Methods of strength assessment have been categorised (McKibbins & Melbourne 2006) as semi-empirical, limit analysis and solid mechanics methods.
Most semi-empirical methods are based on the MEXE (Military Engineering Experimental Establishment) method which evolved from work undertaken in the 1930s for the military to rapidly assess arch bridges. It is often still used as a first pass strength assessment but its use is highly subjective and there are many limitations. It is of little value for any detailed work such as the design of strengthening.
Most conventional bridge assessments are now carried out using computerised versions of limit analysis also known as mechanism analysis. In its simplest form these methods consider a 2D arch comprising a series of blocks of infinite compressive strength, which cannot slide against each other and cannot carry tension. A routine is used to establish the locations of hinges in the span followed by calculations of reaction and then vector algebra to position the resultant line of thrust. The method produces a lower bound solution. In other words, if a load path can be found that lays entirely within the masonry then the modelled arch is capable of sustaining that load even if it is not the true load path.
Limit analysis techniques have proved to be excellent tools for first phase strength assessments but several restrictions exist that are important in the design of strengthening. The most important of these is the inability to calculate strain and displacement. Consequently, it is not possible to determine the distribution of stress at operational load levels, it is difficult to assess the serviceability of bridges and in the case of strengthening, it is not possible to determine the share of load between the existing bridge and the strengthening.
The established technique used to model continuum based phenomena in solid mechanics such as deforma-bility is the Finite Element Method (FEM). Not surprisingly this has also become the most popular solid mechanics method used for arch bridge analysis, and there are numerous well developed industry quality computer programs available.
Like limit analysis most work is carried out using 2D representations, generally plane strain, but 3D shell and solid models are used for special assessments.
Although these techniques can be good for determining displacements, strains and stresses at operational load levels they quite often become difficult to use to predict ultimate strength and damage. This is generally because of the type of solver that is used, normally an implicit solver, and the effort required to ensure internal forces are in equilibrium with external loads, as brittle materials such as masonry soften and redistribute load. The solution to the equilibrium problem is normally to use a hypothetical masonry tensile strength but choosing a suitable value, large enough to achieve equilibrium conditions are met but small enough not to influence the result, can be a challenge.
3 THE FINITE/DISCRETE ELEMENT METHOD 3.1 Description
Numerical techniques have been devised to represent discontinua where body or particle interaction defines overall behaviour (Cundall 1971). Perhaps, the most advanced technique that describes this behaviour is the Discrete Element Method (DEM). The relatively new Finite/Discrete Element Method (FDEM) (Munjiza 2004) is a combination of FEM and DEM and provides a more natural approach to the simulation of many materials and structures. It has been applied to a diverse range of engineering and scientific problems from food processing to rock blasting. Through automated adaptive modelling, even the transition from continua to discontinua and the fracturing and fragmentation process can be represented.
FDEM is aimed at problems involving transient dynamic systems comprising large numbers of deformable bodies that interact with each other. Models involve typically thousands, but in extreme cases millions, of separate Finite Element meshes automatically interacting with each other using DEM contact algorithms. The solution of the continuum equations associated with FEM is well established, the algorithms within DEM less so.
Contact detection and contact interaction lay at the heart of DEM. Contact detection is aimed at identifying discrete elements that can potentially come into contact with each other and eliminating those far away from subsequent contact interaction algorithms. Different algorithms have been developed for different packing densities for example, sparse and moving or dense and static. The chief aim here is to reduce computing effort. Contact interaction applied to the surfaces of discrete elements coupled through the detection process is where interface behaviour is calculated. Here interface laws are applied according to the surface characteristics of the contacting discrete elements for example, frictionless no-tension contact. During the solution of transient dynamic problems of even quite modest size millions of contacts will be detected and resolved.
Another key aspect of FDEM is that the analysis involves all equations of motion, is therefore dynamic and uses an explicit central difference solution scheme (Owen 1980). This involves a time stepping procedure that is conditionally stable, but unlike many conventional FE solvers that use an implicit solution scheme, does not involve computationally intensive matrix factorisation. Solutions are achieved only through the use of very small time steps. The critical time step size below which steps must remain for stability and accuracy is given by the time taken for a stress wave to travel across the smallest finite element. The efficiency of DEM contact detection and the avoidance of equilibrium calculations allows FDEM simulations to predict failure, collapse and post-failure kinematic behaviour.
Masonry is a non-homogenised material, can be regarded as a discontinuum and as such is ideally suited to FDEM. Simply, masonry arch bridges are a special form of masonry structure, which is an important consideration when faced with complex bridge arrangements.
The approach that has been developed for arch bridges, applied using the implementation within the Finite Element computer program ELFEN, uses smeared masonry compressive properties and explicit mortar shear and tensile properties. Each brick or block unit is modelled with a separate finite element mesh and each unit becomes a single discrete element. It has been found that units can also be grouped together (Brookes 1998), a blocky arrangement of four or five bricks glued together, to improve computational efficiency without any loss of accuracy. The masonry arch is then assembled using blocky arrangements in hundreds, possibly thousands of discrete elements. Other bridge parts for example, fill, surfacing and backing are similarly represented although the material models are different.
FDEM arch bridge models will develop failure mechanisms consistent with limit analysis results if these are critical as well as providing displacements, stresses and strains consistent with solid mechanics.
Another key aspect to the use of FDEM and the adopted modelling approach is that representing masonry at a fundamental scale requires only commonly available and basic material parameters to be used in order to accurately characterise bridge structural behaviour. Non-linear material models are used to define the deformable behaviour of the masonry in compression and the fill in tension. A perfectly plastic Von Mises yield criterion is generally used to cap compressive strength, and a Rankine yield criterion used to give a simple no-tension soil model.
The behaviour of mortar, as well as other contacting surfaces such as masonry to fill, is included by using interface material models. Interface models give the
Figure 1. Part of a modelled masonry arch barrel showing FE mesh (left) and DE mesh (right).
surface of discrete elements appropriate mechanical properties. Mortar is represented differently depending on the type of construction. Historic construction involving lime mortar joints is represented using a no-tension Mohr-Coulomb friction relationship. Modern masonry with cement mortar produces masonry with some tensile strength. In these instances good predictions of masonry behaviour can only be made by including mortar tensile strength and a fracture energy formulation to model the development of cracking. Generally, masonry arches are historic constructions and do not include cement mortar.
For most types of masonry the generic material characteristics, compressive strength, Young's modulus, mortar friction and mortar cohesion, necessary for FDEM simulations are readily available (Hendry 1990, BD 21/01 2001, BS 5628-1 2002). They are no more demanding to obtain than those parameters required for conventional limit state analyses. An estimate for Young's modulus for different types of fill in compression is similarly available.
There are no limitations to the geometric arrangements of arches that can be represented with FDEM other than those associated with computational resource. However, models are kept as simple as possible to reflect the confidence in material parameters, geometric arrangement and to be reasonably compatible with the codes of practice through
which all design and assessment work has to be undertaken. Hence, the large majority of simulations are 2D and plane strain.
In assessment and design, live load is generally applied by explicit representation of axle loads using discrete elements. Weight is applied to these elements and the axles moved across the span with a prescribed velocity. As transient dynamic solutions are obtained, regard has to be given to acceleration arising from sudden movement and inertia effects. Consequently, loads are applied smoothly and slowly to ensure static responses are obtained. Permanent loads are introduced through construction sequences, which depending on the barrel shape, may necessitate the use of modelled temporary false work; a process that is always required with real arch bridges.
Although the time required to develop FDEM bridge models exceeds that of comparable limit analysis representations, these models can still be assembled in several hours. Also solution times, which are
continuously tumbling as ever faster computers become available, are modest compared with similar FEM representations with strength analysis completed in around 4 hours for a typical bridge on a 3.6 GHz PC. This includes the calculation of permanent loads and the traverse of a single vehicle. To complete an assessment or design, several axle arrangements have to be considered to be sure the critical case is identified, so this could take several days. With relatively small problem sizes, around 5,000-10,000 degrees of freedom, mass scaling techniques to accelerate the solution process are never used to obtain solutions, but are useful to quickly check the simulation process.
Archtec strengthening has been described as 'Key Hole Surgery' for bridges because of the absence of any major intervention to the arch barrel. Generally construction comprises retrofitting stainless steel reinforcement around the circumference of the arch barrel. The reinforcement is grouted in to holes drilled in to the bridge with a coring rig from the road surface or, alternatively in the case of multi-span structures, from below.
Arches conventionally fail by the development of four hinges leading to a mechanism. The design basis for the strengthening is to locate the reinforcement so
as to provide bending strength at the critical locations thereby resisting the development of the hinges. By providing bending resistance the arch barrel is able to resist the critical loading conditions more efficiently and the peak compressive stresses in the masonry are reduced. A similar procedure is applied to more complex arrangements including multi-span arches although failure mechanisms and anchor positioning requires anchors to be placed in different positions.
Compared with conventional arch bridge strengthening such as concrete saddling and lining, the Archtec service has several practical benefits:
1 Through the use of FDEM a good assessment of existing strength and bridge behaviour is obtained. This allows accurate matching of strengthening to the loading requirements if the bridge is under strength, thus minimising any intervention. Alternatively, strengthening may be avoided.
2 Strengthening is invisible which is particularly important for historic and heritage bridges.
3 Construction is small scale and fast to implement.
4 Disruption to bridge users during strengthening is much less than saddling.
5 A more sustainable solution with lower environmental impact, embodied energy and carbon emissions.
6 Because displacement and strain is predictable assessments and strengthening designs can be based on limits states other than purely ultimate strength.
7 Each anchor installation provides a core of information that can be used to confirm the materials and internal arrangement of the bridge.
8 In many instances all these factors equate to reduced cost.
Archtec services have to be provided within a framework which embraces as far as possible national codes of practice. Unfortunately, outside of the UK, there are few rules to help Engineers assess arch bridges. For example live loading is almost always developed for beam arrangements of bridges where load support is primarily through bending, masonry strength assessment is often permissible stress based, and bridge specific earthquake rules are geared towards steel and concrete construction. In India the railways have a code of practice for masonry arch bridges which impose almost arbitrary performance limits on deflection.
The use of FDEM to simulate arch bridges is a performance based method, useful for limit state assessment and design, but cannot be directly used for rules that have been developed for linear, often inaccurate, working stress approaches. In these instances to satisfy bridge technical authorities hybrid analyses are run along side the more realistic and reliable limit state work. The results allow additional checks to be made with local codes of practice.
The process which has been undertaken to verify the FDEM analytical methods employed by Archtec have included a number of key strands and evaluations as follows:
1 Against conventional methods of arch assessment.
2 Against published data from full-scale tests of unstrengthened arches carried out by others.
3 Against full-scale tests by TRL of bridges strengthened by the Archtec method which were specifically commissioned as part of the verification process.
4 Against the results obtained by monitoring bridges in the field including before and after strengthening comparisons.
Additionally, a philosophy of fixing material parameters for whole series of tests where similar masonry construction has been employed (compressive strength of bricks, mortar type etc.) has been adopted. This makes it impossible to adjust an individual arch analysis to gain better correlation with tests within a series without influencing all others. Similarly, the analysis ofArchtec strengthening follows on from verified and fixed unstrengthened analyses.
A selection of the verification work (Brookes 2003) and recent field trials illustrating the accuracy and flexibility of FDEM arch simulation follows.
In order to test the practical implementation of Archtec, to validate the FDEM method of structural analysis, to help quantify key strength parameters and
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