ICE GRADUATE & STUDENT PAPER COMPETITION 2014

ICE WEST MIDLANDS

28th February 2014

ION CARMONA-MARTINEZ

ICE No.: 71251738

THE EVOLUTION OF MASONRY

BRIDGE ASSESSMENT

METHODOLOGIES

Abstract: The scope of this paper is to compare and contrast the assessment method that the bridge engineers

currently use for the assessment of masonry arch bridges. The paper will help engineers and bridge

inspectors to choose an appropriate method for the analysis of masonry arch bridge structures. Features of

the considered methods are explained and possibilities and limitations for selected cases are discussed.

Some suggestions to choose the best solution are given taking into account desirable simplicity from one

side, and required complexity from the other. Detailed explanation for modelling of defects in finite element

method (FEM) is out of the scope of this paper.

Contents

THE EVOLUTION OF MASONRY BRIDGE ASSESSMENT

METHODOLOGIES .......................................................................... 0

1. Introduction: ...................................................................................... 3

2. The structure of masonry arch bridges:.............................................. 4

3. Methods for analysis of masonry arch bridges ................................... 5

3.1. Empirical rules: ................................................................................. 6

3.2. Pippard’s elastic theory for arch bridges ............................................ 7

3.2.1. The MEXE Method:........................................................................... 8

3.3. Jaques Heyman’s plastic theory for masonry arch bridges................ 10

3.3.1. The mechanism method – rigid blocks analysis: ............................... 10

3.3.1.1. RING 2.0. Software......................................................... 11

3.3.2. Thrust analysis ............................................................................... 11

3.3.2.1. Archie-M software ........................................................... 12

3.4. 3D non-linear finite element analysis ............................................... 13

4. CONCLUSION ................................................................................ 16

5. Bibliography and References........................................................... 17

1. Introduction:

Masonry arch bridges form an integral part of the canal, motorway and railway infrastructure in Europe and

throughout the world. It is estimated that there are over 50,000 masonry bridges in the UK [1] making this

type of bridge the most common one. For this reason, a safe, reliable and established methodology should

be used to assess these bridges.

Before the 19th century most of the bridges were built based on empirical and geometrical rules. Afterwards,

elastic analysis was developed by Pippard [2] and Military Engineering Experimental Establishment (MEXE)

[3] was then used to assess the existing masonry arch bridges. During the early and mid 60’s, the plastic

analysis was proposed by Heyman [8, 9, 10] what led to a change in the way we understand the behaviour

of masonry bridges. Due to the computer revolution, several programs were developed based on the

Heyman’s theorem, and since the year 2000 3D finite element analysis has became popular [17, 18].

In this paper, the author will explain the basics on masonry arch assessment methodologies, and an

example based on his own experience will be commented [20].

2. The structure of masonry arch bridges:

Figure 1 describes the principal elements of a masonry arch bridges. Main structural element of an masonry

bridge is the arch barrel and the fill. Arches transfer load to the abutments providing horizontal and vertical

resistance for the arch.

The fill above the arch is considered as structural element that distributes a live load and increases the

stability of the arch by its weight and by providing a passive reaction against large movements. Spandrel

walls on each side of the barrel contain the fill material as well as provide additional stiffness for the arch.

Wing walls retain the fill behind the abutments and may also increase their stability. Waterproofing layers of

clay materials were often applied in order to improve durability of the arch barrel. This layer is very often

damaged due the ageing.

Historical arch bridges generally have relatively shallow foundations compared to modern standards usually

made of concrete or stone masonry. The piles have often rotted over the decades or centuries resulting in

weakened support and movement in the abutments and piers [1].

Figure 1: Typical structure of a double-span masonry arch railway bridge [1, 4]

3. Methods for analysis of masonry arch bridges

The methods shown below will be described on the paper. Depending on the employed method, the results

are shown as distribution of internal forces, prediction of failure modes, indication of the location of plastic

hinges or providing the ultimate bridge capacity.

These methods can be classified as follows:

1. Empirical rules

2. Pippard’s Elastic theory for arch bridges

2.1. MEXE method

3. Heyman’s Plastic theory for masonry arch bridges

3.1. Mechanism method

3.1.1. RING 2.0 software

3.2. Thrust line analysis

3.2.1. ARCHIE-M software

5. 3D non-linear finite element analysis

3.1. Empirical rules: Prior to the application of statics analysis to masonry arches, initiated by La Hire [5] in the first half of the

18th century, the design process of arch bridges involved the use of empirical rules, which were based on

simple geometrical relations. The safety of the structure was based on past experience providing the

dimensions of several bridge components (span, rise and thickness of arch, width and height of piers, etc.).

Although empirical rules are hardly justified from a mechanical point of view, most of them are revealed to be

efficient.

Thickness of the arch: The thickness at the crown (t) is related to span (s) through different mathematical relationships. Many

empirical equations were proposed, mainly during the 19th century. The most well-known expressions are

listed in Table 1 for deep arches only.

Table 1: Historical empirical rules for crown arch thickness [5]

Date Author Deep arch Shallow arch

15th cent. Alberti t = s/10 -

1714 Gautier (s>10m) t = 0.32 + s/15 - 1777 Perroner t = 0.325 + 0.0035s t = 0.325 + 0.0694ρ 1809 Gauthey (s<16m) t = 0.33 + s/48 - 1809 Gauthey (16m<s<32m) t = s/24 - 1809 Gauthey (s>32) t = 0.67+ s/48 - 1809 Sganzin t = 0.325 + 0.3472s - 1845 Dejardin t = 0.30 + 0.045s t = 0.30 + 0.025s 1854 L’Eveille t = 0.333 + 0.033s t = 0.33 0.033√s 1862 Rankine t = 0.19√R - 1870 Dupuit t = 0.20√s t = 0.15√s 1885 Croizette-Desnoyers t = 0.15 + 0.20√ρ - 1855 Lesguillier t = 0.10 + 0.20√s T = 0.10 + 0.20√s 1914 Sejourne t = 0.15 + 0.15√s - - (Average) T = 0.289 + 0.075s s: span ; R: radius of the circle passing through the crown and intrados springing ; ρ: curvature radius

Width of piers:

The minimum geometrical value of pier width for semi-circular arches is given by the sum of the thickness of

adjacent arches at springing. Furthermore, it is possible that aesthetical aspects and hydrodynamic effects

had also been empirically considered in the establishment of the width of piers.

3.2. Pippard’s elastic theory for arch bridges Pippard worked in conjunction with Baker, Chitty, Ashby and Tranter [2] on arch bridge problems from 1936

to 1968. He conducted a series of experiments on arches and concluded that the arch behaved elastically

within a limiting load and failed by 4-hinged mechanism.

The arch behaved elastically until formation of the first hinge or crack was created. Then it failed in a similar

manner by 4-hinged mechanism. Pippard noticed that after the first hinge occurred, there was a significant

amount of reserve strength in the arch before collapse.

The elastic method plays a great role in rough estimation of masonry arches. The derivation of that

estimations starts with a two pinned arch with horizontal forces keeping the arch in place. A force P is

applied at the top of the arch. The set up can be seen in Figure 2.

Figure 2: Nomenclature for Pippard's analysis [2]

Pippard made a series of assumptions in order to simplify the analysis:

1. The arch is assumed to be parabolic, with a span‐to‐rise ratio of 4.

2. The arch is assumed to be pinned at the abutments, i.e. two‐hinged arch.

3. The dispersal of loading applied at the surface of the fill was assumed to occur only in the transverse

direction with a 45° load spread angle.

4. Pippard considered the case of a single point load applied at the mid‐span.

5. The effective width of the arch was taken as twice the fill thickness at the crown, b = 2h

6. The fill was assumed to have no structural strength and to only impose vertical loads on the arch.

The fill was assumed to be of the same density as the arch ring, 22 kN/m3.

7. The limiting compressive stress was taken to be fc = 1.50 N/mm2 and the limiting tensile stress was

taken to be ft = 0.75 N/mm2.

The maximum allowed load P becomes.

� =256�ℎ�

� + 128��ℎ� �28� −

121 −

ℎ + �4� �

25� + 42

�

Therefore, he derived an expression relating the span, rise, thickness and fill depth over the crown. This was

a simple approach to assess arch bridges and was used during the wartime. The expression was then

modified by the Military Engineering Experimental Establishment (MEXE) [3] in the form of a nomograph and

is currently recommended by the Department of Transport in its Departmental standard (BD21/01) [6].

3.2.1. The MEXE Method: This method was based on the two hinged elastic analysis work by Pippard that was then calibrated with

both field and laboratory tests from the 1930s [3].

The method was most predominately used in World War II as a way to quickly classify the load carrying

capacity of older masonry arch bridges. However, since that time, the MEXE method has still been used as a

way to load rate masonry arch bridges.

The Modified Axle Load (MAL) depends equally on the arch and backfill thickness although the ring

thickness has a significantly greater influence on the arch behavior than the backfill. The modification factors

are introduced without taking account of the arch geometry; the backfill depth, ring thickness, and even the

mortar thickness could have differing influences on arches with different geometries.

The method comprises of the primary calculation of:

���������������� = 740�� + ℎ� �!.# $%&$'$($)$(

Where,

d = thickness of arch barrel adjacent to the keystone, [m]

h = average depth of fill at the quarter points of the transverse road profile between the road surface

and the arch barrel at the crown, including road surfacing, [m]

L = span, [m]

Fsr : span/rise factor

Fp : profile factor

Fm : material factor

Fj : joint factor

Fcm : condition based, to be determined onsite

Details of the values of these modification factors may be found in the current departmental standard

(BD21/01) for the assessment of highway bridges and structures. As can be seen from the above

formulations, the MEXE calculation is very user friendly, but does not allow for many inputs to be changed.

Also, arbitrary assumptions must be made about the condition factors with little quantitative basis. Some

guidance is given in the Advice Note BA 16/97 (Department of Transport 1997) [7], but it is primarily left to

the interpretation of the user.

Advantages: [3]

� Quick calculation of maximum allowable load.

� No complex computer modeling required.

Disadvantages and geometric limitations:

� Span L ≤ 20 m.

� Rise at the crown rc ≥ 1/4 of the inside span L.

� Depth of fill 30 cm ≤ h ≤ 105 cm between the upper surface of the arch crown and the underside of

the sleepers.

� It is based on point load at crown. Mechanism may form with point load at 1/4 or 1/3 point prior to

overstress of masonry.

� Allowable axle load is modified by factors to account for material condition and existing arch

cracking.

� No clear guidance on how to handle conditions of support movement or slender piers.

� User does not readily understand the background of the formulation.

� Effects of support movement are only modeled with a single modification factor.

� Changes in hinge location and points of maximum stress cannot be determined.

� Assumes a relatively high allowable tensile strength of the arch.

3.3. Jaques Heyman’s plastic theory for masonry arch bridges

From 1960’s Jaques Heyman [8, 9, 10] authored several books and articles on the topic of masonry

analyses. His methodologies revolved around plastic analysis techniques. Heyman, who had also previously

researched the development of plastic hinges in steel structures, expanded and refined the ideas of Pippard

to masonry arches. He applied the idea of hinge formation in unreinforced masonry structures. As part of his

studies and analysis, he outlined the following simplifying assumptions:

� Masonry units have an infinite compressive strength: fc = ∞

� Masonry units behaves as rigid body: E = ∞

� Joints transmit no tension: ft = 0

� Masonry units do not slide at the joints

As a result of the assumptions, the bounding theorems of plasticity can be applied to determine the ultimate

load of a masonry arch. Plasticity theories incorporate two theorems: a lower bound, or equilibrium solution,

and an upper bound, or mechanism solution.

The lower bound solution can also be described as the yielding solution. For an arch, the yield condition can

be modeled by superimposing the thrust line onto the shape of the arch. The point of yield is the limit where

the thrust line stays just within edges of the arch.

The upper bound theorem is a mechanism based approach. A mechanism, or deformed hinge model, is

drawn of the arch. Then virtual work can be used to assess the failure load of the arch. The collapse load

factor is the spot at which both the upper and lower bound solutions are satisfied by a specified load factor.

Heyman’s work provided the foundation for many future analysis works and methods. He introduced the

concept of mechanism based failures in masonry arch structures and noted the benefits of understanding

hinge locations.

3.3.1. The mechanism method – rigid blocks analysis:

The mechanism method is based on the upper bound plastic analysis. [8, 11, 15]

Masonry arch collapse loads can be determined by analysing the arch as a mechanism instead of an elastic

structure. Pippard’s studies identified how hinges form in arches and their effects on collapse. However, his

analysis still relied on elastic methods. Heyman [8] realised the effects of hinges on the collapse load of

masonry arches. He suggested that the researcher should identify the possible hinge point locations. Then

calculate the forces and stresses in the indeterminate structure. The locations of the hinges should then be

adjusted based on the calculations and the process continued iteratively until the location of the hinges

stabilises [12].

3.3.1.1. RING 2.0. Software

This is a computer software created by Gilbert (2001) [13, 14]. It is based on the mechanism method using

the rigid blocks approach. The program analyses a 2D model of a bridge structure built of arch barrels,

supports and backfill giving possibility for creation of multi-span bridges. A particular feature of the software

is an analysis of multi-ring arch barrels enabling separation of rings. The calculation algorithm employs an

efficient linear programming technique for solution of virtual works equations. As the result of an analysis is

obtained the minimal multiplier for the live load of a given scheme triggering formation of a mechanism.

The program provides presentation of a failure mode in the form of a hinge or sliding mechanism. Exact

locations of hinges or sliding surfaces of the limit state are indicated and vectors of the mobilised soil

pressure are graphically presented (Figure 3). A proposed backfill influence is taken into account as a

stabilising the arch.

Figure 3: Ring 2.0. software output

3.3.2. Thrust analysis The thrust analysis is based on the lower bound limit plastic analysis. [8, 11, 15]

Heyman [8] recommended determining the smallest possible arch thickness of which the thrust line with

assumed hinge locations would fit within. That thickness was then compared to the actual arch thickness. He

considered the ratio of the two values to be the geometric factor of safety for the arch. From his work, he

developed the “quick analysis” method. It is based on an arch with inputs of dimensionless parameters and a

point load P. The equation is based on a failure occurring with hinges at each of the springings, under the

live load, and at the crown.

3.3.2.1. Archie-M software An example of a computer software applied to analysis of masonry arch bridges is Archie-M developed by

Harvey (2001) [16]. A theoretical basis of the program is the thrust line analysis and to be precise it is the

zone of thrust approach employing a finite masonry strength. This application ca be used for modelling multi-

span arch bridges together with supports and backfill.

Calculation are carried out on a static scheme of a 3-pin arch where locations of the pins is chosen as the

most likely one for a given load pattern. In respect of that the program presents a response of the structure

to any value of loads in the form of the most probable zone of thrust location in the profile of the arch (Figure

4).

It also graphically indicates location of the hinges. The condition for the structure to be in equilibrium is that

the zone of thrust exists in the interior of the cross sections along the whole arch. The limit state giving the

load carrying capacity can be obtained increasing the live load value up to the moment when the zone of

thrust begins touching the edge of the arch profile in the fourth point.

The program provides a numerical presentation of results for each arch segment including internal forces

and the zone of thrust position. A very helpful option is an automatic location of the worst position for a given

loading vehicle.

The backfill in Archie-M is a continuous body providing a soil-structure interaction like dispersal of load, at

rest, passive and active earth pressure.

Figure 4: Archie-M software output

3.4. 3D non-linear finite element analysis The methods mentioned earlier have been widely used for analysing arch bridges because no complicated

computations are required. However, these methods were based on many assumptions which might affect

the overal results of the ultimate bridge capacity.

The main advantage of the finite element analysis is the opportunity to model a whole structure including

spandrel walls, wing walls and abutments. Not only the behaviour of the arch is considered but the fill and

abutment are also analysed to check their structural vulnerability. Additional modes of failure are possible to

predict like longitudinal cracking of the arch barrel or spandrel walls separation.

In terms of loads, further to static loads, dynamic and seismic loads can be implemented. This also allows for

differential settlement modelling in order to know the soil-structure interaction [17].

The results provided by application of 3D modelling involve three-dimensional image of stress distribution in

the whole structure or its deformation (Figure 5) [18].

Figure 5: Von Mises stress output from F.E. software - Abaqus

The author [20] was involved in a project where a analysis of a 7 span masonry arch was analysed. The

bridge was curved on plan and 120m long. It carries a rail traffic live load. In a previous inspection, large

cracks were noted across the quarter point and the spandrel walls. (Photo6).

A initial investigation was carried out in order to know what caused the cracks. The author used most of the

analysis techniques mentioned above and concluded that the bridge was stable under dead and live load.

The cracks shown at the Figure 6 could not be explained according to the classical assessment

methodologies.

Figure 6: Existing photos on the bridge

For these reason a 3D nonlinear finite element analysis was carried out. The author modelled the masonry

bridge under dead and rail live load using Abaqus [19] finite element software. However, the results of the

F.E. model were almost the same as the classical methods – the bridge was stable.

After another on-site inspection the author noticed that the bridge had large differential settlement at the

middle piers. This large differential settlement was modeled as imposed vertical displacement on the finite

element model. The result obtained showed that the cracking were created due to the differential settlement

an not for and excesive load on the track.

Looking at the picture 7, cracking can be noted on the red and green areas. Both green areas and real

photos showed similar cracking pattern at the same places.

Figure 7: Plastic model and cracking pictures are linked

Finally, 3D nonlinear finite element analysis is very reccomendable to model complex bridge geometry. In

addition to this, new structural deffect can be considered that classical method are not able to analyse.

4. CONCLUSION

To sum up, numerous methods has been employed during the last decades to assess the capacity of the

masonry arch bridges. Starting from the easy-to-use geometrical rules to the most sophisticated finite

element software.

From the author´s point of view [20] a summary in the table 2 can be found which suggested the best

method depending on the simplicity, the accuracy, the time and the cost of the analisys.

Table 2: Summary of masonry arch bridge analysis methods

Simplicity Accuracy Time Cost Advantages Disadvantages Pippard’s Elastic method

2 2 1 1 Quick analysis Conservative assumptions

MEXE method 1 3 1 1 Well know Conservative assumptions Archie-M software

3 4 2 3 Visual solution No differential settlement analysis

Ring 2.0. software

3 4 2 3 Visual solution No differential settlement analysis

3D F.E. analysis 5 5 5 5 Most accurate solution

High cost and time required

Score: 1 minimum – 5 maximum Obviously, to choose the analysis mothodoly depends on the engineers experience and judgement. The

paper tried to summarise the main assumptions and limitation of these method providing a guidance for a

bridge engineers.

5. Bibliography and References

[1] Orban Z, Gutermann M. Assessment of masonry arch railway bridges using non-destructive in-situ testing

methods. Engineering Structures. 2009.

[2] Leon J, Espejo SR. Load test to collapse on the masonry arch bridge at Urnieta. 2007

[3] Oliveira DV, Lourenco PB, Lemos C. Geometric issues and ultimate load capacity of masonry arch

bridges from the northwest Iberian Peninsula. Engineering Structures. 2010.

[4] Pippard AJS, Tranter E, Chitty LC. The mechanics of the voussoir arch. J Inst Civil Eng 1936;4:281-306.

[5] Military Engineering Experimental Establishment, Classification of masonry arch Bridges (Christchurch:

Military Engineering Experimental Station, 1952)

[6] Department of Transport (2001b): The Assessment of Highway Bridges and Structures, Design Manual

BD 21/01, part of Design Manual for Roads and Bridges Volume 3, Section 4. Department of Transport

Highways Agency, London, UK.

[7] Department of Transport (2001a): The Assessment of Highway Bridges and Structures, Advice Note BA

16/97. Department of Transport Highways Agency, London, UK.

[8] Heyman, J. 1982 The masonry arch. Ellis Horwood, Chichester, United Kingdom.

[9] Heyman, J. (1966). “The stone skeleton. Structural Engineering of Masonry. Architecture”. University of

Cambridge, ISBN-13 9780521629638, ISBN-100521629632.

[10] Heyman, J. (1996). “Arches, Vaults and Buttresses, Masonry Structures and their Engineering

Valorium”. ISBN 0860785971.

[11] Gilbert, M. & Melbourne, C. 1994 Rigid-block analysis of masonry structures. The Structural Engineer

72, 356–360.

[12] Livesley, R. K. A. 1978 Limit analysis of structures formed from rigid blocks. International Journal for

Numerical Methods in Engineering 12, 1853–1871.

[13] Gilbert, M. (2001): RING: A 2D Rigid Block Analysis Program for Masonry Arch Bridges. Proc. 3rd

International Arch Bridges Conference, Paris, pp. 109-118.

[14] LimitState Ltd. Ring Theory & Modeling Guide, Version 2.0k. Sheffield, UK

[15] Gilbert M. Limit analysis applied to masonry arch bridges: state-of-the-art and recent developments.

[16] Archie M 2.0.7. Obvis Ltd, Exeter, 2001

[17] Boothby TE, Fanning PJ. Three-dimensional modelling and full-scale testing of stone arch bridges.

Computers & Structures. 2001.

[18] Frack SA, Bretschneider N, Gebhardt M, Slowick V, Marx S. Simulation of cracking in masonry arch

bridge. VIII International Conference on Fracture Mechanics of Concrete and Concrete Structures.

[19] Abaqus/cae Use’s Manual. Version 6.8.

[20] Carmona Martinez I. Application of non-linear finite element analysis to a seven span masonry arch

bridge. Master thesis. UPV. 2012