S Denton, T Christie, J Shave, A Kidd
1
PD6694-1: RECOMMENDATIONS ECOMMENDATIONS FOR THE T HE DESIGN OF STRUCTURES SUBJECT TO TRAFFIC LOADING TO EN19971 S Denton, Parsons Brinckerhoff , Bristol, UK T Christie, Parsons Brinckerhoff , Bristol, UK J Shave, Parsons Brinckerhoff, Bristol, UK A Kidd, Highways Agency, Bedford, UK
Abstract This paper gives the background to the development of the provisions of PD PD 6694-1. It gives guidance on the application of PD 6694-1 where it is considered that further explanation ma y be helpful and identifies recommendations in PD 6694-1 6694-1 which involve design principles principles or procedures significantly different from those used in past practice. The paper covers the clauses in the PD 6694-1 relating to actions, spread foundations, buried structures and earth pressure on gravity gravity retaining structures and bridge abutments. Traffic surcharge and integral bridges are covered in detai l in companion papers, for which references are provided.
Notation The same notation is used as in the Eurocodes and PD 6694-1. Other symbols are defined within the clause in which they occur. The Clause numbers used in the headings of this paper are the Clause numbers in PD 6694-1 to which the text refers.
Introduction The recommendations given in PD 6694-1 (hereafter referred to as “the PD”) apply to structures that are subject to traffic surcharge and and other traffic loading. The recommendations therefore specifically relate relat e to the rules and partial factors given for "bridges" as opposed to "buildings" in the Eurocodes. Many of the principles described can however be applied to earth retaining structures that are not subject to traffic loading. BS EN 1997-1:2004 does does not specifically cover aspects a spects of design of some types of highway and rail structures such as integral bridges bridges and buried structures. Complementary design recommendations and guidance is therefore included in the PD. For highway structures, PD 6694-1 replaces BD 30/87 [2] (Earth Retaining Structures), BD 31/01[3] Buried Structures, BA 42/96 [1] Integral Bridges and BD 74/00 [5] Foundations. The design of reinforced earth structures is neither covered in BS EN 1997-1:2004 nor in the PD.
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Basis of Design (4) Dispersion of vertical load through fill (4.4) The justification for the 30 method of dispersing vertical loads is given later in this paper in relation to buried structure ( 10.2.7). The use of the 30 dispersion method may however be unsafe when the vertical pressures arising from it are favourable. For example, where sliding resistance is dependent on the load on the base slab, it may be unsafe to assume that part of the weight of the surcharge traffic behind the abutment is supported on the base slab because other dispersion modes including vertical soil arching can occur which may result in the vertical load being supported on the ground behind the base slab while the horizontal surcharge effect is still applied to the wall. When analysing the foundations for bearing pressure the vertical pressure on an abutment base slab due to the traffic surcharge may be favourable or unfavourable. In some cases the additional pressure may increase the toe pressures, but in other cases it may apply a large enough restoring moment to reduce the toe pressure. If the effect of the vertical pressure from traffic surcharge is favourable in respect of bearing pressure, it may be prudent to ignore it.
Model Factors on horizontal earth pressure at ULS (4.7) Following the publication of BS EN 1997-1:2004, concern was expressed that the ULS partial factors were significantly lower than those used in pre-Eurocode standards for bridge design. In particular it was seen that the effective ULS partial factor fL. f3 specified for horizontal earth pressure in BD 37/01 [4] equalled 1.5x1.1 = 1.65 compared with a F of 1.35 for the critical STR/GEO limit state, Design Approach 1, Combination 1 partial factors in the Eurocode (i.e. Set A1 in BS EN 1997-1:2004 and Set B in BS EN 1990:2002). This would mean that structures designed to BS EN 1997-1:2004 could be less robust than those designed in the past. To address this concern, the PD states that where it is required to maintain the same levels of safety as were applied in the past, a model factor Sd;K may be applied to the horizontal earth pressure (effectively to K a or K 0). The recommended value of the model factor was based on the ratio of the pre-Eurocode factors to the STR/GEO Combination 1 factor, namely 1.65/1.35 = 1.22 (rounded down to 1.2), to give similar design values for earth pressures. Its effect was examined for other ultimate limit states verifications. For sliding and overturning, BD 30/87, 5.2.4.2[2] references CP 2[8] in which it says, in relation to sliding: “…a factor of safet y of approximately 2 should be applied…” and “…the angle of friction below the base is equal to the angle of friction of the soil beneath the foundation”. On this basis, the required heel length Bheel for an abutment of height Z is given by: Bheel = 2 K a;k { Z 2/(2tan ' For a Eurocode design using the model factor S;dK :
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Bheel = Sd;K K a;d{ Z 2/(2tan ' cv)} From this is can be shown that, using the model factor and the relevant values of the partial factors, the Eurocode value of Bheel will not be less than the pre-Eurocode value if tan cv is not greater than about 0.9tan ' . In practice tan ' cv is almost invariably less than 0.9tan . For sliding resistance of an undrained foundation CP 2 [8] uses a similar method to the Eurocode. For the CP 2 [8] method with a factor of safety 2 on sliding: 2 H = Bcu where B is the base length, Z is the height of the wall and the horizontal action H = K a;k Thus, B = 2 K a;k { Z 2/(2cu)} In the Eurocode, for a retaining wall subject to permanent act ions and the model factor, G Sd;k H = Bcu/ M where in Design Approach 1, Combination 2 G = 1 and M = 1.4. B = 1.4 Sd;K K a;d{ Z 2/2cu} From this it can be shown that based on k = 33 for the backfill and the relevant values of the partial factors, the Eurocode base length will be approximately 5% longer than the CP 2 [8] base length if the model factor is included, and approximately 13% shorter if the model factor is not applied. The above comparisons apply to retaining walls subject to permanent earth pressure onl y. When surcharge, braking and acceleration are applied, the pre-Eurocode base lengths will theoretically be relatively longer. In practice though, the Eurocode surcharge action is so much larger than the pre-Eurocode surcharge action that it is unlikely that base slabs subject to the Eurocode surcharge will be shorter than base slabs designed in the past. Bearing resistance is frequently governed by settlement requirements at SLS for which the ULS model factor is irrelevant. For ultimate bearing resistance it is less easy to make a direct comparison between Eurocode and pre-Eurocode designs because of the number of different acceptable pre-Eurocode design methods available. Specimen comparative calculations have however shown that if the model factor is applied to the horizontal earth pressure, the Eurocode designs for bearing resistance will usually be comparable with pre -Eurocode designs. In relation to overturning, CP 2 [8] says "...in gravity walls the resultant thrust should not fall outside the middle third of the base, and for other types of wall a factor of safety of at least 2 against overturning is required". Overturning is not usually an issue with conventional gravity walls and abutments because the bearing resis tance under the toe will normally become critical before the structure overturns and the length of heel required to provided sliding resistance is usually sufficient to give an adequate restoring moment. Overturning could however become an issue with a mass gravity wall se ated on rock or a concrete slab and propped or keyed into the slab to prevent sliding as shown in Figure 1.
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Figure 1
For the above structure, considering overturning about A at ULS, the Eurocode effectively requires that the maximum design overturning moment should not be greater than the minimum design restoring moment: Y H G;soil;sup Sd;k MK ≤ X V G;conc;inf where H and V are characteristic actions and MK = K a;d/ K a;k 1.11 at EQU and 1.25 at STR/GEO combination 2 if ' is about 33o. The overall factor of safety is XV /YH which equals ( G;soil;sup Sd;K MK )/( G;conc;inf ). This equals (1.05 x 1.2 x 1.11/0.95) = 1.47 at EQU and (1.35 x 1.2 x 1.0/0.95) = 1.70 at STR/GEO Combination 1. These values reduce to 1.23 and 1.42 respectively if the model factor Sd;K is not applied. However, it can be shown that if this structure was designed to comply with the "middle third" rule at SLS then the factor of safety would automatically be 3.0. From the above comparisons it can be seen that the 1.2 ULS model factor compensates for the difference between the Eurocode and pre-Eurocode values of ULS partial factors in relation to earth pressure, sliding resistance and ultimate bearing resistance, and it is irrelevant in regards to settlement and overturning except in the unusual situation where a structure such as that shown in Figure 1 is not designed to comply with the middle-third requirement at SLS. The Eurocode surcharge loading for highway structures is substantially more onerous than the HA and HB surcharge used in the past, and as this will result in stronger rather than weaker structures, the 1.2 model factor is not required to be applied to the effects of traffic surcharge loading. The PD does not offer an opinion as to whether the pre-Eurocode standards were unduly conservative. The option to use the model factor is for designers and clients who wish to maintain past levels of safety in their earth retaining structures.
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Pile Foundations (6)
This clause is based on the recommendations in BD 74/00 Annex B [5].
Gravity Retaining Structures and Bridge Abutments (7) Earth pressures on retaining walls and abutments with inclined backfill (7.2.3 and 7.2.4) Active pressure on walls with long heels ( < ) (7.2.3) Figure 2(a) shows a structure retaining fill with a surface inclined at an angle . Figure 2(b) shows a plot of the horizontal thrust on plane CE as its inclination, increases from to 90o. As can be seen the thrust reaches a peak when = . In Appendix 1 to this paper it is shown that this peak thrust on CE (and therefore the horizontal thrust on the whole structure) is the same as the thrust on the vertical virtual face CD when on that face equals . This means that provided is less than , the total horizontal thrust on the structure can be found by calculating the horizontal thrust on CD taking = . Appendix 1 also provides a proof of the equation for given in the PD (which is the same as the corrected expression given in Clayton and Milititsky[9]). Values of based on this equation are shown in Figure 3. is not however used in the pressure calculation except to determine the minimum length of BC, (ABcot ), for which the = method is valid.
= E1
D
E2
E
e n a l p l a u t r i v l a c i t r e V
A
Critical thrust on structure K a; CD2/2 =
= ’
e = B
e
K a on CD = K a;
+ ’ – when =
90o
C
t s u r h t l a t A n C o z i r o H
1
E C
E C
2
E C
D C
900
Inclination of planes ( )
(a) Retaining wall with inclined fill
(b) Variation of horizontal thrust on virtual planes of differing inclinations when on these planes equals
Figure 2. Horizontal thrust on inclined backfill p lanes
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Example 1 (Long heel) A 6m high wall with a 4m heel supports backfill inclined at 20 o. Find the active characteristic horizontal thrust applied to the structure if = 18 kN/m 3 and ' = 35o: When = = 20o, K a from PD 6694-1, Table 4 equals 0.30 Height of virtual face CD = 6 + 4tan20 o = 7.46m Horizontal thrust = K aCD²/2 = 18x0.3x7.46²/2 = 150.3kN/m width.
Figure 3. Values of
Active pressure on walls with short heels (7.2.4) When > , (that is when BC is less than ABcot as in Figure 4(c)) the critical inclined virtual face CE is interrupted by the back face of the wall at H as shown in Figure 4(c) and the results described in the previous paragraph will no longer apply because AH in Figure 4(c) is a soil-to-wall surface and over that length will be w rather than ' . When = , as in Figure 4(b) on CD = . When = 90o, as in Figure 4(d), on CD equals w because CD is then coincident with BA and w is applied over the full height of the wall. When lies between and 90o, the effective value of on CD therefore lies between and w, and the effective value of K a on CD (that is K a;CD) will lie between K a; and K a;w as described in the PD. To determine the value of K a on CD for an intermediate position with < < 90, the shaded triangle AHG in Figure 4(c) was considered as a C oulomb wedge with = w on AH, and HCFG was considered as a four-sided wedge with = ' on all soil-to-soil faces. Using this model for a number of values of α it was found that although the oretically as α increased from to 90o the plot of K a;CD followed a curve between K a; and K a;w , in practice it was simpler and marginally conservative to assume that K a;CD increased linearly with from K a; to K a;w in this range
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Figure 4(e) plots the value of the thrust on plane CA and on the whole structure as the wall BA moves towards CD and increases from less than to 90 o. The arc PQ shows the theoretical increase in effective K a;CD as increases from to 90o and the chord PQ shows the linear variation in K a;CD assumed in the PD. F
D
D
E A
E
G
D
A
F
A,D
w
A
=
=
=
w
=
w < <
H = ’
B
e C
B
(a) <
B
C
C
B,C
(c) < <90o
(b) =
(d) = 90o
Critical inclined planes as BA approaches CD ≤ ≤ 90o K a;CD increases from K a; to K a;w
≤ K a;CD = K a; Horizontal thrust on structure equals K a;CD CD2/2 Horizontal thrust on CA as increases
Theoretical Q on CD = on CD =
Linear approximation
w
P
on CD = t s u r h t l a t n o z i r o h
(a)
<
(b)
(c)
(d)
> 900
Values of (e) Horizontal thrusts as BA moves towards CD Figure 4. Abutments supporting inclined backfill
In the above paragraphs it is assumed that w is less than . If w is greater than then K a; is greater than K a;w and K a;CD theoretically reduces from K a; to K a;w as increases from to 90o. As this effect is small and only significant when is very close to 90 o, the PD
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recommends that when w is greater than , K a on CD should be taken as K a; for all values of .
Example 2 (Short heel) A 6m high wall with a 1.5m heel supports backfill inclined at 20 o. Find the characteristic active pressure applied to the structure if = 18 kN/m 3 and ' = 35o and w is taken conservatively as 10o: = tan-1(6/1.5) = 76 o = from Figure 3 (above) (for = 20o and ' = 35o) = 70.8o. > When = = 20o, K a; from PD 6694-1, Table 4 equals 0.302 When = w = 10o and = 20, K a;w by wedge analysis or other means = 0.322 Increase in K a;CD = ( K a;w - K a;){( - )/(90- )} = (0.322-0.302){(76-70.8)/(90-70.8)} = 0.005 K a;CD = 0.302+0.005 = 0.307 Height CD = 6 + 1.5tan20 =6.55m Horizontal thrust = K aCD²/2 = K a6.55²/2 = 18x0.307x6.55²/2 = 118.5kN/m width
Walls with both and greater than 80o It can be seen from the equation for that when approaches ' , approaches 90o. It has been found that in the unusual situation when both and are greater than 80o (i.e. steep backfill with a very small heel), the value of K a;CD starts to increase towards K a; when α is approximately 80o (i.e. less than ). The PD covers this situation by artificially limiting the maximum value of to 80 o in 7.2.4. This limitation only affects the heel length at which the short heel effect becomes critical. It does not affect the value K a;. Compaction pressures (7.3.4) The pressure distribution shown by Clayton and Milititsky [9], Ingold[10] and others is reproduced in Figure 5(a). In practice the position of (A) usually occurs only a short way below the surface and it is simplest and only slightly conservative to consider the combined active and compaction pressure having the value top given in the PD and being constant from ground level to the level at which active pressure equals top as shown in Figure 5(b).
S Denton, T Christie, J Shave, A Kidd
z c = K a
9
2 P
top
0
z c
h = zK p
top =
z
hc
=
2 P
k a
h
= 'hrm =
z
A 1
2V
2 P
hc =
top K d
B zK a Active h = pressure
(a) Clayton[9] & Ingold[11]
h
= top
zK d Active h = pressure (b) PD 6694-1
Figure 5. Compaction pressures
Movement Required to Generate Passive Pressure (7.5) BS EN 1997-1:2004, Annex C effectively defines the relationship between mobilised passive pressure and wall movement by three points and one gradient (see BS EN 1997-1:2004, Annex C, Figure C.4). The three points are K o and zero movement, 0.5 K p and a movement of v2/h and K p and a movement of v/h. By the nature of passive pressure the gradient of the pressure/movement curve at full passive pressure is horizontal. This is illustrated in Figure 6. In devising a formula to replicate the curve illust rated in Figure C.4 in Annex C it was considered adequate to assume that the relationship was l inear between K o and K p/2 and the curve between K p/2 and K p was a cubic curve passing through the K p/2 and K p points and having a horizontal gradient at K p. The equation given in the PD achieves this. Inevitably in some cases the gradient of the curve at K p/2 is not identical to the gradient of the K o- K p/2 line. The errors resulting from this are considered to be small compared to the tolerance on v p/h and v2/h given in BS EN 1997-1:2004, Annex C.
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100% 90% 80%
B
70% 60% p K / K
A
50% 40% 30%
Hambly[10] values (see Table 1) 20% Ko
10% 0% 0%
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
V/h
Figure 6. Plots of K /K p against v /h using the empirical equation in the PD clause 7.5
Plots A and B in Figure 6 are based on the maximum and minimum values of v/h for K = K p and half K p given in section (a) of BS EN 1997-1:2004, Table C.2. It is clear from the range of figures quoted In Table C.2 and Table1 below, that the relationship between movement and earth pressure can be very variable. The values of the wall rotations required to mobilise full passive and half passive pressure given in BS EN 1997-1:2004, Annex C are very large compared to those given by Hambly [10]. For example, to develop a "conventional" half K p behind a 10m high wall the deflection at the top would be between 110mm and 200mm according to Annex C compared with about 50mm according to Hambly [10].
Hambly Annex C to BS EN1997-1:2004
Rotations required to mobilise Half K p 0.5% (approx)
Rotations required to mobilise Full K p 3% (approx)
1.1% - 2%
5% - 10%
Table 1 Comparison of rotation required to mobilise passive pressure
Traffic surcharge (7.6) The background to the surcharge model is described by Shave et al
Integral Bridges (9)
See accompanying paper Denton et al [14].
[13]
.
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Buried Concrete Structures (10)
This section (Clause 10) is based on BD 31/01 [3] updated in line with the Eurocode requirements.
Superimposed permanent load (10.2.2) The model factors Sd;ec to be applied for superimposed permanent load are taken from Figure 3.1 in BD 31/01 [3]. Dispersal of vertical loads (10.2.7) The dispersal of vertical loads through fill is given in the PD as 30 compared to 26.6 (2:1) in BD 31/01 [3]. The 30 value is taken from BS EN 1991-2:2003, 4.9.1, NOTE 2. The Boussinesq equation given in clause 3.2.1 (iii) of BD 31/01 [3] has been omitted from the PD as it underestimates the pressure when there is a rigid plane, such as the roof of a buried structure, located a short distance below ground level. The method in Table 2.1 of Poulos and Davis[12] gives pressures below a point load which are marginally lower than those found using the 30 dispersion method, as can be seen from Figure 7. t i n u r e p e r u d s a s o e l r p l a c i t r e V
1
30 degree method
0.8 0.6
Poulos and Davis
0.4 0.2
Boussinesq
0 -2.5
-1.5
-0.5
0.5
1.5
2.5
Horizontal distance
Figure 7. Comparison of the “30o” pressures with pressures based on the Poulos & Davis[12] and Boussinesq equations for 1 m fill depth.
Longitudinal road traffic actions (10.2.8) Traffic surcharge (10.2.8.1) The treatment of traffic surcharge for buried structures using the simplified model given in PD 6694-1, 7.6 is explained in Table 5, Note B of the PD. Its background is given by Shave et al[13]. Braking and acceleration (10.2.8.2) The reduction of the braking and acceleration actions with increasing depth of eart h cover given in PD 6694-1, 10.2.8.2 has been retained from BD 31/01 [3] in the absence of data to justify a revaluation. Longitudinal joints (10.5) The 0.15 H c limitation of deflection on segmental units is taken from BD 31/01 [3].
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Soil Structure Interaction Analysis of Integral Bridges (Annex A to PD 6694-1) See accompanying paper Denton et al[14].
Tables of Earth Pressures for Buried Structure (Annex B) Maximum pressure (B.2 and tables B.1 and B.2) The values of the earth pressure coefficients given in the tables in Annex A are based on the assumption that the maximum characteristic at rest earth pressure coefficient taking temperature and strain ratcheting into account, is 0.6, as in BD 31/01 [3]. This may be considered as backfill with a characteristic ' of 30 subject to an enhancement factor, F enh, of 1.2 to allow for temperature, strain ratcheting and other unfavourable effects. In addition, at ULS, the pressure coefficients are subject to the model factor Sd;K (also equal to 1.2) described in 4.7. Using the values of M given in the UK National Annex to BS EN 1997-1:2004 the values of the earth pressure coefficient K max given in Tables B.1 and B.2 in Annex B are derived as follows:
d (inc M) K 0 = (1-sin d) K 0 x F enh K max = K 0 F enh sdk
Characteristic
EQU
30o 0.5 0.6 0.6
27.7o 0.53 0.64 0.77
STR/GEO Comb 1 30o 0.5 0.60 0.72
STR/GEO Comb 2 24.8o 0.58 0.70 0.84
As traffic surcharge is not considered to be affected by temperature and strain ratc heting, and as Sd;k is not applied to traffic surcharge (see 7.6) the design value of the at rest pressure coefficient for traffic surcharge given in PD 6694-1, Annex B Tables B.1 and B.2 is simply the value of K o given in the table above.
Minimum pressure (B.3 and Table B.3) For members which are critical with minimum horizontal earth pressure, Table B3 is relevant. The minimum characteristic earth pressure coefficient of 0.2 is as in BD 31/01[3]. This is the characteristic value of K a for k = 30 multiplied by a reduction factor, F red = 0.6. The values of K min;d given in the tables are based on the equation: K min;d = F red(1-sin d )/(1+sin d ) where d = tan-1{(tan 30)/ M *} and M * = 1/ M . This gives approximately the same result as taking F red = 1, k as 38 with K a;k based on / = 0.66 from BS EN 1997-1:2004, Figure C.1.1 and M = as before. Figure C.1.1 gives values of K a as low as 0.13 for = 45o and = and it would therefore be prudent to ignore active pressure altogether if it was considered that backfills with high values of ' were relevant.
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Active pressure (B.4 and Tables B.4, B.5 and B.6) When braking or acceleration actions are applied, active pressure based on a characteristic ' of 30 is applied on the active face and K max is applied to the passive face as in Tables B.1 and B.2, except that as the single source principle in not applied at EQU and the K max actions are favourable in resisting longitudinal actions, the characteristic value of K max (= 0.6) is applied at EQU in Tables B.4 and B.5. Where it is necessary to increase the pressure on the passive face above the K max pressure to resist the longitudinal traffic actions it should be noted that movements related to specific values of K given in Table C.2 of Appendix C of BS EN 1997-1:2004 are substantially greater than those given by Hambly[10] (see Table 1 above).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
[14]
BA 42/96 Amendment No. 1(2003) The design of Integral Bridges, The Stationary Office, London BD 30/87 Backfilled retaining walls and bridge abutments, The Stationary Office, London BD 31/01 The design of buried concrete box and portal frame structures , The Stationary Office, London BD 37/01 Loads for highways bridges, The Stationary Office, London BD 74/00 Foundations, The Stationary Office, London BS EN 1991-2:2003 (incorporating Corrigenda December 2004 and February 2010) , Eurocode 1, Part 2, Traffic loads on bridges, BSi, London, UK BS EN 1997-1:2004 (incorporating corrigendum February 2009) Eurocode 7: Geotechnical design – Part 1: General rules, BSi, London, UK CP 2 (1951) Earth retaining structures, BSi, London, UK Clayton, C. R. I and Milititsky, J. (1986) Earth pressure and earth retaining structures Surrey university Press, London Hambly, E.C. (1991) Bridge Deck behaviour , 2nd edition, London: E& FN Spon Ingold, T.S. (1979) The effects of compaction on retaining walls. Géothechnique 29(3), 265-283 Poulous, H.G and Davis, E. H. (1974) Elastic solutions for soil and rock mechanics John Wiley & Sons, Inc, London Shave, J, Christie, T. J. C, Denton, S. and Kidd, A. (2010) Development of traffic surcharge models for highway structures, in Proceedings of Bridge Design to Eurocodes – UK Implementation, Ed. by S. Denton, Nov 2010, ICE, London. Denton, S, Riches, O, Christie, T. J. C. and Kidd, A. (2010) Developments in integral bridge design, in Proceedings of Bridge Design to Eurocodes – UK Implementation, Ed. by S. Denton, Nov 2010, ICE, London.
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Appendix 1 Derivation of Earth Pressures on Walls With Long Heels and Sloping Backfill This appendix provides a derivation of the critical wedge angle , the friction angle to be used on the vertical virtual face, and the resulting earth pressure coefficient K a, for a retaining structure such as that illustrated in Figure A1(a) where the backfill is sloping at an angle and the heel of the wall is long enough that > . Figure A1(b) shows the critical wedge ECF and the forces acting on it at the boundaries, R1 and R2, and the self-weight W .
(a)
(b)
Figure A1. Earth pressures with sloping backfill
Figure A2 comprises the Mohr’s circle for the critical wedge ECF as it reaches a critical state simultaneously along CE and CF (represented by points e and f ). Point d represents the stress at the vertical plane CD.
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Figure A2. Mohr’s circle for the critical wedge
The angle between planes EC and ED is (90- ), so this transformation of axes requires a rotation on the Mohr’s circle from point e to point d of eod = 2(90 ) (A1) ˆ
Triangle geo is right angled, hence
eo g = 90 '
(A2)
d oh = 90 '2(90 ) = 2 90 '
(A3)
ˆ
From (A1) and (A2), ˆ
The external angle of triangle gdo is therefore: od j ˆ
=
2 90 '
By considering triangles djo and gjo and using the sine rule, sin oj sin(d jo) = R = R sin(2 90 ' ) sin ' Leading to: sin = sin(2 90 ' ) sin ' Rearranging (A6) gives ˆ
=
sin 1 90 ' sin 1 2 sin '
(A4)
(A5)
(A6)
(A7)
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Now consider the equilibrium of the wedge component triangles EDC and CDF as shown in Figure A3.
Figure A3. Equilibrium of wedge component triangles
The forces R1 and R2 act at the 1/3 points of CE and CF, because there is a linear stress distribution along these lines. These points are vertically below the centroids of triangles EDC and CDF. The force R3 acts at the interface CD at an angle . For equilibrium the 3 forces for each triangle must intersect at a point. These intersection points must therefore be the 1/3 points on the boundaries CE and CF, as shown in Figure A3. Using similar triangles, the angle of the force R3 must be identical to the slope of the backfill 1, or: = (A8) From (A7) and (A8),
=
1 1 sin 90 ' sin 2 sin '
(A9)
By summing the angles around point O in Figure A2 to 360 degrees it can also be demonstrated that the angle e is related to by the expression in (A10):
e = 90 ' =
sin 1 90 ' sin 1 2 sin '
(A10)
Knowing the critical value of as given in (A9), the ratio of the horizontal to the vertical earth pressures may be determined from the Mohr’s circle in Figure A2:
1
An alternative derivation of (A7) considers the triangles of forces for the component triangle EDC and the full wedge ECF shown in Figure A3 and demonstrates that when = the horizontal thrust is equal in each case; however the derivation is more complex and so not included here for space reasons.
S Denton, T Christie, J Shave, A Kidd
1 h v
=
sin ' 1 sin '
17
sin2 ' = sin2 '
1 sin ' sin2 ' 1 sin ' sin 2 '
(A11)
However, while the ratio of stresses defined in (A11) could be thought of as an earth pressure coefficient, the values obtained from (A11) are not the same as K a defined in the conventional way as in (A12), based on the total horizontal force H acting on a vertical plane of height h and assuming a vertical earth pressure of z , where is the soil density and z is the distance below ground level.
H (A12) 1 2 h 2 H may be determined by considering the equilibrium of the wedge component triangle CDF in Figure A3, from which: K a
=
W 2
H = tan
1
(A13)
tan(e ' ) The weight W 2 is calculated based on the area of the triangle CDF: W 2
=
1 2
h 2
cos e cos sin(e )
(A14)
Combining (A8), (A12), (A13) and (A14) results in the following expression for K a:
K a
cos e cos
=
sin(e ) tan
tan(e ' ) 1
(A15)
A comparison of expressions (A11) and (A15) shows that they give almost identical values up to a slope angle of about half , but as the slope approaches the values diverge, with (A15) giving higher values. This difference is due to the way that K a has been defined in (A12), which is convenient for design purposes, but this definition of K a is not strictly the same as the ratio of horizontal and vertical pressures when the backfill is sloping. ’
’
The values for K a presented in Table 4 of PD6694-1 have been calculated from (A10) and (A15) for various values of and ’. Equation (A9) may be used to check whether > . If this is not satisfied (i.e. the heel is short) then the derivation above is not correct; the angle will lie somewhere between 0 and , and the thrust on the wall will need to be increased as described in PD6694-1.
