Engineering Structures 40 (2012) 556–565

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Engineering Structures

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Seismic response of stone masonry spires: Analytical modeling

Matthew J. DeJong ⇑Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK

a r t i c l e i n f o a b s t r a c t

Article history:Received 13 September 2011Revised 8 February 2012Accepted 1 March 2012Available online 19 April 2012

Keywords:MasonrySpireRockingEarthquakesAnalytical modelingCollapseStructural dynamics

0141-0296/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.engstruct.2012.03.010

⇑ Tel.: +44 1223 330278; fax: +44 1223 332662.E-mail address: mjd97@cam.ac.uk

Stone masonry spires are vulnerable to seismic loading. Computational methods are often used to predictthe dynamic linear elastic response of masonry towers and spires, but this approach is only applicableuntil the first masonry joint begins to open, limiting the ability to predict collapse. In this paper, analyt-ical modeling is used to investigate the uplift, rocking and collapse of stone spires. General equations forstatic equilibrium of the spire under lateral acceleration are first presented, and provide a reasonablelower bound for predicting collapse. The dynamic response is then considered through elastic modalanalysis and rigid body rocking. Together, these methods are used to provide uplift curves and singleimpulse overturning collapse curves for a complete range of possible spire geometries. Results are usedto evaluate the historic collapse of two specific stone spires.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Masonry structures have been proven vulnerable to earth-quakes, but their assessment remains a challenge. Traditionally,these structures were assessed with geometric methods, eitheranalytical or experimental, to confirm their stability. While thatapproach has proven effective for static loading, dynamic responseduring earthquakes requires further considerations. Modern engi-neers have turned to computational modeling, often modal analy-sis using finite element methods, to analyze the dynamic response.In general, such an approach has sparked a debate about the utilityof finite elements, and specifically linear elastic modal analysis,when evaluating dynamic response of historic masonry structureswhich are discontinuous by nature.

This paper focuses on the dynamic response of the stone ma-sonry spire, a structural typology which is pertinent to this debate.Because spires are tall and slender, elastic modal amplification maybe important. However, because stone masonry often has noappreciable tensile strength (all stones within these structuresare free to separate when normal compressive forces are lost),modal analysis may be limited as significant modal amplificationmay not be realized when contacts change. Regardless, an initialstatic analysis is important because it provides a benchmark forcomparison of dynamic analysis results. The stability of masonryspires under static dead load and wind load is addressed by Hey-man [1] in the context of ultimate load theory, assuming (i) infinite

ll rights reserved.

compressive strength, (ii) zero tensile strength, and (iii) no slidingbetween masonry units. These assumptions provide the foundationfor the geometric analysis methods mentioned previously, butmust be re-evaluated in the case of dynamic loading. Heyman [1]also assumes that octagonal spires can be modeled as conicalshells, and eloquently explains the importance of the commonlyobserved metal ties which anchor the (solid) spire tip to resistoverturning during high winds.

The dynamic response and collapse of masonry spires has beengiven relatively little attention. Previous studies concentrate on theonset of block separation (or cracking) due to elastic response,making use of the finite element modal analysis methods men-tioned above, but do not address the prediction of post-damage re-sponse or collapse. In this study, the dynamic response is evaluatedusing linear elastic beam theory, but is also addressed in the con-text of rocking structures. While the literature on rocking is exten-sive, the dynamics of spires has not been evaluated from thisperspective.

Housner [2] provides the fundamental formulation for investi-gating the rocking response. Zhang and Makris [3] provide a criticalcontribution regarding the response of rocking objects to cycloidalpulses which can dominate earthquake ground motions and gov-ern overturning collapse. More recently, Dimitrakopoulos and De-Jong [4] provide analytical equations for fully non-dimensionaloverturning envelope plots for cycloidal pulses, which are of gen-eral use for any rocking structure with given parameters. Numer-ous others have investigated different aspects of the singlerocking block, but these studies provide the fundamental basisfor the approach taken herein.

M.J. DeJong / Engineering Structures 40 (2012) 556–565 557

DeJong et al. [5] built upon the work of Zhang and Makris [3],using the idea of dominant cycloidal pulses to predict the experi-mental seismic response of discrete block arches. The single pulsewas indeed found to be dominant for several past earthquake timehistories, and single degree of freedom (SDOF) rocking theory waseffectively applied to a more complex multiple block structure. De-Jong and Ochsendorf [6] further evaluated the use of SDOF rockingtheory to predict multiple block response, and evaluated the en-ergy input and rocking amplification into rocking systems.

There are numerous instances of seismic damage to masonryspires, both historically and recently. This paper aims to providea framework to understand this damage and the dynamics of spiresin general, and to predict seismic capacity. In the process, the aimis to evaluate the limits and merits of two analysis approaches:elastic modal analysis and rigid rocking response. A purely analyt-ical investigation is presented herein, while the companion paper[7] investigates the dynamic response of spires using computa-tional (discrete element modeling) and experimental methods.

The analytical modeling approach is divided into three aspects.First, static analysis is used to obtain a reference point for lateralstability. Second, the linear elastic dynamic response is evaluated.Third, the rigid rocking response is considered. Finally, all threeapproaches are used to investigate the partial spire collapses of:(1) Lion’s Walk Congregational Church in Colchester, UnitedKingdom, and (2) Christchurch Cathedral in New Zealand. Aconical shell is assumed to be representative of the octagonal spire(as in [1]), and the three assumptions of ultimate load theory aretaken [1].

2. Static analysis

As a starting point, static analysis is useful to determine theminimum horizontal base acceleration (if applied for infinite dura-tion) necessary to cause overturning of a complete (un-cracked)hollow spire, modeled as a conical shell. For the geometry inFig. 1a, the fraction (kh) of gravitational acceleration (g) requiredfor overturning of the hollow conical shell is:

kh ¼3rb

H¼ 3 tan / ð1Þ

The conical shell has a relatively low center of gravity (H/3), andis therefore more resistant to overturning than a solid rectangularprism. However, assuming masonry structures have no tensilecapacity, diagonal cracks may reduce stability when lateral loads

x

y

H

rb

CM

H/3

λ

mg

(b) (a)

φ

Fig. 1. Definition of maso

are applied [8]. In reality, the location of these cracks may be lim-ited by interlocking of blocks and metal tie rods, but assuming thata diagonal crack can form at any angle b and may initiate at anyheight ho from the top of the spire (Fig. 1c), the fraction of gravita-tional acceleration (g) required for overturning of the cracked hol-low spire is:

kch ¼3rb

H

1p2 g3 � 2

p2 � 23

� �g2 � 2gþ 2

g3 � 2g2 þ 2

!ð2Þ

where g = hc/ho is the normalized crack height (see Fig. 1c).In reality, the tip of many spires is constructed with layers of so-

lid stone in addition to the metal tie rod, so the spire tip is effec-tively solid. Thus, it is also useful to consider the case where thecrack initiates just beneath this solid tip (dotted line in Fig. 1c),again at any angle b. In this case, the fraction of gravitational accel-eration required for overturning of the cracked solid tip spire is:

kcst ¼3rb

H

13 ðg� 1Þ3 þ g t

rb� 1

2p2 g2 þ 1p2 þ 2

3

� �g� 1

� �14 ðg� 1Þ3ð3gþ 1Þ þ g2 t

rb

32 g� 2� � ð3Þ

Throughout this paper, the range of typical spire geometriesused by Heyman [1] will be considered: rb/H = 0.07–0.14 andrb/t = 12–18, where t is the thickness at the base. For this rangeof geometry, Fig. 2 presents the stability predicted by Eqs. (2)and (3). The effect of the solid tip is evident. All plots in Fig. 2aconverge if kc� is normalized by kh, providing the generic curvesin Fig. 2b.

Fig. 2 demonstrates that for completely hollow spires (no solidtip), the stability decreases consistently with the crack height. Notethat for the limiting case of no crack (g = 0), kch = kh. Also note thattheoretically the worst case crack would initiate from near the tipand propagate down to the base. This may not be realistic, as blockinterlocking would limit the crack angle. Regardless, even if thecrack height is half of the entire spire height, the stability is re-duced by �30% compared to the uncracked spire.

The solid tipped spire behaves rather differently. The lateralcapacity is relatively constant whether the crack initiates from50% to 80% of the total height. In this same region, the capacityis significantly lower than that of the hollow cracked spire. Thus,while the solid tip improves stability under wind loading, it canactually decrease stability under earthquake loading. This can onlybe seen when considering the cracked spire, because the un-cracked solid spire (g = 0) would increase stability by a third (seeFig. 2b).

O

hmg

R

α

(c)

O

λc_mcg

mcg hc

β

ho

htip

nry spire geometry.

0

0.2

0.4

0.6

0.8

1

0.5 0.7 0.9 1.1 1.3 1.50

0.2

0.4

0.6

0.8

1

0.1 0.2 0.3 0.4 0.5 0.6

0.07

0.10

0.14

λc-

η

λc- / λh

(a) (b)

Fig. 2. Quasi-static lateral acceleration required for overturning of a cracked conical spire: (a) effect of slenderness (rb/H), (b) all curves converge when normalized by theuncracked lateral capacity kh (solid line = solid spire tip, dashed line = hollow spire tip).

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1

Mode 1

Mode 2

Mode 1 (approximate)

y H

( )x y

Fig. 3. The first two bending mode shapes of a conical shell (Eq. (9)), and theapproximate first mode shape (Eq. (10), k = 2.2).

558 M.J. DeJong / Engineering Structures 40 (2012) 556–565

3. Elastic modal analysis

The dynamic response of masonry structures involves twostages: an initial elastic stage, during which the entire structure re-mains in compression, followed by a ‘rocking’ stage, during whichmasonry units separate and regain contact. For masonry spires,which are inherently slender, neither of these stages can be com-pletely ignored. Low amplitude earthquakes could potentiallycause modal amplification which could lead to minor damage,while large amplitude earthquakes could cause significant rockingwhich could lead to complete collapse. The elastic stage will be ad-dressed first, followed by the rocking stage in the next section.

3.1. Natural frequencies and mode shapes

Masonry spires are often situated above stocky (stiff) masonrystructures, so only the dynamics of the spire itself are initially con-sidered. The elasticity of the sub-structure could be included if de-sired. Due to the slender nature of spires, Euler–Bernoulli beamtheory will be used to estimate natural frequencies and modeshapes. The mass per unit height, m(y), and bending stiffness,EI(y), of the spire can be written as:

mðyÞ ¼ 1� yH

� �mb

EIðyÞ ¼ 1� yH

� �3EIb

ð4Þ

where mb is the mass per unit height at the base, Ib is the secondmoment of area at the base, E is the Young’s Modulus, and y isthe vertical coordinate with origin at the spire base (see Fig. 1a).

As written in Eq. (4), the mass and bending stiffness vary simi-larly to the cantilevered solid wedge beam analyzed by Nagulesw-aran [9], who derived the natural frequencies and mode shapes fornumerous non-uniform beams with varying boundary conditions.Assuming that the spire can be modeled as a cantilevered beam,its natural frequencies are found by solving the following [9]:

X1ð1ÞdX2ð1Þ

dY� dX1ð1Þ

dYX2ð1Þ ¼ 0 ð5Þ

where:

X1ðYÞ ¼ Y þ lY3

4�32 �2þl2Y5

6�52 �42 �32 �2þl3Y7

8�72 �62 �52 �42 �32 �2þ � � �

X2ðYÞ ¼ Y þ lY2

3�22 �1þl2Y4

5�42 �32 �22 �1þl3Y6

7�62 �52 �42 �32 �22 �1þ � � �ð6Þ

where Y ¼ 1� yH, and:

l ¼ mbx2nH4

EIbð7Þ

The natural frequencies of the system can then be written as:

xn ¼

ffiffiffiffiffiffiffiffiffiffiffiffiX2

nEIb

mbH4

sð8Þ

where Xn = 5.32, 15.2, and 30.0 for the first three modes, and onlythe first four terms in Eq. (6) are considered. Naguleswaran [9] pro-vides nearly an identical solution by including the first six terms,but such accuracy is unnecessary for the problem at hand.

The mode shapes can then be found by solving [9]:

�xðYÞ ¼ X1ðYÞ �dX1ð1Þ

dXX2ðYÞ

�dX2ð1Þ

dYð9Þ

The fundamental mode shapes for the first two modes are de-picted in Fig. 3. Alternatively, Lord Rayleigh’s principle can be usedto estimate the mode shape and compute the corresponding natu-ral frequencies. Assume a mode shape of the form:

�xðyÞ ¼ yH

� �kð10Þ

where �xðyÞ is the horizontal modal translation at height y, and k is aconstant. According to Lord Rayleigh’s principle, the fundamentalnatural frequency is approximated by:

�x1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR H0 EIðyÞ d2�x

dy2

� �2dyR H

0 mðyÞð�xðyÞÞ2dy

vuuut ð11Þ

The best approximation of the mode shape considered in Eq.(10) was obtained for k = 2.2, which minimizes the fundamentalfrequency �x1. The corresponding mode shape compares reason-ably well with the actual mode shape derived by Naguleswaran

d

d /4

Elevation View

b

b / 3

Resultant Force

Section A-A

A A

(b) (a)

Fig. 4. (a) Assumed linear stress distribution for the maximum moment where tension does not occur, and (b) the location of the corresponding resultant for two differentcross section shapes.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8

0.07

0.10

0.14

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4

0.07

0.10

0.14

Sa [m/s2]

h / H

Sa / (gλh)

(a) (b)

Fig. 5. (a) Spectral acceleration required to cause tension or loss of contact at a given height (h/H) of an uncracked hollow spire, and (b) normalization of (a).

M.J. DeJong / Engineering Structures 40 (2012) 556–565 559

[9] (Fig. 3). Modal analysis using Eqs. (10) and (11) can now be ap-plied to determine the point at which elastic oscillation may beginto cause damage to the structure.

3.2. Modal analysis

The modal properties derived Section 3.1 can be used to deter-mine the point at which tensile forces within the structure wouldbe required to prevent loss of contact between blocks. For typicalsolid rectangular cross-sections (as in Fig. 4b, right), the resultantforce must act within the ‘middle-third’ of the cross-section to pre-vent tensile stresses (assuming a linear stress distribution). This‘middle-third rule’ is well-documented for masonry structures.However, for a hollow circular cross-section, the resultant forcemust act within the middle half of the cross-section to prevent ten-sion, again assuming a linear stress distribution. In other words,the resultant will act through the quarter point at the limit whentension would begin to occur (see Fig. 4). Thus, the maximum spec-tral acceleration which does not cause tension at a given height of

0

0.2

0.4

0.6

0.8

1

0 2 4 6

h / H

hollow tip, uncracked

solid tip, cracked

Sa [m/s2]

(a) (

Fig. 6. (a) Comparison of the hypothetical modal uplift of an uncracked hollow spire andand the quasi-static uplift (Eq. (3) and Fig. 2b) predictions for a diagonally cracked solid

the spire, occurs when the overturning moment equals the resist-ing moment about the quarter point.

Considering the first mode response of an uncracked hollowcone, the overturning moment at a height h above the base is:

Mover ¼ CSa

Z H

hmðyÞ�xðyÞðy� hÞdy ð12Þ

where C is the typical modal participation factor:

C ¼R H

0 mðyÞ�xðyÞdyR H0 mðyÞð�xðyÞÞ2dy

ð13Þ

Additionally, the resisting moment about the quarter point atheight h is:

Mres ¼rðhÞ

2

Z H

hmðyÞdy ð14Þ

where r(h) is the radius at height h. From Eqs. (12)–(14), the spectralacceleration required to cause tension at a given height h is:

0

0.2

0.4

0.6

0.8

1

0 2 4 6

Sa [m/s2]

quasi-static uplift

modal uplift

b)

a diagonally cracked solid tip spire; (b) comparison of the hypothetical modal uplifttip spire.

(a) (c) (b)

H

htip

ho

hc

O

O

β

α

R

βmax

R

O

α

c.m. O’

θ < 00 < θ

0 < θ

Fig. 7. Definition of geometry for: (a) the rocking conical shell, (b) diagonal cracking below the spire tip, and (c) rocking of the cracked spire.

560 M.J. DeJong / Engineering Structures 40 (2012) 556–565

SaðhÞ ¼Mres

CR H

h mðyÞ�xðyÞðy� hÞdyð15Þ

The spectral acceleration resulting from Eq. (15) is plotted inFig. 5 for the range of spire slenderness under consideration. Theeffect of spire thickness is not shown because it is negligible. InFig. 5b, the results are normalized by kh (Eq. (1)), which effectivelyeliminates the geometric differences. The vertical line atSa/(gkh) = 0.5 represents the quasi-static horizontal accelerationnecessary to cause tension in the structure if no modal amplifica-tion is considered. Thus, Fig. 5b demonstrates, as expected, thatmodal amplification can significantly decrease the base accelera-tions needed to cause tension in the structure, and tension will firstoccur between 80% and 90% of the total height.

The analysis above provides an important acceleration limit, be-low which tension and therefore damage will not occur. Unfortu-nately, this is as far as modal analysis can take us, and this maybe well below the acceleration which could cause collapse. Forexample, even if it was incorrectly assumed that the modes didnot change despite loss of contacts within the structure (and thatdiagonal cracking could not occur), then uplift would begin to oc-cur at twice the acceleration values shown in Fig. 5b (see Fig. 6a).Further, one could consider the limiting uplift for a cracked spirewith a solid tip. In this case, the assumption that the mode shapesare unchanged until a diagonal crack occurs and the spire uplifts isstill not valid. However, the results provide another benchmark forcomparison. Assuming that the top 15% of the spire is solid andthat rb/H = 0.1, the spectral acceleration required to cause upliftis compared to the result for the uncracked hollow spire in

Pul

se a

ccel

erat

ion,

Ap

[g]

Pulse duration, Tp [s]

No impact collapse

One impact collapse

No rocking

Rocking and Recovery

Governing Failure Curve

(a)

Fig. 8. Response of rocking structures to a sin

Fig. 6a. The results indicate that the cracked solid tip slightly de-creases the spectral acceleration required for uplift, but it signifi-cantly reduces the height at which overturning would occur.

Finally, it is useful to compare the hypothetical modal resultswith the quasi-static uplift solution (Eq. (3) and Fig. 2b) for thediagonally cracked solid tip spire. Fig. 6b demonstrates that modalamplification, if it were possible until the instant of uplift, wouldonly slightly reduce the minimum acceleration required to causecollapse, and collapse would occur at approximately 50–70% ofthe spire height. However, it must be noted that the quasi-staticsolution considers the ground motion directly, while the modal re-sponse considers the spectral acceleration, which will vary consid-erably depending on the (elastic) properties of the structure.

It should be re-iterated that the results in this section consideronly the first mode response of the structure. While the first modewill dominate the response in many cases, higher modes could beincluded in a similar fashion if necessary.

4. Rigid rocking analysis

If the earthquake loading induces a large enough response, thespire would uplift and begin to rock, and the elastic natural fre-quencies would be completely altered. In this section, the abilityof rocking to cause collapse is investigated.

4.1. Analytical formulation

First, consider a rigid conical shell on a rigid foundation. Theconical shell will begin to rock (Fig. 7a) when the overturningmoment exceeds the resisting moment, which occurs at a base

0

2

4

6

8

10

0 1 2 3 4

increasing cv

ppT

tanp

A

gα

(b)

gle sinusoidal base acceleration impulse.

0

2

4

6

8

10

hollowsolid

p

h tip

0

2

4

6

8

100 5 10 0 2 4

2 m tip (hollow)

2 m tip (solid)

4 m tip(hollow)

4 m tip (solid)

h o

p

(a) (b)

Fig. 9. Frequency parameter for a rocking spire with H = 10m, rb/H = 0.1, and rb/t = 15 where the spire cracks: (a) horizontally at any height, (b) diagonally below the tip.

M.J. DeJong / Engineering Structures 40 (2012) 556–565 561

acceleration of acrit = khg, where kh is defined in Eq. (1). Once rock-ing commences, the response can be analyzed with a similar ap-proach to that which Housner [2] used for the rigid rockingrectangular block. Assuming that the cone rocks back and forthon its base (the orientation of positive rocking, h > 0, is shown inFig. 7a), and that spinning of the cone about its vertical axis doesnot occur, the equations of motion are:

I0€hþMgR sinðþa� hÞ ¼ �M€ubR cosðþa� hÞ ! h > 0

I0€hþMgR sinð�a� hÞ ¼ �M€ubR cosð�a� hÞ ! h < 0

ð16Þ

where M is the total mass of the cone, R, a, and h are defined inFig. 7a, IO is the mass moment of inertia about the rotation pointof the cone, and €ub is the horizontal base acceleration. For a com-plete hollow cone (not diagonally cracked) which rocks about pointO in Fig. 7a:

IO ¼54

Mr2b þ

16

MH2 ð17Þ

Assuming small angles, Eq. (16) can be rewritten in the form:

€h� p2h ¼ �p2 €ubg þ a� �

! h > 0

€h� p2h ¼ �p2 €ubg � a� �

! h < 0ð18Þ

where p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMgR=IO

pis the frequency parameter of the block. While

the rocking cone (or block) has no fundamental natural frequencybecause the rocking frequency is dependent on the rocking ampli-tude [2], the frequency parameter is of interest. It represents thein-plane pendulum frequency of the cone if it was hung from itsrotation point (point O in Fig. 7a), and is of great importance tothe rocking response.

0

0.2

0.4

0.6

0.8

10 2 4

0

0.2

0.4

0.6

0.8

10 0

p2ho / g

h c /

h o

α

Fig. 10. Rocking parameters for a spire with rb/H = 0.1 and rb/t = 15 which c

For the complete hollow cone, the frequency parameter isdescribed by:

p2 ¼ gH

ffiffiffiffiffiffiffiffiffiffiffiffiffi19þ j2

q54 j2 þ 1

6

� � ð19Þ

where j = rb/H. A similar equation can be written for a solid cone:

p2 ¼ gH

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

16þ j2q2320 j2 þ 1

10

� � ð20Þ

Still following the formulation of Housner [2], the impact whichoccurs each time the block returns to its initial position can bemodeled by a coefficient of restitution, cv, defined as the ratio ofthe angular velocities before and after impact:

cv ¼ 1�MR2

IOð1� cos 2aÞ ð21Þ

For the hollow cone, the coefficient of restitution is defined as:

cv ¼ 1�19þ j2

54 j2 þ 1

6

ð1� cos 2aÞ ð22Þ

Again, a similar equation could be written for the solid cone. Asexpected, the coefficient of restitution is a function of the slender-ness only, not the scale of the spire.

Eqs. (18)–(22) now describe the response of the conical shell tohorizontal base motion in general. However, because ground mo-tion impulses often govern overturning collapse, this study will fo-cus on the response of the spire to single period sinusoidal baseimpulses. Investigating short duration rocking is also appropriatebecause numerous consecutive rocking cycles would degrade the

.2 0.4

0

0.2

0.4

0.6

0.8

10.7 0.8 0.9 1

cv

racks diagonally below a solid (solid line) or hollow (dashed line) tip.

0

0.2

0.4

0.6

0.8

10 2 4

0.07

0.10

0.14

0

0.2

0.4

0.6

0.8

1

0 2 4

0

0.2

0.4

0.6

0.8

1

0 2 4

0

0.2

0.4

0.6

0.8

10 0.2 0.4 0.6

0

0.2

0.4

0.6

0.8

10.6 0.8 1

p2ho / g

h c /

h o

α / φ (1- cv) / (κ sinα )

p2ho / g

h c /

h o

α cv

0

0.2

0.4

0.6

0.8

10 5 10

Fig. 11. Rocking parameters for spires of varying rb/H which crack diagonally below a solid tip.

562 M.J. DeJong / Engineering Structures 40 (2012) 556–565

integrity of the spire, reducing the validity of the rocking conicalshell model. The collapse of a spire due to repeated accelerationimpulses may be important for earthquakes which do not have adominant acceleration impulse, but this would cause local slidingand dislocation of blocks and is not feasible to analyze analytically.Computational methods which could deal with this type of loadingare considered in the companion paper [7].

4.2. Impulse rocking response

The overturning of the rigid rocking block due to single periodcycloidal base acceleration impulses was defined by Zhang andMakris [3], among others. The single period sinusoidal accelerationimpulse is more detrimental than a single period cosine impulse ofsimilar characteristics, so only the single sinusoidal impulse re-sponse will be considered here. Dimitrakopoulos and DeJong [4]defined the non-dimensional collapse envelopes for a variety ofdifferent coefficients of restitution, and these envelopes will be uti-lized in the plots throughout this paper.

Fig. 8a presents the collapse envelope for a single rocking blockin conceptual (dimensional) form. Four possible responses to a sin-gle sinusoidal impulse of base acceleration are theoretically possi-ble, and the regions are labeled accordingly in Fig. 8a. For largeamplitude impulses: the block immediately overturns without im-pact for long duration impulses; the block rocks in one direction,impacts and fails in the other direction for medium duration im-pulses (one impact collapse), and the block rocks but does notoverturn for short duration impulses (rocking and recovery). Forsmall amplitude impulses, rocking does not occur.

Fig. 8b presents the collapse envelopes in non-dimensional form.Specifically, three dimensionless groups define the response [4]:

Ap

g tan a; pTp; cv ð23Þ

where Ap and Tp are the sinusoidal impulse amplitude and duration,respectively. The only geometric parameters necessary to com-pletely define the response are p, a, and cv. In general, it is criticalto note the relationship between scale and the frequency parameterp for rocking structures:

p / 1ffiffiffiffiffiffiffiffiffiffiffiscalep ð24Þ

In other words, as the scale decreases, p increases, and thedimensional collapse envelope (Fig. 8a) would shift to the left,making the block more vulnerable to impulse overturning. Onthe contrary, as the scale increases, p decreases, and the block ismore stable. This confirms the well documented fact that smallblocks are less stable than large blocks of equal geometry (e.g. [2]).

4.3. Rocking of a conical shell

Consider first a conical spire which is not permitted to crackdiagonally. Such a spire has the remarkable characteristic that, ifthe spire is permitted to crack horizontally at any height, then asingle spire contains a wide range of conical shells of different scalebut exactly the same geometric proportions (if the thickness is ig-nored). For example, consider a 10 m tall spire with rb/H = 0.1 andrb/t = 15. Regardless of the height of the rocking portion above thehorizontal crack, a and cv would be unchanged, but the frequencyparameter (Eq. (19)) would vary as shown in Fig. 9a. Recall that dy-namic stability decreases as p increases. Thus, while a given im-pulse may not be long enough to overturn the entire spire, itmay certainly overturn the top portion. The frequency parameter

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

λ

hc / ho

10 m

2 m

Puls

e A

ccel

erat

ion,

Ap

[g]

Pulse Duration, Tp [s]

cs,min

Fig. 12. Geometry and collapse envelopes for a spire (H = 10 m, htip = 2 m, rb/H = 0.1) with a varying diagonal crack height (hc/ho) below a solid tip.

M.J. DeJong / Engineering Structures 40 (2012) 556–565 563

for a solid cone is also plotted in Fig. 9a, as the spire tip may be so-lid and may overturn alone. It is important to note that a squaremasonry tower is different. If it cracked horizontally above thebase, then the aspect ratio would increase making the structuremore stable, even though the frequency parameter would increase,causing the dynamic stability to reduce.

4.4. Rocking of a cracked conical shell

For masonry structures, diagonal cracking must be allowed to oc-cur below the tied tip, which may be solid or hollow (see Fig. 7b).The diagonal crack is assumed to initiate at the base of the tiedtip, but may propagate downward at any angle less than bmax. (Inreality, this angle would be limited by block interlock as mentionpreviously). In this case, Eqs. (18) and (21) can still be used to pre-dict the dynamic response, assuming that rocking in each directionis defined by symmetric mechanisms (see Fig. 7c). Perfect mecha-nism reflection is extremely unlikely, but effectively captures thegeneral response (further considered in the companion paper [7]).Similarly, the use of a coefficient of restitution to estimate the en-ergy dissipated is only a rough approximation of a very complicatedtransition of mechanisms. The actual transition would likely dissi-pate more energy than assumed by this impact formulation.

For the diagonally cracked spire, all parameters (p, a, cv) varydepending on the height of the rocking portion ho; the equationsfor these parameters become cumbersome so they are omitted.Consider the frequency parameter first. In Fig. 9b, the height ofthe rocking portion of the spire is plotted with respect to the fre-quency parameter. The spire geometry from Section 4.3 is as-sumed, and in addition a metal rod is assumed to tie the top 2 or4 m of the spire. The tie rod effectively limits the frequencyparameter, and therefore prevents overturning of the tip duringearthquakes, not just high winds, as expected. Fig. 9b also demon-strates that p is maximum when the diagonal crack extends only ashort distance below the base of the tied tip.

The effect of the crack height hc on a and cv must also be consid-ered. In Fig. 10, the normalized crack height is plotted with respectto all three parameters for rb/H = 0.1. Note that the frequency

Table 1Case study geometries.

Tower

Height (m) Width (m)

Lion’s Walk 18 4.2Christchurch 27 8.2

parameter is presented in non-dimensional from so a single curverepresents all combinations of tip and crack heights, and that theprevious results for the horizontally cracked cone are given byhc/ho = 0. Fig. 10 demonstrates the counteracting effects causedby a crack initiating lower down the spire. While an increase in pshifts the collapse envelope (Fig. 8a) to the left, decreasing stabil-ity, an increase in a shifts the collapse envelope up, increasingstability. Thus, while the frequency parameter indicates minimumstability when the crack height is 20–30% of the entire rocking por-tion, both a and cv indicate minimum stability when rocking occursabout the base of the entire spire (maximum hc/ho).

These counteracting effects must be considered to determinethe minimum capacity, but first it is useful to consider alternatespire geometries. Fig. 11 (top row) presents the same rockingparameters for a range of rb/H. The slenderness of the entire spirehas a considerable effect on the values of a and cv of the rockingportion, as expected, but has a remarkably small effect on the fre-quency parameter. In addition, varying the thickness of the spirehas a minimal effect on these curves (not shown). Thus, a and cv

were normalized to develop single curves which define the rockingparameters for a spire of any geometry. The result is shown inFig. 11 (bottom row). Note that cv is not easily normalized forany spire geometry, but can be directly calculated from the otherdimensionless parameters, using an alternate form of Eq. (21):

cv ¼ 1� 2MRIO

rb sin a ¼ 1� 2p2ho

grb

hsin a ð25Þ

Now that the rocking parameters have been determined for anyspire, the counteracting effects of a larger or smaller crack heightmust be reconciled. By plotting collapse envelopes similar toFig. 8a for a range of crack heights hc, a governing collapse envelopefor the entire spire can be generated. Consider again the 10 m spirewith rb/H = 0.1 and with a solid tip of 2 m. The possible cracking ofthe spire is depicted in Fig. 12, along with the correspondingdimensional collapse envelopes. Note that only the bottom portionof the governing single impact collapse curves are plotted forclarity.

Spire

Wall thickness (m) Height (m) rb/H

0.6 17 0.091.25 27 0.14

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

λ

Puls

e A

ccel

erat

ion,

Ap

[g]

Pulse Duration, Tp [s]

hc / ho

Fig. 13. Collapse envelopes for Lion’s Walk Congregational Church for varyingdiagonal crack heights (hc/ho) below the tied tip.

564 M.J. DeJong / Engineering Structures 40 (2012) 556–565

For a horizontal crack just below the solid spire tip (hc/ho = 0),the collapse envelope significantly overestimates the capacity ofthe diagonal cracked spire. The new collapse envelope for the spireshould instead be defined as the minimum envelope of all thecurves in Fig. 12b. Note that hc/ho = 0.2 provides the maximumshift of the curve to the left (due to maximum p), hc/ho = 0.8 pro-vides a maximum shift downward (due to minimum alpha), andhc/ho = 0.4 provides the best single curve minimum of all the com-bined collapse envelopes.

5. Case studies

The formulations above will now be used to evaluate the ob-served damage to two different spires. In both cases, the spire tipcollapsed, and the spires were reconstructed. However, the differ-ence in epicentral distance likely led to a different response in eachcase.

5.1. Lion’s Walk Congregational Church, United Kingdom

Lion’s Walk Congregational Church was completed in 1863. In1884, an earthquake caused the top 20 ft of the spire to collapse[10]. The epicentral distance to the church was less than 5 km,and the earthquake was relatively short. The earthquake damagewas recorded by Dr. Alexander Wallace, who clearly describeswhat is effectively the governing single impact mode of rockingcollapse [11]:

‘‘next they observed the fall of chimneys and the fall of the spire ofthe Congregational Chapel. It was clearly noticed that the fall ofchimneys to the south-west proceeded the fall of the spire; in fact

0

0.2

0.4

0.6

0.8

1

0

Puls

e A

ccel

erat

ion,

Ap

[g]

(a) (b)

Fig. 14. Christchurch Cathedral: (a) 1888 earthquake damage (photo acknowledgement:envelopes for varying diagonal crack heights (hc/ho) below the tied tip.

they fall one after another; those furthest to the south-west fellfirst. Moreover the debris of the spire and of the chimneys nearlyall over Colchester has tumbled on the north east sides of the build-ings, pointing to the conclusion that something like a wave ofupheaval was felt approaching from the south-west, and causinga fall in the opposite direction.’’

The approximate geometry of the tower and spire are given inTable 1. According to reconstruction drawings, the top 7–8 ft ofspire was tied together with a metal rod. The static modeling inSection 2 (Eq. (3)) predicts that overturning could occur at constantlateral accelerations as low as 0.16 g. However, it is reiterated thatthis may be limited by stone interlock and the duration of theacceleration, for which the dynamics must be considered.

Assuming that the tower beneath the spire is relatively stiff dueto its thicker walls and buttressing, and that the majority of theelastic oscillation occurs in the spire, Fig. 5a indicates that tensionwould begin to occur at spectral accelerations as low as 0.08 g, atapproximately 10 ft (3 m) from the top. Note that the spectralacceleration could be considerably higher than the base accelera-tion. Again note that higher acceleration would be necessary toinitiate rocking. Assuming conservatively that the mode shapedoes not change after damage, then uplift would occur atSa � 0.16 g, which matches well with the static solution despite acompletely different mechanism.

Finally, the rocking analysis in Section 4.4 yields the governingenvelope of collapse curves shown in Fig. 13. The results indicatethat an acceleration pulse with a magnitude of 0.3 g and a periodof 0.5 s could have overturned the spire. This acceleration pulsewould need to occur at the height of overturning, not at the base,and therefore could result from elastic amplification in the structurebelow. While earthquake records are not available, such a pulse isfeasible given the small epicentral distance. Thus, the results con-firm observational evidence that a single dominant pulse could havecaused overturning. Furthermore, the required pulse for overturn-ing could have been smaller if local damage and displacementsoccur (see [7]), and if vertical ground motion were considered.

5.2. Christchurch Cathedral, New Zealand

Images of the severe damage to Christchurch Cathedral wereprolific after the recent earthquake on February 22, 2011, whichcaused the complete collapse of the spire. However, this was notthe first seismic event which damaged the spire. The top 7.8 m ofthe stone spire collapsed in the Amuri earthquake of September1, 1888 [12] (Fig. 14a). The spire was rebuilt, and again lost itstip in the Cheviot earthquake of November 16, 1901 [12]

1 2 3 4

0

0.2

0.4

0.6

0.8

λ

Pulse Duration, Tp [s]

hc / ho

Alexander Turnbull Library, Wellington N.Z., Ref. No. PAColl-7985-72), (b) collapse

M.J. DeJong / Engineering Structures 40 (2012) 556–565 565

(Fig. 14a). Following the Cheviot earthquake, the top portion of thespire was rebuilt using Australian hardwood sheathed in copper,with a counterweight hanging inside. Thus, the spire whichcollapsed in 2011 was not actually completely stone, but a combi-nation of stone and timber.

The cathedral was originally constructed between 1864 and1881, when the spire was completed [12]. The original designerswere concerned about earthquake loading, revising the designmultiple times with earthquakes in mind [13]. In the end, the spirewas constructed of stone, with the top 27 ft tied together with me-tal rods [12]. An openwork iron cross stood atop the spire and wasalso anchored by the rods. The approximate geometry of the spireand tower is summarized in Table 1. The damage to the spire dur-ing the 1888 Amuri earthquake is considered here (Fig. 14a). Theepicenter of the earthquake was approximately 100 km northwestof Christchurch, where the shaking lasted 40–50 s [14].

The static modeling predicts that overturning could occur atconstant lateral accelerations as low as 0.26 g. The elastic modalanalysis Section 3.2 (Fig. 5a) indicates that tension would beginto occur at spectral accelerations as low as 0.11 g, at approximately13 ft (4 m) from the top. However, this maximum occurs in thepre-tensioned (tied) portion of the spire, so loss of contact wouldfirst occur at the base of the tied portion, at Sa � 0.13 g. Again(incorrectly) assuming that the mode shape does not change afterdamage, then uplift would occur at Sa � 0.25 g.

The rocking analysis in Section 4.4 yields the governing enve-lope of collapse curves shown in Fig. 14b. Fig. 14a does indicatediagonal cracking of approximately hc/ho � 0.25, which is close tothe hc value which corresponds to the maximum p value. However,a considerable acceleration pulse with a period above 1.5 s wouldbe required to overturn the spire. It is unlikely that a single accel-eration pulse of this magnitude occurred at �100 km from the epi-center. Thus, if the metal tie was not comprised, it would haveeffectively prevented overturning due to direct rocking. Instead,due to the longer epicentral distance and earthquake duration, itis possible that modal amplification led to continued rocking justbeneath the tied portion, which may have caused local displace-ment and damage and led to collapse. Local displacements anddamage due to continued rocking can better be considered throughcomputational methods, as considered in the companion paper [7].

6. Conclusions

Predicting collapse of masonry spires requires investigation ofboth elastic oscillation and rocking. Considering pure elastic re-sponse and pure rigid body rocking provide important bench-marks, and consideration of the uncoupled interaction betweenthese provides further insight to actual dynamic behavior anddamage. In particular:

� Static analysis provides an effective lower bound of the acceler-ations which could cause collapse, and an important benchmarkfor dynamic analyses.

� Modal analysis was used to identify the spectral acceleration atwhich normal contact is lost during elastic oscillation, and thecritical height at which this occurs. Assuming a linear stress dis-tribution, this occurs when the resultant exits the middle-halfof the cross section. This is well below the acceleration at whichfull uplift of a portion of the cone would occur.� If a spire cracks diagonally during an earthquake, a portion of

the spire may begin to uplift. The slenderness and scale of thecracked portion may vary considerably, and have counteractingeffects for overturning. Thus, numerous rocking envelopes arerequired to predict the possible overturning of a spire due to aprimary pulse which may dominate an earthquake groundmotion.� The case studies indicate that a single impulse could feasibly

have caused direct overturning of the spire of Lion’s Walk Con-gregational Church, but the metal ties in Christchurch Cathe-dral, if effective, likely would have prevented directoverturning from rocking without significant local damage.

The analytical rocking model herein is limited when long dura-tions of shaking cause local damage and dislocations to individualstones. Computational methods may provide insight into rockingamplitudes which will cause local displacements that compromisethe structure. However, while these methods may be useful forspecific case studies where site-specific material properties, stere-otomy, and geometry are known, the analytical formulation hereinprovides a generalized framework to understand and predict theseismic response of stone spires.

References

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earthquakes. Bull Seismol Soc Am 1963;53(2):403–17.[3] Zhang J, Makris N. Rocking response of free-standing blocks under cycloidal

pulses. ASCE J Eng Mech 2001;127(5):473–83.[4] Dimitrakopoulos EG, DeJong MJ. Revisiting the rocking block: Closed form

solutions and similarity laws, Proc of the Royal Society A. http://dx.doi.org/10.1098/rspa.2012.0026.

[5] DeJong MJ, De Lorenzis L, Ochsendorf J. Rocking stability of masonry arches inseismic regions. Earthquake Spectra 2008;24(4):847–65.

[6] DeJong MJ, Ochsendorf JA. Dynamics of in-plane arch rocking: an energyapproach. Proc ICE – Eng Comput Mech 2010;163(3):179–86.

[7] DeJong MJ, Vibert C. Seismic response of stone masonry spires: computationaland experimental and modeling. Eng Struct 2012;40:566–74.

[8] Ochsendorf JA, Hernando JI, Huerta S. Collapse of masonry buttresses. ASCE JArchitect Eng 2004;10(3):88–97.

[9] Naguleswaran S. A direct solution for the transverse vibration of Euler–Bernoulli wedge beams and cones. J Sound Vib 1992;172(3):289–304.

[10] Blaxill EA. A history of the Lion Walk Congregational Church, Colchester, 1642–1937. Liverpool: Benham; 1938.

[11] Hanning P. The great English earthquake. London: Robert Hale Limited; 1976.[12] McKenzie GM. The history of Christchurch Cathedral, New

Zealand. Christchurch: Alex Wildey; 1931.[13] Lochhead I. A dream of spires: Benjamin Mountfort and the gothic

revival. Christchurch: Canterbury University Press; 1999.[14] McSaveney E. Historic earthquakes – the 1888 North Canterbury earthquake.

Te Ara – the Encyclopedia of New Zealand; 2009. <http://www.TeAra.govt.nz/en/historic-earthquakes/4>.