Numerical analysis of a gravity retaining wall: the case of Volterra's urban walls.

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D

IPARTIMENTO DI

I

NGEGNERIA

C

IVILE E

I

NDUSTRIALE

CORSO DILAUREAMAGISTRALE ININGEGNERIAEDILE E DELLECOSTRUZIONICIVILI, CURRICULUMCOSTRUZIONICIVILI

T

ESI DI

L

AUREA

M

AGISTRALE

N

UMERICAL

A

NALYSIS OF A

M

ASONRY

R

ETAINING

W

ALL

:

THE

C

ASE OF

V

OLTERRA

’

S

U

RBAN

W

ALLS

Relatore Accademico:

Dott. Linda Giresini

Relatore Accademico:

Dott. Mario Lucio Puppio

Relatore Esterno:

Dott. Giorgia Giardina

Candidato:

Gianmarco Passera

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This thesis was almost entirely carried out at the Department of Architecture and Civil Engineering of the University of Bath (United Kingdom), where I conducted my studies for about six months. I would like to express my sincere gratitude to my advisor Dr. Giardina for the continuous support during my stay in Bath, for her patience, motivation, enthusiasm and immense knowledge.

Besides, I would like to thank my Italian supervisors Dr. Giresini and Dr. Mario Lucio Puppio from the Department of Industrial and Civil Engineering of the University of Pisa (Italy) for their encouragement, insightful comments and for offering me the opportunity of developing this thesis in a challenging and rewarding such as the University of Bath. Their guidance helped me in attaining the achieved results.

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In the entire European continent, but particularly in Italy (the country where I am from) there is an inestimable amount of architectural works of historical interest, almost all of which are made of masonry. Yet, a general lack of knowledge about this material exists. Its study, indeed, did not arouse particular interest in researchers, who only recently, starting from the 90s, began to study its behaviour, creating models and establishing constitutive laws capable to describe its functioning.

Exactly on the need to deepen the study of masonry is focused the present thesis, which takes the cue from a recent collapse occurred in the medieval urban walls of Volterra, a small town in the heart of Tuscany (Italy) the 31st January 2014, due to the heavy rainfall that hit the area during that night.

The event offered a valid inspiration for research: as the structure was an ancient retaining wall, in addition to studying the ancient masonry behaviour, it also offered the possibility of assessing its interaction with a saturated soil. As it is known, indeed, that of hydrogeological risk is another very current problem in the Italian peninsula, that suffers the lack of care given to this issue over the years.

The main object of the present thesis was to constitute a numerical model able to describe the behaviour of the collapsed structure and, through the analysis, to understand the causes that led to its collapse. The greatest faced difficulty was the total absence of specific data concerning the mechanical strength features of the masonry. Its chaotic texture and its irregular nature, therefore, necessitated a parametric analysis of the failure, in which the procedure was to vary the parameters that most affect the breaking of such a structure: the maximum tensile strength and the fracture energy.

The analyses results made it possible to distinguish the failure mechanisms that occur when these parameters change. Proceeding by comparing the results obtained with the evidences that were found during the collapse, it was possible to identify a possible mechanical features

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range of the treated retaining wall. It follows that these values, taken with due care according to the assumptions made, can provide a good starting point for any safety analysis of the entire Volterra’s surrounding walls, as they are built both with the same materials and with the same construction techniques.

PROBLEM: Collapse of an historical masonry retaining wall

QUESTION: How did it collapse?

AIM: Masonry Modelling

CONCLUSION: Through a parametric analysis, the wall behaviour when

varying the mechanical properties and the stressing forces is obtained Knowing how this wall

collapsed, it is possible to prevent similare scenarios

Which means are available to understand it?

It is needed to understand the methods and

mechanisms of collapse

If a model with a behaviour congruent to what actually happened is found,

it is possible to answer the question

The numerical instrument is effective and can be reused in similar scenarios

It is obtained that the wall collapse according to a specific mechanism The actual failure Mechanism occurred allows to restrict the retaining wall mechanical

parameters range, providing significal data

for any subsequent security analysis of the

entire city walls

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Collapse of Masonry Retaining Walls

A general knowledge of the retaining walls structure is useful in two aspects. First of all, it is a prerequisite to understand the structural behaviour of this kind of constructions, their failure mechanisms and the deformations that precede the collapse. Secondly, having a complete knowledge of this kind of walls, the inspections on the existing structures can be carried out in a more efficient way.

1.1 Static and Foundation Instability of Masonry Retaining

Walls

1.1.1 General

When a retaining wall has a too thin section and the static equilibrium cannot be maintained, the wall loses its stability and it starts to move forward. The movement can cause a decrease in thrusts that allows the stability to be restored, or it can continue until the wall is no longer able to retain the backfill. In the latter case the static failure occurs.

This kind of failures are preceded by the formation and the development of cracks at the top of the embankment and by the overhanging of the wall top. From the observations and experiments carried out by Hope and Burgoyne [A.-S. Colas et al. (2010)], which will be discussed in the Section (1.3), masonry retaining walls are liable to these modes of collapse. In the following sections, the distribution of earth and groundwater pressure behind old masonry walls that can generate a failure mechanism is first considered. This is followed by considerations on the impact of different aspects on the retaining walls static instability. Afterwards, the instability due to insufficient wall section is investigated. A retaining wall, indeed, can collapse as a result of an over-stressing of the foundation. This can often occur when the wall stands on the crest of a wet slope.

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1.1.2 Thrusts Acting on a Retaining Wall

To begin this evaluation of masonry retaining walls, the geotechnical assessment of the thrusts acting on these kind of structures is of fundamental importance.

Thrust Coefficient

The first person to be scientifically involved, between the ’20s and the ’30s of the last century, in the thrusts acting on a retaining wall assessment is the founding father of the geotechnical engineering: Karl Terzaghi. He first identified the so-called thrust coefficient, K, defined as the ratio between the effective horizontal stress (σ_h0) and the effective vertical stress at a generic depth under the level of the ground (σ_v0).

K= σ

0

h

σ_v0 (1.1)

It can be observed that when the wall does not undergo displacements (δ /H = 0), the soil exerts a pressure on the retaining structure and the thrust coefficient is not null. If, instead, the wall is allowed to move, it is necessary to distinguish between to cases: when the wall moves away from the ground, the thrust coefficient will be lower and it is named active thrust coefficient, Ka; while when the wall is pushed towards the ground, the thrust coefficient will be greater and it is called passive thrust coefficient, Kp(Fig.1.1, 1.2). If the horizontal displacement of the wall are neglected, instead, the ratio between the effective horizontal stress and the vertical one is the resting thrust coefficient, K0.

Backfill Rigid Wall

Active Passive

Figure 1.1: Wall Displacement Mechanism

The reason why the thrust coefficient varies is the progressive development of shear stresses as the wall moves from its resting initial position. In the active case, the tangential stress τ acts in order to contrast the soil weight W (Fig.1.3), the angle that the sliding surface creates with the horizontal is greater than 45◦.

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+0.020 KP KA H +0.015 +0.010 +0.005 -0.005 4 K0 3 2 1

Figure 1.2: Thrust Coefficient Diagram

Sliding Surface W Displacement n Active Case W Displacement Passive Case Sliding Surface n

Figure 1.3: Soil Breaking Conditions

In the passive case, the tangential stress τ is in opposition with the thrust that the wall exerts on the ground (Fig.1.3).

Considering the problem in the Mohr plane, in the active case the horizontal stress σ_h0 decreases, while the vertical stress σ_v0 remains constant, as shown in Fig.1.4.

v

f

= c'+ ' tan '

' =

1

'

h

' =

3

'

Figure 1.4: Mohr Diagram of Active Thrust

As known, the stress-path is inclined of −45◦ (discharge compression). In the case of passive thrust, instead, the stress-path will be inclined of an angle of +45◦, the vertical stress will remain constant and the horizontal stress will increase, as shown in Fig.1.5.

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h

f= c'+ ' tan '

' = 1' v' = 3'

Figure 1.5: Mohr Diagram of Passive Thrust

rigid wall that rotates around its lower extremity. In the reality, however, the retaining walls are rarely rigid and they develop strains under the action of the applied loads. Nevertheless, they usually do not rotate around the lower extremity, but can likely translate and rotate around their top or around any other point.

Coulomb’s Theory

At the end of the XVIII century, two ”concepts” of fundamental importance in the soil mechanic were introduced by Coulomb. First, the material strength is divided in its two components, namely the cohesion (independent of applied forces and only function of the sliding surface extension) and the friction (proportional to the compression force acting on the breaking surface). Secondly, the equilibrium of the soil rigid detached part is considered bordered by a sliding surface, where the forces considered are:

• The soil weight, W.

• The forces due to the cohesion and to the sliding surface friction.

• The force that opposes the sliding, PA.

Imposing the translational equilibrium in the direction parallel and perpendicularly to the sliding surface and considering that the parallel forces on that surface must have a null resultant, Coulomb obtains the expression of the force acting on the wall that opposes the sliding. He recognizes that it is necessary to determine the sliding surface for which the maximum value of the force PAis obtained.

Given the triangular soil prism shown in Fig.1.6, it is possible to write the translational equilibrium along the sliding surface and the perpendicular direction.

   W· sin(α) − PA· cos(α) = T W· cos(α) + PA· sin(α) = N (1.2)

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W T N L H H 3 PA

Figure 1.6: Coulomb’s Theory Wedge

Taking into account that:

N· tan(ϕ0) + c0L= T (1.3) and that: W = 1 2γ h 2_cot_(α) (1.4)

It is possible to obtain the active thrust general expression:

P_A=

W− [tan(α) − tan(ϕ0)] −_cos(α)c0L

tan(ϕ0) tan(α) − 1 (1.5)

Cancelling the derivative with respect to α, it is obtained the sliding surface sloping angle that maximizes the active thrust:

α = π 4+

ϕ0

2 (1.6)

that, once substituted in PA, gives the active thrust maximum with respect to all the sloping possibilities: P_A=1 2γ h 2_tan2₍π 4 − ϕ0 2 ) − 2c 0_htan(π 4 − ϕ0 2 ) (1.7)

It must be noted that to the Coulomb’s theory is applied the above mentioned Terzaghi’s modification, so that the strength parameters, in terms of effective stresses, are considered.

It is likewise possible to obtain the expression for the passive reaction.

P_P= 1 2γ h 2_tan2 π 4 + ϕ0 2 + 2c0htan π 4+ ϕ0 2 (1.8)

The Coulomb’s method is thus applied to the case of thrust produced by an homogeneous and horizontal backfill, featured with cohesion and friction, on a vertical wall and in the absence of friction between the wall and the soil. Thanks to Terzaghi, then, the method is

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applicable even if the backfill is saturated, using the soil lighted weight.

Muller-Breslau Correction The method described, developed by Coulomb, has been re-sumed and modified by Muller-Breslau with the aim of removing the restrictive hypotheses. In the case of Fig.1.7, with a topographic sloping surface having a generic inclination angle

P

A

W

R

'

i

P

A

R

W

Figure 1.7: Muller-Breslau Solution - Soil With Friction, Without Cohesion

i, an homogeneous soil only equipped with friction, a face with an inclination of β with the respect to the vertical and a friction angle between the wall and the soil of δ , the inclination of the sliding surface is known and it is possible to obtain the active thrust expression:

P_A= 1 2γ h

2_K

A (1.9)

where the coefficient KA is expressed as:

K_A= sin 2_{(α + ϕ)cos(δ )} sin(α) sin(α − δ ) · h 1 + q sin(ϕ+δ )·sin(ϕ−β ) sin(α−δ )·sin(α+β ) i2 (1.10)

The active thrust PAis inclined with respect to the wall of an angle δ .

Rankine-Bell Theory

In the 1857 Rankine came to the same solution of Coulomb by finding the solution applicable to a soil mass in breaking conditions, while the above mentioned theory was examining a soil wedge delimited by a single breaking surface. Operating according to this methodology, it is possible to make reference to the Mohr’s stresses plan, by assuming:

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• Horizontal or inclined ground level;

• Undefined vertical wall;

• Plan breaking surface.

The assumption that the direction in which the soil thrust is acting on the wall is parallel to the slope of the backfill surface leads to the fact that the wall-soil friction is null if horizontal, while if the ground floor is inclined of an angle i, it will result of δ = i.

f= c'+ ' tan ' v0' c' tan ' c' ' ' A' P'

Figure 1.8: Soil Stresses in the Mohr plane

Referring to the Fig.1.8 and making simple geometric considerations, it is possible to write:        σ_v00 +σA0 2 + c0 tan(ϕ0₎ · sin(ϕ0) =σ 0 v0−σ 0 A 2 σ_v00 +σP0 2 + c0 tan(ϕ0₎ · sin(ϕ0) =σ 0 P−σ 0 vo 2 (1.11)

in which, developing and rearranging it is obtained:

σ 0 A= σ 0 v0· KA− 2c0 √ K_A (1.12) σ 0 p= σ 0 v0· KP+ 2c0 √ K_P (1.13)

where the active and passive thrust coefficients are given by:

K_A=1 − sin(ϕ 0₎ 1 + sin(ϕ0) = tan 2 π 4− ϕ0 2 (1.14) K_P=1 + sin(ϕ 0₎ 1 − sin(ϕ0) = tan 2 π 4+ ϕ0 2 (1.15)

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• The horizontal stress in conditions of active thrust σ_A0 is lower than the horizontal stress in condition K0(σ

0

ho), while in conditions of passive thrust σ

0

P is significantly greater (KA< K0<< KP).

• The Rankine-Bell equations allow to determine the limit condition at each depth and not only along the sliding surface. Hence, it is not only existing a single sliding surface, but an indefinite number of breaking surfaces.

• The plane of the breaking surfaces in active conditions π 4+ ϕ0 2

and in passive con-ditions π 4− ϕ0 2

coincides with the sliding surface using the Coulomb’s theory in the same conditions.

Cohesion The cohesion contributes to the soil stability, as shown in the Eq. 1.12 and 1.13, by reducing the active thrust and increasing the passive strength (Fig.1.9).

v0' A' P' A K v0' KP 2c' KA 2c' KA

Figure 1.9: Active and Passive Thrusts Diagram

The passive thrust diagram assumes a trapezoidal shape, while negative values can be present, in the nearness of the surface, in the active thrust one. This fact, however, is not realistic because the soil in not able to withstand tensile stresses.

Undrained Conditions In the presence of groundwater, on the structure the water thrust is also acting, so that the soil thrust calculation must be carried out in terms of effective stresses. In fine sized soils, the long term condition (drained) is the most unfavourable one, i.e. the thrust is greater. These thrusts can be calculated with the following expressions in function of the total stresses:

σA= σv0− 2cu (1.16)

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In this case the formulations are already keeping in consideration the hydrostatic component, so that it must not be added up.

Wall-Soil Friction

Contrary to what previously expressed in the described theories, experimental tests show that the sliding surface is actually curvilinear. This can be explained referring to the Fig.1.10.

A

Sliding Surface B

Figure 1.10: Wall-Soil Friction Evaluation

In the case of element A, close to the ground floor and far away form the wall, the friction is negligible (even if present), while it is definitely present in the element B. This leads to a change in the plans origin and the breaking surface slope with respect to the horizontal results less inclined (Fig.1.11).

A

B

Figure 1.11: Wall-Soil Friction Evaluation in the Mohr Plane

Considering the wall-soil friction δ , and consequently assuming curved sliding surfaces, leads to slightly lower active thrust coefficients. It leads also to an overestimation of the passive thrust, creating problems in the structure security evaluation. A way to take into account the

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friction is to rely on the solutions present in the literature. One of these is the so-called Navfac Abacus, which, however, being not useful for the purpose of this thesis, is not detailed here.

Groundwater Thrusts

The thrust (Pw) exerted by the groundwater in static conditions on the wall is basically equal to:

Pw= 0.5 · γw· H02 (1.18)

with H’ the groundwater level height measured from the wall base. The construction of the forces polygon must take into account both the water thrust acting on the possible sliding surface (U) and the one acting on the wall (Pw). The magnitude U results equal to:

U = 0.5γwH 02

cos(90 − α) (1.19)

For this reason, with the aim of safeguarding the retaining wall, it should be equipped with a drainage system, in order to reduce the groundwater thrust. Usually it consist in a vertical drain, composed by a coarse material highly permeable, as much as possible devoid of fine-grained soil, shielded by a fabric-non fabric and equipped on the base with a slotted tube which receives water and lets it flow. It is therefore possible to calculate the pressure u along the sliding surface using the following expression:

u0_A= u_A+ γw(zA− z

0

A) (1.20)

and being uA∼= 0, the expression can be considered as:

u0_A= γw(zA− z

0

A) (1.21)

The groundwater resultant must be calculated for different possible sliding surface, until the maximum value of the whole thrust is found. In a soil of an average permeability, as we will see in the Chapter 3, it is possible that the rain produces an increase in the overall thrust due to the filtration forces which are established during the ,rainwater flow. A simplified method for the evaluation, with a quick estimation, of the water thrust magnitude on the potential sliding surface makes use of the so-called Garay’s Diagram, of which there is a qualitative representation in Fig.1.12. It provides the effective thrust as a fraction of the total hydrostatic thrust generated by a groundwater level equal to the wall height.

The estimation is made as a function of the sloping α (with respect to the vertical) assumed by the sliding surface, supposed flat. Obviously, with the growth of α, the water thrust

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F

Figure 1.12: Garay’s Diagram

increases. It must be taken into account, however, that, after a certain value, the soil thrust decreases, so that it is necessary to figure out the couple given by the water thrust and the soil thrust that produces the maximum thrust.

Surcharge Thrust

The retaining wall backfill can be subjected to external loads and, as examined in the case study of Chapter 3, they may be of fundamental importance in the calculation of the total thrust acting on the structure. Usually, the external loads determine an horizontal stress increasing on the retaining structure and, in order to evaluate them, it is used an elastic solution. The most common load configurations, and those are taken into account in this dissertation, are the punctiform and the ribbon ones, which produce a thrust increase equal to σ_A0 = K_A· ∆σ_z0. The vertical stress increase (∆σ_z0) is evaluated with the linear elasticity theory, so that the following solution can be considered valid [Greco (2006)] and [LoPresti (2015)]:

∆σz= 3Pz3 2πR5 (1.22) with R =px2_{+ y}2_{+ z}2_{, and:} ∆σz= q π· [α + sin(α) · cos(2δ + α)] (1.23)

The equation 1.22 regards a concentrated load and it refers to Fig.1.13, while the equation 1.23 regards an uniform load and it refers to Fig.1.14. As a simplification, an infinitely extended overload acting with a magnitude q, can be considered as a further soil elevation and with a height that goes above the ground plane. This is the method used in the Chapter 5 and the equivalent soil height is evaluated as He= q

γ0, producing an active thrust equal to σ

0

A= KA· q and constant along the entire wall height.

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Figure 1.13: Concentrated Load

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1.1.3 Factors Affecting the Static Stability of Masonry Retaining Walls

Among all, the main factors affecting the stability of a masonry retaining wall are four [Chan (1996a)]:

• The retained soil strength;

• The retaining wall geometry;

• The slope inclination;

• The groundwater level behind the wall.

The historical method to evaluate how these parameters affect the design is to consider the maximum height-base width ratio for which no stresses are developed at the base (Fig.1.15).

'= B H 35 '= 40 '= 45 = 2 3 ' = 1 2 '

Range of recommended by the design guide on retaining walls

Usual Range of B H 10 0 20 30 40 4.0 3.0 2.0

Figure 1.15: Height/Base Width Ratio vs. Soil Strength Parameters

Among the four, the soil resistance plays a non-fundamental role in the structural stability, with an incidence of only the 10%[Chan (1996a)]. As expected, instead, and as underlined in the case study developed in the following chapters, the parameter that result to have the main incidence is the groundwater level. To provide an order of values, when the groundwater level reaches the full height of the wall, it is necessary that the wall section is 170% thicker [Chan (1996a)], even if it is very rare for the level to reach those heights (Fig.1.16).

The backfill inclination angle is very important as well: up to a gradient of 30◦, it does not affect much the instability, but after that value, the slope influence is exponential (Fig.1.17).

Regarding the normal range of front face slope angle of retaining walls, the variation in the thickness required when the angle changes is very small. Regarding the rear slope

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B H B 10 0 20 30 40 3.0 2.0 H i No Base Tension

Figure 1.16: Height/Base Width Ratio vs. Ground Slope at the Crest of Wall

B H B 0 0.5 1.0 H No Base Tension 3.0 2.0 1.0 m mH

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B H B 75 H 0 1 2 3 4 5 6 7 8 9 10 11 12 80 85 90 95 100 105 110 = 65° = 75° = 85°

Figure 1.18: Sensivity of Height/Base-width Ratio Against Wall Geometries

angle, instead, small variations in the slope angle cause large differences in the retaining wall thickness required (Fig.1.18).

If the rear face is inclined towards the backfill, great improvements in stability are naturally achieved and a wall can be thinner with just a few percentage of leaning. This is the case of the Japanese solution [Chan (1996b)]. Naturally, it is not possible to know the rear slope angle of a wall just with a surface inspection, even if it is known that usually walls with leaning rear faces have gentle sloping front face.

Another shear strength parameter that has not been considered is the cohesion. Very often, only cohesion can be sufficient to guarantee the stability and can be found in in-situ decomposed materials and in a smaller amount in unsatured backfills, as a result of soil suction. This is usually the reason why some ancient masonry retaining walls stand with a relatively weak width. The cohesion, anyway, can be cancelled by the soil saturation and therefore it is not possible to rely on it if the backfill soil is liable to be saturated.

The amount of friction at the back of a retaining wall depends on the downward displace-ment of the soil with the respect to the wall. Under ordinary circumstances, the friction angle existing between the wall and the retained soil (δ ) varies between 2₃ϕ and ϕ [Facciorusso (2011)]. If the wall stabilizes, δ decreases and in the extreme case in which the wall sinks more than the backfill soil, δ can be negative.

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1.1.4 Factors Affecting the Foundation Stability

The bearing capacity of a foundation plan depends on the soil strength, on the groundwater position, on the applied loads conditions, on the buried depth of the foundation, on its distance from the slope and on the slope inclination. For a retaining wall, the characteristic loads are governed by the wall configuration, by the height and by the backfill features. The greatest effect of the groundwater on the soil properties is that of reduce the soil density and its strength. But for the soil strength parameters derived from normal laboratory tests on saturated samples, the effect of saturation has already been taken into account.

For the particular foundation configuration that will be considered in the case study of the present thesis, the stability is independent from the height of the wall. But, on the contrary to the stability of the wall, where the influence of the soil has less influence, the effect of the soil strength on the foundation stability is very relevant, especially in the case of a groundwater high level. With a not uncommon frictional soil shear strength of 35◦, for instance, the maximum toe slope angle is reduced to 8◦for high groundwater situation [Lancellotta (2008)].

When the distance of a retaining wall from the slope increases, the foundation stability quickly raises. The increase of the bearing capacity due to the wall foundation burying, instead, is small. A deep foundation, however, is less likely to be undermined by a surface slip of the toe slope.

1.2 Structural Stability of Masonry Retaining Walls

1.2.1 General

In the Guide on Retaining Wall Design [Brand et al. (1982)], it is specified that naturally, in addition to static instability, a retaining wall must be even verified for the eventuality of a structural failure. Usually, in the project of a reinforced concrete retaining wall, the calculation and the design are carried out in order to get the specific amount of the necessary reinforcement, because the compression stresses are usually reduced compared to the material strength capacity. However, for the masonry retaining walls made of bricks loosely bounded together, the structural strength may be exceeded, and therefore the likelihood of structural failure must be inspected.

1.2.2 Masonry Strength

When subjected to a stress combination, a material can fail under compression, tension, shear or for local buckling. The collapse probability under specific stress conditions depends on the material strength and on the value of applied loads.

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Compressive Strength

The compressive strength directly depends on the units and mortar strengths, as well as on the shape and size of blocks. To deal with the effects of block shapes, the British Building Standards [Cobb (2008)], specifies four classes of stone blocks, each with the following characteristics:

• Ashlar: Blocks finely squared in very regular shapes, with a width and height not lower than 200 mm or 1/3 of the length. Surface irregularities, besides, must not exceed 2 mm;

• Coarse Ashlar: Similar to the previous ones, but with irregularities not exceeding 20 mm;

• Squared Rubble: Blocks that are squared and are picked to approximate cubes with a not exceeding 200 mm height;

• Random Rubble: Stone blocks with an irregular shape and an height not exceeding 150 mm.

The Table 1.1 shows the compressive strength of masonry, composed by units of different shape and surrounded by mortar. The intact strength of units is taken equal to 100 MPa and an average block height of 350 mm is adopted in the estimation.

Mortar Strength Ashlar

[MPa] Coarse Ashlar [MPa] Squared Rubble [MPa] Random Rubble [MPa] 2.5 12.5 8.7 7.5 1.4 1.0 11.7 8.2 7.0 1 Dry Packed 10.1 7.0 6.0 0.3

Table 1.1: Allowable Compressive Strength of Masonry Walls.

The problem of old masonry retaining walls is that that table only applies to the face layer of blocks, but the behaviour of the core materials behind the face layer is very complex to be understood because it depends on the size and the shape of the elements, as well as the manner in which they were placed [Luciano & Sacco (1997)]. If randomly dumped into position, the core material would behave as a granular material and it would be necessary a lateral pressure in order to maintain the core equilibrium against the vertical stresses.

Tensile Strength

Historically designers have always considered the masonry as non-resistant to tensile stresses. As the tensile strength derives mainly from the mortar cohesion, a tensile strength across an

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unbounded surface of a dry packed masonry should approach zero[Page (1978)]. Anyway, in the presence of headers through the failure surface, a little tensile strength exists. The intensity of this strength depends on the tensile resistance and on the cross-sectional area of the headers, on the depth length and on the friction between the headers and the masonry units.

Shear Strength

The Mohr-Coulomb shear failure criterion is the one used to define the shear strength along the joints of masonry. According to Mohr-Coulomb [de Normalisation et al. (2005)], the cohesion component of the shear strength ranges from 0.15 [MPa] to 0.35 [MPa] in function of the used mortar strength. An uniform value of µ = 0.6, corresponding to a friction angle of 31◦, must be adopted regardless of the used mortar strength[Atkinson et al. (1989)]. The Table1.2 shows the expected shear strengths according to the modified criteria at different magnitudes of compression across the shear surface. The values in the table are taken as estimation values.

Normal Stresses [MPa] Mortar Designation Mortar Strength 0 0.50 0.1 0.15 0.2 I 11 0.083 0.113 0.143 0.173 0.203 II 4.5 0.083 0.113 0.143 0.173 0.203 III 2.5 0.083 0.113 0.143 0.173 0.203 IV 1.0 0.036 0.066 0.096 0.126 0.156

Table 1.2: Shear Strength of Masonry Wall - Movement Along Joints

The shown criterion is applicable when the shear insists along a plan surface. If the stone units are bounded so that the shear is possible along an irregular surface, the shear strength will be different and it will be similar to the one of a smooth rock joint and a rough one. When the shear occurs along a non planar surface, otherwise, the surface irregularities introduce an additional frictional strength part [Patton et al. (1966)].

1.2.3 Stresses in a Stone Rubble Retaining Wall

The next step in the evaluation of the structural stability of an ancient masonry retaining wall is to check out which are the stresses intensities acting on it. Dealing with a masonry wall, an accurate analysis of stress distribution would require a detailed knowledge of the units scheme, as well as of its mechanical features. But the mathematical issue and the series of endless equations necessary to solve a similar problem would be definitely too hard to be solved. Therefore, as a first approximation of the general case, the assumptions that the masonry materials are homogeneous, isotropic and elastic are adopted, although it is known that they

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are not truthful hypothesis. However these assumptions are adequately acceptable if the units dimensions are much smaller than the walls dimensions, as in this case there is a macroscopic uniformity.

1.2.4 Possible Modes of Structural Instability of Stone Rubble

Retain-ing Walls

Compressive Failure

A reference value of maximum capacity that can be accepted with a sufficient safety level against the overturning is 60 · H[kPa] [Hendry (2001)] and depends on the retaining wall height H, expressed in meters. When this reference value is compared with the allowable wall compressive strength, it is possible to find out the approximate allowable retaining wall height.

Moreover, if weathered units are used in the wall construction, as in case of some bad quality random rubble walls, the allowable height is even lower. Finally, if the masonry has blocks of not suitable shape (Figure 1.19), it may be possible that some local issues occur even if the compressive stresses are much smaller than those actually supported by masonry [Tedesco et al. (2017)].

Figure 1.19: Local Stresses due to a Wrong Arrangement of Random Rubble Blocks

Tensile Failure

When Hope (1845) and Burgoyne (1853) carried out their destructive loading tests on masonry retaining walls (Section 1.3), they all have independently observed that a triangular fragment of masonry was remaining at the walls lower inner corner. That was attributed by the authors to the fact that the masonry was failing along the repose material angle [Harkness et al. (2000)]. This assumption, however, is not realistic because a material such as masonry, regularly composed and laid by hand, does not have a repose angle. A more exhaustive explanation can

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be seen in Figure 1.20, that shows the displacements distribution of a retaining wall subject to loads which induce overturning [A. Colas et al. (2010)].

Figure 1.20: Failure Pattern: Toppling of a Brick Wall [A. Colas et al. (2010)]

In the inner lower corner of the wall, the stress acts in an inclined direction and it can have a negative sign producing a tensile stress. This inclined stress can induce a diagonal cracking along the mortar beds, as shown if Fig.1.21 and the units below this cracking lane detach from the main body of the wall, remaining as a triangular panel when the structure overturns.

Figure 1.21: Effect of Inclined Tensile Stress on Masonry

Whether the wall can take this tension or not depends on the horizontal bounding of the structure. If the masonry layers are well bonded, the stress will not affect the integrity of the wall. If, instead, they are poorly linked, the stress will separate the masonry in two different sub-vertical parts. This is usually the case of untied stone rubble walls with small size core materials[Ramalho et al. (2008)]. If these masonry column are adequately high, the most external one can buckle when subjected to vertical compressive stresses, so that the wall can not any longer withstand the vertical loads and can structurally fail. This kind of instability is

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associated to the bulging of the lower part of the wall and the tendency to buckle is a function even of the units state and from the orientation of the mortar beds. If the latter are irregular and not perpendicular to the compression stresses lines, they are more likely to collapse. If the wall is tied, the stone headers prevent the formation of instability by reducing the length of free inflection of the layer, thus preventing the failure for buckling instability.

Shear Failure

The shear movement usually occurs along the continuous horizontal joints of the masonry, along which the shear resistance is smaller[Yokel & Fattal (1976)].

In the presence of an high groundwater level, the shear force will be greater, but the vertical compressive stress at the location of maximum horizontal shear would decrease. As a result, it will happen that the safety factor regarding the local slip will decrease and it will be more probable that this kind of failure occurs. In this process, some of the shear stress will be redistributed at the front of the wall, in which it is present further shear resistance capacity.

If a shear displacement occurs, it will be located in the mortar bed between the stone blocks. The amount of displacement depends on the mortar bed’s shear modulus and on the stresses intensity. As a result of the displacement produced by the shear, at each level of the beds, the wall moves forward, with higher values at the bottom of the wall and with values decreasing moving up.

dn dn+1 dn>dn+1 Profile of Deformed Wall 1 Course of Masonry Profile Wall Due to Shear Displacements

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1.2.5 Structural Instability Involving Cored Wall Structures

It is necessary to take account even the effect induced by the presence of cored structures of masonry on the various mode of structural instability. The structural behaviour of core materials varies depending on the contained material sizes and on the way in which they have been put in place.

Figure 1.23: Collapsed Volterra’s Retaining Wall - Evidence of Core Material

The mechanisms by which the core material influences instabilities are complicated and difficult to be evaluated. Therefore, in order to make a simplification, it is preferable to discuss about its structural performances under different stresses as a whole.

Gravel Size If the core material has a gravel size, its behaviour will be equivalent to that of a granular material, with a frictional internal angle of approximately 35◦[I. Bishop & Koor (2000)]. Under compressive stresses, a lateral pressure will be induced on the face layer (1.2.2). Overall in the case of retaining walls with a sensitive height, this lateral force must be taken into account because it induces substantial stresses and, unless the face layer of masonry is not adequately connected to the core, it is not able to resists to the induced lateral stresses. As a result, the face layer will bulge out and even collapse.

The gravel material in the core would be too weak to withstand the internal shear in vertical direction, so that it is possible that an internal slip take place. This can bring to a loss of resistance against the overturning and the wall may collapse. Anyway, not all the layers are subjected to this phenomenon at the same way. Indeed, at the top and at the bottom the layers are bounded and the result is a more accentuated bulge at half height than at the extremes. If, on the other hand, the face layer is strong enough to provide a bond against the dilation,

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the shear strength of the gravel will be much higher and the stability of the wall could be maintained. The better way to maintain the structural integrity is to adapt the link between the surface layers and the core, so that the material cohesion is increased and the constrain against dilatation is improved.

Large Size If the core material consists of large size elements randomly dumped into position, it will also behave like an isotropic granular material, it will have the same problems of a gravel material, but, at the same time, it will have an internal material shear angle that can reach 70◦[I. Bishop & Koor (2000)] and so, in general, it creates less problems. It follows that the horizontal thrusts generated will be around 1/10 of the previous ones.

1.2.6 Masonry as a Material

The parameters defining the mechanical features of the masonry are the following:

fc: Average compression strength;

τo: Average shear strength;

f_t: Average tensile strength;

E: Average value of the normal elastic modulus;

G: Average value of the tangential elastic modulus;

W: Specific weight of the masonry;

• Tensile and compressive strain capacity.

The mechanical parameters values(minimum and maximum) and the average specific weight are provided by the Italian technical standards for buildings [N. N. T. per le Costruzioni (2008)] for different types of masonry. The Table1.3 shows the average values for the types of masonry closest to our study case. The values refer to the conditions of mortar with scarce features, absence of recourse, simply juxtapose or badly connected layers, non consolidated masonry and proper texture (in case of regular items).

The masonry types we refer to are [Ortolani (2012)]:

• A: Blank stones masonry with limited facing and inner core;

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Figure 1.24: Type A. Figure 1.25: Type B. Masonry Type f_c h N cm2 i τo h N cm2 i f_t h N cm2 i E h N mm2 i G h N mm2 i w h kN m3 i Type A 200-300 3.5-5.1 0-35 1020-1440 340-480 20 Type B 260-380 5.6-7.4 0-45 1500-1980 500-660 21

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The NTC2008 also state that in case the masonry has better features than the above eval-uation elements, the mechanical characteristics will be obtained by applying the following homogenization coefficients. This approach was useful in the present thesis for a first evalua-tion of the homogeneous retaining wall and as a rough calculaevalua-tion of the masonry features values.

• If the mortar has good characteristics it must be applied the coefficient 1.4 for the Type A masonry and the 1.3 for the Type B, both to the resistance parameters ( fmand τo) and to the elastic modules (E and G);

• If the joints are thin (< 10mm) it must be applied a coefficient 1.2 exclusively to the Type A masonry, both to the resistance parameters ( fmand τo) and to the elastic modules (E and G). In case of shear resistance, the percentage increase to be considered is the half of what is considered for compressive strength. In case of natural stone masonry, it is advisable to check that the texture is proper on the entire thickness of the facing.

• In case of edging it is applied a coefficient of 1.2 (type A) and 1.1 (type B) just to the resistance parameters ( fmand τo); for the data at disposal, it does not appear that the masonry present listatures;

• In the presence of cross-linking elements between the wall layers: the coefficient 1.5 (Type A) and 1.3 (Type B) are only applied to the resistance parameters ( fmand τo); this kind of coefficient makes only sense in case of historical masonry.

• Weak core: If the masonry is made up of several layers with an inner core and if it is large in comparison to the layers or if it is made of poor material, it is appropriate to reduce the strength and strain parameters by homogenizing the mechanical features in the thickness. In the absence of accurate assessments, the above mechanical parameters can be penalized through the coefficient 0.8 (type A) and 0.7 (type B);

• In case of consolidation by injection of binding blends the coefficient 1.7 is applied to the type A masonry and the coefficient 1.5 to the type B.

1.3 Full Scale Tests on Masonry Retaining Walls

In winter 1834, the only full-scale experimental study ever performed was completed by Sir John Burgoyne. Four drystone masonry retaining walls were built and tested, each of which was 6.096 m (20 ft) long and 6.096 m high. Although the mean thickness of all four walls was the same, the cross-section of each walls was different. The masonry consisted of roughly

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squared granite blocks, laid dry without mortar. The geometries of the walls vary as shown in Fig.1.26.

Figure 1.26: Four Specimens Tested By Burgoyne [Brady & Kavanagh (2002)].

The space behind each wall was backfilled with uncompacted loose earth with a bulk density of 1390 [kg/m3]. The tests were carried out by gradually increasing the upstream load acting on the retaining walls in order to investigate for which load configuration they were no longer able to withstand the thrusts. Always referring to the Figure 1.26, in the case of wall A, the masonry was built up as the backfill was placed, until the full height of 6.096 [m] (20ft) was reached with no sign of distress. Wall B also stood following placement of the backfill to the total height, although an outward movement at the top of 63 mm, together with some slight fissures on the wall face, occurred. Walls C and D both fell when the backfill had reached a height of 5.182[m] (17ft). Sir Burgoyne even gave a graphic description of the fall pattern of the wall C and D (see Fig.1.27).

The most useful result to be drawn from these experiments is clearly the evaluation of which is the optimal shape of a masonry retaining wall. The fact that a wall with an inclined internal face withstands less than one with an inclined external face gives an important indication of one of the possible reasons that contributes in causing the case study failure, subject of this thesis. The results obtained from the experimental study exposed were afterwards confirmed

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(a) Wall C: The diagram exhibits the form the counter sloping wall assumed when falling.

(b) Wall D: The diagram exhibits the form the rectangular wall assumed when falling.

Figure 1.27: Failures Graphic Representation [Harkness et al. (2000)]

by some recent studies developed by Harkness et al. (2000), Powrie (1996), Zhang et al. (2004), Brady & Kavanagh (2002), which, through different models types, have certified the Burgoyne’s results from a scientific point of view.

1.3.1 Field Trials Reproduction with a Numerical Analyses

The main of these studies is that performed by Harkness et al. (2000), who assessed the behaviour of these four retaining walls by a numerical modelling that uses the discrete element code UDEC. The results have been afterwards compared with the field trials. By using appropriate soil and wall mass densities, strength and stiffness, it was possible to reproduce the field behaviour observed and a reasonably close agreement was obtained.

The constitutive model chosen was the elastic/Mohr-Coulomb plastic model and it was used for both the masonry blocks of which the wall was constructed, the natural bedrock and the soil backfill. Cross sections through each of the walls modelled are given in Fig.1.26, and a typical discrete element mesh is shown in Fig.1.28.

The dimensions of the walls are the same as in Burgoyne’s experiments. Generally, smaller elements were used for the wall blocks and the soil, and larger elements for the natural rock face and base. To enable the stress distribution around the toe of the wall to be investigated, particularly small elements were used in this area. Basically, the results of the analysis become more precise as the element size is decreased, but the number of elements that can be used is limited by the computing time available. In all of the analyses, it was assumed that the pore

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Figure 1.28: Used Mesh [Harkness et al. (2000)].

water pressures throughout the backfill were zero at every stage. The soil strength and density were chosen as the primary variables. The internal stability of the wall could also depend on the joint normal stiffness Knand shear stiffness Ks. In the analyses the soil was considered with a density of 1550 [kg/m3], its bulk modulus (K) was taken as 1 MPa for the top layer, increasing linearly with depth at a rate of 0.5 MPa per 0.3048 m (1 ft). The angle of friction of the soil φ0 was estimated on the basis of Burgoyne’s description to be in the range 25 − 28◦, and the angle of sliding friction between blocks of granite was taken as 45◦. The maximum angle of friction δ between the wall and the soil was assumed to be equal to the soil strength φ0 .

For walls A and B the stability was achieved at the full backfill height of 6.096 m. The horizontal displacement at the top of wall A was 12.8 mm, compared with 32.6 for the wall B: that indicates that the shape B was probably closer to the failure than wall A. For walls C and D, the horizontal displacement stabilized at each loading stage until a backfill height of 4.877 m was reached. When the height of the backfill was increased at 5.182 m, the calculated horizontal displacement for both of these walls started to accelerate, indicating instability and impending collapse.

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performance and ultimate stability. Of the four configurations tested the parallel sided wall that leant into the backfill offered the greatest degree of stability. This is partly because the horizontal component of the earth pressure is a minimum in this case, and partly because the masonry is placed where its mass has the greatest stabilizing effect. A wall with a sloping front face is more stable than a vertical wall with parallel sides and the same cross-sectional area, because while the lateral force that must be resisted is the same in each case, the lever arm of the centre of mass of the masonry is greater in the case of the battered wall. A wall with a sloping back is marginally less stable than a vertical wall, because the effect of the batter in this case is to increase the thrust of the soil, and therefore the overturning moment, more than the restoring moment due to the weight of the wall. In carrying out a limit equilibrium analysis of a drystone masonry retaining wall, it is necessary to take into account the possibility that crack will open within the masonry, as otherwise the potentially critical failure mechanism may be missed.

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Chapter 2 Numerical Modelling

2.1 Masonry

The masonry structural analysis is a composite and complex problem and, in order to have a correct representation of its behaviour, various strategies have been developed, depending on the analysis accuracy requirements and on the need for simplification. In a general framework, the masonry analysis depends essentially on three aspects [Senthivel & Lourenc¸o (2009)]:

• Modelling scale: the modalities to be used for carrying out a structural analysis change according to whether the local mortar-unit interactions and the relative damage mecha-nisms are to be known, or if an entire structure is considered and its global behaviour is requested to be analysed.

• Geometric configuration: the scheme chosen to represent a masonry structure strongly depends on the geometry, which allows simplifications, in case of symmetries or partic-ular elements spatial features.

• Constitutive law: the relationship between stress and deformation that describes the material behaviour can be more or less complex, depending on the objective of the analysis and the accuracy to be obtained.

Modelling Scale

The first aspect that needs to be explored is the one concerning the detail scale to be used. The masonry is a composite material and for a numerical modelling of its behaviour the choice can fall on three distinct modelling scales [Roca et al. (1998)]:

• Detailed micro-modelling: units and mortar are considered separately and are repre-sented through continuous elements, with the addition of discontinuous unit-mortar

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interface elements to describe the interaction between the two elements. Unit

Interface

Mortar

Figure 2.1: Micro-modelling

In this first approach the two components are considered separately and all the distinct inelastic characteristics are evaluated. The interface element represents a potential breaking-sliding plane, to which is assigned a proper stiffness in order to describe the interaction behaviour and to avoid the elements interpenetration.

This kind of procedure is suitable for describing the local behaviour of the material, because it manages to capture the various breakage modes due to the interaction between unit and mortar components. If the main goal is understanding what is locally happening in a masonry structural element, then, it is necessary to chose this modelling type. Since the computational duty for large structures is so high that it does not allow the use of this modelling technique to analyse entire buildings, another similar but simplifying approach has been devised , in order to extend the analysis to elements of larger than a simple portion of masonry; i.e. the simplified micro-modelling.

• Simplified micro-modelling: the units are expanded in order to maintain the geometry unchanged and are separated from one another by discontinuous interface elements in which the mortar features and its behaviour are also condensed.

Unit Joint

Figure 2.2: Simplified Micro-modelling

In analogy to the previous case, even this one is a discrete modelling. With this approach, however, the units are not separated by mortar and interface elements as distinct entities, but they are concentrated in a sort of ”average” interface. to keep the geometry unchanged, the units are slightly larger than their actual size. The discontinuity

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elements placed at the interface are presented as sliding/fracture surfaces which provide a joint behaviour description. This leads to a slight loss in the results accuracy, since the mortar Poisson’s coefficient can no longer be considered. Thus, a more synthetic representation of the material is obtained and the interface behaviour is described in terms of relations between stresses and relative displacements of the two opposing units faces. The assumption is that an element can only deform in a normal and tangential direction to its plane.

• Macro-modelling: the masonry is considered as a single continuous medium character-ized by homogeneity and anisotropy, whereby the individual components are no longer distinguishable.

Continuous

Figure 2.3: Macro-modelling

In the two previous approaches, the masonry behaviour is described through the geomet-ric and mechanical features of its components. The discrete representation is suitable, however, to a detailed modelling: if the damage, indeed, is presented as a localized phenomenon, the micro-model choice for a structural analysis will surely be the most desirable one. On the contrary, if the structural damage is widespread and if, as in the study case treated in (RREEFF), its dimensions are big enough to consider the stress state as homogeneous, then it is possible to resort to a continuous model operating with the macro-modelling [Lourenc¸o (2002)]. In this approach, the material is idealized as a homogeneous medium, without any distinction between mortar and unit, and it is the constitutive law to determine the stress and strain relationships: in this way the computational effort with respect to micro-modelling is considerably reduced. The main defect is that the characteristic masonry shear collapse can not be included in the macro-model because the geometries of mortar and units are not discretized. The macroscopic mechanical properties to be assigned to this ideal material can be determined in two ways:

– through a phenomenological approach: by performing experimental tests with tensile compressive and shear stresses on representative masonry elements, it is

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possible to obtain the overall masonry mechanical properties to be assigned to the model [Andreaus & di Paolo (1988)].

– through homogenization techniques: since the masonry is represented as a base cell that repeats continuously, moving from the mechanical properties of its constituent elements, it is possible to calculate with sufficient accuracy the continuous features with an homogenization techniques.

Compared to discrete modelling, it is evident that there is a loss in accuracy, since the model deals with average stresses and strains. However, even if synthetic, this analytical approach represents a right compromise between precision and efficiency. But even with this approach, some difficulties are present, due overall to the complexity of modelling the inelastic and anisotropic behaviour of the continuous. In order to describe this attitude, some of the constitutive laws proposed in the continuum mechanics will be reported in the following paragraphs.

Given the scale of the problem analysed in this thesis, the hardly identifiable characteristics of the considered masonry and, above all, the irregular texture that characterizes the retaining wall in question, it has been opted for this last type of modelling to analyse the problem by means of the numerical finite element method, with which the software Diana Fea, used as modelling software, operates.

Constitutive Laws

The constitutive law formulation strongly depends on the material assumption made, with reference to its strength and strain features. The choices essentially concern the reversibility or not of strains, the need to modify the stiffness during the load history and the possibility of strength degradation after a certain stress state is reached. Briefly, the constitutive laws capable to describe masonry can be divided into three main types [Anthoine (1995)]:

• Non-tensile resistant laws: being clear and known that the masonry is not characterized by a good tensile strength, the simplest constitutive law that can be proposed is that in which the material tensile strength is completely neglected. In this kind of bond, therefore, the deformations associated with the reaching of the elastic limit are reversible and correspond to null stresses (Figure 2.4).

The elastic stiffness matrix and the strength domain do not undergo modifications during the loading history and the behaviour is described only by these two elements.

• Plastic laws: that assumes that the material has a resistance limit after which permanent strains are formed. The total strains are so composed by a reversible elastic contribution

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E

Figure 2.4: Representation of the non-tensile resistant law

(εe) and an irreversible inelastic one (εp), as shown in Figure 2.5.

p c

Figure 2.5: Representation of a plastic law.

The elastic stiffness matrix does not undergo changes during the loading history, while the plastic strain process depends on the load path and not only on the applied stress value, due to its dissipative nature. Therefore, in order to know the load relationship that bind together stresses and strains it is necessary to write them in incremental or differential form. Once the yield point has been reached, i.e. when the value for which plastic strains begin to develop, the stress-strain relationship can be of three types: hardening, softening and perfect plasticity.

In particular, the type that represent the masonry properly is the softening one, as the material start to lose is strength capacity when the yield limit is reached.

It must be kept in mind that the plastic theory is born to describe the behaviour of plastic materials, but, since the masonry is almost a brittle material, it is sensitive to hydrostatic pressure, so the yield criteria used must take this aspect into consideration. An example

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of this is the Mohr-Coulomb criterion, which evaluates the masonry yield limit and it is discussed in detail in 4.7.2.

• Damage laws: unlike the plasticity theory, the damage mechanics in the continuous can get the micro-cracks that arise and develop in the material stressed beyond the yield stress and that have, as primary effect, that of reducing the material stiffness, especially under tensile stresses (Figure 2.6).

c

Figure 2.6: Representation of a damage law.

As for plastic strain, the damage also occurs with an irreversible process: manifesting as a fracture that develops due to concentration of defects produced by the material stresses. The damage mechanic is distinguished from the fracture one mainly in the conception of the cracking process. While in the fracture mechanic the material degradation is considered along a line and, therefore, the stress depends only on the strain that exist in the same point, in the damage mechanic the degradation due to the cracking growth is described through a ”continuous damage variable”, which represent a quantitative measure of the degradation of the features that describe the macroscopic behaviour of masonry. In order to develop a proper constitutive damage model, therefore, an appropriate definition of this variable is required, and it is generally based on the concept of fracture energy.

The damage and plasticity model The damage and plasticity models are born to combine the plasticity theory and the damage mechanics in a single model, in order to take into account both the irreversible contributions due to plastic strains and those related to the material damage processes. The two theories are, indeed, complementary and capable to establish a specific type of constitutive law: the elasto-plastic damage model, in which the stiffness degradation is combined with the plasticity theory [Creazza et al. (2002)].

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continuum composed by a quasi-brittle material, where the tensile and compressive strength of the material in the two directions are different. For the definition of the constitutive law used for describing the masonry retaining wall discussed in this thesis, the reference is made to the section 2.2.5.

2.2 Numerical models in DIANA FEA

2.2.1 Non-linear finite element method - types of elements

In the finite element method a structure is subdivided into a number of separate ”elements”, each of them with certain properties. The behaviour of the elements is described by means of displacements and forces in the nodes. This discontinuity is followed by the assembly in which the elements are literally ”knotted” together. Afterwards, the boundary conditions, i.e. the supports and the external loads are added and a set of equilibrium and compatibility equations is set up for the entire system. At this point, the software is introduced to solve this system of equations, which results in nodes displacements and support reactions. From these displacements and reaction the elements strains and stresses can finally be calculated, by which a complete insight into the mechanical of the structural behaviour is obtained.

The Figure 2.7 shows the two kind of elements used in the thesis model. It concerns a two-dimensional continuum element and a linear line shaped interface element. The continuum element has been used for masonry units and is based on stress-strains relations.

y x y x u u xy xx yy u v t n

Figure 2.7: Finite elements applied in this research. a) Plane stress distribution with Gauss integration scheme, b)Interface element with Lobatto integration scheme.[Manie (2017)]

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three strain components εxx, εyy, γxy, so that the stress vector can be described as:

σ = [σxxσyyτxy]T (2.1)

and the strain vector as:

ε = [εxxεyyγxy]T (2.2)

The interface element has been used for separating the singular model elements and in order, overall, to model properly the behaviour between soil and structure. With this kind of elements, stress-displacement relations are used instead of stress-strain ones. The considered stress is either a normal stress σ perpendicular to the interface or a shear stress τ along the interface, with a displacement µ normal to the element and the shear displacement along it. The collection of these components is called the interface stress vector τ:

τ = [σ τ ]T (2.3)

and the interface displacement vector µ:

µ = [µ ν ]T (2.4)

2.2.2 Solution procedure for non-linear material behaviour

It is known that only in the case of very low stress levels a linear-elastic model can suffice, but, in the treated case non-linear models able to simulate crack formations, their shear and crushes are needed. The non-linear behaviour of stone-like materials as masonry is usually characterised by softening (Figure 2.15)and can be considered an intermediate trend between the two extremes represented by the perfect plastic and the completely brittle behaviour. This quasi-brittle behaviour occurs in both tensile and compressive stress and is caused by the gradual break down of the weakest links of an heterogeneous material, after which the micro cracks finally link up to form a macro crack or a shear plane.

The linear relation requires an incremental-iterative solution procedure in which the load is applied step by step and equilibrium are carried out in each load increment. The used procedure is described in the following section.

2.2.3 Structural Non-Linear Analysis

The structural analysis of an engineering problem is fundamentally divided in two main categories: static and dynamic. Both of them, in turn, can be carried out in a linear or non-linear field. The structural non-linear behaviour, expressed by the non-linearity of the

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equilibrium equations with a non-proportional relationship between the actions and the stress state, is due to several reasons. In this dissertation the non linearities which occur in the study-case exposed in the Chapter 5 are only mentioned, i.e. the geometrical non linearity due to the second order effects, the non-linearities due to the typical feature of the material (cracking for masonry and the non linear effects of the constitutive law), and the boundary conditions of the structure (constrains)[Lenza et al. (2012)]. In particular it is given:

• Material Non-Linearity: The structural material undergoes a strain that increases in a non-linear proportion with the increase of externally acting loads.

• Geometrical Non-Linearity: The strains that affect the structure can not be considered small. Hence, it is necessary to consider the equilibrium of the deformed configuration.

• Constrains Non-Linearity: The boundary conditions with the increasing of acting loads.

In a non-linear field, acting on the structure with loads increasing their magnitude with time, the load-displacement diagram basically evolves following a non-linear trend as shown in the Fig.2.8. PA P q qA PB P q qB P =C P q qA P +A P B q =C q A+ q B

Figure 2.8: Non-linear Trend of a Load-Displacement Diagram

In other words, at each instant, the structure presents a different deformed configuration, whose points evolve following the material constitutive bond. This means that, in order to obtain reliable results, the temporal evolution of the structure as a whole must be considered, from the initial state until the collapse. The determination of a single equilibrium configuration in a particular load disposition is no longer enough, as it does not allow the assessment of the structural response under the variable load level.

Furthermore, one of the most important things to consider is that, unlike the linear analysis, it is no longer possible to apply the principle of the effects superposition. The consequence is that every single combination must be analysed separately, resulting in a longer computing effort.

It should be specified that for the stress and strain analysis of a static or dynamic structural assessment only the finite element formulation allow a correct structural modelling, as it allows

Numerical analysis of a gravity retaining wall: the case of Volterra's urban walls.

D

I

C

I

T

L

M

N

UMERICAL

A

NALYSIS OF A

M

ASONRY

R

ETAINING

W

ALL

:

THE

C

ASE OF

V

OLTERRA

’

S

U

RBAN

W

ALLS

Relatore Accademico:

Dott. Linda Giresini

Relatore Accademico:

Dott. Mario Lucio Puppio

Relatore Esterno:

Dott. Giorgia Giardina

Candidato:

Gianmarco Passera

Contents

Collapse of Masonry Retaining Walls

1.1

Static and Foundation Instability of Masonry Retaining

Walls

1.1.1

General

1.1.2

Thrusts Acting on a Retaining Wall

= c'+ ' tan '

' =

'

' =

'

P

W

R

'

i

P

R

W

1.1.3

Factors Affecting the Static Stability of Masonry Retaining Walls

1.1.4

Factors Affecting the Foundation Stability

1.2

Structural Stability of Masonry Retaining Walls

1.2.1

General

1.2.2

Masonry Strength

1.2.3

Stresses in a Stone Rubble Retaining Wall

1.2.4

Possible Modes of Structural Instability of Stone Rubble

Retain-ing Walls

1.2.5

Structural Instability Involving Cored Wall Structures

1.2.6

Masonry as a Material

1.3