1 RETAINING STRUCTURES
(outline) to Lecture no.5
1. Applications, types of retaining structures and scope of calculations
Retaining structures, if they include also steep soil embankments and fortifications, are one of the oldest engine- ering structures in history, next to roads and bridges - they are even linked to them. In the following, only "very long" constructions are analyzed, i.e. in a 2D state of displacements.
1.1. Types of retaining structures
The main task of retaining structures is to keep the land level discontinuity ∆H as small as possible on a hori- zontal section, unlike unsupported embankments and excavations. In other words, they serve to safely transfer forces from soil pressure, mainly horizontal or close to horizontal ones. It is impossible to imagine any serious investment in an urban area without limiting the occupied surface - as a rule, limiting it to a minimum, i.e. using vertical walls on the plane of building foundations: with a foundation slab of 40mx20m and a foundation depth of 10m, an unsupported excavation would normally have to be in the order of 80÷100mx 60÷80m, not to mention water inflow problems; the secondary function of the retaining structures is also a reaction against water pressure (and limiting its inflow), which is associated with a subsoil fault in excavations.
The third function is also to transfer vertical forces to the subsoil, if the loads can be very large. Examples are diaphragm walls and bridge abutments.
The retaining structures are generally divided into structures embedded in the subsoil and retaining walls, while the role of the latter is usually limited to the transfer of soil pressure and possibly groundwater pressure.
1.2. Structures embedded in the subsoil
Fig.1
In general, the work of a construction embedded in the subsoil resembles the behavior of a vertical beam, but the Winkler model (this time with horizontal springs) has even less use here, because of its linearity. If we consi- der that the y(x) horizontal displacement at depth x is proportional to the resultant reaction of the subsoil r(x) in this cross-section1, this proportionality would be acceptable only for small displacement values - quickly the resultant subgrade reaction becomes almost constant, independent of increasing displacements; determining such ultimate displacements that occur at depth x is very difficult.
With a diaphragm wall-width of 60, 80 or 100 cm, its ability to transfer vertical loads is very high, which is also
1although this is not the case, because neighbouring cross-sections also affect
Because of their length, they are also commonly called walls, but it is necessary to distinguish the specificity of their interaction with the subsoil, which is very different from "traditional" retaining walls.
First of all, they reach at least a few meters below the bottom of the excavation, and sometimes their founding level below the bottom is even greater than the depth of the excavation itself. In this situation, the primary stabilizing role is played by the reaction of the subsoil in front of the construction, below the bottom of the excavation. The great height of these walls (in a vertical direction, e.g. at least of the order of a dozen meter) makes bending deformations of the walls are the most significant. In particular, on certain sections at ancho- ring, ceiling or strut, and under the bottom of the excavation, the walls may move towards the ground and not towards the excava- tion; there is a certain analogy with the "waving" of a beam on an elastic subsoil, depending on its (relative) rigidity.
due to friction and adhesion on the vertical outer surfaces of the wall; if, however, the transfer of vertical forces from a wall-head, to the interacting bottom plate (combined with the wall) is also considered, the possibilities of transferring vertical forces are enormous.
1.3. Massive retaining walls
Fig.2
The use of weak structural materials (wall with lime mortar joints, unreinforced concrete) yields immediately the basic conclusion - there must be no significant tensile stress in these walls2.
Let x denote a certain depth below the top of the wall, bx is the width of the horizontal cross-section of the wall at this depth x, the dead weight Gx [kN/m] and the soil pressure Ex [kN/m] are taken from the parts above the depth x. The (inclined) backfill pressure Ex causes also a significant moment in horizontal cross-sections, so these cross-sections at each wall height x must be very wide, and the vertical dead weight of the wall Gx very significant, so that the eccentric resultant load Wx does not leave the cross-section core (ex < bx/6).
The symbol B denotes the width of the base of the wall foundation (inclined, measured along the contact with the subsoil), Fig. 2; the symbol bx means the width of the wall in horizontal cross-section at depth x, while the eccentricity ex applies to this cross-section. The yellow dot marks the center of the cross-section, and the white one the point where the resultant loading Wx is applied3.
Shaping of the vertical cross-section of the wall.
• The wall cross-section is thick, usually B > 0.6÷0.8∆H, the offsets are short (especially the right one), the press-down zone directly underneath the bearing with P force requires a strengthening.
• In addition to the thick shape, the second characteristic feature in Fig. 2 is the slope of the foundation base, a certain not very large angle |α|≥ 0 (like 1:5, 11o = 0.2 rad) is applied, which was found even in some ancient constructions in the form of stair-like steps of clinker bricks or stony blocks; this (averaged) slope α significantly improves the stability of the wall, because it makes the resultant load W approaches the normal direction to the base.
• The offset on the right side of the foundation does not matter and may not present. The offset on the left side is very important. Its length allows "a good balance" of the structure; in principle, the offset should be chosen so that the center of the base (yellow dot) hits exactly under the resultant force W, in the vector form W = P + E + G. If there is no bridge span-beam and no bearing loaded by the force P, then a “small”
eccentricity to the left side is recommended, Fig.2, like eB∼ 0.1÷0.2m. The point is that the wall should reveal a clear tendency to move out of the backfill, because then minimal earth pressure E (active
2Stony material can hardly be called "weak" - in this case, joints are filled with weak mortar, e.g. lime, sometimes only clay, or are not filled at all (the Inca stone retaining walls from perfectly fitted blocks)
3The centers of the bx sections, i.e. the yellow dots, define along the height the so-called wall core (±bx/ 6), white dots draw along the wall the so-called pressure line, which should be everywhere inside the wall core, if the material does not carry any tensile forces.
They are also called gravitational walls, because their principle of working is based mainly on the very high G weight, [kN/m]. They were made for hundreds of years as masonry, then concrete, and all the time for
thousands of years as stony ones. Many of these historic structures have survived in good condition until modern times.
The height ∆H of massive walls can be even 6÷8m and sometimes more, they can carry additional concentrated loads P, e.g. from a bridge span, so the wall becomes an abutment.
The dimensional proportions in Fig. 2 are quite realistic, the self-deformations of this type of walls are obviously negligible.
α
Ex
P
Gx
x Wx
x bx
B W N
T
∆H
hf
3
pressure) occurs. However, this eccentricity value should not be “large”, because then the bearing capacity would be significantly reduced, B’= B-2eB <B.
• The inner surface of the wall in contact with the backfill is inclined and the earth pressure E is also inclined (upwards, above the normal); this increases the (generally favorable) normal N component in the base, but first of all it reduces the (certainly destabilizing) shearing T component, Fig. 2.
• The depth of foundation hf is defined as the shortest distance from the bottom left corner of the foundation to the ground surface from the left, Fig. 2; it should be greater than the depth of freezing of soil (frost-heave zone) and takes into account the depth of river bottom washout, if it is an abutment.
Walls of this type are also made nowadays, although usually with some variation - the required great self-weight is provided by a thin-walled cuboid multi-chamber, reinforced concrete box with chambers filled with backfill or other ballast material.
1.4. Light retaining walls
They are also called slab walls or cantilever walls, being made of reinforced concrete, their thickness generally does not exceed 30÷40 cm, minimum 20 cm. Low walls can be made of pre-cast RC-elements and then have the shape of an angle.
Generally, light walls are less high in ∆H than massive walls, and if not, they have vertical reinforcing ribs located every few meters along the wall, creating a 3D monolithic structure with the wall itself; the width of the foundation B is often even greater, because the plate on the right works also as an anchoring element.
As thin-walled constructions, they are not intended for transferring vertical loads P, unless these forces are applied locally to the built-in columns (ribs) interacting with the wall
The dimensional proportions in Fig.3 are quite realistic, and the SLS self-deformations of this type of wall (deflection of the cantilever part, strains) should be checked by calculation.
Contrary to appearances, the wall of Fig.3 is not very different computationally from the wall of Fig.2.
The kinematics of the wall as a rigid body - settlement, displacement to the left, slight rotation to the left - causes that in the concave angle between the wall and its foundation slab, a rigid soil wedge is formed, approximately triangular, bounded by the BC line. This wedge is drawn by a straight line, from the corner C to the intersection with the wall (B) or the surface of the backfill, which can happen for a "very wide" right foundation slab. For this purpose, a certain angle θ measured from the horizontal level is used, depending on backfill’s angle of internal friction ϕ:
= + + (1) where ε = a backfill slope near the wall, ωε = an angle to be found from the equation sin(ωε) = sin(ε)/sin(ϕ).
The rigid wedge formed from the backfill is like being "attached" to the reinforced concrete wall and it is treated as its integral part; similarly as the rigid triangular wedge of soil, which occurs in the bearing capacity limit state
.
Fig.3 EAB
EBC
B
C D W G
+
A
ECD
θ
Backfill
Backfill
Natural noncohesive soil
∼0
for shallow foundations (GEO). In this way, the inner wall surface in Fig.3 consists of 3 design sections A-B-C-D, and each of them is loaded by a different earth pressure E [kN/m] - in terms of value and direction. In this situa- tion, the earth pressure along the broken-profile A-B-C-D wall can be calculated using the Coulomb-Poncelet method, similarly as for a massive wall.
The dead load (weight) also breaks down into two G components, because the unit weights of the reinforced concrete and the backfill in the rigid wedge are different. As a rule, stabilizing forces acting in front of the wall on its left are ignored, allowing a possibility of local excavation in front of the wall - in a small section and not deeper than the lower corner of the foundation (∼0 in Fig.3).
After determining the resultant vector W on the founding level (point of application, value, direction), one proceeds in the same way as in the case of a massive wall, i.e. the eccentricity eB is checked, followed by two GEO stability conditions according to EC7-1:
• for sliding on the founding level (upper subsoil layer),
• for bearing capacity (note that the subsoil may be layered).
Shaping of the vertical cross-section of the wall and its reinforcement.
STR calculations of the structure itself are different, of course, because there are practically no such calcula- tions for the massive wall.
• The slope of the wall towards the backfill improves the aesthetic impression and is not a major complication of formworks. There are also, for example, walls vertical in the lower part and an inclined surface only in the upper part; then it may happen that the profile will consist of 2, 3 and even 4 intervals.
• Light retaining wall has usually a minimal thickness of 20cm or 25cm at the upper end, at the foundation slab similarly, but towards the central joint (wall-slab rigid connections) these thicknesses increase up to 30cm, 35cm, sometimes more.
• The joint is the most important bearing part of the wall; maximal moments occur here, the tendency to open the angle between the wall and the slab, scratches, water penetration and potential corrosion of
reinforcement; note that corrosion of reinforcement in the joint leads inevitably to wall failure. In Fig.3 a chamfering is shown, which is reinforced using additional bars anchored in the wall and in the slab - to strengthen the joint.
• The cantilever wall is designed for bending (less often also for shear) as a cantilever, fixed in the foundation slab - this time ignoring the rigid soil wedge, so calculating the continuous inclined load e(l) [kPa] along the entire cantilever. The foundation slab is designed as two separate cantilevers, on the left and on the right, both fixed in the wall.
• The main reinforcement in the wall is, of course, on its inner side, the main reinforcement of the left
cantilever of the foundation slab is in its lower part, in the right (longer one) – in its upper part. Because the bending moment in the wall splits into two moments applied to the cantilevers of the slab, so in each of them the reinforcement cross-section can be less than the maximal in the wall. The moments in the upper part of the wall are very small and reinforcement can be reduced - probably in the upper part close to the minimal one, e.g. alternately:
- shorter rebars φ12, from the foundation level to a height of e.g. 2.0 m every 25 cm (no more than every 30 cm),
- and longer rebars φ12, from the foundation level to the top of the wall every 25cm (no more than every 30cm),
in total, φ12 rebars at the bottom of the wall are every 12.5cm.
Mainly in English-language literature, other design shapes are used that are considered simpler, i.e. cheaper:
only vertical and horizontal elements, with constant thickness are used, Fig.4. It would be possible to use again the Poncelet wedge from the corner C as in Fig.3, but a trapezoidal area bounded by the vertical BC line is more popular, and the earth pressure of the backfill acts along the one vertical section BD. Such a design situation is associated with the name of Rankine, because the solution he received is suitable for calculating the EBD earth pressure along a vertical "wall" B-C-D.
The wall spur in the shallow vertical trench, especially easy to make in a cohesive soil, serves to improve the stability against sliding of the wall, first of all. The use of the spur causes here to appear another rigid wedge of soil which is "attached" to the construction, this time a small triangle under the real foundation; de facto, the wall itself creates an inclination of the foundation base, which is always considered together with the rigid wedge, as a whole.
5
In this case, three weights G and an additional vertical force Q (calculated from the continuous load q [kPa]
along the interval AB) will occur.
Comparative calculations show that the Poncelet wedge method (Fig.3) and the Rankine trapezoid wedge method (Fig.4) give very similar results - after bringing the forces to the foundation base.
In the case of Fig.4, attention should be paid to drain carefully the backfill behind the wall, as water could collect behind the wall on the impermeable (cohesive) soil; we are talking about rainwater, including surface run-off:
• the disadvantage of the horizontal foundation slab – in the context of possible and even assumed slight rotation to the left - is water accumulation around the rigid joint (chamfering), i.e. a greater risk of corrosion of concrete and steel,
• low water levels above the foundation, but permanent, may also cause weakening of the subsoil composed of (sensitive) cohesive soils,
• even more serious problem for the wall statics is a back-water, which is to be prevented by longitudinal drainage system (marked symbolically from the right and left of the foundation) and drainage pipes passed through the wall in its lower part, Fig.4.
The large water damming significantly increases destabilizing forces and with high probability – if not taken into account in the stability calculations - could lead to damage or overturning of the wall; note that the BD = 4.0m, the pressure increases even more than 2-fold4. Such a situation took place, among others in Wroclaw, right after the 1997 food, on the quay at the University of Wroclaw:
1) the flood water level raised above the point A in Fig.4,
2) there is no threat if water level is high but on both sides of the wall, 3) on the left side of the wall, water in the river lowered within a few days,
4) on the right side of the wall, water remained in fully saturated backfill and was lowering very slowly, because of the inefficient drainage,
5) breakage of the wall under additional water pressure took place along a distance of several dozen meters.
4 no water table: E = ½⋅Ka⋅γ⋅H2 = ½⋅0,3⋅18,5⋅4,02 = 44,4kN/m;
potentially maximal water table: E = ½⋅Ka⋅γ’⋅H2 + ½⋅1⋅γw⋅H2 = ½⋅0,3⋅9,5⋅4,02 + ½⋅1⋅10,0⋅4,02 = 102,8kN/m.
Natural cohesive soil
Fig.4
EBD
B
C
+
G A
W D
Backfill
Q
2. Introduction to earth pressure calculations 2.1. Symbols
Fig.5
Example:
Determine the value of the angle x, at which each of the three forces E in Fig.3 act, if transferred to the wall foundation level, i.e. calculate the tangential (shearing) component T yielding from E.
Answer: x = δ + β - α and next T = T⋅cos (x). Of course, δ and β are usually different for each interval5. 2.2. Simplified kinematics of the retaining wall
Fig.6
Note that this is still a great simplification, because the formula does not distinguish the order of components:
fA /H = 2 ‰ and fB /H = 4 ‰ in practice is not the same as fA /H = 4 ‰ and fB /H = 2‰. Moreover, for f < 0 the term fA is much more important than fB, but for f > 0 it is the opposite, although not so clearly,
see EC7-1 Tab.C.1 for non-cohesive and Tab.C.2 for cohesive soils.
5the same angle of inclination of the base α also appears in the GEO bearing capacity condition, but without a sign and there is always α ≥ 0, which requires a comment. Of course, this is the absolute value of the angle α, the sign of angle α in GEO is irrelevant. The situation is clear which way to tilt the base - so that the resultant load W becomes perpendicular to the base, in practice - approaches the perpendicularity, and not opposite.
E
e(z) H
E(ρ)
ρ Ep
Eo
Ea
ρa 0 ρp
A vertical rigid retaining wall of a height H is considered, which can be horizontally shifted without friction to the right (f > 0) or to the left (f < 0); for constant H we define dimensionless displacement ρ = 2f/H.
The resultant earth pressure E [kN/m] is the result of con- tinuous horizontal stress e(z) [kPa] along the wall surface.
In Fig. 6 it is assumed that the wall surface is smooth and therefore the vectors are horizontal - perpendicular to the vertical wall. A typical graph of function E =E(ρ) has the shape as in Fig.6 - generally, it is well confirmed by experience.
If the wall can rotate, then things get complicated;
the horizontal displacement of its upper edge at A is denoted as fA and analogously fB at the bottom. The Polish code PN-83/B-03010 introduces a more versatile
definition: ρ= (fA +fB)/H.
A
B
+
x z A
B q
β
ε τ σ
• Sign convention:
counter-clockwise angles are positive,
and the following linear displacements are positive (Fig.5)
• backfill parameters (noncohesive, usually ϕ>30o),
ε = ground-surface slope [o], measured from horizontal, usually ε ≥ 0, |ε| < ϕ, about +20o in Fig.5,
q = uniform loading on the ground surface [kPa], usually vertical, sometimes just q=0,
β = wall surface slope [o], measured from vertical, about +25o in Fig.5,
δ = external friction angle on the wall surface [o], measured from the formal to the wall surface
τ /σ = tgδ, |δ| ≤ ϕ for cohesionless soils, usually δ≠0.
• slope of the foundation base α [o, rad], measured at the corner B from vertical, about -9o in Rys.5,
• 3 angles in Fig.5 are positive, only α < 0.
δ
α
7
2.3. Types of soil pressure
There are two limit values (extreme, asymptotic) in Fig.6, i.e. Ep and Ea, as well as an important Eo "at rest"
for ρ=0, whereas: Ea < Eo, Eo << Ep:
• the limit state for pushing the wall into the soil (f > 0, Ep) is called the passive earth pressure, it happens for ρ≥ρp >>0.
• the limit state for moving the wall out of the soil (f < 0, Ea) is called the active earth pressure, t happens for ρ≤ρa <0.
• the state “at rest” (f ≅ 0, Eo) corresponds to oedometric conditions; it is a hypothetical situation when the wall was mounted into the soil without causing any changes in the initial state of stress, the so-called geostatic ones; then eo(z) = Ko⋅σz = Ko⋅γ⋅z. The lateral pressure coefficient Ko can be estimated on the basis of the theory of elasticity as Ko = ν/(1-ν) using the Poisson ratio ν; it is even more popular and reliable to use the Jaky formula (purely empirical, without deeper theoretical background)
Ko≅ (1-sinϕ) ⋅√OCR, where OCR stands for the overconsolidation ratio, OCR ≅ 1 for normally consolidated soils. For simple cases, both methods give convergent results, e.g. for a medium sand
ϕ=30o, ν=1/3 there is indeed 1-sin30o = (1/3)/(2/3) = 0.50.
It is a matter of fundamental importance what are values of dimensionless displacements ρp and ρa for which the limit states can be reached. There are typical subjective problems associated with asymptotic behavior of functions:
• estimation of ρa is less controversial: omitting signs, it is from 5mm/m for compacted noncohesive soils to even over 50mm/m for plastic cohesive soils,
• more provisional is the estimate of ρp: it is from 30mm/m for compacted cohesive soils up to even 200mm/m for plastic cohesive soils.
For heavily consolidated soils (OCR >> 1), the situation may be quite different, since for example Ko >> 1 and the values of ρa are much greater.
Conclusions:
in practical geotechnical bearing capacity situations (including retaining walls), the active limit state is very often present, because the displacements of structures needed for this are very small; with the passive limit state it is the opposite, usually there are intermediate earth pressures, somewhere between Eo and Ep, because
serviceability limit state conditions SLS do not allow such large displacements of soils and structures.
2.4. Friction on the wall surface
Fig.7
Fig. 7 presents a concrete block sliding on the surface of a non-cohesive soil, this can be easily reformulated to soil sliding on the concrete surface of the wall:
• or a perfectly smooth surface δ=0o, the force can be applied only perpendicularly,
• for a “real smooth” surface δ∼±1/3⋅ϕ, e.g. after removing smoothly finished formworks,
• for a “real rough” surface δ∼±2/3⋅ϕ, e.g. after removing rough wooden formworks (boarding),
• for a perfectly rough surface δ = ±ϕ.
Roughness of the wall surface is of great importance in static calculations, which EC7-1 correctly appreciates, because it affects the inclination of the earth pressure vectors E [kN/m] in Fig.3. However, it has as well an im- pact on the length of the vector itself, reducing the Ea values and
increasing (significantly) Ep values. For cohesive soils, the angle δ for the resultant force in Fig.7 is called the angle of external friction, this is a measure of surface roughness, |δ| ≤ ϕ. If mentally assumed δ > ϕ, so sliding of the "block" in Fig.7 would occur immediately just below the contact surface, i.e. in the subsoil, because there this angle certainly cannot exceed the angle of internal friction ϕ (in the standard Coulomb theory). In cohesive soils, δ > ϕ is a general law (another file on the WWW).
δ
δ +
Polish code PN-83/B-03010 recommends reduction of δ by approx. 15% for the passive pressure.
Finally, a very important question of the correct choice of the angle δ sign (in the accepted sign convention):
• if in Fig.7 the heavy block moves to the right (or soil to the left), then δ> 0,
• if in Fig.7 the heavy block moves to the left (or ground to the right), then δ <0.
Referring to Fig.2,3,4,5, the wall with the "attached" rigid triangle or trapezoid moves/rotates slightly to the left, at the same time a wedge is formed behind the wall, which has its weight so slightly moves down, therefore the tangential force due to friction on the wall surface is downwards, this means the angle δ measured from the perpendicular direction must be positive; in other words, the resultant E is above the normal to the surface, as correctly indicated in all drawings. These topics are also analyzed in detail in the Coulomb-Poncelet theory - in another file on this website (the symbol δ2 used there means the same as δ). For a massive wall with an angular internal surface profile (on the backfill side), the angle δ would probably be the same on each of the linear sections of the A-B-C-D profile, because there is the same concrete and the same backfill. For the analogous multilinear surface in Fig.3 this is not so, because the BC interval is only virtual, there is no contact of soil with concrete, but there is "a contact of soil with soil". Since this is an internal section in the backfill, not an external section in contact with concrete, the internal friction angle ϕ is used along the interval BC, not the external friction angle. Briefly speaking: correctly selected angles for a light wall with a rough surface
should be δ=+2/3⋅ϕ on the sections AB,CD and δ=+ϕ on section BC.
Final Note:
another sign convention for angles, and even the usual mirror image of Fig.2,3,4,5, completely changes the signs of angles δ, ε, β, thus "overturns" the results of calculations of forces E; be careful if using other sources, textbooks etc. where signs may be different.
2.5. Safety margins
For design purposes, there are two basic approaches to assure safe results for design purposes.
In a simplified traditional version, Ea and Ep are calculated with a very simplified but conservative assumptions that δ=0o, so as for perfectly smooth walls. Characteristic values of all parameters are in use and such are the results in terms of Ea and Ep. In this way, there is a hidden margin of safety for walls, because the destabilizing Ea is overestimated and the stabilizing Ep is underestimated. Years of practice are evidences that this method is generally safe and acceptable, in such sense that the results are similar to ones in the more general method of partial safety factors. There are some exceptions, however - note that this margin of safety can be illusive, for example for steel sheet-pile walls (Larsen) driven into saturated very fine loose sands, where the δangle has in fact relatively small values.
Methods recommended by the Eurocode EC7-1 use realistic δ-values from p.2.4 and next Ep is reduced (divided) by a resistance partial safety factor and Ea is increased (multiplied) by a load factor.
Unfortunately, the case seems vague, because EC7-1 suggests several interpretations (not to be confused with deign approaches DA, which are clarified by national regulations, like DA-2 * and DA3 in Poland) - one can analyse impacts or effects of impacts, as illustrated in another place on this website (GEO stability against sliding).
Overlooking of symptoms of a security risk often results from not considering some important patterns of loss of stability, e.g. from the unforeseen water damming or the global loss of stability of the slope (method by
Fellenius, Bishop or others) - which is not the subject of Lecture 5.
2.6. Earth pressure reduction
Instead of a summary – find the analysis which solutions can reduce the destabilizing soil pressure on the wall (most of the ideas just lower the value of Ka coefficient, but not all of them):
1) allow the wall to move slightly out of the backfill (active pressure as minimal backfill thrust),
9
2) use a backfill with a greater ϕ6,
3) reduce the slope angle ε, even locally behind the wall; reduce the load q on the backfill,
4) reduce β if possible, shape the wall with β < 0; if β ≤ ϕ-90o, then no wall would be needed at all, 5) the wall surface should be as rough as possible, δ ≅ ϕ,
6) reduce the backfill unit weight γ … although it is not very effective and may be in conflict with point 2), 7) use two walls as a cascade with H=2m each, instead of one with H=4m 7,
8) use relief cantilevers - the dashed black line indicates soil pressure on a smooth wall without such a shelf.
Details can be found in PN-83/B-03010,
as well as in a publication entitled "Czy istnieje …”
on my website in a separate file.
Example:
Draw a probable plot of soil pressure ea(L) [kPa] along a bi-linear profile of the very rough wall surface.
Pay special attention to angles to the normals, proportion of values and discontinuities at the corner B.
Solution:
δ >> 0 is the same on both intervals; all depends on β angles.
On the left - βAB > βBC, so ea values are greater on AB than on BC (pt. no.4).
On te right βAB < βBC, so ea values are greater on BC than on AB (pt. no.4).
The increase of ea(L) along each interval is faster on less steep parts.
Testing questions:
1. The self-weight of the wall G in Fig.3 is stabilizing for two reasons when checking the stability of a wall against sliding. For which ones?
2. If a rough rigid wall is pushed into the soil, is the angle δ positive or negative? (signs and figures as above).
3. Why for β≤ϕ-90o "no wall would be needed at all "?
6 In particular - apply geosynthetic reinforcement in the backfill; the angle ϕ = 30o of the backfill will increase a little, but the reinforced soil takes macro-properties of some "cohesion" and may have parameters similar to ϕ = 33o and at the same time c = 30kPa; as the effect, the (not very high) wall may be unnecessary at all, or rather it will transform into a lining of
reinforced soil.
7 like 2⋅½⋅Ka⋅γ⋅22 < ½⋅Ka⋅γ⋅42, since 2⋅4 < 16.
A
B
C
A
B
C
