21 March, 2024 13 min reading
Based on the thesis by: Lucy Jane Davis, M.Sc. in advanced masters in Structural Analysis of Monuments and Historical Constructions.


The main aim of this thesis, by the author Lucy Jane Davis is to provide correction factors for nonlinear 2D solutions which cover the transverse fluctuations for the case study of the Poniklá Bridge located in Bohemia in the Czech Republic and help to make load rating models more accurate and easier to carry out with less input data.





Masonry arch bridges are a common bridge typology around the world. This type of bridge has existed for millennia as transport for people, livestock, materials and as aqueducts for water carriage.

Today, many of these masonry arch bridges are used for modern needs such as for cars and railroads.

A number of these bridges are over 100 years old, and they need to be load rated for continual use. Many different methods to assess the load-carrying capacity of masonry arch bridges exist, but there is no single adapted standard used widely. Modern codes typically include a procedure for the load rating of existing bridges yet cannot always be applied to masonry arch ones.



Masonry Arch Bridges


Masonry arches have long been one of the most important building tools in history, allowing builders to create larger structures with less material. The arch has greatly developed over time along with the changes in technology, knowledge, and materials.

The first example of a masonry arch, a corbel arch, to today’s creative works in examples such as thin shelled concrete vaults prove its development over.

Masonry was the preferred choice for ancient construction due to its high compressive strength and the availability of local materials. Masonry arches have the ability to provide great interior spans but come with the cost of the horizontal thrust they produce as well as the difficulty in construction. Well-designed centering to support the voussoirs during construction was necessary to complete a project.


The masonry arch bridge has existed for thousands of years and is faced today with different traffic and loading conditions than before. It is important to understand the stability and behaviour of these structures in order for their continued use and safety today.



Limit Analysis Methods and Historical Development


Limit analysis is the systematic approach to determining the plasticity and stability of masonry arches using geometry and began to be developed in the 1930s.

Limit analysis is a powerful tool that can be used to determine the capacity curve of a structure.

However, as it can only provide the maximum capacity, it should be used as a complimentary to any computer-based analysis and not used to predict the failure mechanisms or cracking behavior of the structure.



Finite Element Modeling


Finite element methods (FEM) are a useful numerical tool used in structural analysis. Masonry can be modeled in a few different ways.

  • Micro-modeling
  • A simplified micro-modeling approach, also known as meso-modeling
  • Macro modeling

Limitations of finite element analysis are:

  • the constitutive material models available in FEM software,
  • the uncertainty of many material and geometrical properties in existing structures,
  • the uncertainties involved in modeling the soil to structure interaction.

However, these models are the closest estimate of reality that can exist.



Discrete Element Modeling


The discrete element method (DEM) is a numerical modeling approach that utilizes an idealization of masonry as a “discontinuum” elements. Masonry is represented as an assembly of discrete elements that represent masonry units as blocks or particles while joints are modeled as zero thickness elements.

For the analysis of masonry arch bridges, the most commonly used method for structural analysis using DEM is the distinct element model. The distinct element model is based on the following assumptions:

  • Blocks are either rigid or deformable;
  • A soft contact approach is used;
  • An explicit solution procedure in the time domain is used for either static or dynamic analysis.





For a bridge to be used in service for local roads, railroads, and highways it must be load rated to determine the live load capacity of the structure. This ensures that the bridge can carry the load of various vehicles.



Types of Load Rating


The three main categories of methods of load rating of masonry arch bridges that exist today include:

  • Semi-Empirical methods, including the MEXE method;
  • Equilibrium based methods, including mechanisms and limit analysis;
  • Numerical models, including finite element (FE) analysis.

Eurocode EN1990 and EN1991-2

Bridge load rating analyses can be run based on the Eurocode EN1990 and EN1991-2.

The ultimate limit states (ULS) and the serviceability limit states (SLS) are both considered in EN1990.

  • The ultimate limit states are defined as the limit states concerning the safety of the structure and the safety of people.
  • Serviceability limit states concern the functioning of the structure under normal use, the comfort of people, and the appearance of the structure especially concerning deflection and extensive cracking.



Military Engineering Experimental Establishment (MEXE)


The MEXE method is a very prominent method used for masonry arch bridges, and many load carrying capacities have been determined using this semi-empirical formula.

The MEXE uses a calculation of the provisional axle load (PAL) and compares it to the capacity of a standard arch barrel to complete the assessment. The equation is based on the arch span, thickness, and depth of fill.


One issue with the MEXE method is that it may be unconservative when lower spans are considered.

With this being said, the MEXE method is simple and quick method as the parameters can be determined with a visual inspection. Additionally, as it is a single formula, the cost of computation is low especially as compared to finite element or discrete element methods.



AASHTO Manual for Bridge Evaluation (MBE)


Load rating of existing bridges, including masonry arch bridges, in the United States is completed in accordance with the AASHTO MBE.

According to the MBE, bridges are either rated:

  • at the Inventory Level (results in a reported live load level that a bridge can carry without damaging the structure for an indefinite amount of time)
  • or the Operating Level (describes the maximum permissible load that a structure is allowed to be subjected to safely


Masonry arch bridges are evaluated at the Inventory Level while other existing bridges are evaluated at the Operating Level.



Czech Guide


The Czech Ministry of Transport developed a manual for the load rating of masonry arch bridges.

This guide uses semi-empirical formulas and introduces a new limit state, the repeated load limit state (RLLS) in order to create a simplified manual that can be used by civil engineers with a standard background, based on five main concepts:

  • Does not account for the compliance or failure of abutments, piers, and the foundation.
  • The load rating of the bridge is determined by the ULS and the RLLS.
  • A direct, semi-empiric formula is applied to the bridge. The formula is derived using a least squares approach from solutions of the bridge data using linear and nonlinear finite element analysis.
  • Criteria for the load rating can be determined from a linear numerical analysis.
  • Proposes two procedures, one using numerical analysis and one using a semiempiric formula to determine the load rating. The use depends on each application individually.



SMART (Sustainable Masonry Arch Resistance Technique)


The SMART is an assessment procedure designed for the load rating of masonry arch bridges that considers:

  • the construction,
  • material properties,
  • limit states,
  • present and past actions,
  • analysis,
  • modes of failure.



Limit Analysis Based Methods


Equilibrium based methods, such as those based on the study of limit analysis, provide the basis for many existing computational tools used in the analysis of masonry arch bridges. The most common computer software using this type of analysis methods are:

  • Archie-M
  • RING
  • CTAP
  • Elasto-Plastic Model



Numerical Methods


Another solution to structural analysis is by the use of numerical methods. Numerical methods include finite element methods (FEM), discrete element analysis (DEM) as well as other models that approximate solutions using numerical approximation.



Load Rating Methods – Conclusion


The most used methods for load rating currently used are the empirical methods including the MEXE and limit state analysis-based methods such as RING and ARCHIE-M.



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History and Information


The two-lane bridge is located near the village Poniklá in the Czech. It is a three-span masonry arch bridge with a total length of 53.3 m over the Jizera River in the Liberec Region.


The bridge was built in 1857 and is still used as a second-class road bridge. It is a valuable structure with industrial heritage that is currently planned for reconstruction works.

The bridge was considered not fit for service, according to the requirements from the 2016 main bridge tours, the current requirements in the Czech Republic Bridge Management System, requirements from the investor, as well as the standards of the Czech Republic.

A reconstruction of the Poniklá bridge was carried out in the mid-1930s:

  • a layer of concrete to the top of the stone arches was added,
  • the pavement and backfill were removed, the concrete layer added, and several damaged ashlars were replaced,
  • in addition to adding concrete, drainage pipes were inserted at the foot of the arch.




As no original documentation of the Poniklá Bridge remains, geometrical properties are from a survey done for planned reconstruction works:

  • The bridge has a total length of 53.3 m with three spans of 14.07 m, 14.26 m and 14.07 m.
  • The clear span of the arches is 11.4 m and the free height of the bridge above the water is 5.45 m.
  • The radius of the arch is 9.81 m, and the thickness of the arch voussoirs is 0.8 m.
  • The bridge substructure and the arch vaults and spandrels are composed of sandstone blocks assembled with mortar. The backfill material is unknown and consists likely of low strength excavation material.



Material Properties


The bridge is a masonry construction built using sandstone units. A diagnostic survey and non-destructive testing campaign were carried out in June 2019. Non-destructive tests were complete on the mortar joints at 16 locations to determine the compressive strength of the mortar.




Modeling Assumptions


  • The FEM models were created using a macro-modeling approach.
  • The model would be considered only using linear properties
  • As the geometry determined from the construction company included inconsistencies throughout the width of the bridge, an average of the values for the arch span width was taken. The geometry used in the finite element model is shown in Figure 12. This geometry is consistent throughout the 2D and 3D models.



Elastic Properties and Modulus of Elasticity


The material properties used in the case study of the Poniká Bridge are shown in Table 3.

The constitutive law used for all materials is isotropic linear elastic. Masonry behaves nonlinearly and can be represented using different material models such as the no tension model or the smeared crack model. However, the comparative analysis will be completed using a linear elastic analysis and therefore the use of isotropic linear elastic material properties is sufficient.



Boundary Conditions


Different boundary conditions were considered for models consisting of a single span as well as the models depicting the entire bridge in order to best represent reality.

In every model, the boundary conditions at the base of the piers are considered to be a rigid support. The support used is a line support in the 2D model and a surface support in the 3D model.

No tests have been completed to determine the soil structure beneath the piers, but a geological survey completed in 2019 determined the bedrock within the region to have high strength, so it can be assumed that the bedrock has sufficient stiffness beneath the bridge as well.

Boundary conditions at the horizontal supports between the spans constrain horizontal movement while allowing freedom in the other directions. This boundary condition simulates the existence of the other spans while recognizing that in reality there will be some displacement in the x-direction as well.



Loading Conditions


The loading condition used to assess the load-carrying capacity of the Poniklá bridge was EN1991-2: Load Model 1. Dead loads and live loads were considered.



The load combination used was the combination of the design dead load and the design live load.

Superposition is utilized as the structure is considered in the linear range, so the combined effects could be added.





As a 2D FEM analysis is a useful tool to be used within load rating of masonry arch bridges, it is necessary to understand the level of difference with respect to a 3D FEM analysis. By using a comparative analysis between a 2D finite element model and a 3D finite element model of the same bridge, this can be considered.

Finite element models were constructed of the Poniklá Bridge using the modeling software RFEM, developed by Dlubal. RFEM is a finite element analysis software designed for modern use and structural analysis of structures with various material types.

First, initial assumptions were made. This includes the definition of constitutive models, boundary conditions and element types that can be considered in both 2D and 3D. In this way, the models are dependent on the same assumptions and the two models can be considered analogous.

The third type of model considered in this analysis is a 2D nonlinear model created in RFEM, considering material nonlinearity in the masonry arch. In this case, the masonry material of the arch was considered as a material without tension using the setting for surfaces “without membrane tension” in RFEM. The purpose of the 2D nonlinear model is to compare the results to the 2D linear model, where tension is acceptable within the material model.





Outside of RFEM, a non-commercial FEM software could be used to represent a 2D nonlinear model of the Poniklá Bridge. This model uses a program developed by the CTU Faculty of Civil Engineering.

This model has been created in order to represent the Poniklá Bridge according to code requirements that may be used to legally defend the Poniklá Bridge’s current safety conditions in order to prevent its future demolition.

The material model adopted for the masonry is a no tension material model. This model was chosen as it is used to reflect the real behavior of a stone arch with limited input information due to its simplicity. Geometry follows geometry used in previous models, however, only the arch, backfill and pavement are modeled neglecting the piers in favor of boundary conditions.



Application of 3D/2D Ratios


The results from the 2D no-commercial finite element model can be updated using the 3D/2D correction factors, the ratios determined in Section 5.4. By multiplying the 2D results by the respective factor, the results can consider 3D effects. Table 16 shows this calculation and its results.






The global bridge model can be used to verify the boundary conditions used within the 2D non-commercial model.

The single-span non-commercial model considers boundary conditions at the span extents restricted from displacement in the x-direction. This is not representative of reality because of movement in adjacent spans which cause displacement in the x-direction. The displacements resulting from the global bridge RFEM models can be used as an imposed displacement in the non-commercial model to more represent reality.






The goal of this analysis was to determine the x-displacement of the model at the extents of each span to verify the boundary conditions used in the 2D nonlinear model created with the CTU inhouse software. This was completed by running a linear elastic analysis of the global bridge model and determining the u-x displacement at each of these locations. The extents of the span were considered from the midpoint of one pier to the midpoint of the next pier, the same definition used in the single span models. Results from the 2D linear RFEM model can be seen in Figure 48, scaled up 2000x to display the deformed shape.





This work determines a method that can be used to adjust values from a two-dimensional analysis to consider three-dimensional effects. By understanding the differences, lower-cost analysis methods can continue to be run with confidence and a greater level of safety.

This study analyzed a three-span masonry arch bridge in the Czech Republic using finite element analysis.

  • The analysis produced ratios quantifying the difference between the 2D and 3D models, to give a range of values.
  • These ratios can be used as correction factors to adjust values from the 2D models through simple multiplication of the extreme stresses or displacements.
  • The correction factors produced can be summarized in Figure 51 and Figure 52.



  • The 2D and 3D models for in the comparison were analyzed linear elastically. Even in the Ultimate Limit State, most of the bridge behaves linearly and the nonlinear behavior experienced is local. For this reason, it can be expected that the ratios determined in this comparative analysis can be extended to nonlinear models. However, further investigation must be done in order to verify this assumption.


More detailed analysis techniques can be applied, and a nonlinear analysis technique used. It is recommended to include nonlinear material models as well as consideration of distribution effects in the backfill in further analyses.

Further recommendations to expand upon the work presented here also include the analysis of additional case studies. In addition to a simple arch bridge, skew bridges and ones with more complicated geometry can be analyzed as well. This can expand the knowledge of the difference between 2D and 3D models.



Load Rating of the Poniklá Bridge


The results from the analysis of the various models (Table 22) used to determine the load rating of the Poniklá Bridge, according to the ULS, show that the bridge is safe under this assessment except for the updated 2D non-commercial model with the 3D/2D ratios. However, this value does not include the existing concrete layer which strengthens the bridge.


  • These results support the safety of the structure.
  • Most importantly, the final value highlighted in green in Table 22 is shown to be safe.
  • It is recommended that the results reported in this work contribute to the argument of the preservation of the bridge.


Final recommendations for the preservation of the bridge include:

  • the replacement of the pavement,
  • removal of vegetation growing on the bridge
  • not its demolition.