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RES — A numerical program for reinforced-soil slopes based on the rigid-plastic theoretical model
Geotextiles and Geomembranes 12 (1993) 435-439
RES - - A Numerical Program for Reinforced-Soil Slopes Based on the Rigid-Plastic Theoretical Model
D. Le~niewska Institute of Hydroengineering, IBW PAN, Gdafisk, Poland
ABSTRACT A brief description of a rigid-plastic model of reinforced soil is presented with an account of its application -- program RES. which may servefor numerical analysis of reinforced slopes. The program is able to solve the classical differential-boundary-value problem of slope-bearing capacity for reinforced and unreinforced soil in a wide range of soil and reinforcement parameters. The solution is obtained with a method of characteristics and contains the slope ~ bearing capacity (the value of the failure load), a definition of slip line. responsiblefor construction damage, and the total stresses in the plastic region of the slope.
1 INTRODUCTION From the viewpoint of continuum mechanics, reinforced soil is a twocomponent, homogeneous composite, showing macroscopic anisotropy. The macro-behaviour of this composite depends on the mechanical properties and interactive contribution of each component, Sawicki (1983a, b, c). In practice, reinforced soil means soil with some kind of uniform, regularly distributed inclusions, which may be metal strips or rods, fibers, geogrids, geotextiles, geomembranes, etc. One of the possible theoretical models that may be used to investigate this material is a rigid-plastic model of reinforced soil proposed by Sawicki (1983c). 435 Geotextiles and Geornembranes 0266-1144/93/$06.00 © 1993 Elsevier Science Publishers Ltd, England. Printed in Great Britain
D. Le,~niewska
436
2 RIGID-PLASTIC MODEL OF REINFORCED SOIL The basic assumptions of the rigid-plastic model of reinforced soil are as follows: (i) soil is rigid-plastic and obeys the Coulomb-Mohr criterion: the associated flow rule is assumed to be valid; (ii) the reinforcement is rigid-plastic and has strength only in tension, characterized by its tensile strength and a unit vector parallel to the reinforcement's geometrical orientation; (iii) both components of reinforced soil are perfectly mixed, which means that it is possible to select from the total volume of reinforced soil a representative elementary volume, containing soil and reinforcement, much smaller than the total volume; (iv) the mechanical behaviour of reinforced soil is described by two kinds of tensors: macrostress and plastic macrostrain tensors cr and e and microstress and plastic microstrain tensors a ~, a r and e ~, # , which are the respective averages over the representative elementary volume and its parts that are filled by each constituent; (v) there is no sliding between the soil and its reinforcement: ~. =
er
=
/~s
(1)
(vi) the relation between the macrostress tensor and microstress tensors is defined by: G
=
r/r G r + r / s G s
(2)
where r/~ and Or are the volumetric ratios of soil and reinforcement. respectively.
3 THE PROBLEM OF THE BEARING CAPACITY OF A REINFORCED-SOIL SLOPE The rigid-plastic model of reinforced soil, together with the method of characteristics, allows solution of the differential-boundary-value problem of the bearing capacity of a reinforced slope in the plane strain state (Legniewska, 1988). A mathematical solution of this problem consists in solving sets of hyperbolic-type differential equilibrium equations, eqns (3), with the assumption of a limit state of stresses inside a certain volume of reinforced slope:
Numerical program .for reinforced-soil slopes 00" x 19x +
[90"y
437
197Y.x-v Oy
-
0
-
)"
(3)
19~'xv
19y + 19x
The solution of this problem is important from an engineering point of view, because it contains the value of the limit load, leading to the failure of the structure, the geometry of the plastic region, and the total stresses in this region, determined by the characteristics net. It is worthy of note that the theoretically determined limit load and failure mode (the shape of the plastic region limited by the active slip line) may be easily verified experimentally in model or full-scale tests, and this has already been done for several prototypes with good results by Thamm et al. (1990) and Krieger et al. (1992).
4 GENERAL DESCRIPTION OF THE PROGRAM RES The structure of the program is presented in Fig. 1. To determine the solution of the bearing-capacity problem for a reinforced slope, the program needs the following data: (i) the geometry of the slope -- height h and inclination/3: (ii) the soil properties m cohesion c, angle of internal friction q~,and unit weight y: (iii) the reinforcement parameters -- strength in tension R, volumetric ratio r/r or number of reinforcing elements per cross-section of the slope n for geotextile reinforcement, and reinforcement inclination 0. The physical meaning of the parameter R depends on the kind of reinforcement used. For metal rods, it determines the maximum values of stress, which cause breakage of the reinforcing element in a uniaxial tension test. For geotextiles, geomembranes, and related products, the tensile force is related to the allowable reinforcement elongation. The RES program was written to be used as a tool in the theoretical analysis of experimental slopes, reinforced by different types of materials (including geosynthetics). In practice, the program could be a helpful part of a complete design system. It calculates an internal stability of construction, which enables the correct slope or wall geometry and parameters of fill and reinforcement to be chosen. The program is also
D. Le,4niewska
438
I INPUT DATA l
I
J
SOIL
[
]
SLOPE GEOMETRY
REI NFOeCEMENT
R,n
i
I I
.~,.,
I
[
=.0,1ONs I
CALCULATIONS
RESULTS
I
1
J
STRESSES X [m] 000
000 000 000 000 000 000 000 000 000 000 000 000
000 000 000
Y [m] 0.0000 0'09110"18220"27330"36440"45550'54660-63770-72880-81990-91101"00211"0931 1"18421"27530'0504
coordinates of points creating characteristics net
(x,y)
numericat calculations: method of characteristics
[ distribution of failure load
SIGMA [kPa] 0.0000 0'4286 0"8571 1" 2 8 5 7 1' 7143 2"1429 2"5714 3'0000 3"4286 3"8571 4-2857 4"7143 5"1429 5"5714 60000 0"2966
62"4 62-4 62'4 62"4 62'4 62'4 62'4 62.4 62-4 62'4 62"4 62"4 62"4 62.4 62"4 64'2
Fig. !. Structure of RES program.
stip
"
Numerical program for reinforced-soil slopes
439
helpful in planning laboratory and field experiments, especially if they lead to failure. It is possible to determine the best way of loading the slope models under given laboratory conditions. The RES program needs about 200 kB of memory, calculations are very fast (the analysis of one reinforced slope takes about half a minute on an AT-286 PC), it can present most important results in graphic form, and it is easy to operate, even for users not conversant with the theory. The RES program has limitations of a physical and a numerical nature. The first limitation relates to the assumption of a rigid-plastic model of reinforced soil. Other limitations are that the program can be used only for uniformly reinforced slopes, when distances between reinforcing elements are significantly less than slope dimensions and that it cannot calculate any displacements. Numerical analysis of many reinforced-slope examples has shown that it is necessary to determine reasonable limits for parameters R, n, h, and }, and for relations between them. Experimental results could be helpful, but the n u m b e r of test results reported is still insufficient. The RES program may be obtained from the Institute of Hydroengineering, IBW PAN, 80-953 Gdafisk-Oliwa, Ko~cierska 7, Poland. Tel. (058) 52-20-11, Tlx. 0512845 ibw pl, Fax (058) 52-42-11.
REFERENCES Krieger, B., Le~niewska, D. & Thamm, B.R. (1992). Messungen an einer geotextilbewehrten Stutzwand mit Betonfertigteilelementen ais Aussenhaut, submitted for publication in K-Geo92 Conference. Le~niewska, D. (1987). Statics and kinematics of reinforced soil. Ph.D. thesis, Institute of Hydroengineering, Ko~cierska, Poland. Sawicki, A. (1983a). Axisymmetrical elasto-plastic behaviour of reinforced earth, Int. J. Num. Anal. Meth. in Geomech., 7, 493-8. Sawicki, A. (1983b). Engineering mechanics of elasto-plastic composites. Mech. Mater., 2, 217-31. Sawicki, A. (1983c). Plastic limit behaviour of reinforced earth.J. Geoteeh. Engng. Div., Proc. ASCE. 109, 1000. Sawicki, A. & Legniewska, D. (1991). Stability of fabric reinforced cohesive soil slopes. Geotext. & Geomemb. 10 125--46. Sawicki, A., Le~niewska, D. & Kulczykowski, M. (1988). Measured and predicted stresses and bearing capacity of a full-scale slope reinforced with nails. Soils & Foundations, 28(4), 47-56. Thamm, B.R., Krieger, B. & Le~niewska, D. (1990). Full-scale test ofa geotextilereinforced soil wall. In Proceedings of the International Reinforced-Soil Conference, Glasgow, UK, ed. A. McGown, K.C. Yeo & K.Z. Andrawes. Thomas Telford, London.
RES - - A Numerical Program for Reinforced-Soil Slopes Based on the Rigid-Plastic Theoretical Model
D. Le~niewska Institute of Hydroengineering, IBW PAN, Gdafisk, Poland
ABSTRACT A brief description of a rigid-plastic model of reinforced soil is presented with an account of its application -- program RES. which may servefor numerical analysis of reinforced slopes. The program is able to solve the classical differential-boundary-value problem of slope-bearing capacity for reinforced and unreinforced soil in a wide range of soil and reinforcement parameters. The solution is obtained with a method of characteristics and contains the slope ~ bearing capacity (the value of the failure load), a definition of slip line. responsiblefor construction damage, and the total stresses in the plastic region of the slope.
1 INTRODUCTION From the viewpoint of continuum mechanics, reinforced soil is a twocomponent, homogeneous composite, showing macroscopic anisotropy. The macro-behaviour of this composite depends on the mechanical properties and interactive contribution of each component, Sawicki (1983a, b, c). In practice, reinforced soil means soil with some kind of uniform, regularly distributed inclusions, which may be metal strips or rods, fibers, geogrids, geotextiles, geomembranes, etc. One of the possible theoretical models that may be used to investigate this material is a rigid-plastic model of reinforced soil proposed by Sawicki (1983c). 435 Geotextiles and Geornembranes 0266-1144/93/$06.00 © 1993 Elsevier Science Publishers Ltd, England. Printed in Great Britain
D. Le,~niewska
436
2 RIGID-PLASTIC MODEL OF REINFORCED SOIL The basic assumptions of the rigid-plastic model of reinforced soil are as follows: (i) soil is rigid-plastic and obeys the Coulomb-Mohr criterion: the associated flow rule is assumed to be valid; (ii) the reinforcement is rigid-plastic and has strength only in tension, characterized by its tensile strength and a unit vector parallel to the reinforcement's geometrical orientation; (iii) both components of reinforced soil are perfectly mixed, which means that it is possible to select from the total volume of reinforced soil a representative elementary volume, containing soil and reinforcement, much smaller than the total volume; (iv) the mechanical behaviour of reinforced soil is described by two kinds of tensors: macrostress and plastic macrostrain tensors cr and e and microstress and plastic microstrain tensors a ~, a r and e ~, # , which are the respective averages over the representative elementary volume and its parts that are filled by each constituent; (v) there is no sliding between the soil and its reinforcement: ~. =
er
=
/~s
(1)
(vi) the relation between the macrostress tensor and microstress tensors is defined by: G
=
r/r G r + r / s G s
(2)
where r/~ and Or are the volumetric ratios of soil and reinforcement. respectively.
3 THE PROBLEM OF THE BEARING CAPACITY OF A REINFORCED-SOIL SLOPE The rigid-plastic model of reinforced soil, together with the method of characteristics, allows solution of the differential-boundary-value problem of the bearing capacity of a reinforced slope in the plane strain state (Legniewska, 1988). A mathematical solution of this problem consists in solving sets of hyperbolic-type differential equilibrium equations, eqns (3), with the assumption of a limit state of stresses inside a certain volume of reinforced slope:
Numerical program .for reinforced-soil slopes 00" x 19x +
[90"y
437
197Y.x-v Oy
-
0
-
)"
(3)
19~'xv
19y + 19x
The solution of this problem is important from an engineering point of view, because it contains the value of the limit load, leading to the failure of the structure, the geometry of the plastic region, and the total stresses in this region, determined by the characteristics net. It is worthy of note that the theoretically determined limit load and failure mode (the shape of the plastic region limited by the active slip line) may be easily verified experimentally in model or full-scale tests, and this has already been done for several prototypes with good results by Thamm et al. (1990) and Krieger et al. (1992).
4 GENERAL DESCRIPTION OF THE PROGRAM RES The structure of the program is presented in Fig. 1. To determine the solution of the bearing-capacity problem for a reinforced slope, the program needs the following data: (i) the geometry of the slope -- height h and inclination/3: (ii) the soil properties m cohesion c, angle of internal friction q~,and unit weight y: (iii) the reinforcement parameters -- strength in tension R, volumetric ratio r/r or number of reinforcing elements per cross-section of the slope n for geotextile reinforcement, and reinforcement inclination 0. The physical meaning of the parameter R depends on the kind of reinforcement used. For metal rods, it determines the maximum values of stress, which cause breakage of the reinforcing element in a uniaxial tension test. For geotextiles, geomembranes, and related products, the tensile force is related to the allowable reinforcement elongation. The RES program was written to be used as a tool in the theoretical analysis of experimental slopes, reinforced by different types of materials (including geosynthetics). In practice, the program could be a helpful part of a complete design system. It calculates an internal stability of construction, which enables the correct slope or wall geometry and parameters of fill and reinforcement to be chosen. The program is also
D. Le,4niewska
438
I INPUT DATA l
I
J
SOIL
[
]
SLOPE GEOMETRY
REI NFOeCEMENT
R,n
i
I I
.~,.,
I
[
=.0,1ONs I
CALCULATIONS
RESULTS
I
1
J
STRESSES X [m] 000
000 000 000 000 000 000 000 000 000 000 000 000
000 000 000
Y [m] 0.0000 0'09110"18220"27330"36440"45550'54660-63770-72880-81990-91101"00211"0931 1"18421"27530'0504
coordinates of points creating characteristics net
(x,y)
numericat calculations: method of characteristics
[ distribution of failure load
SIGMA [kPa] 0.0000 0'4286 0"8571 1" 2 8 5 7 1' 7143 2"1429 2"5714 3'0000 3"4286 3"8571 4-2857 4"7143 5"1429 5"5714 60000 0"2966
62"4 62-4 62'4 62"4 62'4 62'4 62'4 62.4 62-4 62'4 62"4 62"4 62"4 62.4 62"4 64'2
Fig. !. Structure of RES program.
stip
"
Numerical program for reinforced-soil slopes
439
helpful in planning laboratory and field experiments, especially if they lead to failure. It is possible to determine the best way of loading the slope models under given laboratory conditions. The RES program needs about 200 kB of memory, calculations are very fast (the analysis of one reinforced slope takes about half a minute on an AT-286 PC), it can present most important results in graphic form, and it is easy to operate, even for users not conversant with the theory. The RES program has limitations of a physical and a numerical nature. The first limitation relates to the assumption of a rigid-plastic model of reinforced soil. Other limitations are that the program can be used only for uniformly reinforced slopes, when distances between reinforcing elements are significantly less than slope dimensions and that it cannot calculate any displacements. Numerical analysis of many reinforced-slope examples has shown that it is necessary to determine reasonable limits for parameters R, n, h, and }, and for relations between them. Experimental results could be helpful, but the n u m b e r of test results reported is still insufficient. The RES program may be obtained from the Institute of Hydroengineering, IBW PAN, 80-953 Gdafisk-Oliwa, Ko~cierska 7, Poland. Tel. (058) 52-20-11, Tlx. 0512845 ibw pl, Fax (058) 52-42-11.
REFERENCES Krieger, B., Le~niewska, D. & Thamm, B.R. (1992). Messungen an einer geotextilbewehrten Stutzwand mit Betonfertigteilelementen ais Aussenhaut, submitted for publication in K-Geo92 Conference. Le~niewska, D. (1987). Statics and kinematics of reinforced soil. Ph.D. thesis, Institute of Hydroengineering, Ko~cierska, Poland. Sawicki, A. (1983a). Axisymmetrical elasto-plastic behaviour of reinforced earth, Int. J. Num. Anal. Meth. in Geomech., 7, 493-8. Sawicki, A. (1983b). Engineering mechanics of elasto-plastic composites. Mech. Mater., 2, 217-31. Sawicki, A. (1983c). Plastic limit behaviour of reinforced earth.J. Geoteeh. Engng. Div., Proc. ASCE. 109, 1000. Sawicki, A. & Legniewska, D. (1991). Stability of fabric reinforced cohesive soil slopes. Geotext. & Geomemb. 10 125--46. Sawicki, A., Le~niewska, D. & Kulczykowski, M. (1988). Measured and predicted stresses and bearing capacity of a full-scale slope reinforced with nails. Soils & Foundations, 28(4), 47-56. Thamm, B.R., Krieger, B. & Le~niewska, D. (1990). Full-scale test ofa geotextilereinforced soil wall. In Proceedings of the International Reinforced-Soil Conference, Glasgow, UK, ed. A. McGown, K.C. Yeo & K.Z. Andrawes. Thomas Telford, London.