal arch (a/h = 0,e/d = 0),

approximated using the first 3 symmetric modes

Q1, Q3 and Q5 (3-sym-dof), is shown in figure

3. Clearly, the slope of the obtained load-path for

quasi-static increasing load A varies and exhibits a

limit-point. If the load would be further increased at

the indicated limit-point (LP), the arch would have to

snap-through to a downward configuration to attain

equilibrium again. For the theoretical case where the

arch is purely symmetric, the limit-point load ALP in-dicates the loss of stability of the upward configuration.

In applications, however, the arch and its loading will

never be purely symmetric. The effect of such an asym-

metry, here chosen as an asymmetrical geometrical im-

perfection, can be studied by examining the quasi-static

response for an arch with e/d = 0. With e/d = 0,

also the asymmetrical modes are triggered (see section

2.2) and must be incorporated in the model (13). The

initial load-path of the arch with a/h = e/d = 0, us-

ing the first 6 modes (both symmetrical and asymmetri-

cal, 6-dof), is depicted in figure 4 (left). By comparingthis load-path with the initial load-path for the similar

case using 3 symmetrical modes (figure 3), it can be

noted that if that arch is allowed to deform asymmet-

rically, a bifurcation point (B) appears at a load which

is significantly lower than the earlier found limit-point

load. The secondary load-path which bifurcates from

the initial load-path at point (B), corresponds to first

harmonic asymmetric arch shapes. The importance of

this bifurcation point (B), becomes clear when arches

with an imperfection (e/d = 0) are considered, see

figure 4 (right). Clearly, the depicted load-paths for

e/d= 0 have a limit-point which tends for e/d

→0

to the earlier found bifurcation point (B) for e/d = 0.

For e/d = 0, however, no asymmetries are triggered

and the arch will still snap-through at the limit-point

load ALP . Consequently, the limit-point load ALP and

the bifurcation load AB as computed for the perfect

arch can be considered as the static buckling loads for

the (theoretical) case e/d = 0 and the (practical) case

0 < e/d ≪ 1, respectively.

For validation, the results for the semi-analytical

model with 6-dof are compared in figure 5 with FEM

results. Both the FE model for the perfect arch and for

the arch with the small imperfection consist of twenty3-node Timoshenko beam elements known as element

−0.075 −0.05 −0.025 0 0.0250

10

20

30

40

50

60

A [ m / s 2 ]

W mid

stableunstableFEM

e/d = 0.1

e/d = 0

Figure 5. Comparison initial load-path FE model and 6-dof arch

model with a/h = 0.

2-dof 3-dof 6-dof 12-dof

AB [m/s2] 35.323 35.219 35.219 35.219

ALP [m/s2] 2.3·104 58.913 58.908 58.907

Table 2. Bifurcation-point (AB) and limit-point (ALP ) of an arch

with a/h = 0,e/d = 0 using 2-dof, 3-dof, 6-dof and 12-dof.

type 45 (MSC.Marc, 2003). In all FEM analyses kine-

matic relations are used which are valid for large dis-

placements and rotations. The influence of the retained

number of dof’s in the semi-analytical model on the

bifurcation load AB and the limit-point load ALP are

shown in table 2. The results for the semi-analytical

model do not change dramatically if more than 6 dofs

are used and show a good correspondence with the ac-

curate though numerically expensive FEM computa-

tions.

Since the arch will never be purely symmetric in prac-

tice, the bifurcation load AB dominates the static sta-bility behaviour. Unfortunately, the bifurcation load

AB can hardly be influenced by varying the arch shape,

as illustrated in figure 6. The bifurcation load can be in-

creased by 6% (with respect to the arch with a/h = 0)

by setting the shape-factor to a/h = 0.097. The limit-

point load ALP shows a distinct maximum for a/h =0.0384 at which the corresponding snap-through mode

switches between the w -shape and the m-shape. More-

over, the limit-point load is much more sensitive to the

arch-shape and can be increased by 40% by setting the

shape-factor to a/h = 0.0384. However, following the

results of the quasi-static analysis, this critical load has

less practical importance.



−0.05 0 0.05 0.1 0.1530

40

50

60

70

80

90

a/h

[ m / s 2 ]

AB

ALP

Figure 6. Influence arch shape on AB and ALP for e/d = 0.

4 Dynamic Buckling

In this section dynamic buckling of the arch under

drop-shock loading is examined. The half-sine pulse

in acceleration, characterized by the pulse duration T pand the maximum acceleration A, is used to model the

loading during an actual drop-shock

y(t) =

A sin

πtT p

if 0 ≤ t ≤ T p

0 if t > T p. (16)

After the arch is briefly forced by the shock-load pulse,

the arch is no longer subjected to external forces. Con-

sequently, if the effect of the shock-load can be related

to certain initial conditions, the dynamic stability prob-

lem may be approximated by an initial value problem.

Considering Q (0) = 0¯

, the velocities just after the

shock-pulse can be determined by assuming that dur-

ing the (short) shock-loading no deformations occur,

i.e. Q (T p) = 0¯

, by neglecting the effect of damp-

ing during the interval 0 ≤ t ≤ T p and by using the

impulse-momentum theorem

T p0

Fdt = MQ (T p) , (17)

where F = By(t) is the column of forces and M themass-matrix. For the half sine shock pulse (16) this

implies

Q (T p) = M−1B2

πλ, (18)

with only a single load parameter λ = AT p [m/s].

Under the considered assumptions, the drop-shock is

imparted instantaneously into the system as kinetic

energy only. Relating this amount of kinetic energy to

the level of potential energy at some unstable equilib-

rium point, a lower bound for the dynamic bucklingload can be derived, see for example (Hsu, 1966),

(Simitses, 1990) and (Kounadis et al., 2004). This

lower bound is based on the consideration that the total

energy surface of the structure consists of multiple

wells, peaks, saddles and ridges. Each well corre-

sponds to a stable equilibrium state. Peaks and saddles

correspond to unstable equilibrium states. Dynamic

buckling is related to the escape from the initial well

in the total energy surface. In order to make such an

escape possible, the imparted energy in the structure

by the shock-load must overcome a critical amount

to get over the ridge of the well at some point. The

lowest pass along a ridge between two wells is defined

by a saddle. Therefore, the escape at a point very

close to a saddle equilibrium point requires the least

amount of energy to be imparted by the shock-load.

However, for multi-dof systems it is not trivial which

saddle equilibrium point defines the lowest pass from

the initial well. Furthermore, the established lower

bound for the dynamic buckling load by the energy

approach can be very conservative (Ovenshire and

McIvor, 1971; Johnson and McIvor, 1978).

Since the energy approach is less feasible for multi-

dof discretized models of the arch and does not allow

the incorporation of the effect of damping in the analy-

sis, here numerical tools are pursued to analyze dy-

namic buckling of the arch. The critical drop-shock

loads are determined with the aid of the equations of

motion approach (Simitses, 1990). The equations of

motion are (numerically) solved for various values of

some load parameter. The load at which there exist

a sudden jump in the response is called critical load

(Budiansky and Hutchinson, 1964). In order to evalu-ate the response in time for the multi-dof model, some

scalar measure must be chosen. Here, the following

measure is adopted

W mid = max0≤t≤T

|W mid(t)|, (19)

with W mid defined by (15). Also with the numerical

approach, the problem of the arch under drop-shock

loading can be studied as an autonomous initial value

problem with Q (T p) = 0¯

and Q (T p) as defined by

(18). This approach is beneficial, since it reducesthe number of load-parameters from two (A, T p) to

one (λ). However, as stated before, it is based on

the assumption that the energy of the shock-load is

imparted in the structure as kinetic energy only. Since

it is not trivial for which pulse durations (T p) this

assumption is valid, in this paper the response of the

arch under drop-shock loading is simulated by solving

the non-autonomous set of ODEs (13) incorporating

(16) with zero initial conditions. The parameter Ais selected as load parameter to be varied. However,

various (fixed) pulse durations T p will be considered.

For the numerical integration of (13), an integration

routine based on an 8th order Runge-Kutta schemewith automatic step-size control is used. For all results



mode shape

f 1,2 6-dof [Hz] 10.88 24.51

f 1,2 FEM [Hz] 10.71 24.21

Table 3. First 2 linearized eigen-frequencies (f 1,2) and modes for

arch with a/h = 0 and e/d = 0 for 6-dof and FE model.

b [Ns/m] 1 2 4

[Hz] ξ [-] ξ [-] ξ [-]

10.88 0.0453 0.0905 0.1810

24.51 0.0201 0.0402 0.0805

Table 4. Damping ratios corresponding to the first 2 eigen-

frequencies for several values of b.

a relative tolerance of T OL = 1 · 10−8 is used.

However, to assure that the shock-pulse energy is

imparted correctly to the arch, the maximum step-size

while 0 ≤ t ≤ T p is set to T p/50.

According to (Johnson and McIvor, 1978), results for

direct dynamic snap-through buckling, i.e. snapping

occurring during the initial induced oscillations, are

not significantly altered if the time-span of integration

is extended from 0 ≤ t ≤ 3T 1 to 0 ≤ t ≤ 5T 1, where

T 1 is the period corresponding to the lowest linearizedvibrational eigen-mode f 1. Therefore, a time-span

of three periods of the lowest eigen-frequency is an

appropriate time-span to examine direct dynamic snap-

through buckling. For the arch under consideration,

the period corresponding to the lowest undamped

eigen-frequency for a/h = 0 and e/d = 0 appears to

be T 1 ≈ 0.1 [s], see table 3. For the considered range

of the shape-factor a, the first eigen-frequency of the

arch stays nearly constant. Therefore, a time-span of

T = 3T 1 ≈ 0.3 [s] is used in all computations, unless

stated otherwise. Note that the eigen-frequencies

are determined by linearizing (13) around the un-

loaded upward configuration (Q = 0). Moreover,

the eigen-frequencies of the semi-analytical model

are in good correspondence with FE results, see table 3.

In practice the arch will always exhibit some level of

damping. Therefore, in all analyses a small to moderate

amount of damping is taken into account. The damp-

ing ratios of the first two linearized vibrational eigen-

modes, corresponding to the eigen-modes depicted in

table 3, are shown in table 4.

First, results for an arch without an imperfection

(e/d = 0) are discussed. In figure 7, the influenceof the arch shape on the dynamic buckling load is il-

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 80.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

W m i d

AT p [m/s]

a/h = 0

a/h = 0.04

a/h = 0.08

Figure 7. Influence shape factor for T p = 10 [ms], b = 2[Ns/m], e/d = 0.

5 5.5 6 6.5 7 7.5 80.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

W m i d

AT p [m/s]

b = 1 [Ns/m]

b = 2 [Ns/m]b = 4 [Ns/m]

Figure 8. Influence damping for T p = 10 [ms], e/d = 0,

a/h = 0.04.

lustrated for a fixed pulse duration of T p = 10 [ms]

and b = 2 [Ns/m]. The dynamic buckling loads can

clearly be distinguished by the sudden jumps in the de-

picted graphs. Similar as was found for the quasi-static

buckling load of the arch without an imperfection (see

section 3), the shape of the arch also has a distinct influ-

ence on the dynamic buckling load for the perfect arch

(the dynamic buckling load for a/h = 0.04 is approxi-mately 50% higher than the dynamic buckling load for

a/h = 0).

Before the solution may escape from the initial well,

energy will be dissipated due the presence of damp-

ing. To get over a ridge of the initial well, the en-

ergy dissipation caused by damping before the occur-

rence of escape must be accounted for. An increasing

amount of damping will, therefore, result in an increas-

ing dynamic buckling load as illustrated in figure 8 for

a/h = 0.04 and T p = 10 [ms]. For the case with the

lowest level of damping (b = 1 [Ns/m]), the boundary

between the region where no dynamic buckling occursand the region where dynamic buckling does occurs, is



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

AT p = 6.2703 [m/s]

AT p = 6.2763 [m/s]

AT p = 6.2823 [m/s] W m i d

t[s]

Figure 9. Load-parameter sensitivity of the occurrence of dynamic

buckling for b = 1 [Ns/m], T p = 10 [ms], e/d = 0, a/h =0.04.

a/h

A T

p

[ m / s ]

Figure 10. Complex transition between dynamic buckling (black)

and no dynamic buckling (white) for T p = 10 [ms], b = 1 [Ns/m],

e/d = 0.

not indicated by a single sudden jump in the response

measure, but by a small transition region. In this re-

gion the occurrence of dynamic buckling is extremely

sensitive to small variations in the load parameter as il-

lustrated in figure 9. Note that the integration time in

figure 9 is extended from T = 0.3 [s] to T = 0.5 [s]and the sensitivity is thus not due to a too short integra-

tion time.

The space spanned by the shape-factor a and the

load-parameter A for a certain fixed value for T p, can

be divided into a safe region (no dynamic buckling)

and an unsafe region (dynamic buckling). The bound-

ary between these regions is defined as the dynamic

stability boundary. By examining the occurrence

of dynamic buckling at a dense grid of parameter

values in the space spanned by the shape-factor a and

the load-parameter A, it appears that this dynamic

stability boundary can have a very complex geometricshape, see figure 10. The complexity of the boundary

−0.1 −0.05 0 0.05 0.1 0.152

3

4

5

6

7

8

9

a/h

A T

p

[ m / s ]

T p = 5 [ms]

T p = 10 [ms]

T p = 20 [ms]

Figure 11. Dynamic stability boundary for various pulse durations,

b = 2 [Ns/m] and e/d = 0.

between the safe region and the unsafe region may be

compared with fractal boundaries as encountered inthe steady-state analysis of chaotic dynamical systems.

The complex transition between the safe region and

the unsafe region is, therefore, indicated as a fractal

transition. It is noted that the fractal transition only

occurs close to the arch shape where a maximum in

dynamic buckling load appears and only for a low level

of damping (see also figure 8). Similar load-parameter

sensitivities in transient analyses of 1-dof models are

reported in (Thompson and Stewart, 2002; Soliman

and Goncalves, 2003).

The final considered parameter variation for the per-fect arch is the pulse duration T p, see figure 11. Note

that the results are plotted against the product AT p = λ(see (18)) allowing to examine the mutual relation be-

tween A and T p with respect to the dynamic buckling

load. As can be noted, the stability boundaries for the

various pulse durations do not coincide perfectly and,

therefore, if the parameters A and T p are varied inde-

pendently, the dynamic buckling load does not scale ex-

actly with the parameter λ as found for the case where

the dynamic stability problem is approximated by an

initial value problem. Still, the results for the vari-

ous pulse durations match qualitatively, that is for each

value of T p the dynamic buckling load shows a compa-

rable sensitivity to variations in the arch shape.

By comparing the dynamic stability boundaries (see

figure 11) with the static stability boundary for the

perfect arch (ALP in figure 6), a clear correspondence

can be distinguished. However, in the quasi-static

analysis it is shown that for the more practical situation

where e/d = 0, snap-through of the arch will always

occur in an asymmetric shape at a significantly lower

load. The static buckling load corresponding to this

buckling mode could be increased much less by

varying the arch shape.



5 5.5 6 6.5 7 7.50.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

W m i d

AT p [m/s]

e/d = 0

e/d = 0.1

e/d = 0.5

e/d = 1

Figure 12. Influence imperfection for a/h = 0.04, b = 2[Ns/m], T p = 10 [ms].

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

W m i d (No dynamic buckling)

t [s]

e/d = 0e/d = 0.1

Figure 13. Influence imperfection on vibrations for A = 695[m/s

2], a/h = 0.04, b = 2 [Ns/m], T p = 10 [ms].

In figure 12, the influence of a small imperfection

on the dynamic buckling load is illustrated. Com-

pared to the static buckling load (considering a quasi-

static varying acceleration A), the dynamic buckling

load (considering a short half-sine pulse in accelera-

tion) shows a much less fatal sensitivity to small geo-

metric imperfections. The dynamic buckling load isonly decreased less than 10% for the moderate value

of imperfection e/d = 1. The small geometric im-

perfection significantly increases the amplitudes of the

vibrations induced by the shock-pulse prior to the oc-

currence of dynamic buckling, see figure 13.

The imperfection sensitivity of the dynamic stability

boundaries, is illustrated in figure 14. For the depicted

range of the shape-factor a, the dynamic buckling load

decreases only with maximum 10% for the moderate

imperfection of e/d = 1. Furthermore, for a small geo-

metric imperfection, the dynamic stability boundaries

become smooth curves, indicating the disappearanceof the extreme sensitivity of the occurrence of dynamic

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.14.5

5

5.5

6

6.5

7

7.5

8

8.5

9

a/h

A

T p

[ m / s ]

T p = 5 [ms]

T p = 10 [ms]

T p = 20 [ms]

e/d = 0

e/d = 1

Figure 14. Imperfection sensitivity of dynamic stability boundary

for b = 2 [Ns/m] and various values for T p.

buckling to small variations in the load-parameter.

The distinct maximum in the dynamic buckling load

as found around a/h ≈ 0.05 seems insensitive to

the geometric imperfection. Similar smooth stability

boundaries, also with a maximum close to a/h ≈ 0.05,

are found at the lower level of damping b = 1 [Ns/m].

The dynamic stability boundaries show, therefore, a

clear qualitative correspondence with the static stabil-

ity boundary as found for the perfect arch (see figure

6), even for the case e/d = 0. Consequently, for

the arch and load-case under consideration, arch shape

optimization based on maximizing the static buckling

load for the perfect arch would approximately result inthe same shape when optimizing the dynamic buckling

load for the practical case e/d = 0. This is beneficial

since, compared to the nonlinear transient dynamical

analyses, the quasi-static analyses are computationally

significantly less expensive.

5 Conclusions and Recommendations

The objective of this research was to study the effect

of the arch shape on the dynamic buckling load for

a thin shallow arch under drop-shock loading. The

drop-shock loading is modelled by a half sine pulsein acceleration in transversal direction. Based on the

curved beam kinematic model, a discrete multi-dof

model of the arch is derived in which longitudinal

and rotary inertial effects are negelcted. The resulting

semi-analytical model enables variation of the arch

shape (the initial curvature) with a single shape-factor

and an imperfection parameter which controls the

amplitude of a small asymmetry in the arch shape.

By comparing quasi-static responses and modal

analysis results of this model with accurate finite

element modeling results, the accuracy of this model

is validated. First static snap-through buckling of

the arch under a quasi-static varying acceleration isconsidered. It is shown that the arch shape signifi-

enocnmallon_id120

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