Chapter 5. Reinforced Concrete Beams
5.4. Model identification
5.4.1. Pinned-pinned beams
Two sensitivity analyses were developed in this case, one for service phase and other until failure load. The evaluated parameters are those related to materials, concrete and steel, and to geometry. It was respectively varied one standard deviation (σ) from each parameter mean value. In order to compute each standard deviation (σ), the following coefficient of variations (CV) were established [93]: (1) concrete elasticity modulus (Ec): 10%; (2) concrete tensile strength (ft): 20%; (3) concrete compressive strength (fc): 10%; (4) concrete fracture energy (Gf): 10%; (5) concrete compressive strain at compressive strength (εc): 10%; (6) concrete critical displacement (wd): 10%; (7) reinforcing steel elasticity modulus (Es): 5%; (8) reinforcing steel yield strength (σy): 5%; (9) reinforcing steel limit strength (σu): 5%; (10) reinforcing steel limit strain (εlim): 15%; (11) reinforcing steel area (As): 2%; (12) inferior concrete cover (cinf): 20%; (13) superior concrete cover (csup): 20%; (14) beam width (b):
10%; and (15) beam height (h): 10%. Figure 5.10 and 5.11 gives the obtained results.
Figure 5.10. Importance measure (service).
The analysis developed in service phase pointed out for the importance of concrete elasticity modulus (Ec), tensile strength (ft) and fracture energy (Gf), of longitudinal steel reinforcement elasticity modulus (Es) and area (As), and of section width (b) and height (h). In fact, it is reasonable to admit that when submitted to low stresses the beam response is only dependent of concrete elasticity modulus and fracture energy, and of steel elasticity modulus. In this situation the concrete, in tension, reached their tensile strength and started to crack. Therefore, tensile strength is also an important parameter. By developing this analysis, from 15 initial possible parameters, only 7 of them were considered critical, reducing so the computational cost.
Figure 5.11. Importance measure (failure).
The analysis performed until failure load revealed a decrease on the importance of concrete parameters, with exception of compressive strength (fc) that increases. In respect to longitudinal steel reinforcement it is important to point out an increase on the importance of
the increase on the importance measure of inferior concrete cover (cinf). Other critical parameters, previously identified during the sensitivity analysis in service phase, are still considered. Therefore, from 15 initial parameters, only 9 of them were considered, reducing so the computational cost.
Once the numerical model and critical parameters, to be optimized, are identified, the further step is the application of proposed model identification methodology. In this case, the middle span displacement was measured during the laboratory test. This measurement is expressed by the graphic that plots the applied load (F) against the measured displacement (δ). In this situation the fitness function characterizes the approximation between numerical and experimental values for applied loads. One of used tolerance criteria is related to the convergence in the space of fitness value, defined by the threshold value (ε). In order to obtain this value, a division between uncertainty types, respectively, experimental and numerical, is developed.
From experimental uncertainties, it is possible to select: (1) Sensor accuracy (0.10%), which includes not only the displacement transducer precision but also the cable and acquisition equipment losses [68, 154]; (2) Load positioning, which, in this case is considered to be zero as it was perfectly controlled within the developed test; (3) Load intensity (0.10%), that includes not only the load cell resolution but also the cable and the acquisition equipment losses [68, 154]; (4) Environmental effects, which can be neglected due to the fact of being a short term test and so the variations in temperature and humidity are very small to be considered; (5) Vibration noise, that can also be ignored as the test is performed in a static way.
In this case, only the load intensity component will be considered when computing the experimental uncertainty. This component presents a uniform PDF (Type B) and so, according to JCGM [90, 91, 92], it should be divided by √3, obtaining then the result of 5.77*10-2 %. In order to compute the experimental uncertainty it will be necessary to determine the experimental data derivative in respect to this component (∂yexp/∂x = 1.00 kN).
This uncertainty is obtained through equation (5.4) [90, 91, 92],
( )
22 2 2 7 4
exp 1.00 5.77 10 100 3.33 10 exp 5.77 10 kN
u = ⋅ ⋅ − = ⋅ − →u = ⋅ − (5.4)
From numerical uncertainties, it is possible to select: (1) Finite element method accuracy (3.79%), determined by comparing the developed numerical model with other which presents a higher number of load steps [69]; (2) Mesh refinement (6.74%), determined by comparing the developed numerical model with other which presents a more refined mesh [69]; (3) Model exactitude, that can be neglected as the numerical model is developed
according to the experimental test; (4) Considered hypothesis, that are also neglected as all model simplifications (e.g. consideration of supports as point loads) are validated within a global structural analysis.
When computing the numerical uncertainty, both finite element method and mesh refinement components will be considered. These components are represented by a uniform PDF (Type B) and so, according to JCGM [90, 91, 92], they should be divided by √3, obtaining then the result of 2.19 % and of 3.89 %, respectively. In order to determine the numerical uncertainty, the partial derivative of the numerical results in respect to these two components should be computed (∂ynum/∂x = 1.00 kN). This uncertainty is obtained through equation (5.5) [90, 91, 92],
( )
2( )
22 1.002 3.89 100 1.002 2.19 100 1.99 103 4.46 102 kN
num num
u = ⋅ + ⋅ = ⋅ − →u = ⋅ − (5.5)
Once the experimental and numerical uncertainties are computed, it will be possible to determine the fitness function uncertainty. In order to obtain this value, it is necessary to compute the partial derivative of the fitness function in respect to both experimental and numerical data. These values vary with tested beam as they are proportional to maximum applied load (∂f1/∂ynum = ∂f1/∂yexp = 1/max(y1exp) = 4.10 * 10-2 kN-1, for beam 1, and
∂f2/∂ynum = ∂f2/∂yexp = 1/max(y2exp) = 4.00 * 10-2 kN-1, for beam 2). The fitness function uncertainty is respectively computed, for each tested beam, through equations (5.6) and (5.7) [90, 91, 92],
( ) ( ) ( ) ( )
1 1
2 2 2 2
2 2 4 2 2 6 3
4.10 10 5.77 10 4.10 10 4.46 10 3.29 10 1.82 10
f f
u = ⋅ − ⋅ ⋅ − + ⋅ − ⋅ ⋅ − = ⋅ − →u = ⋅ − (5.6)
( ) ( ) ( ) ( )
2 2
2 2 2 2
2 2 4 2 2 6 3
4.00 10 5.77 10 4.00 10 4.46 10 3.20 10 1.79 10
f f
u = ⋅ − ⋅ ⋅ − + ⋅ − ⋅ ⋅ − = ⋅ − →u = ⋅ − (5.7)
The global fitness function value is obtained through the square root of the sum of the square of these components. In order to determine the global uncertainty, the partial derivative of the fitness function in respect to each component should be computed (∂f/∂f1 = ∂f/∂f2 = 1.00).
This uncertainty is obtained through equation (5.8) [90, 91, 92],
( )
2( )
22 2 3 2 3 6 3
1.00 1.82 10 1.00 1.79 10 6.49 10 2.55 10
f f
u = ⋅ ⋅ − + ⋅ ⋅ − = ⋅ − → =u ⋅ − (5.8)
The improvement on global fitness value (∆f) from two generations, separated of a specified gap (n), is given in chapter four. In order to determine its uncertainty, the partial derivative of the improvement in respect to each component needs to be computed (∂∆f/∂fi+n = ∂∆f/∂fi = 1.00). This uncertainty is obtained through equation (5.9) [90, 91, 92],
( )
2( )
22 1.002 2.55 10 3 1.002 2.55 10 3 1.30 10 5 3.60 103
f f
u∆ = ⋅ ⋅ − + ⋅ ⋅ − = ⋅ − →u∆ = ⋅ − (5.9)
As all uncertainty sources are of Type B, a coverage factor (k) of 2 should be adopted [90, 91, 92]. The fitness value criterion establishes that the respective improvement (∆f) should be less than or equal to the threshold value (ε). This value is obtained by multiplying the value from expression (5.9) by factor k. The obtained threshold value for the analysis in service phase is determined in a similar way. These values are further indicated, (5.10),
3 3
7.15 10 0.72%
7.21 10 0.72%
Service Failure
ε ε
−
−
→ = ⋅ =
→ = ⋅ =
(5.10)
This means that, for instance, for model identification until failure load, if the improvement in minimum fitness value of a population from two generations separated of a specified gap (n) is, respectively, less than or equal to 0.72%, the algorithm stops, as the fitness function tolerance criterion is reached. This shows that it is not meaningful to improve the fitness function of a value that is less than or equal to the precision itself.
The evolutionary strategy algorithm in its plus version [29] is further executed. In this case, a parent population (µ) and a parent for recombination (ρ) of 10 individuals, and an offspring population (λ) of 50 individuals were defined. The algorithm will run until one of the established criteria is reached. Other stopping criteria, as the maximum generation’s number (1000), were considered. The generation gap (n), used for the fitness function tolerance criterion, is proportional to this number. It was established that this value is 10% of the specified maximum generation’s number. Therefore, the improvement on minimum fitness value is evaluated from a gap of 100 generations. Once the algorithm stops, a population, constituted by different individuals, is obtained.
The respective algorithm is processed with different starting points. An engineer judgment procedure is developed to determine the most suitable individual, from those previously extracted. This individual is constituted by a set of values, a value for each critical parameter.
Table 5.6 presents the nominal values, and individuals obtained from model identification in service phase and until failure load. In the same table, between brackets, the bias factor, which represents the ratio between the identified and the nominal value for each variable, is also presented. When applying this methodology in service phase, not only the critical parameters, as already identified during the sensitivity analysis, but also their optimal values, may differ from the application until failure load.
From a first analysis of Table 5.6., it is possible to realize that: (1) Obtained value of some parameters, as the concrete tensile (ft) and compressive strength (fc), the section width (b)
and height (h) and the inferior concrete cover (cinf), is lower than the nominal one; (2) The longitudinal steel reinforcement elasticity modulus (Es) value, obtained from this methodology, when applied until failure load, is lower than the nominal one and, both these values are lower than the one from the application of the methodology in service phase;
(3) The concrete elasticity modulus (Ec) value, obtained from this methodology, when applied until failure load, is higher than the nominal one and, both these values are higher than the one obtained from the application of the methodology in service phase; (4) Obtained value of some parameters, as the concrete fracture energy (Gf) and the longitudinal steel reinforcement yield strength (σy) and area (As), is higher than the nominal one.
Table 5.6. Model identification results.
Numerical model Nominal
value
Model identification Service * Failure *
Parameter
Material
Concrete
Ec [GPa] 31.00 24.80 (0.80) 31.44 (1.01) ft [MPa] 2.60 2.09 (0.80) 2.54 (0.98) fc [MPa] 33.00 33.00 (-) 29.87 (0.91) Gf [N/m] 65.00 74.50 (1.15) 65.00 (-)
Longitudinal steel reinforcement
Es [GPa] 200.00 233.27 (1.17) 186.73 (0.93) σy [MPa] 500.00 500.00 (-) 535.83 (1.07) As [cm2] 0.85 1.02 (1.20) 0.91 (1.07)
Geometry
cinf [cm] 1.00 1.00 (-) 0.99 (0.99)
b [cm] 7.50 6.53 (0.87) 7.27 (0.97)
h [cm] 15.00 14.52 (0.97) 14.28 (0.95)
* Bias factor is presented between brackets.
Both analyses indicated that concrete material presents a worse quality than the expected.
The main reasons for that are the difficulties related to the concreting process of small structural elements. However, the analysis performed until failure load indicated a higher elasticity modulus than the predicted. When evaluating the steel reinforcement, obtained results indicated a better quality material. However, the analysis performed until failure load indicated a lower elasticity modulus than the predicted. Both analyses indicated a higher steel area. Obtained value from the analysis in service phase is far from the others. In respect to geometry parameters, obtained values for inferior concrete cover are close to
It is also important to mention that the same collapse mechanism and failure mode was obtained in experimental and numerical tested beams. Figure 5.12 presents the applied load (kN) plotted against the middle span displacement (m) for measured experimental data and for numerical results, obtained by considering the nominal values, and those from model identification in service phase and until failure load. From the analysis it is possible to conclude that the results from model identification until failure load are those that best fit the experimental curve. The numerical results considering nominal values present a higher cracking load and post-cracking stiffness and a lower failure load, while the results obtained by using model identification in service phase, are very similar to experimental data, within the service region, presenting then a higher post-cracking stiffness and failure load.
Figure 5.12. Numerical results.
Table 5.7 presents the minimum fitness function values obtained by considering the nominal values and those from model identification in service phase and until failure load. A first analysis permit to conclude that the fitness function value obtained in service phase is lower than that determined until failure load. This is due to the fact that in service region the numerical results are close to experimental data than in failure region. It is also verified, for both cases, an improvement of this value with model identification. However, it is also verified that this improvement is higher in service region (92.97%) than in failure region (75.90%). The structural behavior in failure region presents a higher nonlinearity.
Therefore, for this region, optimization becomes harder and obtained results are not as good.
Table 5.8 presents the obtained failure load (FR) by considering the nominal values and those from model identification in service phase and until failure load. Obtained error from model identification until failure load is considerably lower than that given by nominal values and by model identification in service phase. In fact, during this later analysis the model identification is only realized in service phase. Therefore, it becomes difficult to predict the failure load. Hence, model identification in service phase, itself, does not give good results.
Additional complementary testes are thus recommended. In respect to model identification until failure load, obtained error is less than 1% which is very good.
Table 5.7. Minimum fitness function values.
Numerical model
Fitness function
Service Failure
Value [%] Improvement [%] Value [%] Improvement [%]
Nominal values 3.38 - 7.70 -
Model identification 0.24 92.97 1.86 75.90
Table 5.8. Failure load (FR).
Numerical model
Failure load
Value [kN] Error [%] *
Nominal values 23.28 6.13
Model identification
Service 26.42 6.53
Failure 24.84 0.16
* Comparing with the real failure load.