discrete element modeling of the walls of the prince … · 2012. 2. 12. · 15th international...

15th International Brick and Block

Masonry Conference

Florianópolis – Brazil – 2012

DISCRETE ELEMENT MODELING OF THE WALLS OF THE PRINCE

OF WALES FORT

Isfeld, Andrea Cathleen1; Shrive, Nigel Graham

2

1 PhD, Student, University of Calgary, Civil Engineering Department, [email protected]

2 PhD, Killsm Memorial Professor, University of Calgary, Civil Engineering Department, [email protected]

The Prince of Wales Fort, located in Churchill, Manitoba was constructed by the Hudson Bay

Trading Company in the early 18th century. This Vauban style, rubble masonry fortification

is the most northerly construction of its kind, and was intended to secure the fur trade in

northern Canada. In the 1920’s the fort received recognition as a National Historic Site by the

Historic Sites and Monuments Board of Canada. At that time, extensive repairs were required

to part of the fort and commenced under the supervision of Parks Canada.

The fort’s northern latitude has left it vulnerable to extreme temperatures and freeze thaw

cycles over the last 250 years, resulting in a gradual breakdown and washout of the mortar

within the escarp walls. Consequently, the walls currently exist as a partially-grouted rubble

core, encased with ashlar face stones. The deteriorating core conditions have caused the walls

to bulge outwards significantly in several areas and fail in others: an extensive restoration

project is currently underway.

In order to understand the current conditions of the wall core, the wall has been modeled

using the software package Logiciel de Mécanique Gérant le Contact (LMGC 90). Five

models with simplified geometry of the wall cross-section have been used to determine the

values of normal and tangential cohesion necessary to create stability within the wall sections.

A relationship between the friction angle and the normal cohesion component required to

obtain stability for each of the five models is presented graphically and related to the

variations in the modes of failure.

Keywords: Stone masonry, discrete element modelling, historic structures, analysis

INTRODUCTION During the last 250 years the Prince of Wales Fort, Figure 1, has been subjected to extreme

weather conditions due to its location on the edge of Hudson’s Bay. Freezing and thawing of

water in the walls, together with intense wind, snow and rain have led to continuing

degradation and even collapse. Recently, ongoing maintenance has been necessary to

maintain the fort as a historic monument. Over the last decade deterioration has become

increasingly rapid, specifically on the north wall. Public Works and Government Services

Canada began a stabilization project in 2003, using either a local dismantling and rebuilding

approach or permanent shoring to prevent further deterioration or possible collapse.


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Several locations have been identified for structural stabilization. Ashlar face stones on each

deteriorated section are individually removed, and the inner core is stabilized using flat stones

and mortar. The size of the face stones, up to 3500 lbs each, makes this process slow and

cumbersome. Due to the harsh climate, the work season is limited to the summer months,

meaning sections of wall are typically dismantled one summer and rebuilt the next. Based on

the initial state of existing degradation, this project was expected to take ten years. However,

the north facing wall, which was previously undamaged, has recently begun degrading at an

alarming rate. Stabilization is underway on one section of the north wall, and other sections

will soon require attention if the current rate of deterioration continues. This type of

reactionary maintenance has been deemed insufficient as a long term solution. An alternative

process to slow the rate of deterioration would allow the current stabilization work to be

completed, and less invasive maintenance to be continued into the future. In order to develop

a preventative approach, the cause of failure must first be understood.

The escarp wall is comprised of a rubble masonry core with ashlar face stones on the exterior

and above the rampart surface on the interior. Below the rampart, the core is of split boulders.

The total height of the wall sections, including the 1.8 m parapet, is 4.8 m. This height is

divided into 10 courses of ashlar masonry of heights ranging between 360 mm and 600 mm.

The length of the stones typically ranges between 300 mm and 1000 mm, and their depths

between 200 mm to 480 mm. A similar construction, for the foundation, continues 2.1 m

underground and is 2.7 m wide. Behind the wall lies an earthen and gravel backfill which

reaches a height of 3 m. A general cross sectional view is given in Figure 2.

The walls are constructed primarily of two local stone types Churchill quartzite, and

dolostone. Compression testing has been carried out on sample of both rock types and the

compressive strengths of 186 MPa and 188 MPa were found for the quartzite and dolostone

respectively ( Heritage Conservation Program (2000)). Due to the high compressive strength

of the stones, it was typical that only the exposed face was cut to have regular edges and

finish. While bearing areas were cut on the top and bottom of the stones, the backs were

typically left uncut, and taper off irregularly. Stones within the core of the wall were left

entirely uncut. Mud (or clay) mortar was used primarily during the first ten years of

construction, and some can still be found within the core of the walls. Lime mortar was used

later when the cut face stones were placed. These mortars are inherently weak and acted

primarily as filler materials when placed initially. The cold climate would have hindered the

Figure 1: Aerial view of Prince of Wales Fort looking south. Photo courtesy of the Canadian

Military History Gateway.


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curing process necessary for the lime mortar to gain significant strength (Heritage

Conservation Program (2000)). As much of the mortar has been washed out, or severely

degraded within the wall, and the stones are strong and very stiff compared to the weak, or

absent, mortar, the behaviour of the masonry will be dominated by the mortar-unit interface,

or stone on stone contact. Rolling and sliding of the stones is subsequently a critical issue

governing the failure of the walls: crushing of the stones has not been observed.

Under current conditions therefore, little of the original mortar remains within the escarp wall,

although some can still be found at the level of the foundation. Thus, the core material has

become essentially “loose-packed”, and the load is now distributed by stones bearing on each

other, rather than on the mortar holding them together. Such action would decrease the

stability of the core system over time, and the large stones would have begun shifting and

applying pressure on the face stones. Simultaneously, the connection between the face stones

and the core material has been reduced due to washout of mortar, with the bond reduction

being exacerbated by the lack of keystones in the initial construction. The consequence

appears to be that the outer wythe is now acting largely independently of the core material and

is subject to lateral pressure causing the gradual bulging of the curtain wall observed at

approximately mid-height. This problem is often focused in areas between gun embrasures,

being directly below the parapet which acts as an additional load. As the outer wythe

delaminates from the core material the parapet loads the outer wythe eccentrically, increasing

the lateral deformations.

CONSIDERATIONS FOR ANALYSIS

Developing an understanding of rubble masonry under washout conditions poses several

major challenges. Water flowing through the wall tends to follow the path of least resistance

leading to high variance in the mortar conditions across any given cross section. This will

create areas of localized mortar absence surrounded by mortar of reduced strength, while

mortar in other areas is unaffected. Further variance in mortar conditions will occur over the

height of the wall according to exposure weathering and ground water runoff. While the face

stones are cut to have a flat base, they taper off into the wall and can be subjected to rolling in

the out of plane direction if the bedding mortar or contact with the core is insufficient.

Figure 2: General wall cross section (Heritage Conservation Program (2000))

Figure 4. Cut boulder, view from left hand s ide (a), top (b) and right hand side (c)

Figure 3. Results of fiberscope probe of South curtain wall (Heritage Conservation Program 2000)


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DISCRETE ELEMENT MODEL As a preliminary step in determining the most suitable method for retrofitting of the Prince of

Wales Fort walls, discrete element modelling has been used to examine the failure

mechanisms. This has been executed using the non-smooth contact dynamics method (NSCD)

in the LMGC90 program (Jean (1999); Jean et al. (2001); Jourdan et al. (1998); Moreau

(1999)). The goal of this modelling has been to look at the bonds within the walls, attributed

to the mortar, and determine if a transition from instability to stability can be made by

increasing the bond strength (the reverse of what we think has happened over time).

NSCD FUNDAMENTALS

Contact dynamics can be used to treat unilateral frictional contacts between rigid bodies, with

the primary computational objective being the determination of the relative velocities. The

principles of dynamics are applied, and time stepping algorithms are used to determine the

evolution of the velocity function.

The system of bodies is defined such that individual bodies neither overlap nor impose

attractive force on each other. This set of relations can be summarized as the Signorini

condition, where is the gap between two bodies and is the normal reaction component

(1)

A friction law can be included to relate the tangential force component to the normal

component. The vectors for the relative velocity , and the reactions, , in the local reference

frame of a particle contact are related to the global values through the transformation vectors

and . In the global system is defined as a position vector, and as the velocity, while

composes the reaction vector. This relationship is combined with the equations of motion,

and the interaction laws, taking the form shown in (2)

A (2)

A time stepping algorithm, similar to the block Gauss Seidel algorithm, is applied. For each

time step values of and are assumed, calculations are carried out based on the assumed

values and continued iteratively until there is sufficient convergence.

MODEL DEVELOPMENT

Five models were created to idealize cross sections of the walls using geometric shapes,

shown in Figure 3. Walls B (b) and C (c) have different cores but the same face-stone

arrangement as wall A (a), while walls D (d) and E (e) have the same core as wall A but

different face-stone arrangements. A detailed explanation of the process used to develop the

geometry of these models is provided in Isfeld & Shrive (2011)


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Figure 3: Walls A (a) to E (e) respectively

LOADS

For this modelling technique, the primary load of interest is self weight as this is the major

loading in situ, and it is the internal conditions, rather than any external loading, that is being

examined in the models. Consequently, loads such as those from the earthen rampart within

the fort, as well as wind and snow were ignored. The stones were therefore given a unit

density of 2000 kg/m3

and assigned rigid body behaviour. The models were run to determine

the effect of normal gravitational acceleration (9.81 m/s2) acting on the stones.

BOND STRENGTH

The Mohr Coulomb contact law was used to simulate the effects of the bond strength. For

this cohesive contact, the normal and tangential components of the contact forces can be

adjusted in order to find a level at which the wall section is stable. Values of the normal,

, and tangential, , components for which the sections are stable were determined by

running the models at different values while maintaining a consistent friction angle, φ,

according to Equation (3)

(3)

FAILURE MECHANISMS OF MODELS

When low values of the normal and tangential force components were input in the model and

was kept equal to 1.0, failure occurred with all wall cross sections. The failures of the

five modelled walls are shown in Figure 4. For Wall A shear planes initially develop within

the wall, and as the material settles along the shear planes the outer wythe is displaced

horizontally. The rectangular shaped face stones allow greater loads to be sustained by these

bonds than with the circular core material, due to the increased bonding surface. Thus there is

little sliding between the face stones. Instead, the face stones rotate on the bed-joints and it is

the rotation of these stones that becomes critical, as the rotation causes a decrease in the

bonding area which in turn, increases the rate of collapse.

The core material in Wall B featured less fine material at the top of the wall than Wall A.

Subsequently the frictional resistance was reduced at this level and the outer wythe could

slide more easily relative to the concrete cap. The face stones then acted monolithically and

thus changed the failure mechanism (Figure 4 (b)): this failure was not representative of the

actual conditions. Additionally, the maximum deformations of Walls B and C (Figure 8 (b)

and (c)) occurred at a lower height than Wall A, and tended to be dependent on the face stone


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geometry rather than the core conditions. This is evident as the rotations occurred at the

location of the narrowest face stone rather than at mid-height. To avoid this dependency in

Models (d) and (e), the core geometry of Model (a) was used. Comparison of the

deformations of Walls A, D and E (Figures 4 (d) and (e)) can be used to determine the effect

of varying the face stone geometry on the failure mechanism of the wall section. It is evident

that when the core geometry from Wall A was used, the deformations were consistently near

mid-height. This reinforces the hypothesis that the failure of the wall sections can be

attributed to the conditions of the core material.

Figure 4: Failure mechanisms of Walls A (a) through E (e)

STABILITY VALUES

A coefficient of friction of 1.0 was used in the models described in Isfeld & Shrive (2011) and

the normal and tangential cohesion values required to achieve stability were determined.

These values were not to be taken as indicative of the actual bond required of a remediation

effort as the models have oversimplified geometry, and the actual wall geometry would have

additional stability due to locking of the angular stones, as well as increased bearing surfaces.

Also, a coefficient of friction of 1.0 is unlikely. Instead, these values were intended to be a

starting point in determining if it is possible to achieve stability in these inherently unstable

cross sections and if so, to determine if a relationship exists between the frictional coefficient

and the normal force component at stability. Logic would suggest that if stability is achieved

at a given frictional coefficient, decreasing this value would lead to a necessary increase in the

normal force component to maintain stability.

The graphical results in Figure 5 summarize the data from numerous runs with each wall

model (A through E) at varied combinations of normal and tangential force components. It

may be seen that the hypothesized relationship is generally confirmed. Some anomalies exist,

and it is thought that these are related to changes in the failure mechanism resulting from

altering the ratio of the tangential and normal force components.

DISCUSSION

For a given model, the failure mechanisms at the point of transition from stability to

instability can be compared. Comparing the various failures as they evolve with time helps to

show how a change in the frictional coefficient affects the behaviour of the wall sections. In

Figure 6, failure images for Walls C (a) and E (b) are presented for varied coefficients of

friction, with the highest cohesion values for which instability still occurs.


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From Figure 6(a) for Wall C with a low coefficient of friction (0.3) failure of the outer wythe

occurs with the second and third stone pivoting relative to one another. However when the

frictional coefficient is raised (at 0.5 and 0.8), it is the fifth and sixth stones that pivot one

relative to the other. For a friction coefficient of 0.8, sliding also occurs in the outer wythe,

between the second and third stone. For Wall E, seen in Figure 6(b), relative rotation of the

third and fourth stones can be seen to be the predominant failure mechanism with the lower

coefficients of friction. However, for a friction coefficient of 0.7 there is additional rotation

between the fourth and fifth stone, while for a coefficient of 0.8 there is only rotation between

the fourth and fifth stone.

Figure 5. Stability curves for Walls A through E


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When considering a clean, homogeneous, slope of coarse-grained soil, increasing the

coefficient of friction, or friction angle, of the soil would increase the slope along which

wedge failure would occur. This is because sliding of this unconstrained soil system occurs

along the angle of repose, which is equal to the friction angle. As the friction angle increases,

the shear resistance at a steady normal load increases, allowing for a steeper slope to occur at

failure.

(a) (b)

Figure 6: Failures of Wall C (a) and Wall E (b) at various coefficients of friction (the

coefficient being listed in the top right corner for each result)

In the wall models, the grain size in the core of the wall is highly variable, and fewer grains

are considered compared to a soil slope problem. Hence, the relationship between the


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frictional coefficient and the angle of the slip plane becomes more complex. As the frictional

coefficient is increased the failure slope within the wall would tend to become steeper.

However, this problem is distinctly dependent on the geometry, as stones can more easily

become wedged in place when the size varies significantly. This wedging of stones can

further shift the failure surface, either increasing or decreasing the slope according to what is

the most favourable geometrically at a given frictional coefficient. When the core material is

examined for the different frictional coefficients used in Figures 6 (a) and (b) it can be seen

how the stone geometry affects the internal slip, and subsequently which stones on the outer

face undergo the greatest displacements.

If a line was interpolated along the boundary of stability and instability in any of the plots in

Figure 5, some points found from the models would not fall on this line. For example, with

Wall C the normal cohesion values that correspond to the coefficients of friction of 0.5 and

0.8 would be higher than anticipated. However, these values have unique failure mechanisms

when compared with the other frictional coefficients. Likewise, for Wall E the location of

face stone rotation shifts from between the third and fourth face stone to the fourth and fifth,

and back again as the coefficient of friction is raised. While only slight variations occur from

the anticipated values, similar variations in the failure mechanisms could be found for the

other three walls.

CONCLUSION

To begin to understand the behaviour of the escarp walls of the Prince of Wales Fort,

modelling of simplified geometries has been undertaken using the NSCD method in the

program LMGC90. Five wall sections were modelled with varying geometry in both the core

material and the outer wythe, with a consistent profile for the back and top of the wall. The

models were run with different coefficients of friction, increasing the cohesion until stability

was achieved. The initial deformations during the failures of the models tended to follow

observations of the lateral bulging of the actual walls at mid height. A transition from

stability to instability occurred when the cohesion was reduced at any ratio of normal and

tangential components. A plot of all results for a given wall model followed approximately

the relationship that reducing the frictional coefficient requires an increase in the normal

cohesion in order to maintain stability. This relationship tended to vary, depending on the

different failure mechanisms that developed as the frictional coefficient was adjusted.

Based on the ability of these unstable cross sections to transition to a state of stability at a

given bond strength, it can be hypothesised that stability of the walls could be obtained by

increasing the interior bonding. It would be feasible to create these conditions by injecting

grout into the walls. However, further testing of this hypothesis will be conducted using more

complex models, including actual geometries and a grout infill.

ACKNOWLEDGEMENTS

The authors acknowledge most gratefully the support of Parks Canada and Public Works and

Government Services Canada, in particular the Heritage Conservation Directorate. The first

author appreciates scholarship funding provided by the Natural Sciences and Engineering

Research Council of Canada. Additional thanks to Dr. Ali Rafiee for his assistance with the

use of LMGC90.


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REFERENCES

Heritage Conservation Program Real Property Services for Parks Canada. Prince of Wales

Fort 1999-2000 Structural Condition Assessment and Recommendations. 2000.

Isfeld A, Shrive N. "Assessing the stability of the core of stone masonry walls using discrete

element modelling", North American Masonry Conference, 2011.

Jean M. "The non-smooth contact dynamics method", Computer Methods in Applied

Mechanics and Engineering. 177(3-4), 1999, pp 235-257.

Jean M, Acary V, Monerie Y. "Non-smooth contact dynamics approach of cohesive

materials", Philosophical Transactions of the Royal Society of London Series a-

Mathematical Physical and Engineering Sciences, 359(1789), 2001, pp 2497-2518.

Jourdan F, Alart P, Jean M. "A Gauss-Seidel like algorithm to solve frictional contact

problems", Computer Methods in Applied Mechanics and Engineering, 155(1-2),

1998, pp 31-47.

Moreau J.J. "Numerical aspects of the sweeping process", Computer Methods in Applied

Mechanics and Engineering, 177(3-4), 1999, pp 329-349.

discrete element modeling of the walls of the prince … · 2012. 2. 12. · 15th international...

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