Assessment

1 Analysis methods

• Semi-empirical Methods: MEXE
• Limit Analysis Methods
• Solid Mechanics Methods

2 Proposed levels of structural analysis for masonry arch bridges

Level of Analysis	Elements of Analysis
Level 1 Basic Analysis	Semi-empirical methods should only be considered as part of the appraisal by inspection where the owner considers this procedure as appropriate. Basic 2D limit analysis methods should be used to assess all structures except significantly skewed bridges, long spans, bridges with unusual geometries and important structures.
Level 2 Detailed Analysis	The bridges excluded from the previous group and those failing its assessment should be analysed using solid mechanics methods. These analyses should be adapted to the available bridge data and combined with site investigations and monitoring as appropriate, if more refined analyses are required to demonstrate the adequacy of the structure to fulfil its purpose. The use of characteristic or worst credible strengths of materials may be used, based on test results from samples taken from the structure. The level of refinement achieved before it is decided that the structure is unfit for purpose will depend on owner needs and constraints.
Level 3 Special Analysis	Where the use of refined solid mechanics methods cannot demonstrate structural adequacy, it may be possible to demonstrate its adequacy and inherent safety by comparison of its safety characteristics with other similar structures using stochastic approaches and probability analysis Actual live traffic loadings of the bridge might be determined statistically and used for analysis The safety criteria of the bridge in question might be assessed and specific relaxations considered if this can be justified and the acceptability of risks clearly demonstrated

3 Significance of defects for assessment purposes

Settlement	MEXE	Rigid Block	FEM/DEM
Vertical differential settlement between adjacent supports Springings stay parallel	Not specifically mentioned but is rolled up in the 'lateral cracks or permanent deformation of the arch which may be caused by partial failure of the arch or movement at the abutments'. This is given a condition factor of 0.6 - 0.8.	No specific consideration	Settlement can be incorporated into the model
Horizontal spread of support. Springings stay parallel	Not specifically mentioned but is rolled up in the 'lateral cracks or permanent deformation of the arch which may be caused by partial failure of the arch or movement at the abutments'. This is given a condition factor of 0.6 - 0.8.	No specific consideration	Settlement can be incorporated into the model
Horizontal inward movement of support. Springings stay parallel	Not specifically mentioned but is rolled up in the 'lateral cracks or permanent deformation of the arch which may be caused by partial failure of the arch or movement at the abutments'. This is given a condition factor of 0.6 -0.8.	No specific consideration	Settlement can be incorporated into the model
Transverse settlement of an abutment or pier a) Rotation b) Local differential settlement	In the absence of any barrel cracking no specific guidance is given. If diagonal cracking is present then this is considered to be dangerous if extensive. A condition factor of between 0.3 and 0.7 is given	No specific consideration	With a 3D model it is possible to model the cracking but this would make the analysis very expensive and could only be justified in special cases. (eg. Listed structure)
Effects of point load actions (slipped units, fan yield-line patterns)	Considered when determining the condition factor	No specific consideration	With a 3D model it is possible to model the cracking but this would make the analysis very expensive and could only be justified in special cases. (eg. Listed structure)
Hinge formation and incremental loss of statical indeterminacy	Not specifically mentioned but is rolled up in the 'lateral cracks or permanent deformation of the arch which may be caused by partial failure of the arch or movement at the abutments'. This is given a condition factor of 0.6 -0.8.	The 2D ultimate limit state load carrying capacity is calculated.	The successive hinge formation can be followed through the analysis. 2 and 3D models can be built.
Shear loading	No specific consideration but should be incorporated into the condition factor.	RING program specifically models the interface between the units to allow for unit slippage and ring separation.	This can be modelled but there are problems with validating the output given that the parameters that govern this mode of behaviour are not well understood nor measurable for an existing structure.
Transverse bending effects: Longitudinal cracking	The position and extent of the cracking influences the condition factor. This varies depending upon the bridge authority. Longitudinal cracks due to differential settlement are considered dangerous by the Trunk Road Overseeing Authorities . If the cracks are greater than 3mm and at less than 1m centres then a condition factor of 0.4 should be adopted. Otherwise, a factor of up to 0.6 may be used. The Network Rail on the other hand offers a range of condition factors from 0.95 for a situation where the longitudinal cracks are outside the centre third of the barrel, less than one tenth of the span in length, to 0.85 for a crack within the centre third of the barrel longer than one tenth of the span in length.	Currently, all rigid block formulations analyse the barrel as a 2-dimensional problem and so do not consider the transverse distribution. Consequently, whether longitudinal cracks are present or not does not affect the calculated carrying capacity per metre width.	Like the rigid block programs, 2 dimensional models will not be affected by the longitudinal cracking. In the 3-dimensional models, however, it is possible to introduce the discontinuities caused by the cracking. This will make the model very complex and the validation of such a complex model would be highly questionable given our current knowledge and material property data.
Transverse bending Spandrel wall separation	MEXE (BA16) suggests that spandrel wall separation does not per se affect the condition factor but that its presence should be considered within the remit of the overall condition factor. The railways cover it in the above classification.	Currently, all rigid block formulations analyse the barrel as a 2-dimensional problem and so do not consider the transverse distribution. Consequently, whether longitudinal cracks are present or not does not affect the calculated carrying capacity per metre width.	Like the rigid block programs, 2 dimensional models will not be affected by the longitudinal cracking. In the 3-dimensional models, however, it is possible to introduce the discontinuities caused by the cracking. This will make the model very complex and the validation of such a complex model would be highly questionable given our current knowledge and material property data.
Material deterioration	Material factors take into account the condition of the material	Engineering judgement is applied to represent the barrel by adjusting the geometry and strength of the barrel	It is possible for a 3-dimensional model to vary the properties of the bridge material throughout its thickness and width and thus represent the actual condition of the bridge. This is very difficult and expensive to do. More usually a 'smear' approach is taken to replicate the actual structure.
Ring Separation	This should be considered very carefully as tests have confirmed the potential crack propagation and accompanying loss of carrying capacity. Ignoring the bottom ring and carrying out an assessment using the reduced effective thickness is not recommended unless it has been shown that the remaining ring is intact.	RING program specifically models the interface between the units to allow for unit slippage and ring separation.	This can be modelled but there are problems with validating the output given that the parameters that govern this mode of behaviour are not well understood nor measurable for an existing structure.

4 Advantages and Limitations of the Analysis Methods:

Main Parameters	Applicability	Advantages	Disadvantages
Method: MEXE
Span Rise at crown and quarter point Arch thickness and crown cover Arch material Backfill material Mortar joint depth and condition General condition factor	Only applicable to spans shorter than 18 m Not applicable for flat or appreciably deformed arches Not applicable for multi-span bridges, although the BR version of MEXE accounted for that using an extra factor	If applied by an experienced engineer, the condition factors may allow to account for effects difficult to model	The only resisting mechanisms considered are the arch and the weight of the backfill The limiting load criterion is not realistic Unnecessary assumptions on geometry and load locations When applied by inexperienced engineers, some modifying factors can be dangerously subjective Its results are assumed to be conservative, but can be over-conservative as well as unsafe Cannot consider the effect of strengthening measures
Method: Heymans Limit Analysis Methods
Arch geometry Compressive strength of masonry (in some models) Masonry and backfill densities	Can be difficult to apply to deep arches Can be difficult to apply to shallow arches with large spans Can be difficult to apply to bridges with complex geometries	For simple structures, it can produce safe results from very limited input and at a limited cost These methods are very effective when the engineer has a clear idea of the mechanism by which the structure will fail	For Upper bound methods, if some failure mechanisms are ignored, the method would provide an unsafe prediction For Lower Bound methods, if some kinematically admissible equilibrium states are ignored, the method would provide a conservative prediction. Similarly, if an assumed equilibrium is not possible (because some failure criterion has been ignored) the method will produce unsafe results Cannot consider ring separation Cannot consider snap-through failures Cannot consider the contribution of the spandrel walls.
Method: Discrete and Indiscrete Rigid Block Methods
Arch geometry Compressive strength of masonry (in some models) Masonry and backfill densities Dilatancy Angles of friction (radial and tangential)	Can be applied to multi-ring arches Can be applied to multi-span arches Can produce unsafe results in shallow arches with large spans (bridges where snap-through failure is possible) Some methods might be able to consider skewed arches	Quick and reliable for a significant range of bridge configurations It is a very versatile tool for an experienced engineer	Cannot consider snap-through failure Cannot consider the contribution of the spandrel walls The separation between rings during cannot be reproduced. Instead, the used has to assume whether ring separation will or will not take place Consideration of masonry compressive failure might increase the computational time
Method: Castigliano's Non-linear Analyses
Arch geometry Compressive strength of masonry (in some models) Masonry and backfill densities The deformational and strength parameters of the backfill	Cannot be used with skewed bridges	Simple and easy to use	The prediction of the in-service behaviour can be quite sensitive to the boundary conditions and the initial stress state, which are very difficult to determine Cannot consider ring separation
Method: Finite Element
Arch geometry Initial stress state Compressive strength of masonry (in some models) Masonry and backfill densities The deformational and strength parameters of the backfill Masonry tensile strength and post-yielding response (softening rule, etc.)	These methods provide results on the in-service behaviour of the structure. As such, they can be used to analyse existing defects, their origin and relevance on the safety of the structure Similarly, they can be used to consider strengthening and/or repair options taking into account not only their effect on the capacity of the structure, but on the performance of the structure and its components. This allows to give more consideration to the performance of the structure and therefore consider the long term effects of the methods adopted.	Can be extremely versatile and allow almost any sophistication required The versatility of these methods means that they can be used to explore the benefits of various strengthening options.	The prediction of the in-service behaviour can be quite sensitive to the boundary conditions and the initial stress state, which are very difficult to determine The lack of customized packages means that the preparation of the models can be quite time consuming As the complexity of the model increases, so does the time required to obtain results. Since parametric studies are essential, this option can become too expensive. The results are sensitive to input parameters difficult to determine such as the backfill properties, the masonry strength and the interface between the different structural elements.
Method: Discrete Element
Arch geometry Initial stress state Compressive strength of masonry (in some models) Masonry and backfill densities The deformational and strength parameters of the backfill Masonry tensile strength and post-yielding response (softening rule, etc.) Contact properties	Same as FE	Same as FE, with the advantage of coping better with discontinuities (such as ring separation and spandrel separation). Uses explicit solver which improves convergence.	Sensitive to the boundary conditions and the initial stress state, which are very difficult to determine The lack of customized packages means that the preparation of the models can be quite time consuming As the complexity of the model increases, so does the time required to obtain results. Since parametric studies are essential, this option can become too expensive. The explicit solver requires a rigorous check of the results predicted.