Rocking instability of free-standing statues atop

Soil Dynamics and Earthquake Engineering 63 (2014) 83–91

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Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn

Rocking instability of free-standing statues atop slender viscoelastic columns under ground motion Anthony Kounadis n Academy of Athens, Office of Theoretical & Applied Mechanics, Foundation for Biomedical Research of the Academy of Athens, Soranou Efessiou 4, Athens 11527, Greece

art ic l e i nf o

a b s t r a c t

Article history: Received 6 December 2013 Received in revised form 23 January 2014 Accepted 29 January 2014 Available online 15 April 2014

The highly complex rocking response of free-standing statues atop multi-drum columns underground excitation resulting in insuperable difficulties for obtaining reliable solution is reexamined analytically. This is achieved after simulating the columns by monolithic viscoelastic cantilevers having structural damping, based on experiments, equivalent to the energy dissipation due to impact and sliding of multidrum columns. Subsequently, the conditions of rocking (overturning) instability of free-standing rigid blocks (representing the statues) after their uplift from the top surface of the laterally vibrating cantilevers, are established, including overturning with or without impact. Attention focuses on the minimum amplitude ground acceleration which leads to an escaped motion through the vanishing of the angular velocity and acceleration. Maximization of such a minimum amplitude (implying stabilization) of the rigid block is obtained by seeking the optimum combination of values of the slenderness ratio of the column and its height. Analytically derived results based on linearised analyses are in excellent agreement with those obtained via nonlinear numerical analyses. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Rocking instability Overturning Multi-drum column Ground excitation Viscoelastic column Impact Sliding Free-standing statue

1. Introduction Over the last decades much research has been devoted to the determination of the actual rocking response of freely standing statues atop multi- drum columns which have survived from strong earthquakes in the past [1–6]. Indeed, such columns, with freely supported (between them) drums interconnected to each other through loose studs (from timber or lead), exhibit a rather impressive seismic resistance behavior compared to monolithic columns of the same geometric and material features. This is mainly due to the dissipation of energy (during rocking) related to impact and sliding between drums (simulating the human vertebra).Thus the magnitude of the ground excitation at the top surface of the multi-drum column (carrying the free-standing statue), being substantially reduced (due to energy dissipation) is usually inadequate to overturn the statue. However, the energy loss between consecutive drums decreasing along the height of the cantilever, is extremely difficult to be estimated and only experimentally can be obtained [7]. In this study the rocking (overturning) instability analysis of the free-standing statue atop the multi-drum column is discussed

n

Tel./fax: þ210 6597517. E-mail address: [email protected]

http://dx.doi.org/10.1016/j.soildyn.2014.01.021 0267-7261/& 2014 Elsevier Ltd. All rights reserved.

assuming that the column disposes sufficient resistance against the ground excitation. As such, the falling (owing to overturning) of the statue is not due to the failure of the multi-drum column, but to its rocking (overturning) instability while the multi-drum column is laterally vibrating safely under the ground excitation. Attention is focused on the minimum amplitude of ground acceleration which causes the statue to overturn during the lateral vibration of the multi-drum column in which the statue is freely supported without any link. The rocking instability of the statue occurs after its uplift from the top surface of the column with initial conditions the end conditions of the laterally vibrating column (relative to which the motion of the statue is examined). However, an exact analysis of such complex system (of the multidrum column and the freely standing statue on its top surface), is extremely difficult – if not impossible – to be established. Besides the aforementioned inability to evaluate the energy dissipation between drums the difficulty of such an analysis becomes more pronounced as the number of drums increases. This is clearly shown even in case of rocking analysis of a two-rigid block system [8]. Very recently it was found [7] that for a multi-drum column with n drums the number of possible configuration patterns which may lead to overturning is N ¼3n 1, while the number of the corresponding highly nonlinear 2nd order differential equations of motion is Nn. Namely, for a multi-drum column with 10 drums one has to solve 590480 (!) highly nonlinear differential equations,

84

A. Kounadis / Soil Dynamics and Earthquake Engineering 63 (2014) 83–91

which is practically impossible. These two insuperable difficulties turn the research interest to approximate analyses using various models and computational techniques. In an attempt to approximate this highly complex behavior, as much as possible, one may consider instead of the multi-drum column a monolithic viscoelastic column with coefficient of viscosity in accord to the total dissipation of energy of the multi-drum column (determined experimentally). Due to this total energy loss the magnitude of the ground excitation at the tip of the multi-drum column is substantially reduced for overturning the free-standing statue atop it. In this case, the statue is replaced by a heavy rigid block, whose effect on the buckling load of the monolithic column (and thereafter on its natural frequencies) being very small can be neglected according to the previous analyses [9]. Such a simplification facilitates the dynamic analysis of the column and subsequently the study of the rocking response of the rigid block after its uplift from the top surface of the vibrating column. The objective of this work, being an extension of previous studies [4,8,6,9], is:a)to establish the best approximate solution of the highly complex response of multi-drum columns carrying atop freestanding statues and b) to discuss via a viscoelastic monolithic column the individual and coupling effect of the principal control parameters on the rocking instability of the (separated from the top surface of the monolithic cantilever) rigid block which are the slenderness ratio of the column and its height. Recall, as observed by Housner [1] and shown by Kounadis [7,6], that between two geometrically similar rigid blocks of the same material, the higher is more stable than the lower one. The motivation for this study lies in the need for seismic protection of two very elegant marble statues (Athena and Apollo) standing freely atop two multi-drum columns in the forefront of the Academy of Athens building, one of the most beautiful neoclassic building in the world.

2b

2h

MC, JC

C B

l

EI, m

2ho

A

c2 Mo, Jo

c1

2hs

2bo Fig. 1. Elastically restrained cantilever carrying freely a heavy rigid body at its tip.

C

C y(x,t)

Mc, Jc

Mc g

Mc g

l

Consider the system shown in Fig. 1 composed of a uniform slender cantilever AB (fixed on a foundation with mass M0 and rotatory inertia J0) and a rigid-block freely supported atop of weight Mcg (with gravity acceleration g, mass Mc and rotatory inertia Jc) which is in equilibrium in the vertical direction under its own weight. The foundation is elastically supported (with rotational spring constant c1 and translational spring constant c2) by a rigid ground which is subjected to a horizontal motion Ug (t). The mathematical model of the above system is shown in Fig. 2.

EI, m

EI, m x x

Mo, Jo

Mo, Jo c2

2.1. The system lateral motion

c1

During small bending (periodic) vibrations of the above system as a whole (i.e. before uplift of the rigid block from the top surface of the column) the column AB in addition to a rigid body translation and rotation (due to horizontal ground excitation) is subjected to lateral small deflection (Fig. 2). Hence, denoting by Y (x,t) the lateral horizontal deflection of the cantilever at a height x, the total horizontal displacement of the cantilever, U(x,t), due to ground displacement Ug(t),at a height x is Uðx; tÞ ¼ U g ðtÞ þ Yðx; tÞ þ xY 0 ð0; tÞ;

U 0 ðx; tÞ ¼ Y 0 ðx; tÞ þ Y 0 ð0; tÞ

Mc, Jc Mc g

u(x,t)

2. The column–rigid block system

C

ð1Þ

where the prime denotes derivative with respect to x. Assuming that the rotational restraint of the foundation is sufficiently large, the displacement at the tip of the cantilever of height l due to the rigid body rotation Y'(0,t) l (compared to the bending deflection Y(x,t)) can be neglected [9]. Taking also into account that the compressive load P¼ Mcg is small, compared to the buckling load of the cantilever, its effect on the governing

c2 c1

y

ug(t) Fig. 2. The system of the elastically restrained viscoelastic cantilever and the rigid body at its tip subjected to ground displacement.

equation of the cantilever lateral motion can be omitted [9]; thus Pl2/EI ¼(π2/4) P/PE C0 (PE ¼π2EI/4l2). Then, the governing equation for the lateral small deflection considering an equivalent (to the multi-drum column) viscoelastic cantilever [10,11] is given in dimensionless form by the linear partial differential equation 4

4

η ml ml y0000 ðx; tÞ þ y_ 0000 ðx; tÞ þ y€ ðx; tÞ ¼ u€ g ðtÞ E EI EI

ð0 rx r 1Þ

ð2Þ


85

Fig. 3. Photo of the experimentent of a multi -drum column with five drums and a capital atop subjected to the Parnitha earthquake in 7/9/1999.

where the prime and dot denote partial derivatives with respect to x and t, while x ¼x/l, y(x,t) ¼Y(x,t)/l, ug(t) ¼Ug(t)/l,u(x,t) ¼U(x,t)/l; EI is the flexural rigidity of the cantilever, m is its mass per unit length and η is a viscosity coefficient which can be determined with the aid of experiments. Relevant experiments on multi-drum columns with five drums and a capital on top have been conducted by Drossos et al. [13]. Pertinent experiment is shown in Fig. 3. The associated boundary conditions are At x ¼0 y000 ð0; tÞ ¼ c2 yð0; tÞ

3

M0 l € tÞ; uð0; EI

y00 ð0; tÞ ¼ c1 y0 ð0; tÞ þ

J0 l 0 u€ ð0; tÞ EI

which yields X n 0000 ðxÞ kn X n ðxÞ ¼ 0; 4

4 kn

T€ n ðtÞ þ 2βn T_ n ðtÞ þ ω2n T n ðtÞ ¼ 0

ð7Þ

4 ω2n ml =EI

βn ¼ηω2n/2E

¼ (dimensionless frequency), where (structural damping coefficient). Integration of the 1st of Eq. (7) gives X n ðxÞ ¼ An sin kn x þ Bn cos kn x þ Γ n sinh kn x þ Δn cosh kn x

ð8Þ

The boundary conditions (3) and (4) associated with the free vibrations, after using relation (5), become at x ¼ 0 : X n 000 ð0Þ ¼ c2 X n ð0Þ þ kn M 0 X n ð0Þ; X n 00 ð0Þ ¼ c1 Xn 'ð0Þ kn J 0 X n 0 ð0Þ 4

ð3Þ

00

4

000

4

at x ¼ 1 : X n ð1Þ ¼ kn J c X n 'ð1Þ; X n ð1Þ ¼ kn M c X n ð1Þ

ð9Þ

At x ¼1 y00 ð1; tÞ ¼

3

3

Jc l 0 Mc l € u€ ð1; tÞ; y000 ðl; tÞ ¼ uð1; tÞ EI EI

ð4Þ

3

where c1 ¼ c1 l=EI and c2 ¼ c2 l =EI are the rotational and translational dimensionless spring constants. Eq. (2) along with the boundary conditions (3) and (4) can be solved via the series solution [12] 1

yðx; tÞ ¼ ∑ X n ðxÞT n ðtÞ

ð5Þ

n¼1

where Xn(x) is the dimensionless shape function of the nth mode of free vibration with corresponding dimensionless time function Tn(t).

2.1.1. Free motion Setting the RHS of Eq. (2) equal to zero and then using relation (5), one gets for a periodic motion X n 0000 ðxÞ 4

)

4

ðml =EIÞX n ðxÞ

¼

T€ n ðtÞ

T n ðtÞ þ ðη=ΕÞT_ n ðtÞ

¼ ω2n

ð6Þ

3

where M 0 ¼ M 0 =ml, J 0 ¼ J 0 =ml J c ¼ J c =ml , M c ¼ M c =ml. Conditions (9), due to relation (8), lead to the following system of homogeneous algebraic equations with respect to the integration constants An, Bn, Γn and Δn 2 32 3 α11 α12 α13 α14 An 6 α21 α22 α23 α24 76 Bn 7 6 76 7 ð10Þ 6 76 7¼0 4 α31 α32 α33 α34 54 Γ n 5 α41 α42 α43 α44 Δn where 3

4

α11 ¼ kn =ðc2 kn M 0 Þ; α14 ¼ 1; α24 ¼

α12 ¼ 1; 2

4

α21 ¼ kn ; α22 ¼ kn =ðc1 kn J o Þ;

2 4 kn =ðc1 kn J o Þ;

α32 ¼ kn sin kn þ α34 ¼ kn sinh kn

2 kn J c

2

kn J c

;

;

4

α23 ¼ kn ;

α31 ¼ kn cos kn þ

cos kn

cosh kn

3

α13 ¼ kn =ðc2 kn M o Þ; sin kn 2

kn J c

α33 ¼ kn cosh kn α41 ¼ sin kn

;

sinh kn 2

kn J c

cos kn M c kn

;

;

86


α42 ¼ cos kn þ

sin kn

α44 ¼ cosh kn þ

cosh kn ; α43 ¼ sinh kn þ ; M c kn M c kn sinh kn M c kn

where Λ ¼ ½β2n þðωg ωnn Þ2 ½β2n þ ðωg þ ωnn Þ2 ð11Þ

where M 0 , J 0 , M c , J c , are given in relation (9). The vanishing of the determinant of the homogeneous system of Eq. (10) yields the frequency equation with respect to the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 dimensionless circular frequency k2n ¼ωn ml =EI for given values of the parameters M 0 , J 0 , M c , J c , c1 and c2 . During lateral periodic vibrations, at a certain time, the freely standing rigid block (depending on the magnitude of ground excitation) can be uplifted from the top surface of the cantilever. Τhe subsequent motion of the rigid block (relative to the top surface of the cantilever) and thereafter its rocking instability depends on the initial conditions related to the corresponding conditions of the top surface of the laterally vibrating cantilever which (conditions), in turn depend on the magnitude of ground excitation. Such a magnitude of ground excitation and more specifically its minimum amplitude leading to rocking (overturning) instability can be determined through the establishment of the forced laterally vibration response given below. 2.1.2. Forced vibration By virtue of Eqs.(2), (5) and (6) the differential equation related to the time forced part, Tn(t) is [9,11] T€ n ðtÞ þ 2βn T_ n ðtÞ þ ω2n T n ðtÞ ¼ Z n u€ g ðtÞ .R o R1 1 where Z n ¼ 0 X n ðxÞ dx 0 X 2n ðxÞdx

ð12Þ

Assuming that the cantilever is initially at rest, i.e. yðx; 0Þ ¼ y_ ðx; 0Þ ¼ 0 (implyingT n ð0Þ ¼ T_ n ð0Þ ¼ 0) integration of Eq. (12) gives Z t Zn T n ðtÞ ¼ n e βn τ ð13Þ u€ g ðτÞeβn τ sin ωnn ðt τÞdτ ωn 0 pffiffiffiffiffiffiffiffiffiffiffiffi where βn is the structural damping coefficient and ωnn ¼ ωn 1 η2 . where η ¼ η=ηcr with ηcr the critical viscosity coefficient for which βn ¼ωn yielding η ¼βn/ωn. Assuming the dimensionless ground acceleration € ¼ αg sin ðωg t þ ψÞ uðtÞ

with ¼ αg =l

ð14Þ

valid for –ψ/ωg rt r(2π ψ)/ωg, otherwise u€ g ðtÞ ¼ 0, one can write Z t Z t eβn τ sin ðωg τ þ ψÞ cos ωnn τdτ : u€ g ðτÞeβn τ sin ωnn ðt τÞdτ ¼ αg 0

n

sin ωn t αg

Z

0

t

βn τ

e 0

sin ðωg τ þ ψÞ sin ωnn τdτ cos ωnn t:

For convenience one can also write 9 Rt β τ n sin ðωg τ þ ψÞ cos ωnn τdτ ¼ I 1 cos ψ þ I 2 sin ψ = 0e Rt β τ n sin ðω τ þ ψÞ sin ωn τdτ ¼ I cos ψ þI g 3 4 sin ψ ; n 0e where Rt I 1 ¼ 0 eβn τ sin ωg τ cos ωnn τdτ; Rt I 3 ¼ 0 eβn τ sin ωg τ sin ωnn τdτ;

I2 ¼ I4 ¼

Rt

βn τ 0e Rt β τ n 0e

ð15Þ

ð16Þ

9 cos ωg τ cos ωnn τdτ = cos ωg τ sin ωnn τdτ ; ð17Þ

By virtue of relations (16) and (17) after evaluating the integral Ii (i ¼1,2,3,4), Eq. (13) gives αg Z n 2ωg βn ½e βn t cos ðωnn t þ ψÞ cos ðωg t þ ψÞ Λ þ ðω2n ω2g Þ sin ðωg t þ ψÞ ωg þ e βn t n ðω2g þ ω2n 2ωnn2 Þ sin ωnn t cos ψ ðω2n ω2g Þ cos ωnn t sin ψ ωn

T n ðtÞ ¼

ð18Þ n

The last equation after omission of damping (βn ¼0, ωn ωn ) is simplified as follows [9] T n ðτÞ ¼ αg

Zn ωg t þ ψÞ cos ψ sin ω t sin ψ cos ω t : sin ðω g n n ωn ðω2n ω2g Þ

ð19Þ The total horizontal displacement at the end (tip) of the cantilever is given by uð1; tÞ ¼ ug ðtÞ þ yð1; tÞ

ð20Þ

Clearly by virtue of relation (5) 1

yð1; tÞ ¼ ∑ X n ð1ÞT n ðtÞ

ð21Þ

n¼1

where Tn(t) is given in relation (18) with corresponding shape function X n ð1Þ ¼ An sin kn þ Bn cos kn þ Γ n sinh kn þ Δn cosh kn

ð22Þ

where An, Bn, Γn and Δn are obtained from Eq.(10). Setting in Eq. (10) the arbitrary integration constant Δn ¼1, one can find Bn; ¼ –1–α11 Αn –α13 Γ n

ð23Þ

where Αn and Γn are evaluated from the following system: " #" # " # α22 α24 An kn α22 α13 kn α22 α11 ¼ α32 α34 Γn α31 α32 α11 α33 α32 α13

ð24Þ

which in turn yields

9 An ¼ D1n ½ðα33 α32 α13 Þðα22 α24 Þ þ ðα22 α13 kn Þðα32 α34 Þ =

Γ n ¼ D1n ½ðα22 α24 Þðα32 α11 α31 Þ þ ðkn α22 α11 Þðα32 α34 Þ ;

ð25Þ

where Dn ¼ ðkn α22 α11 Þðα33 α32 α13 Þ ðkn α22 α13 Þðα31 α32 α11 Þ The initial conditions for which the rigid block shown in Fig. 4a,b can be uplifted from the top surface of the cantilever are associated with the tip conditions of the cantilever, given by 1

1

n¼1

n¼1

y0 ð1; tÞ ¼ ∑ X n 0 ð1ÞT n ðtÞ; y_ 0 ð1; tÞ ¼ ∑ X n 0 ð1ÞT_ n ðtÞ

ð26Þ

Since at the initiation of rocking u€ g ¼λαg ¼αg sin ψ and λαg ¼ ¼g tan αEgα [4] it is clear that the minimum amplitude ground acceleration (causing the above uplift) is αg 1 ¼ gα sin ψ

ð27Þ

where α is the angle of the semi-diagonal R with respect to the vertical (Fig. 4a,b ). This uplift may occur either before the ground excitation Tex expires (i.e. for 0 ot rTex) or after the expiration of the ground acceleration (i.e. for t ZTex). Both intervals of time are very important for the rocking instability of the freely supported rigid block which may overturn either with or without impact, regardless of whether or not impact occurs prior or after the ground excitation expires. Although according to Shenton [14] impact may be associated with sliding, this effect is neglected assuming that the friction between drums is sufficiently large to exclude sliding.

2.2. The rigid block motion Since the slender cantilever undergoes small amplitude (periodic) vibrations its top surface (end cross-section) can be assumed as horizontal. The very small deviations of the top surface of the


87

where u€ g ¼ u€ g ðτÞand Χn (1) are given in relation (14) or (31) and € (22), while TðτÞ can be computed via Eq.(18) or T€ n ðtÞ þ 2βn T_ n ðtÞ þ ω2n T_ n ðtÞ ¼ 0 (see Eq.(7)), where t¼τ/p. Eqs.(32) and (33) are valid before the rigid excitation expires (i.e. 0otrTex). For the subsequent response of the block (i.e. tZTex) one should calculate Tn(t) via Eq. (7) i.e. " T_ n ðT ex Þ þ βn T n ðT ex Þ T n ðtÞ ¼ e βn ðt T ex Þ sin ωnn ðt T ex Þ ωnn ð34Þ þ T n ðT ex Þ cos ωnn ðt T ex Þ ðt 4 T ex Þ pffiffiffiffiffiffiffiffiffiffiffiffi n where βn is the structural damping coefficient and ωn ¼ ωn 1 η2 . Using Eqs. (22) and (34) one can calculate the rotation and angular velocity of the tip of the cantilever θo ðT ex Þ ¼ y'ð1; T ex Þ, θ_ o ðT ex Þ ¼ y_ 'ð1; T ex Þ which constitute the initial conditions for the free motion rocking response of the rigid block [9]. Note that the separation of the rigid block from the top surface of the cantilever occurs via rotation about the axis either OS for θ(t)o 0 or O'S' for θ (t) 40 (see Fig. 1). Indeed, if the direction of the ground motion coincides with one of the principal axes of the square base of the rigid block, its resistance against overturning is the lowest one compared with the resistance in any other direction of motion. This is so, because the resisting moment (against overturning) of the weight Mcg multiplied by the lever with respect to axes OS or O0 S0 becomes minimum when the direction of motion coincides with one of the above principal axes. Fig. 4. Rigid body motion under ground excitation: (a) for negative (counterclockwise) and (b) for positive (clockwise) rotation.

2.3. Criteria for rocking instability

cantilever from the horizontal plane (i.e. y0 (1,t)a0) constitute the initial (imperfection) conditions of the free-standing rigid block on the top surface of the cantilever. The response of the rigid block (whose motion is studied relative to the cantilever) as a 1-DOF system is governed by a 2nd order differential equation with respect to the angle of rotation θ measured from the vertical [4,6] as shown in Fig. 4a,b. Hence, at any time t0 (after the initiation of the uplift of the block from the cantilever) the initial conditions are

Attention is focused on the minimum amplitude ground acceleration leading ( at a certain time tn) to rocking (overturning) instability, through an unstable equilibrium point with zero _ n Þ ¼ 0 [6].For a slightly greater angular velocity, θðt n Þ ¼ 7α, θðt amplitude an escaped motion occurs with continuously increasing angle |θ(t)| beyond |α|.Hence such a point is an inflection point of _ n Þ ¼ θðt € n Þ ¼ 0.Assuming that overturning the curve θ(t) vs t, ie θðt instability occurs after the free vibrations of the damped cantilever (with very small amplitude around θðt n Þ ¼ 7 α) are practically damped out, the conditions for overturning instability are

θ0 ðt 0 Þ ¼ y'ð1; t 0 Þ; θ_ 0 ðt 0 Þ ¼ y_ 'ð1; t 0 Þ:

θðt n Þ ¼ 7α;

ð28Þ

The governing nonlinear differential equations of motion of the rigid block for the cases of Fig. 3a (θ(t)o0) and Fig. 3b (θ(t)40) from the onset of uplift are given by [9] u€ g þ y€ 0 cos ðα þ θÞ sin ðα þ θÞ ¼ 0; ðθðtÞ o 0Þ ð29Þ θ€ þ p2 g u€ g þ y€ o cos ðα θÞ þ sin ðα θÞ ¼ 0; ðθðtÞ 4 0Þ θ€ þ p2 g

ð30Þ

where pffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ¼ 3g=4R; u€ g ðτÞ ¼ αg sin ðωg τ þ ψÞ; 1

ωg ¼ ωg =p; y0 ðtÞ ¼ lyð1; tÞ ¼ ∑ X n ð1ÞT n ðtÞ; τ ¼ pt: n¼1

ð31Þ

Clearly, y0 may take positive or negative values depending on the positive or negative lateral deflection of the cantilever. In some cases of multi-drum columns the corresponding rigid blocks are characterized by α o0.25 rad. Then Eqs. (29) and (30), using t ¼τ/p become 1 € € θðτÞ ¼ α ug l ∑ X n ð1ÞT€ n ðτÞ; θðτÞ g gn¼1 1 € € θðτÞ ¼ α ug l ∑ X n ð1ÞT€ n ðτÞ; θðτÞ g gn¼1

θðτÞ o 0

θðτÞ 40

ð32Þ

ð33Þ

_ n Þ ¼ 0; θðt

€ nÞ ¼ 0 θðt

ð35Þ n

Namely, such a critical state occurs at a certain time t at which the block oscillates for a short time with very small amplitude around θ(tn) ¼ α (θo0) or θðt n Þ ¼ α(θ 40), while both the angular velocity and acceleration vanish (escaped motion through inflection point in the curve θ(t) vs t). For a slightly smaller amplitude (than the corresponding minimum acceleration amplitude) the angle of rotation θ(t) reverses direction (forcing the block towards its initial position) which allows us to assume the above conditions. Conditions (35) are valid for both modes of overturning instability (i.e. with or without impact) either after the expiration of ground excitation (tn 4Tex) or before the ground excitation expires (tn oTex), i.e. during the negative part of the sinus excitation. Namely, conditions (35) correspond to a free vibration regime.

3. Numerical examples and discussion Numerical results will be presented mainly in graphical form for given dimensions of the rigid block, geometric and material properties of an elastically supported concrete cantilever of circular cross-section (carrying on its top surface the above freestanding rigid block). For given values of the parameters M 0 , J 0 , M c , J c , c1 and c2 one can obtain the dimensionless frequencies k2n through the vanishing

88


of the determinant of the homogeneous system (10). Then, on the basis of k2n one can determine the integration constants An, Bn, Γn (after setting Δn ¼1) which via Eq. (8) provide the corresponding shape functions and thereafter Zn from relation (12). Subsequently using Eq. (18) one can calculate implicitly for a given excitation frequency ωg the time function Tn(t) as a function of t and αg , given 1 that due to Eq. (27) ψ ¼ sin ðαg =gαÞ. Thereafter, via Eq. (5) one can determine y(1,t)¼y0(t)/l and then the initial conditions (28) of the separated (from the top surface of the cantilever) rigid block. Furthermore, using the rocking instability conditions (35), one can seek the minimum amplitude ground acceleration [satisfying Eq. (35)] which leads through the vanishing of the angular velocity and acceleration to an escaped motion. The general solutions of Eqs. (32) and (33) are ) θðtÞ ¼ θh1 ðtÞ þ θp1 ðtÞ; θðtÞ o0 ð36Þ θðtÞ ¼ θh2 ðtÞ þ θp2 ðtÞ; θðtÞ 4 0

!

n

n

αg þ ∑ X i Z i C 3 ¼ θ_ 0 ωg þ

i¼1

gð1 þ ω2g Þ

cos ðωg τ0 þ ψÞ

cos ψ n ωi 2 ∑ X i Z ni cos ωi τ0 gωg i ¼ 1 ð1 þ ω2i Þ sin ψ n ωi 3 ∑ X i Z ni sin ωi τ0 ð1 þω2i Þ gω2g i ¼ 1 ! n

αg þ ∑ X i Z i n C 4 ¼ θ0 α þ

i¼1

gð1 þ ω2g Þ

sin ðωg τ0 þ ψÞ

þ

cos ψ n X i Z ni ω sin ωi τ0 ∑ gωg i ¼ 1ð1 þω2i Þ i

þ

sin ψ n ωi 2 ∑ X i Z ni cos ωi τ0 ð1 þω2i Þ gω2g i ¼ 1

The nonlinear equations of motion (29) and (30) can be numerically integrated for given values of the involved parameters.

where

9 > > > > > > > > > > > > > > > > =

θh1 ðτÞ ¼ C 1 sinhðτ τo Þ þC 2 coshðτ τo Þ θh2 ðτÞ ¼ C 3 sinhðτ τo Þ þC 4 coshðτ τo Þ ! n

αg þ ∑ X i Z i n

cos ψ n ωi sin ψ n X i Z ni ∑ X i Z ni ∑ ω2 cos ωi τ sin ωi τ 2 gωg i ¼ 1 gωg i ¼ 1ð1 þ ω2i Þ i ð1 þ ωi Þ > > > ! > > n > > n > > αg þ ∑ X i Z i > n > n n > cos ψ ωi sin ψ XiZi i¼1 > n 2 > τ þ ψÞ ∑ X Z ω τ ∑ ω cos ω τ sin ðω sin θp2 ¼ α þ > g i i i i i > 2 2 2 2 ; gωg i ¼ 1ð1 þ ωi Þ ð1 þωi Þ gð1 þ ωg Þ gωg i ¼ 1 i¼1

θp1 ¼ α þ

sin ðωg τ þ ψÞ

gð1 þ ω2g Þ

where τ0 ¼pt0 is associated with the initial conditions (28) due to the lateral oscillations of the cantilever, whereas X i X i ð1Þ; Z ni ¼

αg Z i 1 ω2i =ω2g

ði ¼ 1; 2; …; nÞ

ð38Þ

The first two or three terms in the above sums usually provide very good accuracy. However, by means of numerical evaluation one can decide whether additional terms are required in order to further improve the accuracy. For the following initial conditions: ) θðτ0 Þ ¼ y0 ð1; τ0 Þ ¼ θ0 ð39Þ _ 0 Þ ¼ y_0 ð1; τ0 Þ ¼ θ_ 0 θðτ by virtue of Eq. (36) in conjunction with Eqs. (37)–(39), we get ! n

αg þ ∑ X i Z ni i¼1

C 1 ¼ θ_ 0 ωg þ

ð40Þ

cos ðωg τ0 þ ψÞ

gð1 þ ω2g Þ

cos ψ n X i Z ni ω 2 cos ωi τ0 ∑ gωg i ¼ 1ð1 þ ω2i Þ i sin ψ n ωi 3 sin ωi τ0 ∑ X i Z ni 2 ð1 þ ω2i Þ gωg i ¼ 1 ! n

αg þ ∑ X i Z ni i¼1

C 2 ¼ θ0 þ α

gð1 þω2g Þ

sin ðωg τ0 þψ Þ

cos ψ n X i Z ni ω sin ωi τ0 ∑ gωg i ¼ 1ð1 þ ω2i Þ i

sin ψ ωi ∑ X i Z ni cos ωi τ0 ð1 þ ω2i Þ gω2g i ¼ 1 n

2

ð37Þ

Regardless of the above nonlinear solution, one can use the linearised equations of motion valid for the stockiness parameter αo0.25, then employ Eq. (36) and subsequently relations (37)– (40) based on the first three modes along with the instability conditions (35). All final results have been also verified via numerical analysis. It is worth noticing that on the basis of the linearised closed form solution, valid for α o0.25, one can successfully validate any nonlinear numerical scheme. 3.1. Numerical data For the sake of comparison of the findings of the present analysis with existing results regarding the rocking instability of a rigid block supported either by a monolithic cantilever or by a rigid surface, the following numerical data are used [9,4]: (a) Rigid block: Stockiness parameter α¼0.25 rad, semidiagonal R ¼1.606 m, b¼R sin α¼ 0.39733 m, characteristic frepffiffiffiffiffiffiffiffiffiffiffiffiffiffi quency p ¼ 3g=4R ¼ 2.14 s 1, horizontal ground acceleration u€ g ðtÞ¼ αg sin (ωgt þψ). (b) Concrete cantilever: Diameter of the column cross-section, dcol ¼ 1.0 m, height of the column, l ¼10 m, density of concrete, ρc ¼ ¼2500 kg/m3. (c) Concrete footing dimensions: 1.5 m 1.5 m, height of foundation H¼ 0.65 m, dimensionless spring constants: c1 ¼ 1000 and c2 ¼ 1000.

Using the above values of parameters we find M 0 ¼0.1863, J 0 ¼0.000415, M c ¼0.250 and J c ¼0.00215. Subsequently via the vanishing of the determinant of Eq. (10) and for c1 ¼c2 ¼1000 we get the eigenfrequencies k21 ¼2.4518, k22 ¼16.2083 k23 ¼39.6346


yo0 obtain simultaneously their maximum values, while y_ 0 becomes zero. Fig. 5 shows for ωg/p ¼ 3.5 and η¼ 0.1 (which is very conservative compared to the loss of energy exhibited due to impact and sliding of an equivalent cantilever consisting of 8 to 10 drums) the variation of accelerations versus time: of the ground u€ g ðtÞ, of the tip of the cantilever y€ o ðtÞ and of the total acceleration € ¼ u€ g ðtÞ þ y€ o ðtÞ. The dashed applied at the base of the rigid block uðtÞ curves in Figs. 6a–c and 7a–c show overturning instability of the rigid block for ωg/p ¼3.5 and ωg/p ¼6 respectively, via an escaped motion [related to the vanishing of the angular velocity and acceleration via θ(tn) ¼ α, Eq. (35)], leading to the minimum amplitude of ground acceleration αg. Τhese critical cases occur after several impacts (with a 10% reduction of the angular velocity of the rigid block after each impact). The solid line in the above figures corresponds to a slightly smaller value of αg due to which the rigid block does not overturn associated with an asymptotic rocking stability; namely, the rigid block undergoing one or more impacts, after a certain period of time, returns to its initial position. As anticipated the plots in the above Figs. 5, 6a–c, 7a–c are very similar to those of Figs. 6 and 7a,b,c, respectively, of Ref. [9] based on the same numerical data and amount of damping. Fig. 8 shows the relation of the ground acceleration amplitude αg/gα vs ωg/p with the latter varying from 1 to 8 [9]. For reasons of comparison in the same plot we have included the relationship αg/gα vs ωg/p when the rigid block is supported by the rigid ground surface [4]. For ωg/p r1.7 the stability of the rigid block regardless of whether the block is supported by the ground or by the tip of the cantilever is practically the same, since for large values of T¼ 2π/ωn the corresponding response is essentially static. In both cases the minimum amplitude ground acceleration (associated with impact) leading to overturning instability is practically identical. For the range of values 1.7 rωg/pr3.2, the rigid block is slightly less stable when supported by the tip of the cantilever with diameter dcol ¼1.0 m than by the rigid ground surface, whereas, for the range of values 3.2 rωg/pr 4.9 the rigid block is more stable when supported by the tip of the cantilever than by the rigid ground surface. For ωg/p¼ 3.5 the minimum amplitude ground acceleration when the rigid block is supported by the tip of the cantilever is 56% greater than the value calculated when the block is supported by the rigid ground surface. For ωg/p4 4.9 (i.e for very

which coincide with those reported by Kounadis[9]. Namely, the values of the eigenfrequencies remain essentially unaffected when the external axial load Mcg is taken equal to zero. Thereafter, with the aid of Eq. (10) one can determine the associated eigenfunction X1, X2, X3 via Eq. (8) as well as Z1, Z2, Z3 via Eq. (12). An iterative procedure is then employed using a range of values of αg [corresponding to ψ via Eq. (27)] in order to calculate Tn(t) (n ¼1,2,3) as a function of time. Subsequently one can establish the total deflection at the tip of the cantilever y0(t)¼ ¼l y(1,t) and its derivatives y00 , ẏ 00 through which one obtains the initial conditions for the rocking response of the rigid block freely supported on the top surface of the column, after its separation from the cantilever top surface. In the following attention is focused on establishing the minimum amplitude ground acceleration αg [which is related to ψ via Eq. (27)] for given values of the excitation frequency ωg. For instance, for ωg/p ¼3 and η ¼0.1 and the above values of the parameters one can find the minimum amplitude ground acceleration αg at any time t during the lateral vibrations of the cantilever. However, more important is the time at which yo and

Fig. 5. Accelerations for ωg/p ¼3.5: of the ground u€ g ðtÞ , of the tip of the column y€ 0 ðtÞ and of the total acceleration applied at the base of the rigid block € ¼ u€ g ðtÞþ y€ 0 ðtÞ[9]. uðtÞ

p /p

yo

= 3.5 no overturning

g

0.1

-0.1 -0.2

overturning

g

1

0.4 2

3

4

5

6

t

rotation yo' of the tip of the column

0.2

returning to the initial (stable) position

rigid block rotation (t)

overturning

-0.2


0.4

89

-0.1

0.1

-0.2

0.2 1

2

3

4

-0.2

5

6

t

-0.4

Initiation of rocking motion

overturning -0.4 -0.6

(t) -0.6

Fig. 6. Numerical solutions predicting the response of a rigid block (with p ¼2.14, α ¼0.25 and ωg/p ¼ 3.5) on top of the column under one-sine pulse ground excitation presented as: (a) θ versus t, (b) θ_ versus t and (c) phase-plane portrait θ_ versus θ [9].

90


p/p

yo 0.1

no overturning

g

overturning

1 -0.1 -0.2

= 6.0

g=

2

3

0.4

4

5

t

0.2

rotation yo' of the tip of the column returning to the initial (stable) position

rigid block rotation (t)

-0.2

-0.1

0.1

overturning -0.2


0.4

-0.4

0.2 1

-0.2

2

3

4

5

t -0.6

overturning

-0.4

Initiation of rocking motion

-0.6 -0.8

-0.8

Fig. 7. Numerical solutions predicting the response of a rigid block (with p ¼2.14, α ¼ 0.25 and ωg/p ¼ 6) on top of the column under one-sine pulse ground excitation presented as: (a) θ vs t, (b) θ_ vs t and (c) phase–plane portrait θ_ vs θ [9].

8

g= 1/sin

important finding that between two geometrically similar columns the taller is more stable than the shorter one. Thus Housner’s observation valid for geometrically similar rigid blocks is extended to the case of columns.Keeping constant λ¼ 40 an increase in height l (e.g. l ¼12 m and dcol ¼1.2 m) implies a more stable column. Keeping constant the height l ¼10 m and decreasing the slenderness ratio (e.g. λ ¼4l/dcol ¼ 40/1.5¼ 2.667) the column becomes also more stable. The results presented in graphical form in Figs. 5–8, as anticipated, are in excellent agreement with those obtained previously [9] using a completely different methodology.

supported on the ground supported on the column :l = 10m supported on the column :l = 15m

10

Slenderness ratio both columns

6

l = 15m

Overturning after impact

4 g/

l = 10m

2 0 0

1

2

3

4

5

6

7

8

g /p.

4. Conclusions Fig. 8. The predicted values of minimum amplitude ground acceleration for overturning instability, αg/αg vs ωg/p [9].

small values of T ¼2π/ωn) the rigid block appears to remain more stable when supported by the rigid ground rather than when supported by the tip of the column. Indeed, for the interval of values of excitation period 0.60 oTo 0.92 the solid green curve, corresponding to the case of the rigid block being positioned by top of the cantilever with a diameter dcol ¼1.0 m, is above the discontinuous curve related to the case of the rigid block supported by a rigid ground surface. Considering the same problem with a column having a larger cross-sectional diameter dcol ¼1.5 m (instead of 1.0 m) with the aid of Fig. 8 one can observe the following: the range of values where the rigid block is appreciably more stable when supported by the tip of the cantilever than by the rigid ground surface is 3.7 r ωg/p r5.7 (instead of 3.2 rωg/p r4.9 associated with the 1.0 m diameter column). For a given block one can find the optimum combination of the values of the slenderness ratio λ of the column and its height l which maximizes the minimum amplitude of ground acceleration. For the problem under discussion λ ¼4l/dcol which for l¼ 10 m with dcol ¼1 m and l¼ 15 m with dcol ¼1.5 m gives λ ¼40. Namely, these two columns are geometrically similar as having the same slenderness ratio λ ¼40. Hence, with the aid of Fig. 8 one can draw the

In this work dealing with the nonlinear rocking instability of free-standing blocks on the top surface of slender elastically supported viscoelastic cantilevers under ground motion, the following findings are the most important for the chosen numerical examples. 1. In this study the actual system of a multi-drum cantilever and a free-standing statue (block) atop is simulated by a system consisting of a monolithic viscoelastic cantilever with an equivalent structural damping analogous to the dissipation of energy of the multi-drum column (due to impact and sliding). The proposed procedure constitutes the best approximate solution which can be established for this highly complex problem of rocking instability. 2. An analytical procedure is presented for the derivation of the equations of motion of the monolithic viscoelastic cantilever and the rigid block supported by its top surface. Then the nonlinear rocking response and the conditions of overturning instability of the freely supported rigid block after its uplift (from the top surface of the vibrating cantilever) are thoroughly discussed. 3. The governing equation of motion of the cantilever-block system is quite simplified after the omission of the negligibly small effect of the axial compression due to the weight of the


4.

5.

6.

7.

8.

9.

10.

rigid block (statue), being compared with the cantilever buckling load. The two modes of overturning instability of the free-standing rigid block on the top surface of the cantilever with or without impact (used in previous analyses), are also applicable herein. Criteria for overturning (rocking) instability of the rigid block associated with minimum amplitude ground acceleration which leads through an unstable equilibrium point with zero angular velocity and zero acceleration to an escaped motion, are properly presented. Such unstable equilibrium point is an inflection point of the curve θ (t) vs t. The absorption of energy (due to the damped bending vibrations) of the viscoelastic cantilever reduces substantially the magnitude of the horizontal ground excitation at the tip of the cantilever. Thus, the remaining amount of the ground excitation is, in general, insufficient to overturn the free-standing rigid block. For a given rigid block (simulating the statue) one can employ an optimum design by determining the best combination of values of the slenderness ratio of the cantilever and its height leading to the maximization of the minimum amplitude ground acceleration which yields overturning instability. For the more important (in practice) intervals of values 3.2 rωg/pr 4.9 (when considering a 1 m diameter column) and 3.7 oωg/p o5.7 (when considering a 1.5 m diameter column) the freely supported rigid block is more stable when supported by the top surface of the cantilever than when supported on a rigid ground. Between two geometrically similar columns (i.e. having the same slenderness ratio) the taller is more stable that the shorter one (extension of Housner's finding for the rigid blocks to slender columns). All results of linearised analysis (for α¼0.25) are confirmed via nonlinear analysis and moreover are in excellent agreement with existing relative results. Clearly on the basis of the closed

91

form linearised solution one can successfully validate any nonlinear numerical scheme.

References [1] Housner GW. The behavior of inverted pendulum structure during earthquakes. Bull Seismol. Soc. Am 1963;53(2):403–17. [2] Spanos PD, Koh A-S. Rocking of rigid blocks due to harmonic shaking. J Eng Mech 1984;110(11):1627–42. [3] Zhang J, Makris N. Rocking response of free-standing blocks under cycloid pulses. J Eng Mech 2001;127:473–83. [4] Kounadis AN. On the overturning instability of a rectangular rigid block under ground excitation. Open Mech J 2010;4:43–57. [5] Kounadis AN, Makris N. (2012) Restoration–preservation in an urban environment and seismic stability of the Statues of Athena and Apollo on the Forefront of the Academy of Athens. Special Issue “Sustainable environment design in architecture, impacts on health”. In: Rassia STh, Pardalos PM (editors), Springer optimization and its applications, vol. 56, Berlin, Heidelberg, New York: Springer; 2012. 277–305. [6] Kounadis AN. Parametric study in rocking instability of a rigid block under harmonic ground pulse: a unified approach. Soil Dyn Earthq Eng 2013;45: 125–43. [7] Kounadis AN. Rocking of slender cantilever consisting of rigid blocks freely supported on each other under seismic excitation. Proc Acad Athens 2012; issue 87A:225–51. [8] Kounadis AN, Papadopoulos GJ, Cotsovos DM. Overturning instability of a tworigid-block system under ground excitation. J Appl Math Mech/Z Angew Math Mech 2012;92(7):536–57. [9] Kounadis AN. Rocking instability of free-standing statues atop slender cantilevers under ground motion. Soil Dyn Earthq Eng 2013;48:294–305. [10] Rogers LG. Dynamics of framed structures. New Work: John Wiley and Sons; 1959. [11] Kounadis AN. 2nd Ed. Dynamics of continuous elastic systems, vol. 1. Athens: Symeon; 1989; 246–9. [12] Kounadis AN. Dynamic response of cantilevers with attached masses. J Eng Mech 1975;101(5):695–706. [13] Drossos V, Anastasopoulos I, Gazetas G. Seismic behaviour of classical columns: an expirimental study. Lab Soil Mech. Res Rep: LSM 2012 (12-01). [14] Shenton HW. Criteria for initiation of slide, rock, and slide-rock rigid-body modes. J Eng Mech 1996;122(7):690–3.