U.S. Department of Transportation
Federal Highway Administration
1200 New Jersey Avenue, SE
Washington, DC 20590
2023664000
Federal Highway Administration Research and Technology
Coordinating, Developing, and Delivering Highway Transportation Innovations
REPORT 
This report is an archived publication and may contain dated technical, contact, and link information 
Publication Number: FHWAHRT13066 Date: August 2013 
Publication Number: FHWAHRT13066 Date: August 2013 
PDF files can be viewed with the Acrobat® Reader®
The use of geosynthetic reinforced soil (GRS) for load bearing applications such as bridge abutments and integrated bridge systems (IBS) has expanded among transportation agencies looking to save time and money while delivering a better and safe product to the traveling public. GRS has been identified by the Federal Highway Administration (FHWA) as a proven, marketready technology, and is being actively promoted through its Every Day Counts (EDC) initiative. FHWA interim design guidance for GRS abutments and IBSs is presented in Publication No. FHWAHRT11026. The guidance includes the procedure and use of the GRS performance tests, also termed a minipier experiment. This report presents a database of nineteen performance tests performed by the FHWA, largely at the TurnerFairbank Highway Research Center. It also presents findings, conclusions, and suggestions regarding various design parameters related to the performance of GRS, such as backfill material, reinforcement strength, reinforcement spacing, facing confinement, secondary reinforcement, and compaction.
A reliability analysis for load and resistance factor design (LRFD) was performed based on the results of this performance testing to determine a calibrated resistance factor for the soilgeosynthetic capacity equation. The results of this analysis can also be used by bridge designers to estimate capacity and deformation of GRS. In addition, an insight into the behavior of GRS as a new composite material due to the close reinforcement spacing is described.
Jorge E. PagánOrtiz
Director, Office of Infrastructure
Research and Development
Notice
This document is disseminated under the sponsorship of the U.S. Department of Transportation in the interest of information exchange. The U.S. Government assumes no liability for the use of the information contained in this document. This report does not constitute a standard, specification, or regulation.
The U.S. Government does not endorse products or manufacturers. Trademarks or manufacturers' names appear in this report only because they are considered essential to the objective of the document.
Quality Assurance Statement
The Federal Highway Administration (FHWA) provides highquality information to serve Government, industry, and the public in a manner that promotes public understanding. Standards and policies are used to ensure and maximize the quality, objectivity, utility, and integrity of its information. FHWA periodically reviews quality issues and adjusts its programs and processes to ensure continuous quality improvement.
Technical Report Documentation Page
1. Report No.
FHWAHRT13066 
2. Government Accession No.  3 Recipient's Catalog No.  
4. Title and Subtitle
Geosynthetic Reinforced Soil Performance TestingAxial Load 
5. Report Date August 2013 

6. Performing Organization Code  
7. Author(s)
Nicks, J.E., Adams, M.T., Ooi, P.S.K., Stabile, T. 
8. Performing Organization Report No.


9. Performing Organization Name and Address TurnerFairbank Highway Research Center 
10. Work Unit No. 

11. Contract or Grant No. N/A 

12. Sponsoring Agency Name and Address
Office of Infrastructure R&D 
13. Type of Report and Period Covered
Technical 

14. Sponsoring Agency Code


15. Supplementary Notes The FHWA Contracting Officer's Technical Representative (COTR) was Mike Adams, HRDI40. 

16. Abstract
The geosynthetic reinforced soil (GRS) performance test (PT), also called a minipier experiment, consists of constructing alternating layers of compacted granular fill and geosynthetic reinforcement with a facing element that is frictionally connected, then axially loading the GRS mass while measuring deformation to monitor performance. This large element load test provides material strength properties of a particular GRS composite built with unique combinations of reinforcement, compacted fill, and facing elements. This report describes the procedure and provides axial loaddeformation results for a series of PTs conducted in both Defiance County, OH, as part of the Federal Highway Administration's (FHWA) Every Day Counts (EDC) GRS Validation Sessions and in McLean, VA, at the FHWA's TurnerFairbank Highway Research Center as part of a parametric study. The primary objectives of this research report are to: (1) build a database of GRS material properties that can be used by designers for GRS abutments and integrated bridge systems; (2) evaluate the relationship between reinforcement strength and spacing; (3) quantify the contribution of the frictionally connected facing elements at the service limit and strength limit states; (4) assess the new internal stability design method proposed by Adams et al. 2011 for GRS; and (5) perform a reliability analysis of the proposed soilgeosynthetic capacity equation for LRFD calibration. ^{(1,11)} 

17. Key Words
Geosynthetic reinforced soil, performance test, minipier experiment, abutment, integrated bridge system, geotextile, capacity, deformation 
18. Distribution Statement


19. Security Classification Unclassified 
20. Security Classification Unclassified 
21. No. of Pages 169 
22. Price 
Form DOT F 1700.7 (872)  Reproduction of completed page authorized 
SI* (Modern Metric) Conversion Factors
Appendix B. Nuclear Density Testing for TFHRS PTS
Appendix C. Deformation Instrumentation Layouts for PTS
Figure 1  Photo. Vegas minipier experiment 
Figure 2.  Illustration. Plan view of Vegas minipier experiment 
Figure 3.  Illustration. Face view of Vegas minipier experiment 
Figure 4.  Illustration. Side view of Vegas minipier experiment 
Figure 5.  Illustration. Reinforcement schedule for Vegas minipier experiment 
Figure 6.  Illustration. Plan view of Defiance County experiment 
Figure 7.  Illustration. Elevation view of Defiance County (DC) test 
Figure 8.  Equation. Ultimate vertical capacity for a GRS composite 
Figure 9.  Equation. W factor 
Figure 10.  Equation. Required reinforcement strength 
Figure 11.  Equation. Confining stress (Wu et al. 2010) 
Figure 12.  Graph. Predictive capability of the soilgeosynthetic composite capacity equation 
Figure 13.  Graph. Predictive capability of the required reinforcement strength equation 
Figure 14.  Graph. Reinforced backfill gradations 
Figure 15.  Graph. LSDS testing results 
Figure 16.  Photo. DC1 GRS PT (before testing) 
Figure 17.  Illustration. Concrete footing on GRS composite, inset from facing 
Figure 18.  Photo. Hollow core hydraulic jacks for PT assembly 
Figure 19.  Photo. TF1 PT setup with reaction frame 
Figure 20.  Photo. TF6 PT setup with reaction frame 
Figure 21.  Photo. TF10 PT setup with reaction frame 
Figure 22.  Photo. TF9 at failure with reaction frame 
Figure 23  Photo. Spherical bearing to apply load to the footing on the GRS composite 
Figure 24.  Illustration. Instrumentation layout for DC tests and TF1 
Figure 25.  Illustration. General additional instrumentation layout TF PT series 
Figure 26.  Graph. Loaddeformation behavior for the Defiance County PTs 
Figure 27.  Graph. Loaddeformation behavior for the Turner Fairbank PTs 
Figure 28.  Photo. Tilting of the footing during TF4 testing 
Figure 29.  Graph. TF4 results 
Figure 30.  Graph. TF1 results 
Figure 31.  Graph. TF5 results 
Figure 32.  Graph. TF6 results 
Figure 33.  Graph. Repeatability of PT at TFHRC 
Figure 34.  Photo. TF11 at failure with S_{v} = 313/16 inches, T_{f} = 1,400 lb/ft, and 21A material 
Figure 35.  Photo. TF3 at failure with S_{v} = 7⅝ inches, T_{f} = 2,400 lb/ft, and 21A material 
Figure 36.  Photo. TF13 at failure with S_{v} = 11¼ inches, T_{f} = 3,600 lb/ft, and 21A material 
Figure 37.  Photo. TF10 at failure with S_{v} = 15¼ inches, T_{f} = 4,800 lb/ft, and 21A material 
Figure 38.  Photo. Rupture pattern for geotextiles in TF6 (q_{ult,emp} = 43,828 psf); the lowest layer of reinforcement is the closet fabric in the picture 
Figure 39.  Photo. Posttest picture of TF6 (S_{v} = 7⅝ inches, T_{f} = 4,800 lb/ft) 
Figure 40.  Equation. MohrCoulomb shear strength 
Figure 41.  Equation. Ultimate capacity of an unconfined GRS PT 
Figure 42.  Equation. Ultimate capacity of a strip footing on slope 
Figure 43.  Equation. Ultimate capacity of a strip footing on a vertical GRS abutment 
Figure 44.  Equation. Ratio of plane strain capacity to PT capacity 
Figure 45.  Equation. Stability Factor 
Figure 46.  Graph. Ratio of plane strain capacity to PT capacity for different stability factors 
Figure 47.  Graph. MohrCoulomb failure envelope for Pham (2009) plane strain GSGC tests 
Figure 48.  Graph. Plane strain capacity to PT capacity for a stability factor of 0.29 
Figure 49.  Illustration. Infinitely Long Unconfined GRS abutment 
Figure 50.  Equation. Stiffness of an Infinitely Long Unconfined GRS abutment 
Figure 51.  Illustration. Solution for strip footing on top of a wall 
Figure 52.  Equation. Vertical displacement of a GRS abutment with a strip footing 
Figure 53.  Equation. Vertical strain 
Figure 54.  Equation. Stiffness of a GRS abutment supporting a strip footing 
Figure 55.  Equation. Vertical displacement of a GRS abutment with a strip footing. 
Figure 56.  Graph. Ratio of plane strain stiffness of a strip footing on top of a wall (S_{GRS}) to that of a PT (S_{PT}) for the case of constant stiffness with depth 
Figure 57.  Graph. Comparison between compacted and uncompacted GRS composites 
Figure 58.  Design service limit for uncompacted sample DC5 
Figure 59.  Graph. Comparison of compacted and uncompacted strains between the DC1 and DC5 tests 
Figure 60.  Graph. Effect of bearing bed reinforcement for TF7 and TF8 
Figure 61.  Graph. Measured lateral deformation at 3,600 psf applied stress for TF7 (no bearing bed reinforcement) and TF8 (2 courses of bearing bed reinforcement) 
Figure 62.  Graph. Measured lateral deformation at 26,600 psf applied stress for TF7 (no bearing bed reinforcement) and TF8 (2 courses of bearing bed reinforcement) 
Figure 63.  Graph. Comparison of opengraded and wellgraded backfills for TF1 and TF2 
Figure 64.  Graph. Stressstrain curves for PTs with CMUs at T_{f}/S_{v} = 3,800 psf 
Figure 65.  Graph. Stressstrain curves for PTs with no CMU facing at T_{f}/S_{v} = 3,800 psf 
Figure 66.  Graph. Capacity of GRS with no CMU facing at various reinforcement spacing for different T_{f}/S_{v} 
Figure 67.  Graph. Capacity of GRS with CMU facing at various reinforcement spacing for different T_{f}/S_{v} Ratios 
Figure 68.  Graph. Capacity of GRS with no CMU facing at various reinforcement strength for different T_{f}/S_{v} ratios 
Figure 69.  Graph. Capacity of GRS with CMU facing at various reinforcement strength for different T_{f}/S_{v} ratios 
Figure 70.  Graph. Stressstrain response for TF2 (CMU facing) and TF3 (No CMU facing) with S_{v} = 7⅝ inches and T_{f} = 2,400 lb/ft 
Figure 71.  Stressstrain response for TF6 (CMU facing) and TF7 (No CMU facing) with S_{v} = 7⅝ inches and T_{f} = 4,800 lb/ft 
Figure 72.  Graph. Stressstrain response for TF9 (CMU facing) and TF10 (No CMU facing) with S_{v} = 15¼ inches and T_{f} = 4,800 lb/ft 
Figure 73.  Graph. Stressstrain Response for TF12 (CMU facing) and TF11 (No CMU facing) with S_{v} = 313/16 inches and T_{f} = 1,400 lb/ft 
Figure 74.  Graph. Stressstrain response for TF14 (CMU facing) and TF13 (No CMU facing) with S_{v} = 11¼ inches and T_{f} = 3,600 lb/ft 
Figure 75.  Graph. Effect of CMU facing on ultimate capacity as a function of reinforcement spacing 
Figure 76.  Graph. Effect of CMU facing on ultimate capacity as a function of reinforcement strength 
Figure 77.  Graph. Calculated confining pressure due to CMU facing at the ultimate capacity 
Figure 78.  Graph. Comparison of predicted capacity and measured capacity 
Figure 79.  Graph. Cumulative distribution function plot for DC and TF PTs 
Figure 80.  Graph. Cumulative distribution function plot for all GRS composite tests 
Figure 81.  Graph. Normalized applied stress versus strain for all PT 
Figure 82.  Graph. Normalized loaddeformation behavior for the DC and TF PTs up to 5 percent vertical strain 
Figure 83.  Graph. Cumulative distribution function for proposed service limit pressure 
Figure 84.  Graph. Loaddeformation behavior for the Turner Fairbank PTs at low strain levels 
Figure 85.  Graph. Normalized loaddeformation behavior for the DC and TF PTs up to 0.5 percent vertical strain 
Figure 86.  Graph. PTs strictly meeting FHWA GRS abutment design specifications 
Figure 87.  Equation. Limit state function for FOSM approach 
Figure 88.  Graph. Reliability index for lognormal R and Q 
Figure 89.  Equation. LRFD format 
Figure 90.  Equation. Resistance factor using FOSM 
Figure 91.  Equation. Coefficient of variation for factored load 
Figure 92.  Equation. Coefficient of variation for resistance 
Figure 93.  Graph. Resistance factor for footings on GRS composites for different dead to dead plus live load ratios and target reliability indices based on PT series 
Figure 94.  Graph. Resistance factor for footings on GRS composites for different dead to dead plus live load ratios and target reliability indices based on all testing to date 
Figure 95.  Graph. AASHTO No. 8 LSDS test results (DC tests) 
Figure 96.  AASHTO No. 8 LSDS deformation test results (DC tests) 
Figure 97.  Graph. AASHTO No. 8 pea gravel LSDS test results (DC tests) 
Figure 98.  Graph. AASHTO No. 8 pea gravel LSDS deformation test results (DC tests) 
Figure 99.  Graph. AASHTO No. 57 LSDS test results (DC tests) 
Figure 100.  Graph. AASHTO No. 57 LSDS deformation test results (DC tests) 
Figure 101.  Graph. AASHTO No. 9 LSDS test results (DC tests) 
Figure 102.  Graph. AASHTO No. 9 LSDS deformation test results (DC tests) 
Figure 103.  Graph. AASHTO No. 8 LSDS test results (TFHRC tests) 
Figure 104.  Graph. AASHTO No. 8 LSDS deformation test results (DC tests) 
Figure 105.  Graph. AASHTO A1a (VDOT 21A) LSDS test results (TFHRC tests) 
Figure 106.  Graph. AASHTO A1a (VDOT 21A) LSDS deformation test results (DC tests) 
Figure 107.  Illustration. Instrumentation layout for DC tests and TF1 
Figure 108.  Illustration. Instrumentation layout for TF2, TF9 
Figure 109.  Illustration. Instrumentation layout for TF3, TF4 
Figure 110.  Illustration. Instrumentation layout for TF5, TF7 
Figure 111.  Instrumentation layout for TF6, TF12 
Figure 112.  Illustration. Instrumentation layout for TF8 
Figure 113.  Illustration. Instrumentation layout for TF10 
Figure 114.  Illustration. Instrumentation layout for TF11 
Figure 115.  Illustration. Instrumentation layout for TF13 
Figure 116.  Illustration. Instrumentation layout for TF14 
Table 1.  Summary of PT conditions 
Table 2.  PT reinforced backfill gradations 
Table 3.  PT backfill gradation properties 
Table 4.  Maximum dry density for PT aggregates 
Table 5.  LSDS testing results 
Table 6.  Geosynthetic reinforcement properties 
Table 7.  PT dimensions 
Table 8.  PT measured results summary 
Table 9.  Parametric study on aggregate size 
Table 10.  Effect of aggregate type results 
Table 11.  Parametric study on compaction 
Table 12.  Parametric study on bearing bed reinforcement 
Table 13.  Parametric study on gradation (T_{f} = 2,400 lb/ft, S_{v} = 7⅝ inches) 
Table 14.  Parametric study on gradation (T_{f} = 4,800 lb/ft, S_{v} = 7⅝ inches) 
Table 15.  Parametric study on reinforcement strength with opengraded aggregates 
Table 16.  Parametric study on reinforcement strength with wellgraded aggregates 
Table 17.  Parametric study for 3,800 lb/ft^{2} T_{f}/S_{v} ratio (with facing) 
Table 18.  Parametric study for 3,800 lb/ft^{2} T_{f}/S_{v} ratio (with no facing) 
Table 19.  T_{f} /S_{v} ratios for each PT 
Table 20.  Effect of CMU facing on stiffness and capacity 
Table 21.  Effect of CMU facing on strain 
Table 22.  PTs meeting GRS strength and service limit design criteria 
Table 23.  Predicted and measured vertical capacity for DC and TF PTs 
Table 24.  Predicted and measured vertical capacity for all GRS composite tests 
Table 25.  Estimation of allowable dead load to limit vertical strain to 0.5 percent using the GRS capacity equation 
Table 26.  Statistics for dead and live loads 
Table 27.  AASHTO No. 8 sieve analysis (DC tests) 
Table 28.  AASHTO No. 8 pea gravel sieve analysis (DC tests) 
Table 29.  AASHTO No. 57 Sieve analysis (DC tests) 
Table 30.  AASHTO No. 9 Sieve analysis (DC tests) 
Table 31.  AASHTO No. 8 Sieve analysis (TFHRC tests) 
Table 32.  AASHTO A1a (VDOT 21A) sieve analysis (TFHRC tests) 
Table 33.  Summary of AASHTO No. 8 LSDS results (DC tests) 
Table 34.  Summary of AASHTO No. 8 pea gravel LSDS results (DC tests) 
Table 35.  Summary of AASHTO No. 57 LSDS results (DC tests) 
Table 36.  Summary of AASHTO No. 9 LSDS results (DC tests) 
Table 37.  Summary of AASHTO No. 8 LSDS results (TFHRC tests) 
Table 38.  Summary of AASHTO A1a (VDOT 21A) LSDS results (TFHRC tests) 
Table 39.  TF2 Nuclear density test results 
Table 40.  TF2 Nuclear density test results 
Table 41.  TF3 Nuclear density test results 
Table 42.  TF4 Nuclear density test results 
Table 43.  TF5 Nuclear density test results 
Table 44.  TF6 Nuclear density test results 
Table 45.  TF7 Nuclear density test results 
Table 46.  TF8 Nuclear density test results 
Table 47.  TF9 Nuclear density test results 
Table 48.  TF10 Nuclear density test results 
Table 49.  TF11 Nuclear density test results 
Table 50.  TF12 Nuclear density test results 
Table 51.  TF13 Nuclear density test results 
Table 52.  TF14 Nuclear density test results 
Table 53.  DC1 PT Data 
Table 54.  DC2 PT Data 
Table 55.  DC3 PT Data 
Table 56.  DC4 PT Data 
Table 57.  DC5 PT Data 
Table 58.  TF1 PT Data 
Table 59.  TF2 PT Data 
Table 60.  TF3 PT Data 
Table 61.  TF4 PT Data 
Table 62.  TF5 PT Data 
Table 63.  TF6 PT Data 
Table 64.  TF7 PT Data 
Table 65.  TF8 PT Data 
Table 66.  Â TF9 PT Data 
Table 67.  TF10 PT Data 
Table 68.  TF11 PT Data 
Table 69.  TF12 PT Data 
Table 70.  TF13 PT Data 
Table 71.  TF14 PT Data 
Abbreviations
AASHTO  American Association of State Highway and Transportation Officials 
CMU  Concrete masonry unit 
DC  Defiance County, OH 
EDC  Every Day Counts initiative 
FHWA  Federal Highway Administration 
GPGM  Poorly gradedsilty gravel 
GRS  Geosynthetic reinforced soil 
IBS  Integrated bridge systems 
LFD  Load factor design 
LRFD  Load and resistance factor design 
LSDS  Large scale direct shear 
LVDT  Linear voltage displacement transducers 
MARV  Minimum average roll value 
POT  Potentiometer 
PT  Performance test 
SRW  Segmental retaining wall 
TFHRC  TurnerFairbank Highway Research Center 
USCS  Unified Soil Classification System 
Symbols
β  Reliability index 
β_{s}  Slope angle 
β_{T}  Target reliability index 
γ  Unit weight of the backfill 
γ_{b}  Bulk unit weight of the facing block 
γ_{d}  Maximum dry density 
γ_{D}  Load factor for dead load 
γ_{L}  Load factor for live load 
γ_{GRS}  Unit weight of the GRS composite 
γ_{i}  Load factor for load component i 
δ  Interface friction angle between the geosynthetic and the facing element for a frictionally connected GRS composite 
ε_{@q=4000psf}  Measured vertical strain at an applied load of 4000 psf 
ε_{@qult}  Measured vertical strain at failure 
ε_{max}  Maximum recorded vertical strain 
ε_{v}  Vertical strain 
ε_{v,compact}  Vertical strain for a compacted GRS composite 
ε_{v,uncompact}  Vertical strain for an uncompacted GRS composite 
λ  Bias, ratio of measured to predicted 
λ_{D}  Bias factor for dead load 
λ_{L}  Bias factor for live load 
λ_{R}  Bias factor for resistance 
ν_{GRS}  Poisson’s ratio of the GRS 
ρ  Vertical displacement 
σ  Applied normal stress 
σ_{c}  External confining stress due to the facing 
σ_{h}  Total lateral stress within the GRS composite at a given depth and location 
τ  Shear strength of soil 
Φ  Peak friction angle 
Φ_{GRS}  Friction angle of the GRS composite 
Ф  Resistance factor 
Ф_{cap}  Resistance factor for capacity 
ω  Optimum moisture content 
a  Footing offset from the edge of the wall face (i.e., setback distance) 
b  Footing width on top of the GRS composite 
B  Base width of the GRS composite 
B_{total}  Total width of the PT with the CMU facing 
c  Cohesion of the backfill 
c_{GRS}  Cohesion of the GRS composite 
C_{c}  Coefficient of Curvature 
C_{u}  Coefficient of Uniformity 
d  Depth of the facing block unit perpendicular to the wall face 
d_{max}  Maximum aggregate size 
D_{10}  Aggregate size in which 10 percent of the sample is finer 
D_{30}  Aggregate size in which 30 percent of the sample is finer 
D_{60}  Aggregate size in which 60 percent of the sample is finer 
D_{85}  Aggregate size in which 85 percent of the sample is finer 
E_{o}  Initial stressstrain ratio 
E_{o,CMU}  Initial stressstrain ratio for tests with CMU facing 
E_{o,noCMU}  Initial stressstrain ratio for tests without any facing 
E_{GRS}  Young's modulus of the GRS composite 
E_{R}  Ratio of stress to strain for the reload cycle 
Mean safety margin 

H  Height of the GRS composite 
K_{ar}  Coefficient of active earth pressure for the backfill 
K_{pr}  Coefficient of passive earth pressure for the backfill 
L  Length of footing/bearing area 
N_{γq}  Bearing capacity factor 
N_{cq}  Bearing capacity factor 
N_{s}  Stability factor 
S_{GRS}  Plane strain stiffness of a strip footing on top of GRS 
S_{PT}  Stiffness of the unconfined GRS column 
S_{v}  Reinforcement spacing 
T_{f}  Wide width tensile strength of the geosynthetic, expressed as the minimum average roll value (MARV) 
T_{req,c}  Required reinforcement strength in the direction perpendicular to the wall face 
q  Applied stress 
q_{@ε=0.5%}  Applied stress at 0.5 percent vertical strain 
q_{@ε=0.5%,predicted}  Predicted applied stress at 0.5 percent vertical strain 
q_{@ε=0.5%}  Applied stress at 5 percent vertical strain 
q_{max}  Maximum applied pressure during testing 
q_{ult,an,c}  Ultimate capacity using semiempirical theory 
q_{ult,emp}  Measured failure pressure 
q_{ult,emp CMU}  Measured failure pressure for tests with CMU facing 
q_{ult,emp no CMU}  Measured failure pressure for tests without any facing 
q_{ult,PS}  Ultimate capacity of strip footing under plane strain conditions 
q_{ult,PT}  Ultimate capacity of the GRS column 
Q  Load 
Q_{D}  Dead load 
Q_{L}  Live load 
Q_{i}  Load component i 
R  Resistance 
V_{dmax}  Coefficient of variation of the maximum aggregate size 
V_{D}  Coefficient of variation of the dead load 
V_{Kp}  Coefficient of variation of the coefficient for passive earth pressure 
V_{L}  Coefficient of variation of the live load 
V_{M}  Coefficient of variation of the model 
V_{Q}  Coefficient of variation of the loads 
V_{R}  Coefficient of variation of the resistance 
V_{Tf}  Coefficient of variation of the reinforcement strength 
W  Factor accounting for the effect of reinforcement spacing and aggregate size 
z  Standard normal variable 