Chapter 5. Reinforced Concrete Beams
5.7. Safety assessment
considering the Bayesian inference, presents the highest index-p values, considering both tested beams. Results from model identification in service phase are the poorer ones.
Figure 5.28 presents the resistance PDF, whose parameter values (mean and standard deviation), are indicated at Table 5.25. It is possible to identify that the obtained resistance PDF with values from model identification until failure load is located among the others. Obtained resistance PDF with nominal values, considering or not the Bayesian inference, are the ones that give the highest mean. The Bayesian inference approach increased the mean and standard deviation of maximum bending moment at fixed support, except for model identification in service phase.
Figure 5.28. Maximum bending moment (MR1*).
Live loads are divided into: (1) sustained or long-term (qlt), defined by a Gamma PDF [93];
and (2) intermittent or short term (qst), defined by an exponential PDF [93]. The applied load (p) is the sum of self-weight with both long and short-term live load component multiplied by the influence length of the beam (Linf), which is, in this situation, 6.0 m. This value is given by equation (5.25). Table 5.26 provides the mean and standard deviation of each PDF.
inf inf
lt st
p= +w q ⋅L +q ⋅L (5.25)
Table 5.26. Probabilistic models.
Parameter PDF µ σ
γconc [kN/m3] Normal 24.00 0.96
γsteel [kN/m3] Normal 77.00 0.77
qlt [kN/m3] Gamma 0.30 0.45
qst [kN/m3] Exponential 0.30 0.57
5.7.1. Pinned-pinned beams
In this case, the resistance model is given by the failure load model (FR), whose parameters are provided at Table 5.21. A model is obtained for each analysis, respectively, considering the nominal values and those from model identification in service phase and until failure load.
In order to compare resistance and loading curves it is necessary to transform this model into a model for maximum bending moment at middle span (MR), through equation (5.26),
(
2 2)
2R R
M = F ⋅ ⋅L (5.26)
This model depends on the beam span (L) which is, in this situation, 1.50 m. A Normal PDF is obtained for resistance. The further step consists in computing the maximum bending moment (MS), through equation (5.27),
(
2)
8MS = p L⋅ (5.27)
A Lognormal PDF is then adjusted to obtained data. A limit state function (Z), which compares resistance and loading curves, is then defined through equation (5.28),
R S
Z=M −M (5.28)
The limit state is exceeded when loading is higher than resistance. The further step consists in generating values for each curve, according to each PDF parameters, and to register the
number of values in which this limit state is exceeded in relation to the total number of evaluated points. The failure probability (Pf) is determined through equation (5.29),
( 0)
Pf =P Z≤ (5.29)
The reliability index (β) is then obtained, considering this value. A detailed description of how this index is computed is given in chapter four. Table 5.27 presents both failure probabilities and reliability indexes for all models.
a) b)
Figure 5.29. Residential building: a) pinned-pinned beams; b) pinned-fixed beams [121].
Through the analysis of this table, it is possible to conclude that obtained β-value considering the values from model identification until failure load is identical to the one considering nominal values. An increase on β-value is verified when considering the values from model identification in service phase. However, and according to Table 5.21, this model presents a low reliability. Accordingly, the most accurate result is the one considering the values from model identification until failure load.
In this example, the building is of class 2 (apartment building – risk to life, given a failure, is medium or economic consequences are considerable) and of class B (normal cost of safety measure), according to JCSS [93]. Therefore, a target reliability index (βtarget) of 3.3 is recommended. This will permit to conclude that the assessed beam is safe.
Table 5.27. Safety assessment.
Numerical model Pf β
Nominal values 2.55 * 10-4 3.48
Model identification (service) 1.00 * 10-4 3.72
Model identification (failure) 1.72 * 10-4 3.51
5.7.2. Pinned-fixed beams
In this situation, due to the fact of being one degree hyperstatic, the collapse mechanism is characterized by two plastic hinges, located at fixed support and beside the point load that is close to the pinned support. Therefore, the limit state function (Z) is composed by two equations, one for each plastic hinge. In this case, the resistance and loading model for maximum bending moment are respectively compared in each equation.
In this case, the resistance model is given by the failure load model (FR), whose parameters are provided at Table 5.24, and by the maximum bending moment at fixed support model (MR1*), whose parameters are given at Table 5.25. A model is obtained for each analysis, respectively, considering the nominal values and those from model identification in service phase and until failure load, considering or not the Bayesian inference.
Therefore, it is necessary to transform these models into a model for maximum bending moment at fixed support (MR1) and beside the point load that is close to the pinned support (MR2). Consequently, the maximum bending moment at fixed support (MR1**) is computed for each generated value of failure load model (FR), according to the static equilibrium equations, (5.30),
**
1 0.125
R R
M = ⋅F ⋅L (5.30)
If MR1** is lower than MR1*, then MR1 = MR1** and MR2 is computed through the static equilibrium equations. If MR1** is higher than MR1*, then MR1 = MR1*. In this case, the load intensity (FR1), necessary to obtain MR1*, is computed through equation (5.31),
* *
1 0.125 1 1 8.0 1/
R R R R
M = ⋅F ⋅ →L F = ⋅M L (5.31)
In this case, MR2 is obtained in other way, (5.32),
( )
( )
2 1 1
* *
2 1 1
*
2 1
0.250 0.1875
0.250 8.0 / 0.1875 8.0
0.250 0.5
R R R R
R R R R
R R R
M F F L F L
M F M L L M
M F L M
= ⋅ − ⋅ + ⋅ ⋅ ⇔
= ⋅ − ⋅ ⋅ + ⋅ ⋅ ⇔
= ⋅ ⋅ −
(5.32)
Both MR1 and MR2 models depend on the beam span (L) which is, in this situation, 1.50 m.
These models are represented by a Normal PDF. The further step consists in computing the maximum bending moment at fixed support (MS1) through the static equilibrium equations, (5.33),
(
2)
1 8
MS = p L⋅ (5.33)
In this situation the maximum bending moment beside the point load that is close to the pinned support (MS2) is computed through the static equilibrium equations. However, if MS1 is higher than MR1, MS2 needs to be computed in other way. In this case, the load intensity (p1), necessary to obtain MR1, is computed through equation (5.34),
(
2)
21 1 8 1 8 1/
R R
M = p L⋅ →p = ⋅M L (5.34)
In this case, MS2 is obtained through equation (5.35),
( )
( )
( )
( )
2 2
2 1 1
2 2 2 2
2 1 1
2
2 1
8 0.0625
8 / 8 0.0625 8 /
/ 8 0.5
S
S R R
S R
M p p L p L
M p M L L M L L
M p L M
= − ⋅ + ⋅ ⋅ ⇔
= − ⋅ ⋅ + ⋅ ⋅ ⋅ ⇔
= ⋅ − ⋅
(5.35)
A Lognormal PDF is then adjusted to obtained data. A limit state function (Z), which compares resistance and loading curves, is then defined through equation (5.36),
1 1
2 2
R S
R S
M M
Z M M
−
=
− (5.36)
The limit state is exceeded when loading is higher than resistance. The further step consists in generating values for each curve, according to each PDF parameters, and to register the number of values in which this limit state is exceeded in relation to the total number of evaluated points. The failure probability (Pf) is determined through equation (5.37),
( 0)
Pf =P Z≤ (5.37)
The reliability index (β) is then obtained, considering this value. A detailed description of how this index is computed is given in chapter four. Table 5.28 presents both failure probabilities and reliability indexes for all models [121].
Table 5.28. Safety assessment.
Numerical model Pf β
Nominal values 1.40 * 10-5 4.20
Nominal values + Bayesian inference 5.00 * 10-6 4.42
Model identification (service) 4.22 * 10-5 3.93
Model identification (service) + Bayesian inference 1.65 * 10-5 4.15
Model identification (failure) 1.97 * 10-5 4.11
Model identification (failure) + Bayesian inference 1.30 * 10-5 4.21
Through the analysis of this table, it is possible to conclude that obtained β-value considering the values from model identification until failure load is lower than the one considering nominal values. A decrease on β-value is verified when considering the values from model identification in service phase. An increase on β-value is verified with Bayesian inference. In this case, and according to Table 5.24 and 5.25, the most accurate result is the one considering nominal values and Bayesian inference.
In this example, the building is of class 2 (apartment building – risk to life, given a failure, is medium or economic consequences are considerable) and of class B (normal cost of safety measure), according to JCSS [93]. Therefore, a target reliability index (βtarget) of 3.30 is recommended. This will permit to conclude that assessed beam is safe.