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Sensitivity quantification of airport concrete pavement stress responses associated with top-down and bottom-up cracking
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ScienceDirect International Journal of Pavement Research and Technology 10 (2017) 410–420 www.elsevier.com/locate/IJPRT
Sensitivity quantification of airport concrete pavement stress responses associated with top-down and bottom-up cracking Adel Rezaei-Tarahomi a,⇑, Orhan Kaya a, Halil Ceylan b,1, Kasthurirangan Gopalakrishnan c,1, Sunghwan Kim d,1, David R. Brill e,2 a
24 Town Engineering Building, Department of Civil, Construction and Environmental Engineering (CCEE), Iowa State University, Ames, IA 50011-3232, United States b FAA PEGASAS (Partnership to Enhance General Aviation Safety, Accessibility and Sustainability) Center of Excellence (COE) on General Aviation, Program for Sustainable Pavement Engineering and Research (PROSPER), 406 Town Engineering Building, CCEE, Iowa State University, Ames, IA 50011-3232, United States c 354 Town Engineering Building, CCEE, Iowa State University, Ames, IA 50011-3232, United States d 24 Town Engineering Building, Institute for Transportation, Iowa State University, Ames, IA 50011-3232, United States e Airport Pavement Technology, FAA Airport Technology R&D Branch, ANG-E262, William J. Hughes Technical Center, Atlantic City International Airport, NJ 08405, United States Received 22 February 2017; received in revised form 28 June 2017; accepted 4 July 2017 Available online 13 July 2017
Abstract The Federal Aviation Administration’s (FAA’s) rigid pavement design standard employs the NIKE3D-FAA software to compute critical pavement responses of concrete airport pavement structures. NIKE3D-FAA is a modification of the original NIKE3D threedimensional finite element analysis program developed by the Lawrence Livermore National Laboratory (LLNL) of the U.S. Department of Energy, and is currently used in the FAA’s FAARFIELD program. This study evaluated the sensitivity of NIKE3D-FAA rigid pavement responses with respect to top-down and bottom-up cracking. The analysis was conducted by positioning a Boeing 777-300ER (B777-300ER)aircraft at different locations (interior, corner, and edge of slab) as baseline while varying other NIKE3D-FAA inputs, including rigid pavement geometric features, mechanical properties of paving and foundation materials, equivalent temperature gradient and thermal coefficient of Portland Cement Concrete (PCC) layers. Several sensitivity charts were developed by examining the sensitivity of critical pavement responses to each input variation. Sensitivity evaluations were performed using a normalized sensitivity index (NSI) as the quantitative metric. Using such sensitivity evaluation, the most significant NIKE3D-FAA input parameters for generating an effective synthetic database that will lower computational cost for future modeling developments were identified. Ó 2017 Chinese Society of Pavement Engineering. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Sensitivity analysis; Airfield concrete pavement; Finite element analysis; Top down cracking
⇑ Corresponding author.
E-mail addresses: [email protected] (A. Rezaei-Tarahomi), [email protected] (O. Kaya), [email protected] (H. Ceylan), [email protected] (K. Gopalakrishnan), [email protected] (S. Kim), [email protected] (D.R. Brill). 1 Fax: +1 515 294 8216. 2 Fax: +1 609 485 4845. Peer review under responsibility of Chinese Society of Pavement Engineering.
1. Introduction The Federal Aviation Administration (FAA) released the FAA Rigid and Flexible Iterative Elastic Layer Design (FAARFIELD) software as the FAA standard for airfield pavement design upon publication of its Advisory Circular (AC) 150/5320-6E. A research version of the FAARFIELD design software (FAARFIELD 2.0) has been
http://dx.doi.org/10.1016/j.ijprt.2017.07.001 1996-6814/Ó 2017 Chinese Society of Pavement Engineering. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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developed, in which the single-slab three-dimensional finite element (3D-FE) response model is replaced by a 4-slab 3D-FE model with initial temperature curling to produce reasonable thickness designs accounting for top-down cracking behavior. However, the long and unpredictable run times associated with the 4-slab model and curled slabs make routine design with this model impractical. To be able to run the 3D-FEM stress computation routine efficiently, the FAA is seeking practical alternatives such as the use of artificial neural networks (ANNs), that have the potential to give a close estimate of the top-down bending stress in a fraction of the time needed to perform a full 3D-FE computation, for combined vehicle and temperature loading in rigid airport pavements. Finite Element Analysis – FAA (FEAFAA) program, which makes use of NIKE3D, as a stand-alone tool for 3D FE analysis, predicts design stresses associated with the top-down cracking mode, typically the maximum tensile stress occurring on the top surface of the Portland Cement Concrete (PCC) slab for a given combination of gear load and initial temperature-induced curling. There are numerous explicit inputs to FEAFAA that need to be considered in developing the surrogate stress response prediction model using ANN. However, it requires significant understanding of pavement analysis input properties that characterize the pavement materials, layers, load position, temperature condition, and joint properties. In addition, this understanding will help determine where additional effort is justified for further analysis and future development of response models. To do so, it is important to perform a sensitivity analysis to determine the sensitivity of different input parameters in the pavement analysis process. Sensitivity analysis (SA) has become a useful tool in analyzing most engineering problems that involve a large number of interacting variables. One of the most common uses of sensitivity analysis is in pavement design and analysis [1–6]. A number of studies have been conducted by the authors in evaluating the local and global sensitivity analyses of pavement performance predicted by the Mechanistic-Empirical Pavement Design Guide (MEPDG) to design inputs through the National Cooperative Highway Research Program (NCHRP) 01-47 project [3,4,7] In this study, SA can help to: Identify critical control points and prioritize additional data collection or research, Focus on those design inputs that have the most effect on airport rigid pavement thickness Identify the most significant input parameters to generate an effective synthetic database that will facilitate lower computational cost for future modeling developments, and Identify significant deviations between the model and the real structure. Preliminary sensitivity studies on the 4-slab 3D FE model employed in FAARFIELD 2.0 have been carried
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out by Chen et al. [8,9] as part of their efforts to identify the critical aircraft gear (single-gear and multiple-gear) loading position that induces the critical tensile stresses. Their study evaluated the effect of elastic modulus and thickness of each pavement layer and the joint stiffness on the critical tensile stresses and the critical top-tobottom tensile stress ratio (t/b ratio). They used threelayered pavement structure (PCC surface, granular subbase, and subgrade) with 25-ft (7.6 m) PCC slab under the restricted loading condition (i.e., A380 aircraft load with an assumed equivalent thermal gradient (ETG) of 1.25 °F/in. (0.2 °C/cm). These studies [8,9] reported that the critical top-to-bottom tensile stress ratio (t/b ratio) was sensitive to the PCC slab thickness and the modulus of the subgrade variation, but it was not sensitive to the variation of subbase thickness, the modulus of PCC, and the modulus of subbase. Further investigations of interest include the use of different cases including a four-layered pavement structure, different loading conditions, and different load locations. The objective of this paper (part of an on-going FAA sponsored study) is to quantify sensitivity of critical stress outputs to various inputs required in NIKE3D at different load locations and case scenarios for a single aircraft type (B777-300ER). A four-layered pavement structure (PCC surface, cement treated base, granular subbase, and subgrade) with 9 slabs was modeled to represent a typical and realistic airport pavement structure. The 9-slab 3DFE model is better for simulating the real continuous jointed slab conditions than 4-slab 3D-FE model. In addition, a 9-slab model is more suitable for accommodation of multiple gear aircraft like B777-300 ER. Three loading locations were selected: slab interior (Fig. 1(a)), slab edge (Fig. 1(b)), and slab corner (Fig. 1(c)). Two loading case scenarios were considered per loading location: (1) mechanical loading only and (2) simultaneous mechanical and thermal loading. A One-at-a-time (OAT) SA was implemented using a baseline limit normalized sensitivity index (NSI) to provide quantitative sensitivity information. The procedure and the results of the sensitivity analysis are discussed in this paper. The discussions in this paper highlights the significant analysis input properties required for generating an effective synthetic database will facilitate the computational efficiency of future modeling developments. 2. Methodology 2.1. Pavement model The analysis has been done for a four-layered pavement structure with 9 slabs by applying a B777-300ER aircraft loading (Fig. 1). The structural layers and topmost layers of the subgrade have standard eight-node solid hexahedral elements. The bottommost layer of elements in the subgrade consists of 8-noded ‘‘infinite” elements to represent the assumed infinite subgrade.
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Fig. 1. Slab configuration and load positions: (a) Interior (b) Edge (c) Corner.
The number of mesh elements for the concrete slab and foundation layers (base, subbase, and subgrade) was 30 at each side. In other words, each slab had 900 elements and 1,922 nodes. The total number of elements for the pavement structure was 12,600 and the total number of nodes was 23,064. Therefore, if a slab with dimension of 25 25 ft. (7.6 7.6 m) is used, mesh size will be 10 10 in (25.4 25.4 cm). The Boeing B777-300ER aircraft gear has three dual (3D) tires in a tandem main gear; its loading on the pavement sections is a 273.6-kN wheel load approximated as a uniform pressure of 1523.7 kPa applied over six rectangular areas of 0.18 m2 each. These areas were placed at a tandem spacing of 1,463 mm and a dual spacing of 1,397 mm. 2.2. Batch run automation To develop an extensive database of input–output records from FEAFAA 2.0, it is required to use automation programs to reduce required time, increase the accuracy and decrease investment of human interaction. A great number of automation programs are available for use. Among them, the AutoIt has been used. Moreover, to make the automation program scheme more efficient in such a way that it will be able to meet a majority of the requirements for FEAFAA’s batch run, a tool was developed using the C# programming language and utilized with AutoIt. The end result was that the combined AutoIt-C# tool was able to automatically perform batch runs, get the outputs, and do the post-processing. Also, a post-processing program was written in C#. This program was able to analyze all output values, calculate maximum and minimum stress responses and deflection responses for each case and find critical response locations. This program can read the output text file of FEAFAA and then gather all information in an Excel spreadsheet. The data are separated for top and bottom of the slab, so the critical responses and their locations can be found separately for top or bottom. Besides, for each input case, the type of critical normal stress is specified as either tensile or compressive. So, using the results of the postprocessor, tensile stresses at the top and bottom of the slab,
which are of interest to this paper, can be determined for SA investigations. 2.3. Sensitivity analysis The sensitivity of the input parameters has been evaluated by considering their effects on the critical responses corresponding to the top-down and bottom-up cracking. Fig. 2 shows the overall SA process employed in this study. The OAT SA has been carried out on the FEAFAA 2.0 software by varying one parameter at a time while holding the others fixed. This analysis helps to identify the most significant inputs in the airport rigid pavement structural analysis by the NIKE3D FE analysis program (which is at the heart of FEAFAA). Inputs that are needed for FEAFAA can broadly be categorized as:
Pavement structure inputs Airplane inputs Loading inputs Joint modeling inputs
The sensitivity of all these inputs cannot be evaluated because some of them are interrelated inputs and some others are just used for FE modeling (Number of elements). There are also some inputs that have been assumed to be not so effective (e.g., Poisson ratio, equivalent boundary stiffness) from the design and analysis perspectives. Based on engineering judgment and pre-parametric sensitivity analysis, constant values were assigned to these inputs. The goal is to evaluate the sensitivity of those input parameters which are more important for analyzing and designing airfield concrete pavements have been evaluated. A detailed summary of ranges of the inputs to be varied as well as constant inputs are shown in Table 1. Each evaluated input was varied within its recommended range to study its effect on critical responses (maximum tensile stress and shear stress at top/bottom of the slab) while assigning base case values to all other input parameters. To consider the effects of load locations and different types of loading,
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Sensitivity Analysis
Initial Triage
Select Base Cases
Special Input Consideration
Identification of Inputs for Further Analysis
Varying Each Input
Local Sensitivity Analysis
Comparing Outputs by Baselines
Sensitivity Metrics
Evaluate and Rank the Input Data Fig. 2. Simplified flow chart for local sensitivity analysis of FEAFAA.
six general cases have been evaluated involving two loading case types. In FEAFAA, joints are modeled as continuous linear elastic shear springs connecting the slabs. The springs act only in the vertical direction, representing the action of dowel bars. To evaluate the sensitivity of stress responses to the joint properties, equivalent joint stiffness was used. The equivalent joint stiffness ranges are shown in Table 1. To determine the number of elements for mesh modeling, a pre-parametric sensitivity analysis was carried out. Fig. 3 shows the changes in stress responses with respect to varying number of elements for slabs (Fig. 3(a)) and foundation layers (Fig. 3(b)). As the number of elements increase, the stresses converge to a specific value. From this figure, it is obvious that for number of elements less than 30 the results are not constant but for number of elements more than 30 the results stay the same. However, increasing the number of elements can dramatically increase the run time of NIKE3D (i.e. about 16 min. for 30 elements and 50 min. for 50 elements). So, an optimum number of 30 elements was selected for meshing the slab and foundation without sacrificing the accuracy of the results and at the same time not too computationally intensive. Furthermore, By using 30 elements, it does not need to refine the mesh near the loading because the slabs’ mesh size (25.4 25.4 cm) is smaller than the tires’ foot-print (60.5 37.6 cm), so the FE model can promise accurate results. 3. Results To present the sensitivity of each parameter, a normalized sensitivity index (NSI) has been adopted as a quantitative metric.
NSI ¼
DY j X k DX k Y K
ð1Þ
where X k = Baseline value of input k, DX k = Change in input k about the base line, DY j = Change in output J corresponding to DX k , Y K = Baseline value of output J. As mentioned earlier, three load locations were considered for sensitivity analysis: interior, edge, and corner. For each location, analysis was carried out for two load types: mechanical loading only and simultaneous mechanical and thermal loading. Four stress types were considered as critical stresses for each load location and load type and used as outputs for the NSI calculation: Maximum tensile stress at the top of the slab (top tensile stress) Maximum tensile stress at the bottom of the slab (bottom tensile stress) Maximum shear stress at top of the slab (top shear stress) Maximum shear stress at bottom of the slab (bottom shear stress). The tensile stresses on top and bottom of the slab are in x-direction that is perpendicular to traffic. Shear stresses on top and bottom of the slab are in xy-direction that is more critical for crack opening. 3.1. Interior loading 3.1.1. Mechanical loading only Fig. 4(a) shows sensitivity of critical stresses to different inputs when the airplane is located at the center of the middle slab (interior loading). The Figure shows that the PCC thickness has the most influence on all responses at the top
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Inputs Category
Inputs
Range Min
Pavement structure inputs
PCC Slab
Cement Treated Base
Granular Subbase
Subgrade
Airplane inputs Loading inputs
Joint modeling inputs
Modulus, psi (GPa) Thickness, in. (cm) Poisson Ratio Modulus, psi (GPa) Thickness, in. (cm) Poisson Ratio Modulus, psi (GPa) Thickness, in. (cm) Poisson Ratio Modulus, psi (GPa) Poisson Ratio
Slab Dimension, ft. (m) Slab Number of Elements Number of Slabs Foundation Number of Elements Airplane parameters Loading Angle Loading position Equivalent Temperature Gradient, °F/in. (°C/cm) Thermal Coefficient, 1/°F (1/°C) Equivalent Joint Stiffness, psi/in (GPa/m)
Base case Baseline
Max
3 106 (20.7) 5 106 (34.5) 7 106 (48.3) 4 106 (27.6) 10 (25.4) 17 (43.2) 24 (61) 14 (35.6) 0.15 2.5 105 (1.7) 1.125 106 (7.8) 2 106 (13.8) 5 105 (3.5) 6 (15.2) 18 (45.7) 30 (76.2) 8 (20.3) 0.15 15,000 (1 101) 45,000 (3.1 101) 75,000 (5.2 101) 75,000 (5.2 101) 6 (15.2) 18 (45.7) 30 (76.2) 12 (30.5) 0.35 3,000 (2.1 102) 16,500 (1.1 101) 30,000 (2.1 101) 3,000 (2.1 102) 0.4 25 (7.6) 30 9 30 B777-300ER: Gross weight = 777,000 (lb), %GW = %95, No. Main Gears = 2, Wheels on Main Gears = 6, Tire Pressure (psi) = 221 0 Interior/Mid Slab Edge/Corner 2 (0.26) 2.5 (0.37) 3 (0.48) 2.3 (0.33) 4.1 106 (7.6 106) 1000 (2.7 101)
5.5 106 (9.9 106) 100000 (27.1)
7 106 (12.2 106) 300000 (81.3)
5 106 (9 106) 144,798 (39.2)
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Table 1 Ranges of inputs for sensitivity analysis of FEAFAA.
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415
400
300
350
250
Stress (psi)
Stress (psi)
300 200 150 100
250 200 150 100
50
50 0
0 0
10
20
30
40
50
0
60
10
Number of Elements
20
30
40
50
60
70
Number of Elements
(a)
(b)
Sensitivity Index
Fig. 3. Sensitivity of number of elements of (a) slabs (b) foundation layers to the stress responses in FEAFAA.
3.0 2.5 2.0 1.5 1.0 0.5 0.0 Slab Thickness
Subgrade Modulus
Base Thickness
PCC Modulus
Subbase Thickness
Base Modulus
Equivalent Joint Stiffness
Subbase Modulus
Top tensile stress
1.60
0.57
0.30
0.28
0.06
0.15
0.12
0.05
Bottom tensile stress
1.01
0.75
0.41
0.40
0.21
0.07
0.00
0.05
Top shear stress
1.23
0.61
0.57
0.33
0.30
0.07
0.05
0.06
Bottom shear stress
1.23
0.60
0.56
0.33
0.30
0.08
0.05
0.06
Sensitivity Index
(a) 3.0 2.5 2.0 1.5 1.0 0.5 0.0 Subgrade Modulus
Slab Thickness
Base Thickness
Thermal Coefficient
Temperature Gradient
PCC Modulus
Subbase Thickness
Base Modulus
Subbase Modulus
Equivalent Joint Stiffness
Top tensile stress
0.12
0.39
0.28
0.64
0.55
0.41
0.28
0.11
0.08
0.01
Bottom tensile stress
2.22
0.90
0.93
0.19
0.25
0.42
0.41
0.09
0.08
0.00
Top shear stress
0.78
1.08
0.42
0.24
0.25
0.25
0.32
0.13
0.07
0.04
Bottom shear stress
0.77
1.09
0.42
0.23
0.24
0.24
0.32
0.13
0.06
0.04
(b) Fig. 4. NSI of different input values vs. critical responses for (a) interior-mechanical loading and (b) interior-mechanical loading with temperature loading.
and bottom of the slab (i.e., top tensile stress, bottom tensile stress, top shear stress, and bottom shear stress). This means that a change in slab thickness results in a significant change in these stress responses.
After the PCC thickness, subgrade modulus has the next highest influence on the stress response. Sensitivity indices of base thickness and PCC modulus show that all stress responses are also sensitive to changes in these input vari-
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ables. The inputs which the responses are insensitive or ‘very low sensitive’ to are base modulus, equivalent joint stiffness, and subbase modulus. Most effective inputs, except slab thickness, have higher NSI for the bottom tensile stress than for the top tensile stress. In addition, NSI of all inputs for top shear stress are closer to those for bottom shear stress. 3.1.2. Simultaneous mechanical and thermal loading Fig. 4(b) depicts the sensitivity analysis results for the case in which interior mechanical loading along with temperature loading were applied to the modeled pavement structure. NIKE3D-FAA as implemented in FEAFAA considers both temperature and concrete shrinkage effects in the modeling for concrete slab curling analysis by introducing the concept of equivalent temperature gradient (ETG) [10]. Because of temperature gradient and shrinkage effects, the initial concrete slab shape affects the responses of the PCC slab to aircraft gear loading. The use of a simple compensation term applied to ETG can change the initial slab shape from circular to catenary, thereby significantly improving the computational results [10]. All analyses related to the temperature loading cases were conducted using catenary shape of slab’s curling. As shown in Fig. 4(b), the bottom tensile stresses are more sensitive to subgrade modulus than the other inputs. The higher sensitivity index of subgrade modulus to tensile stress at the bottom of the slab shows the importance of this input for studying bottom-up cracking in concrete pavement. The figure shows that the bottom tensile stress is also sensitive to base thickness and PCC slab thickness followed by PCC modulus, subbase thickness, ETG, and temperature coefficient. Variations in thermal coefficients and ETG have the most impact on tensile stresses at the top of the slab. Higher sensitivity to temperature loading related inputs for top tensile stresses indicate the importance of curling analysis for top-down cracking model development. Other inputs that strongly effect top tensile stresses include PCC modulus, PCC thickness, and subbase thickness. Unlike the bottom tensile stresses, top tensile stresses show lower sensitivity to the subgrade modulus. Fig. 4(b) indicates that the shear stress is most sensitive to slab thickness and subgrade modulus followed by base thickness, subbase thickness, temperature loading related inputs (ETG and thermal coefficient), and PCC modulus. Fig. 4(b) also shows that changes in base and subbase modulus and equivalent joint stiffness have the lowest effect on all stress responses. 3.2. Edge loading For rigid pavement design, the FAA uses the maximum tensile stress at the bottom edge of the PCC slab as a predictor of pavement structural life. The maximum tensile stress for design is determined using an edge loading condition. Fig. 5 shows sensitivities computed for edge loading
only as well as edge loading in combination with temperature loading. 3.2.1. Mechanical Loading Only Fig. 5(a) displays the sensitivity analysis results for different inputs when the load is centered on one edge of the middle slab. PCC slab thickness has been identified as the most effective input for top and bottom tensile stresses in this case. Top tensile stresses, unlike the bottom tensile stresses, exhibit significant sensitivity to the base and subbase thickness. Variations in modulus of PCC, base and subbase show less sensitivity index for bottom tensile stresses. However, alteration of subgrade modulus has considerable effect on all stress responses. The top tensile stresses exhibit considerable sensitivity to most inputs, but the bottom tensile stresses have considerable sensitivity to just two inputs (PCC thickness and subgrade modulus). It is noted in the Fig. 5 that the inputs having the greatest effect on shear stresses are thickness of PCC, base and subbase; subgrade modulus; and PCC modulus. Similar to previous cases, the stress responses are not sensitive to subbase/base modulus and equivalent joint stiffness (lowest NSI). 3.2.2. Simultaneous mechanical and thermal loading case Analysis results for this case are presented in Fig. 5(b). It can be seen in Fig. 5 (b) that top and bottom tensile stresses exhibit higher sensitivity to PCC slab thickness than other inputs. Temperature loading related inputs (ETG and thermal coefficient) also exhibit sensitivity for both tensile stresses. Most notably, higher NSI values were observed for these inputs for bottom tensile stresses than for top tensile stresses. Similarly, bottom tensile stresses show higher sensitivity to subgrade modulus, base thickness and subbase thickness than top tensile stresses. It can also be noted that top tensile stresses is not sensitive to the subgrade modulus input, while top tensile stresses have higher sensitivity, than bottom tensile stresses, to the PCC modulus . The ETG, thermal coefficient, PCC slab thickness and modulus, subgrade modulus, base thickness, and subbase thickness are all effective inputs for both top and bottom shear stresses. 3.3. Corner loading In this case, sensitivity analysis was conducted for the B777-300 ER gear located at the corner of the slab and. Corner loading is important for analysis and design purposes because of major observed failures that have occurred at this location and load transfer issues arising from the joints. 3.3.1. Mechanical Loading Only Case Fig. 6(a) illustrates how critical stress responses are sensitive to different inputs. As displayed in this Figure, PCC slab thickness exhibited the highest NSI for top and bot-
Sensitivity Index
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3.0 2.5 2.0 1.5 1.0 0.5 0.0
417
Slab Thickness
Base Thickness
Subbase Thickness
Subgrade Modulus
PCC Modulus
Subbase Modulus
Base Modulus
Equivalent Joint Stiffness
Top tensile stress
1.62
1.00
0.73
0.51
0.29
0.15
0.07
0.09
Bottom tensile stress
1.80
0.19
0.07
0.60
0.14
0.02
0.04
0.00
Top shear stress
0.74
0.73
0.33
0.70
0.50
0.10
0.13
0.05
Bottom shear stress
0.72
0.72
0.35
0.71
0.51
0.11
0.14
0.05
Sensitivity Index
(a) 3.0 2.5 2.0 1.5 1.0 0.5 0.0 Slab Thickness
Temperature Gradient
Thermal Coefficient
Subgrade Modulus
Base Thickness
PCC Modulus
Subbase Thickness
Equivalent Joint Stiffness
Base Modulus
Subbase Modulus
Top tensile stress
0.80
0.41
0.44
0.02
0.16
0.36
0.15
0.07
0.02
0.03
Bottom tensile stress
2.54
1.24
0.98
0.98
0.70
0.13
0.34
0.06
0.08
0.07
Top shear stress
0.32
0.49
0.50
0.22
0.29
0.57
0.19
0.13
0.04
0.04
Bottom shear stress
0.33
0.49
0.51
0.22
0.29
0.56
0.20
0.12
0.04
0.05
(b) Fig. 5. NSI of different input values vs. critical responses for (a) edge mechanical loading and (b) edge mechanical loading with temperature loading.
tom tensile stresses. Base and subbase thicknesses have high NSI for top tensile stresses but low NSI for bottom tensile stresses, while subgrade modulus and PCC modulus are effective inputs for both types of tensile stresses. It is also illustrated that shear stresses are sensitive to subgrade modulus, PCC modulus, PCC slab thickness, base thickness and subbase thickness. Moreover, base modulus, subbase modulus, and equivalent joint stiffness have the lowest NSI for all stress responses. 3.3.2. Simultaneous mechanical and thermal loading case Fig. 6(b) shows that PCC slab thickness, among all other inputs, has the highest NSI for top and bottom tensile stresses. Especially, higher NSI values of PCC slab thickness were observed for bottom tensile stresses than for top tensile stresses. Thermal coefficient is the second effective input for bottom tensile stresses. Other inputs which the bottom tensile stresses are sensitive to are base thickness, subbase thickness, and subgrade modulus. Top tensile stresses have low sensitivity to PCC modulus and ETG.
PCC slab thickness and modulus and thermal loading related inputs are the most effective inputs for top tensile stresses. The NSIs of these inputs (i.e., about 0.5) are close to one another. Base thickness and subbase thickness form the next lower tier of effective inputs. While, top tensile stresses have low sensitivity to subgrade modulus. The shear stresses are sensitive to PCC slab thickness, thermal loading related inputs, PCC modulus followed by base thickness, subgrade modulus, and subbase thickness. Similar to other cases, stress responses are not much sensitive to the equivalent joint stiffness and subbase/base modulus (lowest NSI). 4. Discussions of sensitivity results Table 2 summarizes the maximum NSI of each input among all loading conditions for discussion. As presented in this table, the inputs were ranked separately for each stress response. Note that shear stress in Table 2 indicates either top or bottom shear stress since both have almost similar sensitivity levels to all inputs.
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Sensitivity Index
418
3.0 2.5 2.0 1.5 1.0 0.5 0.0
Top tensile stress Bottom tensile stress Top shear stress Bottom shear stress
Slab Thickness
Base Thickness
1.28 1.51 0.58 0.50
1.03 0.19 0.66 0.63
Subbase Thickness 0.76 0.05 0.33 0.36
Subgrade Modulus 0.53 0.60 0.68 0.72
PCC Modulus
Base Modulus
0.34 0.23 0.47 0.46
0.08 0.04 0.15 0.15
Subbase Modulus 0.14 0.02 0.08 0.12
Equivalent Joint Stiffness 0.09 0.01 0.03 0.04
Sensitivity Index
(a) 3.0 2.5 2.0 1.5 1.0 0.5 0.0 Slab Thickness
Thermal Coefficient
PCC Modulus
Base Thickness
Temperature Gradient
Subgrade Modulus
Subbase Thickness
Equivalent Joint Stiffness
Base Modulus
Subbase Modulus
0.59
0.43
0.44
0.17
0.41
0.05
0.14
0.07
0.03
0.03
Bottom tensile stress
2.12
0.64
0.03
0.46
0.07
0.34
0.32
0.11
0.10
0.09
Top shear stress
0.45
0.48
0.54
0.27
0.44
0.19
0.16
0.17
0.03
0.03
Bottom shear stress
0.66
0.38
0.40
0.34
0.32
0.25
0.21
0.14
0.05
0.05
Top tensile stress
(b) Fig. 6. NSI of different input values vs. critical responses for (a) corner-mechanical loading and (b) corner-mechanical loading with temperature loading.
This table shows that all stress responses has the highest sensitivity to PCC slab thickness. For the top tensile stress, the thickness of pavement structural layers are the most effective inputs. It is noteworthy that subgrade modulus has a higher effect on bottom tensile stresses and shear stresses changes. Top tensile stresses are more sensitive to PCC thermal coefficient variations while bottom tensile stresses are more sensitive to the thermal gradient changes. Both have almost similar NSIs for shear stresses. The inputs categorized as insensitive for all stress responses are equivalent joint stiffness, base modulus, and subbase modulus. 5. Conclusions The primary objective of this study was to quantify sensitivity of stress responses to various inputs required in NIKE3D for critical stress outputs at different loading locations and load case scenarios for a B777-300ER aircraft. A four-layered pavement structure (PCC surface, cement treated base, granular subbase, and subgrade) in 9 slabs was modeled to represent typical and realistic airport pavement structure. The OAT SA was implemented
using a baseline limit normalized sensitivity index (NSI) to provide quantitative sensitivity information on each stress response output for different loading conditions. The major conclusions of this study are as follows: All stress responses are most sensitive to PCC slab thickness, followed by base and subbase thicknesses for top tensile stress, and subgrade modulus for bottom tensile stresses and shear stresses. Top tensile stress is more sensitive to thermal coefficient variation than ETG variation while bottom tensile stress shows higher sensitivity to ETG variation than thermal coefficient. The inputs categorized as insensitive for all stress responses under different loading conditions are equivalent joint stiffness, base modulus, and subbase modulus. In the mechanical loading only case under interior loading condition, PCC thickness and subgrade modulus are the most effective input parameters for all stress responses. In the simultaneous mechanical and thermal loading case under interior loading condition, changes in thermal coefficient and ETG, in addition to PCC slab thick-
NSI Shear stress
1.23 0.78 0.73 0.57 0.50 0.49 0.33 0.17 0.15 0.10
Inputs
Slab Thickness Subgrade Modulus Base Thickness PCC Modulus Thermal Coefficient Equivalent Temperature Gradient Subbase Thickness Equivalent Joint Stiffness Base Modulus Subbase Modulus
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NSI Bottom tensile stress
2.54 2.22 1.24 0.98 0.93 0.42 0.41 0.11 0.10 0.09
Inputs
Slab Thickness Subgrade Modulus Equivalent Temperature Gradient Thermal Coefficient Base Thickness PCC Modulus Subbase Thickness Equivalent Joint Stiffness Base Modulus Subbase Modulus
NSI Top tensile stress
1.60 1.03 0.76 0.64 0.57 0.55 0.44 0.15 0.15 0.12 Slab Thickness Base Thickness Subbase Thickness Thermal Coefficient Subgrade Modulus Equivalent Temperature Gradient PCC Modulus Base Modulus Subbase Modulus Equivalent Joint Stiffness
Inputs
Table 2 Inputs ranking for stress responses.
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ness, have the most impact on top tensile stresses. Bottom tensile stresses are more sensitive to subgrade modulus than other inputs, while top tensile stresses exhibited low sensitivity to subgrade modulus. In the mechanical loading only case under edge loading condition, the top tensile stresses exhibit considerable sensitivity for most inputs (except inputs categorized as insensitive) but the bottom tensile stresses have considerable sensitivity just for two inputs (PCC thickness and subgrade modulus). In the simultaneous mechanical and thermal loading case under edge loading condition, top and bottom tensile stresses are sensitive to temperature loading related inputs (ETG and thermal coefficient) in addition to PCC slab thickness. Especially, higher NSIs of these inputs for the bottom tensile stresses were observed than for the top tensile stress. Top tensile stress does not have sensitivity to subgrade modulus. In the mechanical loading only case under corner loading condition, subgrade modulus and PCC modulus variation have effect on top and bottom tensile stresses. In the simultaneous mechanical and thermal loading case under corner loading condition, PCC slab thickness, among all other inputs, has the highest effect on top and bottom tensile stresses followed by PCC modulus and thermal coefficient for top tensile stresses and thermal coefficient for the bottom tensile stresses.
Acknowledgements The authors gratefully acknowledge the Federal Aviation Administration (FAA) for supporting this study. The contents of this paper of this paper reflect the views of the authors, who are responsible for the facts and accuracy of the data presented within. The contents do not necessarily reflect the official views and policies of the FAA. The paper does not constitute a standard, specification, or regulation. References [1] H. Ceylan, S. Kim, K. Gopalakrishnan, C.W. Schwartz, R. Li, Sensitivity analysis frameworks for mechanistic-empirical pavement design of continuously reinforced concrete pavements, Constr. Build. Mater. 73 (2014) 498–508. [2] H. Ceylan, S. Kim, K. Gopalakrishnan, C.W. Schwartz, R. Li, Sensitivity quantification of jointed plain concrete pavement mechanistic-empirical performance predictions, Constr. Build. Mater. 43 (2013) 545–556. [3] C.W. Schwartz, R. Li, H. Ceylan, S. Kim, K. Gopalakrishnan, Global Sensitivity Analysis of Mechanistic-Empirical Performance Predictions for Flexible Pavements, Transport. Res. Rec. 2368 (2013) 12–23. [4] C.W. Schwartz, R. Li, S. Kim, H. Ceylan, K. Gopalakrishnan, Sensitivity Evaluation of MEPDG Performance Prediction, Contractor’s Final Report for National Cooperative Highway Research Program Project 1-47, Transportation research board of the national academies, University of Maryland College Park and Iowa State University, 2011.
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[5] T. Merhej, D. Feng, Parameter sensitivity analysis of airport rigid pavement thickness using FAARFIELD program, Adv. Mater. Res. 243–249 (2011) 4068–4074. [6] A. Guclu, H. Ceylan, K. Gopalakrishnan, S. Kim, Sensitivity analysis of rigid pavement systems using the mechanistic-empirical design guide software, J. Transport. Eng. ASCE. 8 (2009) 555–562. [7] H. Ceylan, K. Gopalakrishnan, S. Kim, C. Schwartz, R. Li, Global sensitivity analysis of jointed plain concrete pavement mechanisticempirical performance predictions, Transport. Res. Rec. 2367 (2013) 113–122.
[8] Y. Chen, 4-Slab Three Dimensional Finite Element Method Model in FAARFIELD, Report for Contract No. DTFACT- 10-D-00008, SRA International, Inc., 2014. [9] Y. Chen, Q. Wang, D. R. Brill, Critical Stress Analysis of Large Aircraft on Airport Rigid Pavement Using FEAFAA, in: Proceedings of 10th International Conference on Concrete Pavements, 2012, 1050–1067. [10] Q. Wang, Y. Chen, Improvements to modeling of concrete slab curling by using NIKE3D finite element program, Transport. Res. Rec. 2226 (2011) 71–81.
ScienceDirect International Journal of Pavement Research and Technology 10 (2017) 410–420 www.elsevier.com/locate/IJPRT
Sensitivity quantification of airport concrete pavement stress responses associated with top-down and bottom-up cracking Adel Rezaei-Tarahomi a,⇑, Orhan Kaya a, Halil Ceylan b,1, Kasthurirangan Gopalakrishnan c,1, Sunghwan Kim d,1, David R. Brill e,2 a
24 Town Engineering Building, Department of Civil, Construction and Environmental Engineering (CCEE), Iowa State University, Ames, IA 50011-3232, United States b FAA PEGASAS (Partnership to Enhance General Aviation Safety, Accessibility and Sustainability) Center of Excellence (COE) on General Aviation, Program for Sustainable Pavement Engineering and Research (PROSPER), 406 Town Engineering Building, CCEE, Iowa State University, Ames, IA 50011-3232, United States c 354 Town Engineering Building, CCEE, Iowa State University, Ames, IA 50011-3232, United States d 24 Town Engineering Building, Institute for Transportation, Iowa State University, Ames, IA 50011-3232, United States e Airport Pavement Technology, FAA Airport Technology R&D Branch, ANG-E262, William J. Hughes Technical Center, Atlantic City International Airport, NJ 08405, United States Received 22 February 2017; received in revised form 28 June 2017; accepted 4 July 2017 Available online 13 July 2017
Abstract The Federal Aviation Administration’s (FAA’s) rigid pavement design standard employs the NIKE3D-FAA software to compute critical pavement responses of concrete airport pavement structures. NIKE3D-FAA is a modification of the original NIKE3D threedimensional finite element analysis program developed by the Lawrence Livermore National Laboratory (LLNL) of the U.S. Department of Energy, and is currently used in the FAA’s FAARFIELD program. This study evaluated the sensitivity of NIKE3D-FAA rigid pavement responses with respect to top-down and bottom-up cracking. The analysis was conducted by positioning a Boeing 777-300ER (B777-300ER)aircraft at different locations (interior, corner, and edge of slab) as baseline while varying other NIKE3D-FAA inputs, including rigid pavement geometric features, mechanical properties of paving and foundation materials, equivalent temperature gradient and thermal coefficient of Portland Cement Concrete (PCC) layers. Several sensitivity charts were developed by examining the sensitivity of critical pavement responses to each input variation. Sensitivity evaluations were performed using a normalized sensitivity index (NSI) as the quantitative metric. Using such sensitivity evaluation, the most significant NIKE3D-FAA input parameters for generating an effective synthetic database that will lower computational cost for future modeling developments were identified. Ó 2017 Chinese Society of Pavement Engineering. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Sensitivity analysis; Airfield concrete pavement; Finite element analysis; Top down cracking
⇑ Corresponding author.
E-mail addresses: [email protected] (A. Rezaei-Tarahomi), [email protected] (O. Kaya), [email protected] (H. Ceylan), [email protected] (K. Gopalakrishnan), [email protected] (S. Kim), [email protected] (D.R. Brill). 1 Fax: +1 515 294 8216. 2 Fax: +1 609 485 4845. Peer review under responsibility of Chinese Society of Pavement Engineering.
1. Introduction The Federal Aviation Administration (FAA) released the FAA Rigid and Flexible Iterative Elastic Layer Design (FAARFIELD) software as the FAA standard for airfield pavement design upon publication of its Advisory Circular (AC) 150/5320-6E. A research version of the FAARFIELD design software (FAARFIELD 2.0) has been
http://dx.doi.org/10.1016/j.ijprt.2017.07.001 1996-6814/Ó 2017 Chinese Society of Pavement Engineering. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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developed, in which the single-slab three-dimensional finite element (3D-FE) response model is replaced by a 4-slab 3D-FE model with initial temperature curling to produce reasonable thickness designs accounting for top-down cracking behavior. However, the long and unpredictable run times associated with the 4-slab model and curled slabs make routine design with this model impractical. To be able to run the 3D-FEM stress computation routine efficiently, the FAA is seeking practical alternatives such as the use of artificial neural networks (ANNs), that have the potential to give a close estimate of the top-down bending stress in a fraction of the time needed to perform a full 3D-FE computation, for combined vehicle and temperature loading in rigid airport pavements. Finite Element Analysis – FAA (FEAFAA) program, which makes use of NIKE3D, as a stand-alone tool for 3D FE analysis, predicts design stresses associated with the top-down cracking mode, typically the maximum tensile stress occurring on the top surface of the Portland Cement Concrete (PCC) slab for a given combination of gear load and initial temperature-induced curling. There are numerous explicit inputs to FEAFAA that need to be considered in developing the surrogate stress response prediction model using ANN. However, it requires significant understanding of pavement analysis input properties that characterize the pavement materials, layers, load position, temperature condition, and joint properties. In addition, this understanding will help determine where additional effort is justified for further analysis and future development of response models. To do so, it is important to perform a sensitivity analysis to determine the sensitivity of different input parameters in the pavement analysis process. Sensitivity analysis (SA) has become a useful tool in analyzing most engineering problems that involve a large number of interacting variables. One of the most common uses of sensitivity analysis is in pavement design and analysis [1–6]. A number of studies have been conducted by the authors in evaluating the local and global sensitivity analyses of pavement performance predicted by the Mechanistic-Empirical Pavement Design Guide (MEPDG) to design inputs through the National Cooperative Highway Research Program (NCHRP) 01-47 project [3,4,7] In this study, SA can help to: Identify critical control points and prioritize additional data collection or research, Focus on those design inputs that have the most effect on airport rigid pavement thickness Identify the most significant input parameters to generate an effective synthetic database that will facilitate lower computational cost for future modeling developments, and Identify significant deviations between the model and the real structure. Preliminary sensitivity studies on the 4-slab 3D FE model employed in FAARFIELD 2.0 have been carried
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out by Chen et al. [8,9] as part of their efforts to identify the critical aircraft gear (single-gear and multiple-gear) loading position that induces the critical tensile stresses. Their study evaluated the effect of elastic modulus and thickness of each pavement layer and the joint stiffness on the critical tensile stresses and the critical top-tobottom tensile stress ratio (t/b ratio). They used threelayered pavement structure (PCC surface, granular subbase, and subgrade) with 25-ft (7.6 m) PCC slab under the restricted loading condition (i.e., A380 aircraft load with an assumed equivalent thermal gradient (ETG) of 1.25 °F/in. (0.2 °C/cm). These studies [8,9] reported that the critical top-to-bottom tensile stress ratio (t/b ratio) was sensitive to the PCC slab thickness and the modulus of the subgrade variation, but it was not sensitive to the variation of subbase thickness, the modulus of PCC, and the modulus of subbase. Further investigations of interest include the use of different cases including a four-layered pavement structure, different loading conditions, and different load locations. The objective of this paper (part of an on-going FAA sponsored study) is to quantify sensitivity of critical stress outputs to various inputs required in NIKE3D at different load locations and case scenarios for a single aircraft type (B777-300ER). A four-layered pavement structure (PCC surface, cement treated base, granular subbase, and subgrade) with 9 slabs was modeled to represent a typical and realistic airport pavement structure. The 9-slab 3DFE model is better for simulating the real continuous jointed slab conditions than 4-slab 3D-FE model. In addition, a 9-slab model is more suitable for accommodation of multiple gear aircraft like B777-300 ER. Three loading locations were selected: slab interior (Fig. 1(a)), slab edge (Fig. 1(b)), and slab corner (Fig. 1(c)). Two loading case scenarios were considered per loading location: (1) mechanical loading only and (2) simultaneous mechanical and thermal loading. A One-at-a-time (OAT) SA was implemented using a baseline limit normalized sensitivity index (NSI) to provide quantitative sensitivity information. The procedure and the results of the sensitivity analysis are discussed in this paper. The discussions in this paper highlights the significant analysis input properties required for generating an effective synthetic database will facilitate the computational efficiency of future modeling developments. 2. Methodology 2.1. Pavement model The analysis has been done for a four-layered pavement structure with 9 slabs by applying a B777-300ER aircraft loading (Fig. 1). The structural layers and topmost layers of the subgrade have standard eight-node solid hexahedral elements. The bottommost layer of elements in the subgrade consists of 8-noded ‘‘infinite” elements to represent the assumed infinite subgrade.
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Fig. 1. Slab configuration and load positions: (a) Interior (b) Edge (c) Corner.
The number of mesh elements for the concrete slab and foundation layers (base, subbase, and subgrade) was 30 at each side. In other words, each slab had 900 elements and 1,922 nodes. The total number of elements for the pavement structure was 12,600 and the total number of nodes was 23,064. Therefore, if a slab with dimension of 25 25 ft. (7.6 7.6 m) is used, mesh size will be 10 10 in (25.4 25.4 cm). The Boeing B777-300ER aircraft gear has three dual (3D) tires in a tandem main gear; its loading on the pavement sections is a 273.6-kN wheel load approximated as a uniform pressure of 1523.7 kPa applied over six rectangular areas of 0.18 m2 each. These areas were placed at a tandem spacing of 1,463 mm and a dual spacing of 1,397 mm. 2.2. Batch run automation To develop an extensive database of input–output records from FEAFAA 2.0, it is required to use automation programs to reduce required time, increase the accuracy and decrease investment of human interaction. A great number of automation programs are available for use. Among them, the AutoIt has been used. Moreover, to make the automation program scheme more efficient in such a way that it will be able to meet a majority of the requirements for FEAFAA’s batch run, a tool was developed using the C# programming language and utilized with AutoIt. The end result was that the combined AutoIt-C# tool was able to automatically perform batch runs, get the outputs, and do the post-processing. Also, a post-processing program was written in C#. This program was able to analyze all output values, calculate maximum and minimum stress responses and deflection responses for each case and find critical response locations. This program can read the output text file of FEAFAA and then gather all information in an Excel spreadsheet. The data are separated for top and bottom of the slab, so the critical responses and their locations can be found separately for top or bottom. Besides, for each input case, the type of critical normal stress is specified as either tensile or compressive. So, using the results of the postprocessor, tensile stresses at the top and bottom of the slab,
which are of interest to this paper, can be determined for SA investigations. 2.3. Sensitivity analysis The sensitivity of the input parameters has been evaluated by considering their effects on the critical responses corresponding to the top-down and bottom-up cracking. Fig. 2 shows the overall SA process employed in this study. The OAT SA has been carried out on the FEAFAA 2.0 software by varying one parameter at a time while holding the others fixed. This analysis helps to identify the most significant inputs in the airport rigid pavement structural analysis by the NIKE3D FE analysis program (which is at the heart of FEAFAA). Inputs that are needed for FEAFAA can broadly be categorized as:
Pavement structure inputs Airplane inputs Loading inputs Joint modeling inputs
The sensitivity of all these inputs cannot be evaluated because some of them are interrelated inputs and some others are just used for FE modeling (Number of elements). There are also some inputs that have been assumed to be not so effective (e.g., Poisson ratio, equivalent boundary stiffness) from the design and analysis perspectives. Based on engineering judgment and pre-parametric sensitivity analysis, constant values were assigned to these inputs. The goal is to evaluate the sensitivity of those input parameters which are more important for analyzing and designing airfield concrete pavements have been evaluated. A detailed summary of ranges of the inputs to be varied as well as constant inputs are shown in Table 1. Each evaluated input was varied within its recommended range to study its effect on critical responses (maximum tensile stress and shear stress at top/bottom of the slab) while assigning base case values to all other input parameters. To consider the effects of load locations and different types of loading,
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Sensitivity Analysis
Initial Triage
Select Base Cases
Special Input Consideration
Identification of Inputs for Further Analysis
Varying Each Input
Local Sensitivity Analysis
Comparing Outputs by Baselines
Sensitivity Metrics
Evaluate and Rank the Input Data Fig. 2. Simplified flow chart for local sensitivity analysis of FEAFAA.
six general cases have been evaluated involving two loading case types. In FEAFAA, joints are modeled as continuous linear elastic shear springs connecting the slabs. The springs act only in the vertical direction, representing the action of dowel bars. To evaluate the sensitivity of stress responses to the joint properties, equivalent joint stiffness was used. The equivalent joint stiffness ranges are shown in Table 1. To determine the number of elements for mesh modeling, a pre-parametric sensitivity analysis was carried out. Fig. 3 shows the changes in stress responses with respect to varying number of elements for slabs (Fig. 3(a)) and foundation layers (Fig. 3(b)). As the number of elements increase, the stresses converge to a specific value. From this figure, it is obvious that for number of elements less than 30 the results are not constant but for number of elements more than 30 the results stay the same. However, increasing the number of elements can dramatically increase the run time of NIKE3D (i.e. about 16 min. for 30 elements and 50 min. for 50 elements). So, an optimum number of 30 elements was selected for meshing the slab and foundation without sacrificing the accuracy of the results and at the same time not too computationally intensive. Furthermore, By using 30 elements, it does not need to refine the mesh near the loading because the slabs’ mesh size (25.4 25.4 cm) is smaller than the tires’ foot-print (60.5 37.6 cm), so the FE model can promise accurate results. 3. Results To present the sensitivity of each parameter, a normalized sensitivity index (NSI) has been adopted as a quantitative metric.
NSI ¼
DY j X k DX k Y K
ð1Þ
where X k = Baseline value of input k, DX k = Change in input k about the base line, DY j = Change in output J corresponding to DX k , Y K = Baseline value of output J. As mentioned earlier, three load locations were considered for sensitivity analysis: interior, edge, and corner. For each location, analysis was carried out for two load types: mechanical loading only and simultaneous mechanical and thermal loading. Four stress types were considered as critical stresses for each load location and load type and used as outputs for the NSI calculation: Maximum tensile stress at the top of the slab (top tensile stress) Maximum tensile stress at the bottom of the slab (bottom tensile stress) Maximum shear stress at top of the slab (top shear stress) Maximum shear stress at bottom of the slab (bottom shear stress). The tensile stresses on top and bottom of the slab are in x-direction that is perpendicular to traffic. Shear stresses on top and bottom of the slab are in xy-direction that is more critical for crack opening. 3.1. Interior loading 3.1.1. Mechanical loading only Fig. 4(a) shows sensitivity of critical stresses to different inputs when the airplane is located at the center of the middle slab (interior loading). The Figure shows that the PCC thickness has the most influence on all responses at the top
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Inputs Category
Inputs
Range Min
Pavement structure inputs
PCC Slab
Cement Treated Base
Granular Subbase
Subgrade
Airplane inputs Loading inputs
Joint modeling inputs
Modulus, psi (GPa) Thickness, in. (cm) Poisson Ratio Modulus, psi (GPa) Thickness, in. (cm) Poisson Ratio Modulus, psi (GPa) Thickness, in. (cm) Poisson Ratio Modulus, psi (GPa) Poisson Ratio
Slab Dimension, ft. (m) Slab Number of Elements Number of Slabs Foundation Number of Elements Airplane parameters Loading Angle Loading position Equivalent Temperature Gradient, °F/in. (°C/cm) Thermal Coefficient, 1/°F (1/°C) Equivalent Joint Stiffness, psi/in (GPa/m)
Base case Baseline
Max
3 106 (20.7) 5 106 (34.5) 7 106 (48.3) 4 106 (27.6) 10 (25.4) 17 (43.2) 24 (61) 14 (35.6) 0.15 2.5 105 (1.7) 1.125 106 (7.8) 2 106 (13.8) 5 105 (3.5) 6 (15.2) 18 (45.7) 30 (76.2) 8 (20.3) 0.15 15,000 (1 101) 45,000 (3.1 101) 75,000 (5.2 101) 75,000 (5.2 101) 6 (15.2) 18 (45.7) 30 (76.2) 12 (30.5) 0.35 3,000 (2.1 102) 16,500 (1.1 101) 30,000 (2.1 101) 3,000 (2.1 102) 0.4 25 (7.6) 30 9 30 B777-300ER: Gross weight = 777,000 (lb), %GW = %95, No. Main Gears = 2, Wheels on Main Gears = 6, Tire Pressure (psi) = 221 0 Interior/Mid Slab Edge/Corner 2 (0.26) 2.5 (0.37) 3 (0.48) 2.3 (0.33) 4.1 106 (7.6 106) 1000 (2.7 101)
5.5 106 (9.9 106) 100000 (27.1)
7 106 (12.2 106) 300000 (81.3)
5 106 (9 106) 144,798 (39.2)
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Table 1 Ranges of inputs for sensitivity analysis of FEAFAA.
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415
400
300
350
250
Stress (psi)
Stress (psi)
300 200 150 100
250 200 150 100
50
50 0
0 0
10
20
30
40
50
0
60
10
Number of Elements
20
30
40
50
60
70
Number of Elements
(a)
(b)
Sensitivity Index
Fig. 3. Sensitivity of number of elements of (a) slabs (b) foundation layers to the stress responses in FEAFAA.
3.0 2.5 2.0 1.5 1.0 0.5 0.0 Slab Thickness
Subgrade Modulus
Base Thickness
PCC Modulus
Subbase Thickness
Base Modulus
Equivalent Joint Stiffness
Subbase Modulus
Top tensile stress
1.60
0.57
0.30
0.28
0.06
0.15
0.12
0.05
Bottom tensile stress
1.01
0.75
0.41
0.40
0.21
0.07
0.00
0.05
Top shear stress
1.23
0.61
0.57
0.33
0.30
0.07
0.05
0.06
Bottom shear stress
1.23
0.60
0.56
0.33
0.30
0.08
0.05
0.06
Sensitivity Index
(a) 3.0 2.5 2.0 1.5 1.0 0.5 0.0 Subgrade Modulus
Slab Thickness
Base Thickness
Thermal Coefficient
Temperature Gradient
PCC Modulus
Subbase Thickness
Base Modulus
Subbase Modulus
Equivalent Joint Stiffness
Top tensile stress
0.12
0.39
0.28
0.64
0.55
0.41
0.28
0.11
0.08
0.01
Bottom tensile stress
2.22
0.90
0.93
0.19
0.25
0.42
0.41
0.09
0.08
0.00
Top shear stress
0.78
1.08
0.42
0.24
0.25
0.25
0.32
0.13
0.07
0.04
Bottom shear stress
0.77
1.09
0.42
0.23
0.24
0.24
0.32
0.13
0.06
0.04
(b) Fig. 4. NSI of different input values vs. critical responses for (a) interior-mechanical loading and (b) interior-mechanical loading with temperature loading.
and bottom of the slab (i.e., top tensile stress, bottom tensile stress, top shear stress, and bottom shear stress). This means that a change in slab thickness results in a significant change in these stress responses.
After the PCC thickness, subgrade modulus has the next highest influence on the stress response. Sensitivity indices of base thickness and PCC modulus show that all stress responses are also sensitive to changes in these input vari-
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ables. The inputs which the responses are insensitive or ‘very low sensitive’ to are base modulus, equivalent joint stiffness, and subbase modulus. Most effective inputs, except slab thickness, have higher NSI for the bottom tensile stress than for the top tensile stress. In addition, NSI of all inputs for top shear stress are closer to those for bottom shear stress. 3.1.2. Simultaneous mechanical and thermal loading Fig. 4(b) depicts the sensitivity analysis results for the case in which interior mechanical loading along with temperature loading were applied to the modeled pavement structure. NIKE3D-FAA as implemented in FEAFAA considers both temperature and concrete shrinkage effects in the modeling for concrete slab curling analysis by introducing the concept of equivalent temperature gradient (ETG) [10]. Because of temperature gradient and shrinkage effects, the initial concrete slab shape affects the responses of the PCC slab to aircraft gear loading. The use of a simple compensation term applied to ETG can change the initial slab shape from circular to catenary, thereby significantly improving the computational results [10]. All analyses related to the temperature loading cases were conducted using catenary shape of slab’s curling. As shown in Fig. 4(b), the bottom tensile stresses are more sensitive to subgrade modulus than the other inputs. The higher sensitivity index of subgrade modulus to tensile stress at the bottom of the slab shows the importance of this input for studying bottom-up cracking in concrete pavement. The figure shows that the bottom tensile stress is also sensitive to base thickness and PCC slab thickness followed by PCC modulus, subbase thickness, ETG, and temperature coefficient. Variations in thermal coefficients and ETG have the most impact on tensile stresses at the top of the slab. Higher sensitivity to temperature loading related inputs for top tensile stresses indicate the importance of curling analysis for top-down cracking model development. Other inputs that strongly effect top tensile stresses include PCC modulus, PCC thickness, and subbase thickness. Unlike the bottom tensile stresses, top tensile stresses show lower sensitivity to the subgrade modulus. Fig. 4(b) indicates that the shear stress is most sensitive to slab thickness and subgrade modulus followed by base thickness, subbase thickness, temperature loading related inputs (ETG and thermal coefficient), and PCC modulus. Fig. 4(b) also shows that changes in base and subbase modulus and equivalent joint stiffness have the lowest effect on all stress responses. 3.2. Edge loading For rigid pavement design, the FAA uses the maximum tensile stress at the bottom edge of the PCC slab as a predictor of pavement structural life. The maximum tensile stress for design is determined using an edge loading condition. Fig. 5 shows sensitivities computed for edge loading
only as well as edge loading in combination with temperature loading. 3.2.1. Mechanical Loading Only Fig. 5(a) displays the sensitivity analysis results for different inputs when the load is centered on one edge of the middle slab. PCC slab thickness has been identified as the most effective input for top and bottom tensile stresses in this case. Top tensile stresses, unlike the bottom tensile stresses, exhibit significant sensitivity to the base and subbase thickness. Variations in modulus of PCC, base and subbase show less sensitivity index for bottom tensile stresses. However, alteration of subgrade modulus has considerable effect on all stress responses. The top tensile stresses exhibit considerable sensitivity to most inputs, but the bottom tensile stresses have considerable sensitivity to just two inputs (PCC thickness and subgrade modulus). It is noted in the Fig. 5 that the inputs having the greatest effect on shear stresses are thickness of PCC, base and subbase; subgrade modulus; and PCC modulus. Similar to previous cases, the stress responses are not sensitive to subbase/base modulus and equivalent joint stiffness (lowest NSI). 3.2.2. Simultaneous mechanical and thermal loading case Analysis results for this case are presented in Fig. 5(b). It can be seen in Fig. 5 (b) that top and bottom tensile stresses exhibit higher sensitivity to PCC slab thickness than other inputs. Temperature loading related inputs (ETG and thermal coefficient) also exhibit sensitivity for both tensile stresses. Most notably, higher NSI values were observed for these inputs for bottom tensile stresses than for top tensile stresses. Similarly, bottom tensile stresses show higher sensitivity to subgrade modulus, base thickness and subbase thickness than top tensile stresses. It can also be noted that top tensile stresses is not sensitive to the subgrade modulus input, while top tensile stresses have higher sensitivity, than bottom tensile stresses, to the PCC modulus . The ETG, thermal coefficient, PCC slab thickness and modulus, subgrade modulus, base thickness, and subbase thickness are all effective inputs for both top and bottom shear stresses. 3.3. Corner loading In this case, sensitivity analysis was conducted for the B777-300 ER gear located at the corner of the slab and. Corner loading is important for analysis and design purposes because of major observed failures that have occurred at this location and load transfer issues arising from the joints. 3.3.1. Mechanical Loading Only Case Fig. 6(a) illustrates how critical stress responses are sensitive to different inputs. As displayed in this Figure, PCC slab thickness exhibited the highest NSI for top and bot-
Sensitivity Index
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3.0 2.5 2.0 1.5 1.0 0.5 0.0
417
Slab Thickness
Base Thickness
Subbase Thickness
Subgrade Modulus
PCC Modulus
Subbase Modulus
Base Modulus
Equivalent Joint Stiffness
Top tensile stress
1.62
1.00
0.73
0.51
0.29
0.15
0.07
0.09
Bottom tensile stress
1.80
0.19
0.07
0.60
0.14
0.02
0.04
0.00
Top shear stress
0.74
0.73
0.33
0.70
0.50
0.10
0.13
0.05
Bottom shear stress
0.72
0.72
0.35
0.71
0.51
0.11
0.14
0.05
Sensitivity Index
(a) 3.0 2.5 2.0 1.5 1.0 0.5 0.0 Slab Thickness
Temperature Gradient
Thermal Coefficient
Subgrade Modulus
Base Thickness
PCC Modulus
Subbase Thickness
Equivalent Joint Stiffness
Base Modulus
Subbase Modulus
Top tensile stress
0.80
0.41
0.44
0.02
0.16
0.36
0.15
0.07
0.02
0.03
Bottom tensile stress
2.54
1.24
0.98
0.98
0.70
0.13
0.34
0.06
0.08
0.07
Top shear stress
0.32
0.49
0.50
0.22
0.29
0.57
0.19
0.13
0.04
0.04
Bottom shear stress
0.33
0.49
0.51
0.22
0.29
0.56
0.20
0.12
0.04
0.05
(b) Fig. 5. NSI of different input values vs. critical responses for (a) edge mechanical loading and (b) edge mechanical loading with temperature loading.
tom tensile stresses. Base and subbase thicknesses have high NSI for top tensile stresses but low NSI for bottom tensile stresses, while subgrade modulus and PCC modulus are effective inputs for both types of tensile stresses. It is also illustrated that shear stresses are sensitive to subgrade modulus, PCC modulus, PCC slab thickness, base thickness and subbase thickness. Moreover, base modulus, subbase modulus, and equivalent joint stiffness have the lowest NSI for all stress responses. 3.3.2. Simultaneous mechanical and thermal loading case Fig. 6(b) shows that PCC slab thickness, among all other inputs, has the highest NSI for top and bottom tensile stresses. Especially, higher NSI values of PCC slab thickness were observed for bottom tensile stresses than for top tensile stresses. Thermal coefficient is the second effective input for bottom tensile stresses. Other inputs which the bottom tensile stresses are sensitive to are base thickness, subbase thickness, and subgrade modulus. Top tensile stresses have low sensitivity to PCC modulus and ETG.
PCC slab thickness and modulus and thermal loading related inputs are the most effective inputs for top tensile stresses. The NSIs of these inputs (i.e., about 0.5) are close to one another. Base thickness and subbase thickness form the next lower tier of effective inputs. While, top tensile stresses have low sensitivity to subgrade modulus. The shear stresses are sensitive to PCC slab thickness, thermal loading related inputs, PCC modulus followed by base thickness, subgrade modulus, and subbase thickness. Similar to other cases, stress responses are not much sensitive to the equivalent joint stiffness and subbase/base modulus (lowest NSI). 4. Discussions of sensitivity results Table 2 summarizes the maximum NSI of each input among all loading conditions for discussion. As presented in this table, the inputs were ranked separately for each stress response. Note that shear stress in Table 2 indicates either top or bottom shear stress since both have almost similar sensitivity levels to all inputs.
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Sensitivity Index
418
3.0 2.5 2.0 1.5 1.0 0.5 0.0
Top tensile stress Bottom tensile stress Top shear stress Bottom shear stress
Slab Thickness
Base Thickness
1.28 1.51 0.58 0.50
1.03 0.19 0.66 0.63
Subbase Thickness 0.76 0.05 0.33 0.36
Subgrade Modulus 0.53 0.60 0.68 0.72
PCC Modulus
Base Modulus
0.34 0.23 0.47 0.46
0.08 0.04 0.15 0.15
Subbase Modulus 0.14 0.02 0.08 0.12
Equivalent Joint Stiffness 0.09 0.01 0.03 0.04
Sensitivity Index
(a) 3.0 2.5 2.0 1.5 1.0 0.5 0.0 Slab Thickness
Thermal Coefficient
PCC Modulus
Base Thickness
Temperature Gradient
Subgrade Modulus
Subbase Thickness
Equivalent Joint Stiffness
Base Modulus
Subbase Modulus
0.59
0.43
0.44
0.17
0.41
0.05
0.14
0.07
0.03
0.03
Bottom tensile stress
2.12
0.64
0.03
0.46
0.07
0.34
0.32
0.11
0.10
0.09
Top shear stress
0.45
0.48
0.54
0.27
0.44
0.19
0.16
0.17
0.03
0.03
Bottom shear stress
0.66
0.38
0.40
0.34
0.32
0.25
0.21
0.14
0.05
0.05
Top tensile stress
(b) Fig. 6. NSI of different input values vs. critical responses for (a) corner-mechanical loading and (b) corner-mechanical loading with temperature loading.
This table shows that all stress responses has the highest sensitivity to PCC slab thickness. For the top tensile stress, the thickness of pavement structural layers are the most effective inputs. It is noteworthy that subgrade modulus has a higher effect on bottom tensile stresses and shear stresses changes. Top tensile stresses are more sensitive to PCC thermal coefficient variations while bottom tensile stresses are more sensitive to the thermal gradient changes. Both have almost similar NSIs for shear stresses. The inputs categorized as insensitive for all stress responses are equivalent joint stiffness, base modulus, and subbase modulus. 5. Conclusions The primary objective of this study was to quantify sensitivity of stress responses to various inputs required in NIKE3D for critical stress outputs at different loading locations and load case scenarios for a B777-300ER aircraft. A four-layered pavement structure (PCC surface, cement treated base, granular subbase, and subgrade) in 9 slabs was modeled to represent typical and realistic airport pavement structure. The OAT SA was implemented
using a baseline limit normalized sensitivity index (NSI) to provide quantitative sensitivity information on each stress response output for different loading conditions. The major conclusions of this study are as follows: All stress responses are most sensitive to PCC slab thickness, followed by base and subbase thicknesses for top tensile stress, and subgrade modulus for bottom tensile stresses and shear stresses. Top tensile stress is more sensitive to thermal coefficient variation than ETG variation while bottom tensile stress shows higher sensitivity to ETG variation than thermal coefficient. The inputs categorized as insensitive for all stress responses under different loading conditions are equivalent joint stiffness, base modulus, and subbase modulus. In the mechanical loading only case under interior loading condition, PCC thickness and subgrade modulus are the most effective input parameters for all stress responses. In the simultaneous mechanical and thermal loading case under interior loading condition, changes in thermal coefficient and ETG, in addition to PCC slab thick-
NSI Shear stress
1.23 0.78 0.73 0.57 0.50 0.49 0.33 0.17 0.15 0.10
Inputs
Slab Thickness Subgrade Modulus Base Thickness PCC Modulus Thermal Coefficient Equivalent Temperature Gradient Subbase Thickness Equivalent Joint Stiffness Base Modulus Subbase Modulus
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NSI Bottom tensile stress
2.54 2.22 1.24 0.98 0.93 0.42 0.41 0.11 0.10 0.09
Inputs
Slab Thickness Subgrade Modulus Equivalent Temperature Gradient Thermal Coefficient Base Thickness PCC Modulus Subbase Thickness Equivalent Joint Stiffness Base Modulus Subbase Modulus
NSI Top tensile stress
1.60 1.03 0.76 0.64 0.57 0.55 0.44 0.15 0.15 0.12 Slab Thickness Base Thickness Subbase Thickness Thermal Coefficient Subgrade Modulus Equivalent Temperature Gradient PCC Modulus Base Modulus Subbase Modulus Equivalent Joint Stiffness
Inputs
Table 2 Inputs ranking for stress responses.
419
ness, have the most impact on top tensile stresses. Bottom tensile stresses are more sensitive to subgrade modulus than other inputs, while top tensile stresses exhibited low sensitivity to subgrade modulus. In the mechanical loading only case under edge loading condition, the top tensile stresses exhibit considerable sensitivity for most inputs (except inputs categorized as insensitive) but the bottom tensile stresses have considerable sensitivity just for two inputs (PCC thickness and subgrade modulus). In the simultaneous mechanical and thermal loading case under edge loading condition, top and bottom tensile stresses are sensitive to temperature loading related inputs (ETG and thermal coefficient) in addition to PCC slab thickness. Especially, higher NSIs of these inputs for the bottom tensile stresses were observed than for the top tensile stress. Top tensile stress does not have sensitivity to subgrade modulus. In the mechanical loading only case under corner loading condition, subgrade modulus and PCC modulus variation have effect on top and bottom tensile stresses. In the simultaneous mechanical and thermal loading case under corner loading condition, PCC slab thickness, among all other inputs, has the highest effect on top and bottom tensile stresses followed by PCC modulus and thermal coefficient for top tensile stresses and thermal coefficient for the bottom tensile stresses.
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