Multicriteria Sensitivity Analysis for Numerical Model Validation of Experimental Data

Sensitivity analysis is a decisive step in experimental and numerical structural mechanics. The analysis of structural model quantifies the importance of each input parameter, potential interaction and effects on structural response. Therefore, this study aimed to help reduce the uncertainty surrounding major variables, providing valuable guidance for conducting future experiments. During the investigation, numerically deterministic sensitivity analysis based on multicriteria model evaluations of load-displacement curves representing actual behavior of the member correctly, were reviewed. Multicriteria model combined the evaluation of peak load, energy dissipation before ultimate loading, and toughness of load-displacement response. The methodology led to a strong sensitivity analysis method, generating an agreement between numerical and experimental responses. Moreover, an investigation of the method was presented for a geopolymer haunch, the numerical model was based on rigid body spring model (RBSM), which enabled precise behavior simulation of reinforced concrete structures. RBSM was refined, enabling in-depth evaluation of stress-strain contours, plasticity index, initial crack formation and crack propagation, as well as RBSM-spring failure modes. The proposed multicriteria sensitivity analysis can be implemented with other simulation methods, such as finite element analysis (FEA) and structural simulation software. The recommended method is applicable to any structural member, where laboratory-tested full-scale specimens were functioning as validation tools. Following the proposed multicriteria sensitivity analysis, experimental load-displacement curves of this study supported the results of numerical RBSM in an acceptable range of error predictions.

Deterministic sensitivity analysis; Experimental validation; Multicriteria model analysis; Rigid body spring model

A numerical model is a method that provides detailed visualizations of how composite elements behave under stress-strain contours, individual evolution, initial cracking and crack propagations, as well as the plasticity index. However, this information complicates the precision instruments and laboratory equipment, which cannot be captured by experimental testing. A validation process for full-scale identical laboratory-tested  specimens should be conducted to evaluate and further improve numerical models until the predetermined level of correctness is achieved. 

2.1. Experimental Works

Flexural beam-to-column specimens with a length of 3.50 m were cast monolithically using conventional concrete with a 28-day cylindrical compressive strength fc' = 31.11 MPa. The concrete was a self-compacting concrete (SCC), according to (Purwanto et al., 2023; Purwanto, 2021). During this study, targeted compression strength of 30 MPa was designed based on job-mix procedure for both conventional and geopolymer concrete to reduce strain disparities between beam-column and haunch under loading. The primary purpose of this method was to obtain the material mix proportions. During the process, the job mix led to conventional mix proportion as shown in Table 1.

Table 1 Conventional concrete mix design properties by weight 

28-day cylinder compression strength led to a strength of 31.11 MPa, which was in the margins of the design. During this study, mix of geopolymer concrete was obtained in an identic manner leading to the proportions as shown in Table 2. Cement paste used in this process was replaced by fly ash and alkaline activator generally known as binder.

Table 2 Geopolymer concrete mix design properties by weight 

The geopolymer concrete used in this study had a strength of 32.52 MPa. The targeted 28-day compressive strength of 30 MPa was exceeded by 8%, which was in the acceptable limits. The substitution of cement paste as a binder in SCGC was a mixture of fly ash class F and alkaline activator, where the activator consisted of sodium hydroxide (NaOH) with molarity of 12 M and sodium metasilicate (Na2SiO3) type Be-52. Superplasticizer was added to SCGC mix to produce highly flowable concrete. Meanwhile, adding small amounts of cement and water functioned as a spark to activate the superplasticizer. According to (Purwanto et al., 2023; Purwanto, 2021), the concrete mix was poured through narrow-casting inlets with a 20 mm to 40 mm dimension. SCGC was the best option to obtain good homogeneity without vibration compaction (Ulhaq and Andayani, 2021). The mix design of this type of concrete was studied extensively (Ghafoor and Fujiyama, 2023; Kanagaraj et al., 2023; Kumar and Raguraam, 2018). An elaborate review of this topic was found in the work of (Thakur and Bawa, 2022).

During the process, the specimens were reinforced by four deformed rebars with a diameter of 13 mm at the tensile zone (4D13) and two rebars with the same diameter at the compressive one (2D13). In comparison, the column was reinforced with three rebars on each side of eight in total (8D13). The beam and column had stirrups with a diameter of 8 mm and a 50 mm spacing (Figure 1). Additionally, the reinforcement had a yield strength fy of 360 MPa, where its ultimate strength fu was 460 MPa, and the haunch had an angle of 26.5°. 

Figure 1 Reinforcement detail of beam-column joint.

The determination of haunch angle was based on the study of influence of angle on effectiveness of the haunch in shifting the formation of plastic joint (Purwanto, 2021). Figure 2a showed the measured strain of tensile reinforcement, while Figure 2b graphically showed the gradient of strain in the beam as a function of haunch length. The study signified that for the prismatic member, formation of plastic joint occurred at beam-column face. Haunch angle of 14⁰ reduced the stain but was unable to relocate plastic joint formation. 26.5⁰ angle clearly showed that the largest stain was located at haunch-tip. Following this discussion, an angle of 26.5⁰ effectively utilized the haunch.

Figure 2 Strain in tensile reinforcement as a function of haunch angle.

Reinforcement configuration was based on the principles of an under-reinforced beam, and considering the spacing in between reinforcement to ensure that concrete was well poured and homogeneously compacted. The haunch beam-column specimen was tested monotonically with a loading-unloading scheme. In addition, the load was applied at the center of the column. Displacement-controlled testing was performed until the specimen achieved the ultimate load-carrying capacity. Relating to this process, experimental test setup was shown in Figure 3. To record the load and the displacement, load cells and LVDTs (linear variable differential transformers) were installed and connected to a data logger.

Figure 3 Experimental test setup (Purwanto et al., 2023)

In experimental studies, non-uniformity of material properties, where dimensions of elements that were less accurate and precise, imperfect configuration and position of longitudinal reinforcement and stirrups, casting systems causing non-homogeneity as well as density of concrete were expected to occur. The loading procedure was also very responsive to vehicle action, vibrating the surrounding testing location. All these factors led to a load-displacement response including imperfections.

2.2. RBSM

Model was discretized into a series of rigid body plane stress elements interconnected by normal and shear springs at individual interfaces (Takeuchi, 2005). Orthotropic plane stress elements were used to model the concrete and steel reinforcement. Figure 4 showed the displacement field of isotropic plane stress element for the concrete element.

Figure 4 Displacement and kinematics of the isotropic plane stress element.

Each element had three degrees of freedom, namely horizontal displacement (ui), vertical displacement (vi), and rotational displacement (qi), respectively. In this study, xy-axis and ns-axis represented the global and local coordinate systems, respectively. Point P was an arbitrary point at the interface between two adjacent elements. After the elements were displaced, this point was separated and attached to adjacent boundaries of the elements represented by P' and P" in Figure 4. The relative displacement between centroid (Gi) of elements was expressed in the normal (dn) and tangential (ds) direction to the interface (Purnomo et al., 2023a). Moreover, stress-strain relationship in the springs was established by reviewing kinematics between the two displaced elements. The determination of spring failure was justified based on Mohr-Coulomb criteria.

Cumulative displacement was evaluated by summing the displacement of each element as described in Equation 1, where ui represented the degrees of freedom, and Bi was defined in Equation 2. In the Equation, li and mi represented the direct cosines, while xiG and yiG signified the centroid of each element. The strain on the springs was calculated based on Equation 3. Stress-strain matrix was expressed by Equation 4, where Dc represented the concrete material matrix as defined in Equation 5, Ec was the modulus of elasticity, uc represented the Poisson's ratio, and h was the orthogonal distance between centroid of elements to the interface line.

(1)

(2)

(3)

(4)

(5)

Steel rebar in this study was modeled as an equivalent orthotropic plane stress element (Purnomo et al., 2023a). Shear response in the element was transferred through dowel action where expressed the dowel effect coefficient. In addition, recommended value of was estimated using Equation 6, where d and am were the diameter of steel bar and approximated zone height (about 100 ~ 200 mm), respectively. Stress-strain matrix was expressed in Equation 7, where Ds represented the material matrix defined by Equation 8.

(6)

(7)

(8)

During this study, the results were evaluated at the location of springs. Simple averaging method was proposed by lumping the results at springs to adjacent nodes at the element interface. Figure 4 showed the results of re-evaluation procedure at the nodes. In the Figure 4, the results in the springs represented the evenly distributed field at the interface between adjacent elements. These values were summed to the common nodes, N1 and N2.

Stress-strain relationship in tension was modeled as a linear function until ultimate followed by a softening linear curve. The initial stiffness of concrete in the compressive zone was accessed up to stress Fc1, estimated to be 50% of the ultimate cylindrical compressive strength. Subsequently, concrete stiffness was reduced until compressive stress achieved Fc2, defined as 95% of the characteristic cylindrical compressive strength. After reaching this magnitude, constant stress was assumed until achieving ultimate concrete compressive strain. Further, linear softening was presumed followed by a residual stress of 20% Fc.

Uniaxial stress-strain relationship of steel rebars under normal stress was assumed to be elastic-perfectly-plastic in compressive and tensile stress. After reaching the yield stress, the material experienced plastic deformation without significantly increasing stress. In the case of shear stress, the stress-strain relationship behaved linearly until yielding and was perfectly plastic later. The preceding uniaxial stress-strain curve described material behavior under simple load conditions. Moreover, individual stress-strain relationship led to a less accurate outcome in a more complex response due to combined stresses such as normal and shear stress. To address this, the modified Mohr-Coulomb criteria were adopted to justify the production condition of springs under combined normal and shear stress.

3.1. Multicriteria Sensitivity Analysis

Sensitivity analysis is a method frequently used in numerical modeling of structures to determine relative importance of each input parameter, potential interactions, and effects on model responses. The analysis is an important step in computational mechanics and earthquake engineering. In addition, sensitivity analysis had long-term applications in structural engineering, specifically in reinforced-concrete structures applications. Majority-based dynamic sensitivity analyses on the properties of the structure. (Cao et al., 2023; Huang et al., 2023; Bagheri et al., 2021; Suwondo et al., 2021) determined sensitivity analysis of the parameters based on geometric characteristics of buildings, composite and prefabricated concrete components. Consequently (Casafont et al., 2024) included the test set-up and experimental result in sensitivity study to accommodate imperfections.

Soil and uncertainty of structural parameter based on sensitivity analysis incorporating seismic responses was studied by (Zhao et al., 2023; Mekki et al., 2022). This investigation used maximum structure displacement as the single parameter and as a function of material as well as soil characteristics. The study incorporated the seismic response to soil and interaction based on seismic peak acceleration. Following this discussion, the method helped studies to reduce the area of uncertainty and highly important variables and further assisted in quantifying uncertainties by accelerating the entire process of results.

In previous years, contingent on potentiality, application, and applied methodology, as well as possible diverse various classifications had been proposed. (Frey and Patil, 2002) classified sensitivity analysis method into three groups, namely mathematical, statistical, and graphical. Model parameter corrections using the uncertainty qualification (UQ) of numerical simulations were an alternative for model validation (Ren et al., 2023). In this study, the deterministic mathematical method was adopted for sensitivity analysis. The mathematical method was a method to evaluate sensitivity of Quantity of Interest (QoI) to the variation of an input parameter that influenced the output of numerical simulations. In this method, QoIs were typically computed numerically using parametric values to represent the entire input range. Three QoIs (see Figure 5) were considered in this study as follows.

1. Peak load (Pu) of the envelope of loading-unloading experimental results according to (Purwanto et al., 2023),

2. Energy enclosed by occurrence of peak load (Eu),

3. Toughness of structure (energy required per volume until the structure collapsed, Tg).

From the three QoIs mentioned, energy enclosed by occurrence of the peak load was found to show the best prediction.

Figure 5 QoIs of strengthened beam-to-column model

Figure 6 Symmetric haunch strengthened half-span beam-to-column RBSM.

This study investigated sensitivity of input parameters by using a deterministic structural simulation applying the RBSM. During the process, RBSM was developed to represent haunch strengthened beam-to-column joint. Steel-reinforced geopolymer concrete haunch was constructed at both column sides under the beam. The beam-to-column joint was turned upside down for the convenience of a test loading setup in the laboratory that could only apply vertical loading downward. This downward load was applied to the upper column to simulate a negative bending moment at the beam members as in the real situation. Modeling only half of beam-to-column joint structure took advantage of symmetry of the problem. Figure 6 showed the haunch-strengthened beam-to-column model that was specifically designed for this study to represent the discretization scheme used by RBSM. 

Tables 3, 4, and 5 showed mechanical properties of normal concrete, geopolymer concrete, and steel bar materials as the input parameters for running RBSM to conduct the sensitivity analysis and validate experimental results. The notations for the concrete in Table 3 and 4 are explained as follows.

• uc = Poisson's ratio

• C = cohesive strength

• f = friction angle

• Ft = ultimate tensile strength

• c1, c2, c3, c4 = residual tensile stress coefficients

• etu = ultimate tensile strain

• Ec1, Ec2 = 1st and 2nd compressive gradients

• Fc1, Fc2 = 1st and 2nd yield stresses

• ecu = ultimate yield strain

while the representations in Table 5 showed the properties of steel bars as

• Es = modulus of elasticity of steel bars

• Fu = ultimate stress of steel bars

• ?s = dowel effect coefficient

Table 3 Mechanical properties of normal concrete

Table 4 Mechanical properties of geopolymer concrete

Before performing sensitivity analyses to input parameter shown in the aforementioned tables, several initial runs had been conducted to investigate the insensitive input parameters to the defined QoIs. Table 6 showed thirteen sensitive input parameters to QoIs for both normal and geopolymer concrete. Therefore, individual values were varied for sensitivity analyses during the study.

Table 5 Mechanical properties of steel reinforcing bars

Table 6 Sensitive mechanical properties of QoIs

Eighty-nine combinations of thirteen input parameters were extensively ran to generate the calibration data set. The results of QoIs (Pu, Eu, Tg) from RBSM were then normalized by experimental values obtained from the studies of (Purwanto et al., 2023). Table 7 showed sensitivity analysis results where the most sensitive input parameters were varied in RBSM simulation.

Figure 7 showed the variances of the input parameters from the normalized QoI values, as the variances were sorted in descending order. The vertical red line signified the normalized value at 1.0, where simulation results of RBSM were close to experimental tests (Purwanto et al., 2023).

Table 7 Results of sensitivity analyses

Figure 7 showed that the two most influencing and sensitive input parameters had the largest span of variance, including ultimate tensile stress Ft, and residual tensile stress coefficient c2. Since QoIs of Pu and Eu originated from the same physical properties of model, c2 had the most sensitivity to individual QoIs. However, toughness of the structure Tg was highly influenced by Ft. Tornado diagrams (Figure 8) were also constructed using the data applied in Figure 7. Moreover, the span of input parameters was centered on QoIs shown as vertical red lines.

Final, the ultimate tensile stress parameter was the most sensitive variable to QoIs using RBSM. The present results could help studies as a guide on how to simulate experimental model by applying RBSM.

Figure 7 Results of QoIs from RBSM by running eighty-nine test cases.

Figure 8 Tornado diagrams of QoIs from RBSM by running eighty-nine test cases.

3.2. Validation to RBSM

Sensitivity analysis is a method frequently used in statistical-based studies, where the random data are usually generated to give QoI defined. Additionally, reliability analysis typically followed sensitivity analysis to develop a statistical model that could predict new occurrences without being analyzed using conventional methods.

During this study, sensitivity analysis results were used to validate QoIs of experimental investigations conducted by (Purwanto et al., 2023). To validate experimental results of geopolymer concrete, haunch was developed to strengthen the connection between beam and column of a building structure. The output of the load-displacement of experimental works was then validated using RBSM simulations.

Table 8 showed the values of interpolated input parameters in which the values of QoIs were equal to 1.0s. In addition, the validation of experimental results could be conducted at three different conditions of QoIs with these interpolated sets of input parameters.

Table 8 Results of sensitivity analyses at QoIs were equal to 1.0 s.

Numerical simulations in RBSM were run to signify predicted results close to experimental work results by using three sets of input parameters, which were shown in Table 8. The corresponding tensile stress-strain relationships which produced each QoI were equal to 1.0 during validation were shown in Figure 9. Meanwhile, Figure 10 showed envelope of loading-unloading scheme was close to the results when Eu = 1.0, which was represented by vertical red lines.

Figure 9 Tensile stress-strain relationships of concrete material validated for RBSM.

Figure 10 Comparison between envelope curves of experimental results and RBSM.

(a) Failure pattern of haunch-strengthened beam-to-column joint.

(b) Normal stresses of concrete in RBSM at Eu = 1.0 condition.

(c) Concrete plasticity index of RBSM at Eu = 1.0 condition.

Figure 11 Comparisons of failure patterns and plasticity conditions between test and validation results of haunch-strengthened beam-to-column joint.

Figure 10 showed the results of sensitivity analysis when Eu = 1.0 was selected to validate experimental results (Purwanto et al., 2023). In addition, Figure 11 signified the representative failure patterns of test and validation results by using RBSM. As identified in Figure 11a, test specimen showed the compressive and tensile failure of beam, which occurred apart from joint with the column. The test also signified that by haunch strengthening beam-to-column, failure location moved farther from column where maximum negative bending moment occurred. Following this discussion, re-location of failure location had already shown that haunch could increase beam-to-column joint capacity.

In Figure 11b, the results of RBSM showed normal stress distribution in concrete matrix where at the end of the haunch, the stress concentration in compression could be observed. This process caused spalling of concrete cover, some cracking patterns were also shown at the bottom of beam that was subjected to tension due to the bending moment. Figure 11c showed concrete plasticity index of RBSM where concrete failures were spreading, which also signified the effectiveness of the proposed haunch strengthening method.

    In conclusion, multicriteria sensitivity analyses were proven efficient in quantifying the importance of input parameter, potential interaction and effects on the behavior of load-displacement response of structure. The information on this unsureness of variables could be used in improving continuing experimental and minimizing uncertainty of the mentioned parameters. Moreover, sensitivity evaluation of Qols efficiently obtained the best value for each of the three sets of input parameters including peak load of the backbone from the loading-unloading envelope Pu, energy dissipation before ultimate load Eu, and toughness until the collapse of member Tg. The method was shown through analysis of a geopolymer beam-to-column load-displacement response of a haunch connection. Following this discussion, multicriteria method effectively handled complex composite structural elements and captured the variations in concrete strength, including the use of both conventional as well as geopolymer concrete with high accuracy. The application signified the usefulness in constructing accurate behavior of composite concrete member with high precision, incorporating the imperfections of actual laboratory-based load-displacement relationship. Relating to this discussion, the proposed method could be generalized and implemented into any FEA software or subroutines of nonlinear simulation programs.

    The authors are grateful to the following institutions for individual financial support, including Institute for Study and Community Service (LPPM), Universitas Diponegoro with the scheme of Riset Publikasi Internasional Bereputasi Tinggi (RPIBT) through the decreed number 609-125/UN7.D2/PP/VII/2024.

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