This paper discusses on the required level of simplicity for suitable modelling of structural concrete. Traditional equilibrium-based approaches (as strut-and-tie models) are too coarse in some cases, as they account for the cracking state of concrete in a sometimes excessively simplified manner. The alternative of complex nonlinear numerical modelling is also not always satisfactory for design as the number of parameters required, their definition and the sensitivity of the structural response to them is complex and requires a high level of experience. Contrary to these approaches, this paper introduces the elastic plastic stress field method. This method is grounded on the theory of plasticity but allows considering deformation compatibility. The results are consistent both in terms of the strength and deformation field of the member. It also has the advantage of requiring only two physical material properties (modulus of elasticity and plastic strength) which can be easily determined by designers.
Este artículo discute sobre el nivel de sencillez ideal para un análisis no lineal de elementos de hormigón estructural. Los métodos de cálculo basados únicamente en condiciones de equilibrio (como los modelos de bielas-y-tirantes) no son siempre adecuados ya que el estado de fisuración del hormigón se considera a veces de una manera excesivamente simplificada. Los análisis no lineales complejos tampoco son siempre adecuados, ya que el número de parámetros requeridos, su definición y la sensibilidad de la respuesta del elemento a sus variaciones requieren una gran experiencia. Como alternativa, se presenta el método de los campos de tensiones elasto-plásticos. Este método se basa en la teoría de la plasticidad pero incorporando condiciones de compatibilidad. Los resultados son coherentes en términos de resistencia y de deformaciones. Además, sólo necesita la definición de dos parámetros mecánicos (módulo de elasticidad y resistencia plástica) que pueden ser fácilmente determinados por el proyectista.
Critical details of a structure are usually governing for the strength of the member. Such critical details of structures can be found most often where geometrical discontinuities occur or near the region where concentrated loads are applied. The prediction of the load-bearing capacity of such details is very important as well as of the associated failure mode, which may be potentially brittle (associated to localization of strains) or ductile (associated to smeared strains and reinforcement yielding).
The assessment of critical details of existing RC structures with sufficient amount of transverse reinforcement is generally performed for tender or executive design on the basis of equilibrium-based models such as strut-and-tie models or rigid plastic stress-fields. These methods have proven over time to give a clear and accurate insight of force transfer and ultimate capacity of structural concrete elements. However, for complicated or unusual details they require a certain level of experience in order to suitably account for the various load-carrying actions. In addition, these methods do not consider compatibility conditions (they refer to rigid-plastic material laws) and therefore cannot accurately account for some phenomena (such as strength reduction due to transverse cracking) or predict the deformation capacity of a structure. Some of these shortcomings, particularly with reference to the estimate of the crushing strength of a compression field accounting for its cracking state, have been overcome by some approaches as the Modified Compression Field Theory or the Rotating-Angle Softened Truss Model. These latter approaches demonstrate that accounting for the actual strain field of concrete (by means of compatibility conditions) is necessary to lead to consistent estimates of the strength and deformation capacity of concrete members.
Other than the equilibrium-based approaches (traditionally associated to hand-made calculations), design can be performed using nonlinear numerical modelling. This technique has been significantly developed in the last decades, leading to advanced computer programs based on different theories and accounting for a large number of effects in concrete and steel. The use of advanced numerical tools for design and assessment of these regions is becoming increasingly popular. This has been particularly encouraged in new Model Code 2010, whose design philosophy refers to the Levels-of-Approximation approach. This allows the use of simple design models for preliminary or tender design (low-order Levels-of-Approximation) but allows refining it for advanced analyses and assessments (higher-order Levels-of-Approximation). The use of numerical tools consistent with the design procedures used for simpler analyses becomes thus encouraged for designers. A question arises then on how should a suitable nonlinear analysis be performed and how simple should it be. Two strategies are possible, the former consists on the use of the most complete and refined available constitutive laws. The latter consists on the use of the simpler and most understandable constitutive laws that, nevertheless, lead to accurate and consistent results.
With respect to the former strategy, these tools generally give a good prediction of deformation and strength capacities as well as of the potential load-carrying actions. However, they often require defining a large number of parameters in order to provide an accurate solution. Defining these parameters is sometimes cumbersome and the response of the program can potentially be rather sensitive to their choice. Furthermore, such analyses are associated to time-consuming analyses for pre- and post-processing of the data and may not be strictly consistent to the design methods used for simpler (low-order Levels-of-Approximation) analyses and their safety format.
An alternative to complex numerical modelling of structural concrete members with sufficient transverse reinforcement was recently developed at Ecole Polytechnique Fédérale de Lausanne. This technique is based on the analysis of elastic-plastic stress fields (EPSF). It shares the ground of equilibrium-based models (being thus consistent to low-order Levels-of-Approximation and their safety format) but allows accounting for compatibility conditions thus allowing to suitably represent the strain state of a concrete member and to refine some of the assumptions of strut-and-tie or rigid-plastic stress fields. For the analysis of the strength only two material parameters are required (modulus of elasticity and plastic strength) as well a strength reduction law accounting for transverse cracking. The method has proven to give excellent estimates of the strength and failures modes for a wide variety of structural concrete members.
The EPSF method has already been verified for several critical details such as deep beams
The EPSF method assumes a reinforced concrete member to be provided with a sufficient amount of transverse reinforcement
Mohr-Coulomb yield surface for plane stress with tension cut-off (consistent with rigid-plastic stress field assumptions with no tensile strength, equivalent for plane stress problems to a Rankine failure surface with no tensile strength).
Consideration of brittleness of high strength concrete through a strength reduction parameter (η
This parameter can be evaluated according to Muttoni
Consideration of concrete strength reduction with transverse cracking (
For the steel a bilinear constitutive law, symmetrical in compression and tension, is adopted with the possibility to account for strain-hardening as shown in
Since no tensile strength is considered, the stiffness of the member is usually underestimated at low levels of load. However, as load increases, the actual stiffness of the member is controlled by the available reinforcement and good predictions of the stiffness and strain state are obtained
It can be noted that no fracture mechanics concepts are in-built in the stress field (tensile strength of concrete is set to zero) and that the design principles are consistent to those of rigid-plastic stress fields or strut-and-tie models (used for preliminary or tender designs). This allows a significant level of consistency in the results (and safety format) obtained from first stages of design to the most refined ones. With respect to its accuracy, since the EPSF incorporates compatibility conditions, at failure, the resulting stress field is compatible with a licit failure mechanism. This provides an exact solution according to the theory of plasticity (lower and upper bound simultaneously), leading to the best potential stress field that can be selected.
The EPSF method can be efficiently implemented through the Finite Element Method (FEM)
where l
The reinforcement is modelled by a two node link element with no transversal stiffness (thus neglecting dowel action). Using the bilinear material model, described in the previous section, and provided a displacement field at the nodes, the axial forces can be obtained.
By introducing a small incremental displacement at each node and direction the stiffness coefficients can be found the stiffness matrix formed. The solution is found through a full Newton-Raphson solver, providing quadratic convergence to the solution. The Elastic Plastic Stress Field (EPSF) has been implemented into an open Java computer pro-gram that can be run standalone or as an applet of a web-page. The access for students at EPFL and practicing engineers is free (http://i-concrete.epfl.ch) and the source code of the program can also be downloaded and modified.
In this paper the results of the analysis with EPSF method of a total of 125 specimens of which 95 reinforced concrete beams and 30 dapped end beams are included. The tests refer to references
All specimens | V_{test}/V_{EPSF} | |
---|---|---|
Rectangular beams | # specimens included | 39 |
Average | 1.02 | |
COV, % | 8.08 | |
T-beams | # specimens included | 28 |
Average | 1.15 | |
COV, % | 9.35 | |
Pretensioned beams | # specimens included | 28 |
Average | 1.08 | |
COV, % | 8.26 | |
Dapped end beams | # specimens included | 30 |
Average | 1.00 | |
COV, % | 10.5 |
In the included beam specimens, various parameters were investigated including the compressive concrete strength
In most test campaigns for dapped end beams the main goal of the research is to investigate a particular reinforcement detail at the corbel, focusing on the vertical and inclined hanger reinforcement at the corner and the main horizontal reinforcement at the corbel. Another point of interest for researchers is the anchorage of the longitudinal reinforcement in the full depth section, which allowed also for the validation of the EPSF method for the modelling of anchorage of reinforcement.
In
The authors took measurements on both side of the beam so the values plotted in the figure are the mean values of the two corresponding symmetrical measurements. By neglecting the tensile strength of the concrete, for low load levels the method overestimates the contribution of the stirrups to the shear resistance of the beam. However, as the load increases a very good agreement between measured and calculated strains and stresses can be observed. In addition, the method tends to underestimate the stiffness in the uncracked stage of the specimens. However, as the load increases the method can accurately predict the stiffness of the specimens and their deformation
An important property of a good mechanical model is that it can predict not only the ultimate load but also the failure mode of an element. For the example presented in
The analysis of zones with high shear forces is additionally complicated if a geometrical discontinuity is present. A good example is the dapped end beams. In the next paragraphs three common failure modes for dapped end beams and their EPSF solutions will be discussed.
The first failure mode is due to bending of the recessed end
The second failure mode is due to a diagonal crack propagating from the bottom of the full depth section. During the testing of specimen DB1N, presented in
where
The third failure mode which will be discussed in this section is due to crushing of the concrete in the compression zone as it is for the example in
This paper shows that for advanced modelling of a reinforced concrete member different strategies may be followed: use of sophisticated constitutive laws or use of simple constitutive laws. Both strategies are possible and have their advantages. For practical purposes, the use of simple constitutive laws seems however more suitable. This is justified as it is consistent to the design basis of simpler procedures, understandable by designers, keeps the same safety format and (provided that the analysis tool is used within its domain of application) yields results with the same level of accuracy as other more sophisticated (yet more complex and more difficult to use) modelling strategies.
To this purpose, and for members with sufficient transverse reinforcement, this paper presents the main assumptions and implementation of the Elastic-Plastic Stress Field (EPSF) method and the results from the analysis of 125 tests from the literature. It is demonstrated by a number of examples that the method can provide a good estimate of the ultimate load bearing capacity and deformations, as well as the mode and region of failure.
Therefore it can be concluded that the good level of accuracy of the EPSF method makes it a promising tool for the analysis of existing structures for engineers and researchers. Also, its simple and physical hypotheses encourage the use of such method for educational purposes.