Lumped damage mechanics as a diagnosis tool of reinforced concrete structures in service: case studies of a former bridge arch and a balcony slab

A BSTRACT . Reinforced concrete structures may need repair in order to ensure the designed durability. The assessment of the structural condition usually is made in loco , but sometimes numerical analyses are required as a low cost and effective preliminary diagnosis. Lumped damage mechanics (LDM) may be used as an alternative diagnosis procedure. In LDM approaches for reinforced concrete structures, the concrete cracking in modelled by a damage variable and the reinforcement yielding is represented by plastic deformations. Therefore, this paper presents LDM as a diagnosis tool to analyse actual structures. Then, the procedure is applied in two case studies. The first one is a former bridge arch tested in laboratory. Such bridge was deactivated after almost three decades in service and some damage were observable. The second case study deals with a reinforced concrete balcony that collapsed after fifteen years of service. In the moment of collapse, only the dead load was acting in the balcony. The results show that LDM numerical response of those structures are quite close to laboratory observations, for the former bridge arch, and to in loco measurements, for the balcony. In this sense, according to the analysed structures, LDM may be used as a diagnosis tool. his paper presents lumped damage mechanics (LDM) as a diagnosis tool to evaluate damage levels (concrete cracking) in reinforced concrete (RC) structures. Differently from all previous studies, where LDM was used in nonlinear analyses, this paper showed that the LDM formulation can be easily applied in practical engineering problems. Therewith, LDM was applied as a diagnosis tool in two different structures: a former bridge arch in China that was tested in laboratory and a RC balcony slab in Brazil that collapsed after 15 years of service. For the former bridge arch, LDM response was quite close to the experimental one and then initial damage values in four different points were estimate around 0.20, concluding that minor repairs might be needed if the arch was returned to service.


INTRODUCTION
einforced concrete (RC) structures require periodical inspection, maintenance and repairs to ensure the designed durability. Inspections can identify damages that, if were not corrected, can lead to partial or total collapse of the structure. When a structure shows signs of damage it is necessary to investigate its causes, which results in a diagnosis that serves as the basis for adopting intervention measures. The assessment of structural damage may occur in loco using non-destructive methods such as visual inspection [1], acoustic emission [2] and ground penetrating radar [3], or destructive methods as explosions [4]. This analysis can also take place in laboratory, using samples of an original structure [5] or with the combination of experimental and numerical analysis, aiming to reproduce the experimental behaviour, as the stress/strain and fatigue [6] or the damage and dynamic properties [7]. Other diagnosis techniques are based only on numerical analysis of the structures such as nonlinear dynamic analysis [8] or on artificial neural networks [9]. One of the most used computational method for modelling the occurrence of damage and monitoring the cracking process in reinforced concrete structures is the finite element method (FEM) [10][11][12][13][14][15][16][17]. However, a disadvantage of these procedures using FEM is the significant computational cost. An efficient alternative to such procedures in order to assess damage in reinforced concrete structures is the Lumped Damage Mechanics (LDM). This theory combines the concepts of the classic damage and fracture mechanics in plastic hinges, which defines the concrete cracking as the damage variable. LDM was firstly proposed to analyse RC structures submitted to seismic loads [18][19][20][21][22]. Later, LDM was expanded for several other conditions, as impact loads [23][24][25] and tunnel linings [26][27][28]. One of the great advantages of using LDM to evaluate damage of reinforced concrete structures is the use of parameters easily obtained by civil engineers in practice, such as cracking moment and the load conditions. In order to ensure that LDM may be applied in actual engineering problems, Flórez-López et al. [29] defined structure integrity according to the damage level of the structural elements. For reinforced concrete elements, a damage value around 0.3 or 0.4 represents the beginning of the reinforcement yield, and a damage of approximately 0.6 means the bearing capacity of the element [29]. In the light of the foregoing, the objective of this paper is the application of LDM as a diagnosis tool of RC structures, using a former bridge arch and a balcony that suffered collapse as case studies. The first case is an arch that was removed from an actual bridge in China [13] and tested in laboratory and the second one is a collapsed balcony of an actual building in Brazil [30]. Then, the damage variable is used as a diagnosis tool, presenting satisfactory results for both analysed structures.

REVIEW ON LUMPED DAMAGE MECHANICS FOR FRAMES AND ARCHES [26-29]
he variables of a structural problem can be categorised in three sets: kinematic, static and internal ones (Fig 1). Such variables are related by three sets of equations, which can be called as kinematic equations, equilibrium relations and constitutive law (Fig 1). Classic finite element analysis usually relates directly the matrix of generalised displacements {U} and matrix of generalised external forces {P}. Note that the approach presented in this paper presents more matrices, defined in this section. Then, considering a structure composed by frame and arch elements in a XZ Cartesian reference [26], such quantities are described by: where u 1 is the node 1 displacement parallel to the X-axis, w 1 is the node 1 displacement parallel to the Z-axis and θ 1 is the node 1 rotation on the XZ plane. Analogously, P un is the applied force on node n parallel to the X-axis, P wn is the applied force on node n parallel to the Z-axis and Pθn is the applied moment on node n on the XZ plane.
The structural equilibrium is given by: being [B]b the transformation matrix and {M}b the generalised stresses of a finite element b, both given as follows: where Mi and Mj are bending moments on the nodes i and j of the element, respectively, Ni is the axial force on the node i of the element, α b is the orientation of the frame element and L b its length; β b , χ b and R b geometrically define the circular arch element (Fig 2). Note that the second member of (3) is the sum of the internal forces of each finite element. The kinematic relations for a finite element are given by: being [B(U)] b the same transformation matrix previously presented in (5-6) considering large displacements and {Φ} the matrix of generalised deformations, which is conjugated to {M} and written as: where φi and φj are flexural rotations due to the bending moments Mi and Mj, respectively, and δi is the axial deformation related to N i . If small displacements and small deformations are assumed, (7) can be rewritten as follows: being [B0] the transformation matrix for the initial condition. The elastic relations of the element are obtained by assuming the deformation equivalence hypothesis, by applying the Castigliano's Theorem [26][27]29] and considering the inelastic phenomena are concentrated at hinges located at the edges of the element (Fig 3): where {Φ p } b is the matrix of generalised plastic deformations, that accounts for reinforcement yielding, and [F(D)] b is the damaged flexibility matrix, both written as: being di and dj the damage variables at hinges i and j, respectively, which account for concrete cracking. The terms Fij 0 are described in the Appendix for frame and arch elements. In order to complete the constitutive law, the evolution laws of internal variables (damage and plastic rotation) must be considered. Thus, consider the complementary energy of the finite element: The evolution law of the damage variables are obtained throughout the Griffith criterion i.e.
where G i and G j are the damage driving moments for the hinges i and j, respectively, and Y(d i ) and Y(d j ) are the cracking resistance functions [28]. The cracking resistance function for reinforced concrete elements [29] is given by: being Y 0 and q model parameters.
The plastic evolution laws for each hinge of the element are given by: being fi and fj the yield functions for the hinges i and j, respectively, and C and k0 model parameters. Note that the model parameters Y 0 , q, C and k 0 can be directly related to the classic theory of reinforced concrete structures, as presented by [20,[27][28][29].

CASE STUDIES
Bridge Concrete Arch [13] he first case study is the failure analysis of two reinforced concrete arch ribs that were part of a 28 years old bridge in China. The geometry of the arches is described in Tab. 1. The failure experiments were made in laboratory where the arches were submitted to static loads [13]. Fig. 4 presents the location of the applied loads (P).  The loads were applied at middle (point E), at the left quarter plus 27 cm (point C) and at the right quarter of the arch (point G), which were applied in eighteen load steps. The two end sections of the arches were fixed. The reinforcement is located near the upper and lower sides of the arch cross section. Three bars with 16 mm of diameter composed each layer. Tab. 2 presents the elastic constants of the materials.
Due the long usage time and problems related to the removal and transportation of the arches, both of them suffered damages at the two ends (nodes A and I). Moreover, at 1/8 distance of the left and right supports (nodes B and H) the arches were also damaged, when they were being dismantling in the field, resulting in two notches. It was obtained that the collapse load (P max ) was equal to 68 kN and 65 kN for the tested arches. Tab. 3 presents the experimental data of the vertical displacements at the measured location, for different load ratios.  Table 3: Experimental data of vertical displacements for Pmax = 68kN [13].

Collapse of a reinforced concrete balcony slab [30]
A balcony of a fifteen years old building suddenly collapsed. The balcony structure severely cracked at the clamped end of the cantilevered slab, rotating approximately 17 degrees, but remained connected to the rest of the structure of the building, as is shown in Fig. 5. The balcony floor structure is a reinforced concrete slab with 12 cm of thickness, 9.32 m long and 2.03 m wide (Fig. 6). The cantilevered slab is clamped in the outer edge of the living room slab, whose thickness is 15 cm, and both are simply supported on the facade beam. The bottom of the two slabs are on the same level, so the difference in thickness between them results in a 3 cm step at the top of the cantilevered slab, located next to the beam (Fig. 7).   Thus, it was concluded that the accident was caused by the abrupt rectification of the steel bars of the negative reinforcement that were installed with double curvature in the form of a step, due to the rupture of the surface layer of concrete by the action of the straightened bars (Fig. 8). Figure 8: Detail of the steel bars before the accident [30].
Collapse of a reinforced concrete balcony slab [30] Since the acting permanent load is 4.96kN/m² during the accident, it was verified that the damage evolution law (Eqn. 14) that G < Y(d), i.e. the loads did not cause the damage in the clamped section. This is corroborated by comparing the cracking moment (15.73kNm/m) with the acting bending moment in the clamped section (10.22kNm/m). To achieve the cracking moment (15.73kNm/m) and the ultimate moment (23.69 kNm/m), the load in the balcony should be 7.63kN/m² and 11.5kN/m², respectively. Based on that and in the conditions observed in loco, the accident occurred due to the action of the straightened bars. In the ultimate condition of the slab section, was observed that the residual thickness was approximately 2 cm, as shown in Fig. 9. The simulation, considering large displacements, reached a damage value of 0.9995 and the horizontal and vertical displacement in the edge of the slab are equal to 13.57cm and 58.99cm, respectively. These values lead to an angle approximately equal to 17.30°, which confirms the in loco observations. Note that the damage obtained in the analysis characterises an inacceptable structural behaviour [29] (Tab. 4), denoting that almost there is no cross section at the clamped end. Assuming that the damage penalises directly the inertia moment [27], the remaining thickness at the clamped end is estimated as 0.95cm, quite close to the field observations. CONCLUSION his paper presents lumped damage mechanics (LDM) as a diagnosis tool to evaluate damage levels (concrete cracking) in reinforced concrete (RC) structures. Differently from all previous studies, where LDM was used in nonlinear analyses, this paper showed that the LDM formulation can be easily applied in practical engineering problems. Therewith, LDM was applied as a diagnosis tool in two different structures: a former bridge arch in China that was tested in laboratory and a RC balcony slab in Brazil that collapsed after 15 years of service. For the former bridge arch, LDM response was quite close to the experimental one and then initial damage values in four different points were estimate around 0.20, concluding that minor repairs might be needed if the arch was returned to service. On the other hand, the RC slab balcony collapsed with only permanent loads where, the reinforcement straightened up due to an incorrect design. By applying LDM, it was noticed that the analysis reached large displacements as the actual balcony, with a rotation near 17º, quite close to in loco observations. The calculated damage was 0.9995, characterising collapse. Also, the residual thickness is estimated near to 1.0 cm, quite close to in loco measure (2.0 cm). Finally, regarding the analysed engineering problems in this paper, it is possible to observe that the LDM may be adequately a diagnosis tool to evaluate the levels of damage in structures, being able to determine the distribution of initial/current damage values and to verify its acceptability and safety for use. Moreover, it is feasible tool for technical analysis of actual structural accidents, such as the case of the RC balcony slab in Brazil.