21 Abdallah I. Husein Malkawi, PhD 22 President - Professor 23 Jordan University of Science & Technology 24 e-mail: [email protected] 25 Tel: +962-2-7201000 Ext. 21111 26
27 28
29 30 31
32 33 34
Number of words: 7750 35 Number of words in Abstract: 255 36 37 38
Keywords: Geogrid reinforced walls, Limit Equilibrium Analyses, equivalent soil cohesion, 39 DIN 4084, utilization factors of retaining structures. 40
41 42
August, 2012 43 44
45 (1) Corresponding Author. 46
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DESIGN OF GEOGRID REINFORCED EARTH WALLS: THE APPARENT 47 COHESION AND TRANSITION OF LIMITS AND CRITICAL SURFACES 48
49 Izzaldin M. Almoh’d (Ayasrah) , Ahmed Ashteyat and Abdallah I. H.Malkawi 50
51
ABSTARCT: 52
The majority of design approaches or methodologies for reinforced soil walls or slopes are 53 based on separately investigating the internal and external stabilities of the system. The internal 54 stability is examined by satisfying the local stability of reinforcements at each level based on 55 the predetermined critical slip plane (line of maximums) and the tributary area of each 56 reinforcing layer. Recent research aimed at incorporating the contributions of the various 57 elements of reinforced earth walls, some of which are mostly based on statistical correlations. 58 The German code of practice for design/analyses of reinforced earth walls and slopes offers 59 slightly different methodology for analyzing the internal stability of the reinforcement. It is 60 mainly based on investigating numerous circular and random slip surfaces, within and beyond 61 the reinforcement zone (internal and external), while accommodating the axial (resistance) 62 forces provided by all reinforcement layers intercepting these surfaces. 63
This paper presents some of the technical and design considerations and possible improvements 64 on design methodology for reinforced soil walls and slopes. Of particular interest is the use of 65 apparent cohesion concept in design of geosynthetic reinforced soil systems and the transition 66 of limit equilibrium states (mobilization of actual state of equilibrium critical surfaces) for 67 reinforced earth walls. The equivalent cohesion concept was used to transform reinforced soil 68 masses into equivalent cohesive soil masses with friction capacity. Cases of analyses with 69 comparisons between reinforced soil walls and the equivalent cohesive masses were performed 70 and the results revealed very similar results between the two systems in terms of the safety of 71 the walls. 72
73
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INTRODUCTION 74
Since the development of the modern concept of soil reinforcement (1), worldwide research 75 and demonstration projects on soil reinforcement have continuously evolved under the 76 sponsorship of several agencies, such as: The U.S. Department of Transportation (2), United 77 Kingdom Transportation and Road Research Laboratory (TRRL) (3), as well as various leading 78 agencies and laboratories in France (4). 79 Currently, there are many methods (5 to 19) that can be used for the analysis and design of 80 reinforced earth walls and slopes. In the United States, the design/analysis of reinforced earth 81 walls is mostly based on the working stress method, limit equilibrium analyses, Load Factor 82 Design (LFD) method, or the recently developed Load and Resistance Factor Design (LRFD) 83 method. The concept of working stress has been widely described and recognized, due to the 84 involvement of a equilibrium states and equations. It’s mainly based on the stress-deformation 85 behavior of the reinforced soil mass, which is assumed to be coherent, under the external and 86 internal applied stresses. These stresses should not exceed the anticipated allowable strength of 87 the mass. Although the existing design methods may provide conservative design results, they 88 have failed to clearly demonstrate the inherent advantages of the optimum reinforcement 89 distributions upon which the design is based. This could be attributed to the simplifying 90 assumptions and, in some instances, the predetermined and fixed critical surface as well as the 91 predetermined reinforcement layout (lengths and spacing). The resulting conservative designs 92 are still acceptable and practical considering the relatively low cost of the reinforcement 93 materials. However, certain conditions may require deviations from the normal and simple 94 design methodology to verify the safety of a reinforced earth wall with unusually distributions 95 or lengths that could be less than the minimum lengths specified by many standards (typically 96 ranging from 0.6H to 0.7H). 97
98 In this paper, the “Equivalent Cohesion” concept will be interpreted to be used for analyses of 99 walls and slopes. The apparent cohesion will be used as a method of transforming the 100 reinforced soil mass into a homogenous soil mass. Additionally, the transition of the limiting 101 line of equilibrium and critical surface will be demonstrated in this context. 102
GERMAN DIN 4084 NORM 103
The current German design norm DIN 4084 (2005) and its supplement (Terrain Rupture and 104 Slope Rupture) issued by the German “Duetsches Institute fur Normug” is based on the partial 105 safety concept according to the European regulations for different load cases. It utilizes the 106 limit equilibrium analyses for internal stability, external stability and the compound (internal 107 and external) stability. Driving forces are factored up (amplified) and resisting forces are 108 factored down (reduced) in order to achieve a defined margin of safety depending on the 109 individual load case. The load cases (LC) are described as follow: 110
- Load Case 1: Permanent or common situations 111 - Load Case 2: Seldom and temporary situation (e.g. construction stages) 112 - Load Case 3: Extraordinary and rare situation (e.g. earthquake) 113
114 The partial factors vary for each load case depending on the likelihood of their occurrence. The 115 partial factors applied to the soil properties and boundary loads, for each load case, are 116 summarized in Table 1. For the geosynthetic reinforcements in permanent or long term 117
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structures, the maximum allowable design strength (Fd) of reinforcement is evaluated based on 118 ultimate (manufacturer’s) tensile strength (Fk) according to the following equation: 119 120
Fd = Fk / (A1 x A2 x A3 x A4 x B) 121 122 Where, (A1) is a reduction factor for creep; (A2) is installation damage factor; (A3 ) is the 123 connections/seams reduction factor; (A4) is the durability reduction factor; and (B ) is a global 124 factor of safety for design. The factors (A1 through A4) are dependent on the material type, 125 design life and type of backfill. They may vary depending on the type, geometry and strength 126 of the geosynethitic. The global factor (B) depends on the assigned load case and is equal to 127 1.4, 1.3 and 1.2 for LC1, LC2 and LC3, respectively. The German and European standards 128 allow for a design life up to 120 years for geosynthetic reinforced soil structures. 129 130 After all partial factors are applied, the design/analyses is performed based on the Limit 131 Equilibrium Analyses methods for internal stability, external stability and the compound 132 stability of an assumed reinforcement scheme for steep slopes or walls. According to the new 133 DIN 4084, different methods and theories such as Bishop’s and Krey’s sliding circles (20, 21, 134 DIN 4084) or block sliding and non-circular surfaces by Janbu (21, DIN 4084) can be used. A 135 common practice is to perform the following stability checks: 136
- Internal and compound stability with Bishop 137 - Internal and compound stability with the block sliding method 138 - External stability with Bishop (Deep seated slip circle) 139
140 For each analyses, the outcome is provided in the form of degree of utilization of the structure. 141 The degree of utilization is an expression of the ratio of the total factored loads to the total 142 reduced resistances produced by both soil and reinforcement. Accordingly, for a safe design for 143 any load case, the utilization factor should not exceed 1.0. 144
APPARENT COHESION CONCEPT 145
The apparent cohesion concept was discussed and described in both theory and laboratory 146 measurements by many researchers Schlosser and Long (1972) and Yang and Singh (1974). 147 Based on this concept (Figure 1), the reinforcement produces an apparent cohesion due to its 148 tensile capacity. This concept may be more applicable for geogrid reinforcements due to their 149 extensibility, strain compatibility with soils, as well as the coverage ratio of the geogrids. 150 However, only the mobilized tensile forces in the reinforcement are responsible for the 151 apparent cohesion. Assuming a layer of cohesionless soil having a thickness (S), with a layer of 152 geogrid reinforcement, spread over the entire width of the soil, having a mobilized tensile force 153 per meter width of geogrid (T), the equivalent (apparent) cohesion (Ce) may be expressed as: 154 155
Ceq = T x Rc/S (1) 156 Where Rc is the coverage ratio of the geogrid reinforcement (in decimals). The equivalent 157 cohesion shall produce about the same contributions to the stability of a given slip surface to 158 that of the reinforcement’s mobilized force (T). The mobilized force (T) is a function of the 159 soil-reinforcement interface strength, level of stress (embedment) and the mobilization length 160 (Lo) of the reinforcement as follows: 161
oDc LfTR )tan( (2) 162
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f is an interface friction reduction factor. The mobilized tensile force per meter width of geogrid 163 shall not exceed the tensile strength of the reinforcement (TD), whether short-term or long-term, 164 depending on the application. Accordingly, for any depth, the minimum length (Lo) required for 165 full strength mobilization is interpreted as: 166 167
)tan( fTRL Dco (3) 168
For the purpose of geogrid reinforced soil walls, the equivalent cohesion (Ce) is likely to vary 169 with depth and location along the grid. Additionally, the horizontal force (Fhi) to be provided 170 by a reinforcement layer (i), can be estimated based on the tributary area of the reinforcement 171 (=S x 1.0) is calculated as: 172 SKaqszSKaqsFhi )()( (4) 173
Where, is the unit weight of the soil, Ka is the active lateral earth pressure coefficient, and qs 174 is the equivalent surcharge. With Fhi < T, the required equivalent cohesion (Ceqi) at a given 175 reinforcement layer (i), can be expressed as: 176
KaqszCeqi )( (5) 177 178 Also, by comparing Eqns. 2 and 4, yields, 179
)tan(
)(
)()tan(
f
SKaqsL
SKaqsLf
o
o
180
181 Which is further reduced to: 182
)tan(
1
f
SKaLo ; = qs/ 6a) 183
184 Assuming no surcharge (qs) is present, eqn. (5a) can be reduced to: 185 186
tan
f
SKaLo ; qs=0.0 6b) 187
Equations (6a and 6b) show that the level of stress effects on the mobilization length of the 188 reinforcement is not significant and that the mobilization length is approximately the same for 189 all reinforcement layers. For typical ranges of (30 to 36 degrees), the mobilization length will 190 range from 0.16 to a maximum of 0.6m. However, this length is adequate to mobilize resistance 191 that is equal to the active horizontal loads. Additionally, the effects of the level of stress on the 192 angle of internal friction and their possible effects on the friction reduction factor are also not 193 accommodated. 194 195 Equation (1) can be generalized for zones within the reinforced earth walls to transform the 196 reinforced cohesionless mass into an equivalent cohesive mass or masses. The equivalent 197 cohesion (Ceq) within a height (h) containing (n) number of geogrids having a long-term 198 strength of Td is given as: 199
h
Tn
h
TRCeq DDc (7a) 200
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For uniformly spaced geogrid reinforcements, this equation can be simplified as: 201
S
TRCeq Dc (7b) 202
In this transformation, the geometry, boundary loads, drainage characteristics, as well as the 203 unit weights and angles of internal friction remain the same. However, the transformation does 204 require that the angle of internal friction of the reinforced soil mass be corrected during the 205 transformation. Say, a reinforced earth wall, shown in Figure 2a, is approximately 10.0m tall 206 consisting of a total of 25 reinforcing grid at 40-cm spacing with 100% Rc. Each of the lower 207 twelve grids provide a long term strength of 26.7 kN/m-width of grid. The remaining 13 grids 208 provide a long-term strength of 18.3 kN/m, each. Assuming the reinforced soil has an angle of 209 internal friction of 34o, the equivalent cohesive mass system (Figure 2b) consists of the 210 following two blocks: 211 212 Lower block: 213
- Thickness = 4.8m. 214 - Equivalent cohesion, Ceq1 = (12 x 26.7)/4.8 = 66.75 kPa. Since the geogrids are 215
uniformly spaced, Ceq1 = 26.7/0.4=66.75 kPa. 216 - Angle of internal friction and unit weight = same as those for reinforced soil. 217
218 Upper block: 219
- Thickness = 5.2m. 220 - Equivalent cohesion, Ceq = (13 x 18.3)/5.2 = 45.75 kPa (=18.3/0.4 kPa) 221 - Angle of internal friction and unit weight = same as those for reinforced soil. 222
223 Transformed mass angle of friction: 224 According to the DIN-4084, the angle of internal friction is reduced by a factor of 1.25 for 225 static and 1.10 for seismic, as follow: 226
tanD = (tan/1.25 (DIN 4084; 2005) (8) 227 228 For other methods, different reduction factors may apply. 229 230 Accordingly, the design angle of friction for the transformed mass is equal to 28.35-degrees. 231 The equivalency of the two systems (reinforced granular soil and the equivalent cohesive soil) 232 will be verified and useful application of this transformation will be demonstrated in the next 233 section. 234
235
TRANSITION OF LIMIT EQUILIBRIUM AND CRITICAL SURFACES 236
Introduction 237 238 The current practices pertaining reinforced earth walls in the United States as well as other 239 parts in the world specify minimum requirements for design of reinforced earth walls. Of these 240 requirements, is the minimum length of main reinforcements, which shall not be shorter than 241 70% of the height of the wall. This is mainly based on experiences, comparative design 242 problems, and the presumptive and predetermined internal slip surface within the reinforced 243 soil mass. For a typical geogrid reinforced soil mass, the portion of the reinforcement located 244
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within the presumptive active wedge is approximately 50% to 55% of the height above the 245 leveling pad. With the reinforcement length equal to 70% of the wall height, about 20% of the 246 reinforcement length will be left in the resistance zone. The critical slip surface is also assumed 247 to start from the toe of the wall (bottom of wall face). Accordingly, longer reinforcement will 248 be needed within the top of the wall reinforcements. 249 250 These assumptions can be argued by many since the pre-determined critical surface location 251 and shape ignores the combined influences of the reinforcement matrix. A good cause for these 252 arguments is the need for shorter geogrid reinforcement lengths within the upper zone of the 253 wall to leave a room for possible future installations, such as utilities (power, cable, water, gas, 254 sewer, etc.). For some cases, it may be strictly required to minimize the excavation works as 255 part of wall construction at the bottom of walls. These arguments are justified by the following 256 considerations: 257
1- For internal stability, the critical surface may intersect the wall facing at any opint 258 within the facing and not necessarily the bottom of the wall. Considering the separate 259 and combined effects of the reinforcements the slip surface may be transformed to any 260 point within the wall facing depending on the reinforcements patterns and intensities 261 (length, spacing and strength). The bottom of wall can be designed such that the lower 262 reinforcements significantly exceed the active loads up a point where the critical surface 263 will be shifted away from the bottom of wall (upwards or downwards) depending on the 264 foundation subsoil and the reinforcements patterns within the upper parts of the wall. 265
2- The current design method check for the minimum resistance length beyond the 266 presumptive active wedge used in internal sizing of reinforcements. Accordingly, it may 267 be needed to stage the design of the wall by dividing the wall into segments to 268 investigate the transitions of the location and shape of the critical surface within the soil 269 mass. 270
271 It is extremely important not to alter the geometry of the reinforced soil mass and surroundings 272 during the stage design and transition. Vertical or inclined walls shall remain vertical or 273 inclined to the same degree. Replacing the upper wall segments by an equivalent surcharge 274 load, results in conservative, sometimes overly conservative, design of lower reinforced earth 275 walls. This is due to (1) the ignoring of the contributions of the reinforcements within the upper 276 wall, (2) increasing the effects of the existing actual loads that are originally and actually 277 located at higher elevations than the elevations analyzed. For instant, assume that the lower 278 6.0m of 10-m tall wall shall be analyzed separately. A wide surface load (surcharge) of about 279 100 kPa is present on the top of the wall (i.e., 10m above the bottom of wall). Also assume that 280 the upper 4.0m of the wall will represented by an equivalent dead surcharge load (qs) equal to 4 281 x = 80kPa. The reinforcements within the upper 4.0m will not be accommodated in the 282 design, and the 100 kPa surcharge will now be located at 6.0m above the bottom of the wall, 283 producing erratically significant effects on the lower reinforcement layers and possible the wall 284 foundation. Similarly, if the lower portion of the is to be excluded from the design/analyses of 285 the reinforced earth wall, the geometry of the bottom wall shall also remain the same. For 286 instant, replacing the bottom wall by a horizontal ground results in less conservative, may be 287 risky and erratic design. 288 289
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A successful equivalency can be accomplished by transforming the reinforced soil mass into an 290 equivalent cohesive soil mass in the manner described earlier (cohesion is based on the 291 predicted reinforcements mobilized tensile forces). The next section of the paper demonstrates 292 the use of these transitions for optimizing the distributions of reinforcements. The results of 293 transformed sections will be compared with actual designs for validation of results. 294 295 SAMPLE ANALYSES 296 Methodology and Software 297 298 Two cases (design examples) will be analyzed using the straight-forward design approach and 299 the equivalent cohesion and transition method. The analyses/design will be performed using the 300 computer program Huesker-stability developed by CIVILSERVE software (GGU GmbH). This 301 software utilizes the limit equilibrium analyses method to check stability and design reinforced 302 soil walls and reinforced soil slopes. The software allows for using different sets of factors for 303 the reinforced soil system elements to comply with different standards (DIN, U.S., BS and 304 others). For analyses performed according to the DIN standard, the results are displayed in the 305 form of utilization factors (which should not exceed 1.0), or according to the BS and U.S. 306 where the results are displayed in the form of factors of safety (FS, which should not be less 307 than 1.30). In this paper, only the DIN 4084 will be utilized for illustration purposes. 308 309 Using this computer program, two cases of designs are presented, the first being a design that is 310 safe and the second being a risky design (critical design according to the DIN). The analyses 311 are performed using the Bishop Circular slip surfaces with tensile elements (20). The analyses 312 cover both internal stability and deep (global) stability. The computer program also calculates 313 the mobilized reinforcement forces and compares them to the long-term tensile strengths of 314 geogrids. It should be emphasized, however, that these examples are presented for comparisons 315 between the straight-forward design and the transition with equivalent cohesive layers. 316 317 Example Case (Safe Design Comparison): 318 319 A 10-meter tall reinforced earth wall is needed to be designed with 1:10 face inclination. The 320 material properties for reinforced fill, retained mass and the foundation soils are as follow: 321
- Reinforced Fill: 322 Unit weight = 20 kN/m3 323
Angle of internal friction, R=34o 324 Cohesion, C = 0.0 kPa 325
- Retained soil and foundation Soil: 326 Unit weight = 20 kN/m3 327
Angle of internal friction, =32o 328 Cohesion, C = 10.0 kPa 329
- Horizontal backfill. 330 - No water table within the depth of influence. 331 - Traffic surcharge, q = 12 kPa. 332 - Seismicity; horizontal and vertical (kh =0.15g; and kv=0.05g, respectively). 333
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- The design should leave a space for installation/placement of future utilities that can 334 be located not closer than 3.0m from the wall within a depth not exceeding 2.5m 335 beneath the upper finished grade of the wall. 336
337 Design/Analyses: 338 The design/analyses are performed using the Huesker-GGU computer program for reinforced 339 soil walls and reinforced soil slopes. The design was performed according to the new DIN 4084 340 standard. An initial design was performed with two cases of reinforcement lengths: (1) uniform 341 length equal to 7.0m, which corresponds to 70% of the wall height and (2) 2.5m long 342 reinforcements within the upper 3.0m and 7.0m long reinforcements for remaining depth. All 343 reinforcement layers were vertically placed at 40-cm spacing. The short-term and long-term 344 tensile strengths of the reinforcements layers were as follow: 345 346
Layers No
Short-term Strength (kN/m)
Long-term, Design Strength (kN/m)
1-7 80.0 33.3 8-25 55.0 22.9
347 For each design, numerous slip surfaces were created using a predefined search grid as shown 348 in Figure 4. For this design, the resulting maximum utilization factors are summarized in the 349 first and second rows of Table 2, where as the graphical results in the form of contours of the 350 utilization factors for this case are presented in Figures 5 and 6. 351
352 The equivalent (transformed) cohesive systems for the lower portion of the wall (from 0.0 to 353 7.0m from the bottom) were then generated utilizing the long-term design strengths and actual 354 reinforcements’ distributions. The transformed sections consisted of 3.0m tall reinforced earth 355 wall with 7.0-long and 2.5m long geogrids underlain by two equivalent cohesive segments 356 created based on the variable strength orders, as follow: 357 358
- Segment-1 (Bottom 2.8m of wall; from 0.0 to 2.8m): Ceq1 = (7layers x 33.3kN/m)/2.8m= 83.25 kPa. - Segment-2 (next 4.0m of wall; from 2.8 to 6.8 m): Ceq2 = (10layers x 22.9kN/m)/4.0m = 57.25 kPa.
359 It should be emphasized that the long-term design forces are not necessarily the actual forces to 360 be mobilized especially at shallow depths. However, for segments 1 and 2, the long-term 361 tensile forces can be actually mobilized considering their embedment depths. For both 362 segments, the angle of internal friction was reduced in compliance with the correction in Eq. 363 (8). The corrected angle of internal friction was calculated to be 28.35o. 364 365 The Huesker-stability results for the transformed sections are depicted in Figures 7 and 8 and 366 the maximum utilization factors are reported in Table 2. By examining the values of maximum 367 utilization factors and the graphical results of the original sections and the transformed sections, 368 the following remarks can be made: 369
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- The maximum utilization factors of the transformed sections are approximately the 370 same as those for the original (entirely reinforced) sections. The differences in these 371 factors are less than 1%. 372
- There is a very good resemblance between the utilization contour maps for the 373 original and the transformed sections. 374
- The locations of the most critical surfaces (center and radius) for the transformed 375 sections are in close proximity to those of the original sections. 376
377 378
Example Case (Critical Design Comparison): 379 380
For the same geometry, loads and conditions given in the previous example, a critical design 381 was also performed with the following reinforcement pattern: 382 383
- Reinforcement spacing is uniform at 60-cm. 384 - Reinforcements lengths and long-term factored (design) strengths are as follow: 385
387 The first design was performed with all layers shown above and a second design was 388 performed with the equivalent cohesive mass for zone 4 and all remaining reinforcements. The 389 results of the two designs are depicted in Figures 9 and 10 and the resulting maximum 390 utilization factors are summarized in Table 2. The results also reveal that the maximum 391 utilization factors are approximately the same. Additional research is being performed to 392 expand the use of the equivalent cohesion concept to directly relate it to the geogrid-soil 393 interface interaction and the coherent gravity concept with tributary areas (Eqns. 3 and 5). This 394 will be extremely important for shallow depths at which the maximum mobilized tensile forces 395 may be significantly less than the long-term design strength of the geogrid, which may lead to 396 misleading equivalent cohesion values. The investigation also aims at verifying the method for 397 analyses/design under seismic conditions. 398 399
SUMMARY AND CONCLUSIONS 400
The use of the “Equivalent Cohesion” concept for reinforced soil walls was demonstrated and 401 verified. The equivalent cohesive masses were produced simply based on the mobilized tensile 402 forces of the geogrid (not exceeding the long-term tensile strength), the reinforcement spacing 403 and depth. The equivalent cohesive masses are anticipated to enable the design of walls with 404 shorter reinforcements to avoid the design code limitations as well as the limitations of some of 405 the computer programs. Along with the equivalent cohesive mass, the transition of limiting 406 equilibrium state (critical surfaces) were also described from the perspective of the German 407 Standard. Two cases were analyzed with two equilibrium conditions (adequately designed and 408
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critically designed systems). The results of fully reinforced masses were compared to partly 409 reinforced with equivalent cohesive masses (transformed systems). The results revealed the 410 adequacy of the apparent cohesive systems with less than 1.0% tolerance. However, this 411 concept may only be implemented for cases where shorter reinforcements may be needed. It is 412 still being expanded to provided a handy and effective way for analyses/design for variety of 413 conditions. 414 The German standard norm (new DIN 4084) was also briefly described and utilized in the 415 analyses of the presented cases. The reason being that it is based on the limit equilibrium based 416 on the formulation by Bishop and other limit equilibrium analyses method. This standard, as 417 such, does not specify a predetermined critical plain (active wedge), and the critical plain or 418 surface is merely dependent on the reinforcements types and layouts. This is believed to result 419 in a less conservative, yet adequate, designs. 420
421
REFERENCES 422
1. Vidal, H. The Principle of Reinforced Earth. Transportation Research Record, 282, 1969, 1-423 16. 424
2. Walkinshaw, J. L. Reinforced Earth Coonstruction. Department of Transportation FHWA 425 Region 15. Demonstration Project No. 18, 1975. 426
3. Murray, R. T. Research at TRRL to Develop Design Criteria for Reinforced Earth. Symp. 427 Reinforced Earth and other Composite Soil Techniques. Heriot-Watt University, TRRL 428 Sup. 457, 1977. 429
4. Schlosser, F.M. Experience on Reinforced Earth in France. Symp. Reinforced Earth and 430 other Composite Soil Techniques, Heriot-Watt University, TRRL Sup. 457, 1977. 431
5. Mitchell, J.K., and Villet, W.C.B. Reinforcement of Earth Slopes and Embankments. 432 NCHRP Report No. 290, Transportation Research Board, Washington D. C., 1987, pp.323. 433
6. Leshchinsky, D., and Perry, E.B. A design procedure for geotextile-reinforced walls. 434 Proceeding of Geosynthetics’87, New Orleans, LA, 1987. 435
7. Jewel, R.A. Reinforced soil wall analysis and design. Proceedings of the NATO Advanced 436 Research Workshop on Application of Polymer Reinforcement in Soil Retaining Structures, 437 1990. 438
8. Bonaparte, R., and Schmertmann, G. Reinforcement extensibility in reinforced soil and soil 439 wall design. Proceedings of the NATO Advanced Research Workshop on Application of 440 Polymer Reinforcement in Soil Retaining Structures, 1988. 441
9. Gourc, J.P, Ratel, A., Gotteland Ph. Design of reinforced soil retaining walls: analysis and 442 comparison of existing design methods and proposal for a new approach. Proceedings of 443 the NATO Advanced Research Workshop on Application of Polymer Reinforcement in 444 Soil Retaining Structures, 1990. 445
10. Christopher, B.R. Deformation response and wall stiffness in relation to reinforced soil wall 446 design. Ph.D. dissertation, Purdue University, 1993. 447
11. Elias, V., and Christopher, B. R. Mechanically stabilized earth walls and reinforced soil 448 slopes design and construction guidelines. Report No. FHWA-NHI-00-043, FHWA, 449 Washington, D.C. March 2001. 450
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12. Ilan Juran and Chao L. Chen, Strain Compatibility Design Method for Reinforced Earth 451 Walls 452
13. J. Geotech. Engrg., 1989, 115, 435. 453 14. Ilan Juran, Halis M. Ider, and K. Farrag, Strain Compatibility Analysis for Geosynthetics 454
457 16. Mauricio Ehrlich and James K. Mitchell. Working Stress Design Method for Reinforced 458
Soil Walls, J. Geotech. Engrg., 1994, 120, 625. 459 17. William J. Neely, Working Stress Design Method for Reinforced Soil Walls, J. Geotech. 460
Engrg., 1995, 121, 818. 461 18. B. R. Christopher, D. Leshchinsky, and R. Stulgis. Geosynthetic-Reinforced Soil Walls and 462
Slopes: US Perspective, International Perspectives on Soil Reinforcement Applications, 463 Geotechnical Special Publication No. 141, 2005, 167, 12. 464
19. Almoh’d, I.A. A new method for the design and analysis of reinforced earth walls. A PhD 465 thesis submitted to the University of Akron, Ohio, 2003. 466
20. Schlosser, F. M. Mechanically stabilized retaining structures in Europe- Design and 467 Performance of Earth retaining Structure. Edited by Philip C. Lambe and Lawrence A. 468 Hansen, New York, 1990. 469
21. Jewel, R.A. Reinforced soil wall analysis and design. Proceedings of the NATO Advanced 470 research Workshop on Application of Polymer Reinforcement in Soil retaining Structures, 471 1985. 472
22. Bishop, A.W. 1954. The use of the slip circle in the analysis of slopes. Proc. European conf. 473 on stability of earth slopes. Vol. 1, Stockholm. 474
23. Krey, H.D. 1926. Active earth pressure, passive earth pressure and load bearing capacity of 475 the subsoil. 3rd ed, Published by W. Ernstund Sohn, Berlin. 476
24. Janbu. 1955. Application of Composite slip surfaces for stability analysis. Proc. European 477 conf., Stockholm, Vol. 3, 43. 478
LIST OF TABLES 479
Tab 1e 1: Partial safety factors for load and soil properties. 480 Tab 1e 2: Summary of design utilization factors for all design cases. 481
LIST OF FIGURES 482
Figure 1: Apparent cohesion in reinforced granular soil (After Schlosser and Long, 483 1972; and Yang and Singh, 1974). 484
Figure 2: Equivalent systems: a) Reinforced soil mass, b) Equivalent cohesive mass. 485 Figure 3 Demonstration of analyses: Generation of utilization contours and most critical 486
lengths and spacing=40cm): Example 1. 489 Figure 5 Stability results for the 10-m tall reinforced earth wall with 7.0m and 2.5m 490
reinforcement lengths (Spacing =40cm): Example 1. 491
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Figure 6 Stability results for the equivalent system consisting of 3.0m tall reinforced earth 492 wall with 7.0m long reinforcement lengths and equivalent cohesive layers 493 (Spacing=40cm): Example 1. 494
Figure 7 Stability results for the equivalent system consisting of 3.0m tall reinforced earth 495 wall with 2.5m long reinforcement lengths and equivalent cohesive layers 496 (Spacing =40cm): Example 1. 497
Figure 8 Stability results (10-m tall critically reinforced earth, wall with variable 498 reinforcement lengths, and uniform spacing=60cm): Example 2. 499
Figure 9 Stability results (10-m tall critically reinforced earth, wall with variable 500 reinforcement lengths, equivalent cohesive soil for top 3.0m, and uniform 501 spacing=60cm): Example 2. 502
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Tab1e 1: Partial safety factors for load and soil properties. Parameter Load case 1 Load case 2 Load case 3
