Seismic Non-Linear Analysis of Damaged Historic

Buildings : Cathedral of the Blessed Sacrament,

Christchurch

K. Ip, J. Lester & A. Brown

Opus International Consultants Ltd, Christchurch, New Zealand.

2016 NZSEE

Conference

ABSTRACT: There is currently no established guidelines and only limited empirical

methods to predict the structural behaviour of historic buildings constructed with stone

masonry and an unreinforced concrete core. The traditional finite element method (FEM)

has its own limitations on modelling element separation as it cannot accurately simulate

the highly nonlinear behaviour of non-homogeneous structural connections (i.e. mortar

joints, un-bonded beam-column joints), the final stage of the softening process for brittle

materials, rocking behaviour and building collapse. This paper presents an approach to

combine the advantage of the continuum method (i.e. FEM) and the discrete method by

using constitutive material models and contact surface algorithms available in the

numerical multi-physics simulation software, LS-DYNA. The study of a damaged

Category 1 listed heritage building, the Cathedral of the Blessed Sacrament in

Christchurch, was used to demonstrate the analysis methodology and provide results from

evaluating the damaged cathedral structure. Using pushover analyses and NLTHA, the

paper will discuss the applicability of using this modelling approach to accurately predict

the performance of such structures.

1 INTRODUCTION

The performance of unreinforced masonry buildings has been widely discussed after Canterbury

earthquake (J. Ingham and M. Griffith, 2011). However, the majority mainly focus on observed

damaged and strengthening strategies which does not cover the evaluation of damaged existing

structure. The Christchurch Cathedral of the Blessed Sacrament is a typical example which has

sustained considerable damage from the Christchurch earthquake sequence, moreover, the remaining

structure had a certain amount of physical change during the process of removing the dome and bell

towers (J. Lester, A. Brown and J. Ingham 2012). These changes have affected the structural integrity

of the building which increase the potential risk of collapse for the existing structure. Without a

suitable approach to assess the performance of the damaged structure, it is difficult to make informed

decisions on whether and how the damaged cathedral can be repaired.

Experimental methods can be used to determine the failure mechanism of the materials, but the global

performance and collapse mechanism of the whole building cannot be estimated by non-destructive

testing. Also, the high cost of large scale structure testing is a prohibitive. Some empirical methods

have been developed to evaluate the local collapse mechanisms of historic masonry buildings in

Europe (D’Ayala & Speranza, 2002). These procedures are a simplified method to predict the specific

mechanism of local failure or overturning of a portion of structure. However, they are not fully

applicable for evaluation of a structure that has sustained considerable damage during an earthquake.

In contrast, the availability of high performance computers allows the numerical simulation to be

carried out efficiently and at relatively low cost. The development of advanced multi-physics

simulation software has enabled the complex structural dynamics to be solved with higher confidence

of correct prediction. Nevertheless, a sophisticated model is difficult to create and the verification of

these high level analyses needs to be reviewed by experienced analysts and engineers. In this paper,

the detailed analysis of the cathedral was used as an example to demonstrate the analysis methodology

and techniques for the evaluation of a damaged historic building.

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2 METHOLDOLOGY

The brittle behaviour and local stability of structural components are the main concerns for the

cathedral. The structural member connection allows separation, sliding and rocking in the joint

interface. The uncertainty of dynamic response also amplified the level of complexity required to

assess the building capacity and risk of collapse under earthquake loads. The continuum method and

discrete method are the most popular approaches and have been widely used in numerical simulation

of masonry structures (Wolfram Jäger, Tammam Bakeer and Peter Schöps 2009).

The finite element (FE) continuum based on plastic theory has a long history of development. A lot of

research has been done on the material constitutive models and the results are compared against

laboratory testing. This gives a higher accuracy and reliability for macro level modelling, which saves

analysis time and computation power. However, this approach is unable to simulate the final stage of

the softening process in the materials, i.e. the full separation of the continuum.

The discrete element methods (DEM) such as contact surface, smooth particle hydrodynamic (SPH) or

applied element method (AEM) are a relatively new discipline in numerical analysis. These methods

are capable of modelling the discontinuous medium by applying a surface interaction law between the

elements. This approach is suitable for collapse analysis, however, extensive detailed meshing, i.e.

modelling the bricks and mortar explicitly, is required and the accuracy is highly dependent on the

fineness of meshes or particles. The computation time using these methods can be much longer than

FEM which may not be a cost-effective option.

The methodology presented in this paper combines both the advantages of FEM and DEM methods by

modelling the structural components, i.e. columns, spandrels and walls, individually using a macro

model and applying a contact surface between the components such as joint interfaces, pre-existing

cracks, weak planes and supports. The smeared crack material model is able to simulate the cracks and

damage in the structural components. The contact surface is used to handle the element separation,

rocking and sliding in order to simulate the building collapse. This approach provides a powerful,

economic and accurate solution. The Winfrith concrete, a well-established smeared crack model that

has been discussed and tested in many publications (Len Schwer 2010) (Youcai Wu, John E. Crawford

and Joseph M. Magallanes 2012), was used to model the stone and concrete. The stable and reliable

contact algorithms, i.e. tiebreak, surface to surface, etc., were used to simulate the friction, cohesion

and energy dissipation in the joint interface (John O. Hallquist, LSTC 2006).

The analysis procedure is divided into three parts as demonstrated in Figure 1. The initial damage

analysis is used to create the structural damage similar to the existing building condition. The damage,

cracks and material softening are then carried over to the pushover or nonlinear time history analysis

as an initial stage situation. The capacity curve is obtained from pushover analysis to determine the

residual base shear capacity and failure mechanism of structure. The time history analysis is used to

verify the dynamic response and identify the local instability of the structure.

Figure 1 - Analysis procedure for the damage assessment

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3 GEOMETRY MODELLING

In order to model the dimensions of complicated geometry accurately, a 3D CAD model was built

based on the point cloud obtained from the site survey using a 3D laser scan. The geometry was then

simplified and optimized by removing all the unnecessary features. Finally, a detailed finite element

meshing was carried out by using a FE-mesher based on the simplified geometry. The work flow of

geometry modelling is shown in Figure 2. Precise and high quality FE meshes are the key to achieving

accurate results and getting the nonlinear model to work reliably.

Figure 2 - The flow of geometry modelling

4 MODELLING DETAILS

An explicit code finite element simulation package LS-DYNA (LSTC 2014) was used to carry out the

nonlinear analysis. The major geometry was modelled with solid elements and the analytical model is

shown in Figure 3. The contact surface with coefficient of friction 0.75, i.e. Rock to Rock, both static

and dynamic, was assigned to the nonhomogeneous connections to model the separation of joints and

capture the friction, rocking and energy dissipation in the interface. The mesh is more refined in the

areas with contact surfaces or a higher stress gradient to provide a better and more stable contact. A

coarser mesh was used away from those areas to reduce the total number of elements. For the roof and

strengthening elements including slabs inside the transept and bell towers, integrated beams and

layered shell elements were used. The analytical elements are summarized in Table 1.

Figure 3 - 3D view of LS-DYNA analytical model

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Table 1. Summary of Analytical Elements

Element

Element Type

Material

Linear or Non-Linear

Columns

Solid

Stone

Non-Linear 8-node Brick

Spandrel Beams/ Walls

Solid

Stone & Concrete

Non-Linear 8-node Brick

Existing Slabs

Solid/Shell

Concrete

Non-Linear 8-node Brick/Integrated Shell

Strengthening Diaphragm

Layered Shell

Reinforced Concrete

Non-Linear Integrated Shell

Strengthening

Frames/Roof

truss

Frame

Steel

Non-Linear Integrated

Beam

No soil datum or Winker’s springs were modelled and therefore soil-structure interaction (SSI) was

not considered in the analysis. The structure was sitting on a rigid surface with the contact surface

defined on the supporting interface. A high coefficient of friction was assigned to the contact surface

to prevent sliding. However, no cohesion or tiebreak was defined and therefore rocking at the

foundation was allowed and captured in the analysis.

Figure 4 - Modelling details of Portico (left) and Nave (right)

The material properties for concrete and stone were based on physical test data where the samples

were obtained from the site. The average value of results was adopted for the expected characteristic

properties in the analysis. The material properties of the timber and reinforced concrete are based on

NZSEE guideline. The basic input values for the materials are shown in Table 2 below.

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Table 2. Summary of Material Input

Material

Density

(kg/m3)

Young Modulus

(GPa)

Poisson's Ratio

Compressive Strength (fc)

(MPa)

Tensile Strength (ft)

(MPa)

Concrete

2025

17.1

0.2

9.42

1

Stone

2025

5

0.2

8.75

0.8

Concrete (for RC)

2400

28

0.25

34.2

2.8

Timber

700

11

0.3

-

-

Three different types of material constitutive model were adopted in the analysis. *MAT_CONCRETE_EC2 and *MAT_HYSTERETIC_REINFORCEMENT were used to model the reinforced concrete. For geomaterial such as stone and unreinforced concrete, *MAT_WINFRITH_CONCRETE was adopted in the analysis. The plasticity portion of the Winfrith concrete model is based upon the shear failure surface proposed by Ottosen (1977). This is a smeared crack model allowing for up to three orthogonal crack planes per element and the cracks can be reviewed in the post-processor. The strain rate option is available but it was disabled in the analysis. Instead of using the default setting for material softening, user defined compression and tension softening curves were assigned to model the post-yielding behaviour of unreinforced concrete and stone. The applied stress-strain curve (i.e. red curve) and hysteresis behaviour (i.e. green curve) of concrete and stone material are shown in Figure 5.

Figure 5 - Winfrith model for Concrete (left) and Stone (right)

5 INITIAL DAMAGE ANALYSIS

The cathedral was damaged mainly by the February earthquake event, however, a certain portion of

the structure was damaged during the process of removing the dome and bell towers. Due to the

complexity of its history and the unclear parameters of building conditions (i.e. soil, material aging,

etc.), using simulation to directly reproduce the existing damage is extremely difficult unless a huge

amount of work and time are spent on research and investigation. A simplified approach was adopted

to model the damage which combined two methods, macro and micro damage modelling.

The macro-damage includes damage such as stone or concrete layer spall-off, disconnection of

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components (i.e. a crack through the whole section) and partial removal of structure which changes the

geometry as well as the structural mechanism of the building (i.e. load path). This damage was

modelled explicitly by removing the solid elements, disconnecting the elements and offsetting the

nodes to create the gap. A friction type contact surface was assigned to the gaps to model the

interaction of components across the cracks. An example of macro-damage modelling is demonstrated

in Figure 6. The structure at the back of the cathedral is badly damaged and not structurally intact.

This part of the building was considered to have little or no contribution to the main structure and

therefore was excluded in the modelling. The temporary stabilization works were also excluded in the

analysis to reflect the actual performance of the existing structure.

Figure 6 - Demonstration of macro-damage modelling at the front of the Cathedral

The global performance of the existing structure, particularly the shear capacity, will be governed by

the aggregate interlock inside the cracks with the influence of axial load. The expected strength of the

materials can be obtained by in-situ sampling but the softening due to pre-existing cracks is unable to

be identified by physical test. Reducing the material stiffness is a common option to model the

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damage, however, the secant stiffness is not fully compliant with the constitutive law of the material

model which is inappropriate for use in nonlinear analysis. An alternative and more direct approach

was achieved by applying a lateral drift to the diaphragms to create a similar crack pattern compared

with observed damage. The unobserved micro-damage (i.e. hairline cracks) will also be produced at

the expected locations as the actual damage follows the same structural mechanism. The example of

visible cracks and micro-cracks are demonstrated in Figure 7.

Figure 7 - Crack width > 1mm (left), Crack width > 0.1mm (Right)

The strength and stiffness degradation due to cyclic loads are less important for the brittle material (i.e.

the post-yielding curve is short), therefore a single cycle push in both directions was adopted to

generate the initial damaged stage for the pushover and time-history analysis. A sensitivity study on

initial damaged stage analysis was carried out and the best result was chosen. An example of damage

mapping is shown in Figure 8.

Figure 8 - Damage mapping in bell tower (left) and transept (right)

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6 NONLINEAR PUSHOVER ANALYSIS

The nonlinear pushover analysis allows us to understand the nonlinear behaviour of the building and

estimate the residual base shear capacity of the existing structure. The seismic load was simulated by

applying a horizontal acceleration to the structure and hence the load distribution is proportional to the

mass. The acceleration was slowly increased using a ramp function to achieve a quasi-static condition

without premature failure and instability.

The push along longitudinal (X) and Transverse (Y) directions was investigated with monitoring

points at the top of the nave and portico respectively for measuring the displacement. The results show

the structure is brittle but it can still resist a certain amount of lateral load (i.e. 0.48g to 0.63g). Local

instabilities were observed in some components (i.e. slipping in the interface, tip over, etc.) before the

base shear reaches the highest value. This indicates the building has a risk of partial collapse before

the structural system reaches the maximum lateral load capacity. The threshold of different stages are

shown in the capacity curves in Figures 9 and 10.

Figure 9 - Capacity Curve X-direction (Nave)

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Figure 10 - Capacity Curve X-direction (Portico)

An average of 80% stiffness reduction was found due to the pre-exisiting damage in the model. The

comparison of initial tagent stiffness between the model with and without micro-damage is shown in

Figure 11. The corresponding fundamental period is about 0.25s and 0.36s in X and Y direction

estimated by modal analysis.

Figure 11 - Comparison of Capacity Curve in X-direction (left) and Y-direction (right)

1010

Records

Max. Base Shear Response

Vx

Vx/W

Vy

Vy/W

(kN)

(g)

(kN)

(g)

CCCC

23900

0.45

16300

0.31

7 NONLINEAR TIME HISTORY ANALYSIS

The nonlinear time history analysis is a supplementary exercise to verify the building performance,

failure mechanism and local instability under dynamic loads. This analysis is not intended to

determine the building capacity which has previously been discussed in the pushover analysis.

The analytical model is large and requires a long computation time for the transient analysis.

Therefore, only a single record was adopted in the analysis due to the time limitation. The chosen

ground motion is the Christchurch earthquake on 22nd February 2011 which caused the greatest

damage to the cathedral. The set of record was selected from the seismograph station just next to the

site (i.e. CCCC, distance within 100 meters). The unscaled ground accelerations were adopted and

rotated to coincide with the global orientation of the site. The input ground motions are converted to a

response spectrum in Figure 12 for reference.

Figure 12 - CCCC Spectral Acceleration Compared to Code Spectra (Z = 0.3, 5%)

The base shear response is plotted in terms of “base shear/total weight” against time to compare with

the input ground acceleration in Figures 13 and 14. The maximum base shear is summarized in Table 3. The limit of local instability and ultimate capacity obtained from pushover analysis are included in the graph as a reference for dynamic response. The maximum base shear response from the analysis is

around 0.45g and 0.31g in the X and Y direction respectively. It can be observed that the resonant

response is suppressed by the rocking behaviour. Also, the mass is not all contributing to the

fundamental mode and the system may provide higher energy dissipation above the 5% damping. This

explains why the response is generally lower than the elastic response spectrum.

Table 3. Summary of Maximum Base Shear

1111

Figure 13 - Base Shear Response in X-Direction

Figure 14 - Base Shear Response in Y-Direction

From the results of nonlinear time history analysis, the input earthquake ground motion caused the

structure to partially collapse, but global collapse of the whole building didn’t occur. This agrees with

the results of the pushover analysis (Figures 9 and 10), where the global collapse is not expected due

to the peak base shear demand not reaching the ultimate base shear capacity. Nevertheless, the

response was sufficient to trigger local instability. The nave, transepts and bell towers have

experienced further damage (around 1% storey drift ratio for 1st floor, refer to Figures 15 and 16) but

no serious damage in addition to the pre-existing cracks was observed. Minor slipping was found in

the interface between the external columns and spandrel beams which affects the verticality, but no

indication of major structural instability or global collapse was evident.

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Figure 15 - Interstorey Drift in X-Direction

Figure 16 - Interstorey Drift in Y-Direction

Furthermore, the mega columns at the front of the portico were severely damaged and some of them

fell down. Without the support from the columns, the upper portion of the portico structure relied on

the steel truss tie back to the roof and the strengthened reinforced slab acted as reinforcement at the

bottom to help the unreinforced spandrel beam redistribute the loads to both ends. Approximately 10%

plastic yield strain in the steel layer of reinforced slab was found in the mid span (Figure 18). These

results demonstrate that the upper portion of the portico structure is highly unstable and has a potential

risk of collapse at high levels of shaking. The damage at the front of the portico and back of the arch

are shown in Figure 17. Similar to the pushover analysis, the arch was highly unstable throughout the

time-history analysis. It was severely damaged and eventually tipped over from the first floor support.

This collapse is likely to damage the supporting structure below first floor level (i.e. columns and

walls) which may in turn affect the stability of the nave and transepts.

Figure 17 - Front (left) and back (right) view of the damaged cathedral in NLTHA

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Figure 18 - Section view of the damaged portico (left), plastic strain in the reinforcement layer of

strengthened slab (right).

8 CONCLUSION

Through the case study of the Cathedral of the Blessed Sacrament, the combined finite element and

discrete element method successfully simulated the complex nonlinear dynamic behaviour of a

damaged masonry structure. The smeared crack material model, Winfrith Concrete, gives good results

in modelling the plastic behaviour of stone and concrete. Also, the orthogonal crack planes can be

displayed in post-processing which allows the user to visualise the damage and compare this with the

site observations. The local instabilities, rocking, and complex structural dynamics in the joint

interface were effectively captured by using a contact surface algorithm.

The macro and micro damage modelling in the initial damage analysis provide a reasonable estimation

of strength and stiffness degradation for the existing cathedral. The crack patterns show a good

correlation with observed damage on site. Also, the estimated stiffness reduction gives a physical

measure of the level of damage. The crack, strength and stiffness degradation histories can be carried

over to the pushover or nonlinear time history analysis as an initial stage situation.

The pushover analysis shows the existing cathedral is a brittle structure with little or no ductility.

Assuming the damping is 5% and the structure is elastic for base shear demand (i.e. µ=1) without

considering the strength reduction factor (i.e. ф=1), the ultimate residual base shear capacity could be

up to 53% NBS for an IL2 building (NZSEE 2006). However, the actual capacity is limited by the

local instability of structural components. This performance was further confirmed by dynamic

analysis.

The time history analysis verified the dynamic response and identified the local instability of structure.

The results show no global collapse occurred, but the portico mega columns and arch completely lost

stability under strong ground shaking. They can be considered as a Critical Structural Weaknesses

which may cause the building to partially collapse.

It is important to note that, the limitations and assumptions of this high level simulation are

emphasized to prevent any misinterpretation and misuse of the analytical model. The initial damage

was not predicted by direct simulation but only an approximation based on sensitivity study, observed

1414

damage and engineer’s judgement including the input from an independent reviewer. Also, the

accuracy of analysis is highly dependent on the constitutive material model which needs to be well

calibrated with the laboratory test results. Sufficient material test samples are required and a full scale

(i.e. shear wall) test is also suggested to calibrate the model and provide verification. Moreover, one

set of time history records is inadequate to determine all the collapse scenarios. To evaluate the risk of

collapse in detail, forming a fragility curve by running multiple sets of records with different levels of

scaled amplitude is highly recommended.

The proposed methodology and analysis procedure outlined a viable method to assess the capacity and

stability of a damaged historic building. Without an established guideline or standard to follow, this

performance-based approach is proposed by the authors as a sound method for the assessment of

similar historic buildings.

9 ACKNOWLEDGEMENTS

The authors would like to acknowledge the peer reviewer, Grant Wiklinson, Ruamoko Solutions for his involvement in the project of Cathedral of the Blessed Sacrament.

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