Determining the design values for the mechanical characteristics of the sol

Calculations will be carried out using the ULS in sustainable and accidental (seismic) situations, calculation approach 1.

➢ Fill soil φ = 40˚; C’ = 0 KN/m²; δ = 40˚ and γ = 17 KN/m³ (see soil backfill)

According to Eurocode 7, the partial coefficients: γ φ ′ = 1.25 ; γ c ′ = 1.4 ; γ δ = 1.25 and γ γ = 1.25 φ ′ d = 33.8 ; c ′ d = 0   KN / m 2 ; δ d = 33.8 .

➢ Foundation soil ö ′ f d n = 30 ; C ′ = 10   KN / m 2 ; δ f d n = 30 and γ d n = 14.3   KN / m 3 .

According to Eurocode 7, the partial coefficients: γ φ ′ = 1.25 ; γ c ′ = 1.4 ; γ δ = 1.25 and γ γ = 1.00 . φ ′ d , f d n = 24.79 ; c ′ d = 7.142   KN / m 2 ; δ d ′ f d n = 24.79 .

The components used in the pointed masonry retaining wall are of various types and origins. It is built of rubble stone and cement mortar (pozzolanic type CEM IV/B class 32.5, grey in color and produced in Uganda).

Weight walls are the most common type of retaining structure, and for this reason several approaches have been developed that focus on their dimensioning [29] . Weight walls are usually dimensioned using one of two approaches: one based on seismic lateral pressures or the other based on the permanent displacement of the structure. The first approach, which is commonly used, consists in applying the M-O method, adopting the same value of pseudo-static acceleration for the inertia forces of the active corner and all the masses of the structure. Eurocode 8 [28] (NF EN 1998-5-7.3.2.2 clause (4)) proposes, in the absence of specific studies, to calculate the horizontal (k_h) and vertical (k_v) seismic coefficients affecting all masses.

k h = α × s r (31)

k v = ± 0.5 ⋅ k h (32)

(if a v g a g is greater than 0.6)

k v = ± 0.33 ⋅ k h (33)

(if a v g a g is less or equal to 0.6).

a_vg: Vertical component of the design acceleration a_g.

r: safety factor, which takes on the values indicated according to the type of retaining structure. For walls less than 10 m high, the seismic coefficient must be assumed to be constant over the entire height. It is given in accordance with NF EN 1998-5.

S: characteristic parameter of the soil class defined in standard NF EN 1998-1-3.2.2.2, involved in determining the shape of the soil’s elastic response spectrum.

α = α g g (34)

(α ratio of the design acceleration a_g for class A soil to the acceleration g of gravity) with

α g = γ i × α g R (35)

α g R : Accélération maximale de référence (PGA);

γ i : Coefficients d’importance des accélérations sismiques selon l’EC8 (NF EN 1998).

The design values for the effects of actions E_d and resistances R_d and their combinations in seismic situations are determined in accordance with Eurocode 0 (EN 1990) [30] and Eurocode 7 [31] (EN 1997). Calculations focus on reliability in relation to the “non-collapse” requirement (ultimate limit state) and the “damage limitation” requirement (service limit state). The various values of the partial coefficients are given in Eurocode 7 (EN 1997-1). Partial factors for actions (and their effects) should normally be equal to 1.0 in accidental situations (EN 1997-1-2.4.7.1). However, the use of one approach or another for each type of structure also depends on one country or another in the Euro zone. In the case of the Democratic Republic of Congo, approach 1 could be used in the calculations.

² Verification of slidingstability

The general stability equation for the sliding stability of the wall on its base becomes:

V E d ≤ V R d + R p d (36)

V_Ed: Design value of the horizontal load (or horizontal shear force) resulting from the weighted horizontal actions acting on the wall. In the case of Figure 5, this is:

V E d = ( k h ⋅ W ) d + ( P A E ) d ⋅ cos ( δ d + θ ) + U d ⋅ cos θ (37)

(P_AE)_d: dynamic thrust calculation value total land;

U_d: calculation value for the resultant of hydrostatic and hydrodynamic thrusts;

R_pd: calculation value for the resultant of hydrostatic and hydrodynamic thrusts;

V_Rd: design value of the foundation’s resistance to sliding on the ground;

R: calculation resultant of uplift force due to water pressure under the base.

V R d = N E d ⋅ tan δ d (38)

V_Rd value for calculating the normal force on the horizontal basis.

In the case of Figure 5, we establish the following equalities:

N E d = ( 1 − k v ) ⋅ W d + ( P A E ) d ⋅ sin ( δ d + θ ) + U d ⋅ sin θ − U R d (39)

V R d = N E d ⋅ tan δ d (40)

V R d = N E d γ R h ⋅ tan δ d k γ φ ′ (41)

where δ d k : Angle of friction of the soil-structure interface under the base of the footing, which can be assessed in accordance with Eurocode 7.

² Checking the bearing capacity of the foundation soil (punching)

The general stability equation becomes for the punching stability of the wall on the foundation soil [28] [29] :

q E d ≤ q R d (42)

where: q_Ed is the design pressure exerted by the structure on the foundation soil and q_Rd, the bearing capacity of the foundation soil. The design pressure q_Ed, per metre of wall length, is calculated using the following equation:

q E d = N E d ( B − 2 ⋅ e ) × 1 (43)

e: eccentricity of the vertical load resultant V_Rd with respect to the wall’s axis of symmetry;

B: design width (base) of the weight wall’s surface foundation.

The bearing capacity of the subsoil q_Rd can be calculated using known relationships, considering the excess caused by the intensity, inclination and eccentricity of the seismic action.

² Checking rollover stability

Verification of overturning stability consists in ensuring that the resultant of the destabilizing moments M_Ed,dst in relation to the downstream lower edge (point ‘O’ in Figure 5) is less than or equal to the resultant of the stabilizing moments M_Ed,stb with respect to the same edge:

M E d , d s t ≤ M E d , s t b (44)

The forces that cause the destabilizing moments, in relation to Figure 5, are:

­ The horizontal component of total dynamic thrust: ( P A E ) d ⋅ cos ( δ d + θ ) ;

­ The horizontal component of the resultant of hydrostatic and hydrodynamic thrusts:U_d. (for a vertical wall, θ = 0 ˚ ( cos θ = 1 );

­ The horizontal inertia force due to the weight of the wall: ( k h ⋅ W ) d ;

­ Lifting force due to water pressure under the base: U_Rd.

The forces that contribute to the stabilizing moments, on the other hand, are:

­ The vertical component of total dynamic thrust: ( P A E ) d ⋅ sin ( δ d + θ ) ;

­ The vertical component of the resultant of hydrostatic and hydrodynamic thrusts: U d ⋅ cos θ ;

­ The resultant of the wall’s weight and vertical inertia force: ( 1 − k v ) ⋅ W d ;

­ The resultant of the total stop, if any: ( P P E ) d .

² Verification of the structure’s internal (structural) stability

The study of the structural stability of load-bearing walls is a question of material strength and is specific to each type of wall in terms of the forces (and displacements) acting on it. The stability of concrete and reinforced concrete load-bearing walls is assessed in relation to the provisions of Eurocode 2 [32] whereas the provisions of Eurocode 6 apply to masonry walls [33] .

The complete calculation of a retaining wall is a rather laborious task, since the dimensioning of the structure and its verification require a succession of long and iterative calculations. To achieve the right results as quickly as possible, it’s important to pre-dimension the wall’s geometric characteristics as accurately as possible.

The selected site leads us to consider the following values according to Figure 6:

➢ A useful height of H = 4.5 m;

➢ A plug in the ground t = 0.6 m;

➢ Free height h = 5.1 m;

➢ A top width b: From Figure 6, we have:

b = h 12 (45)

With

h = 12 b (46)

B ∈ [ h 3 ; h 2 ] Considering

B = h 3 (47)

with

h = 3 × B (48)

(1) and (2), we have

12 × b = 3 × B (49)

The wall is equipped with a network of barbicans for rapid evacuation of seepage water into the backfill.

Verification of sliding stability

The destabilizing actions are P A h γ and P A h q . They are weighted by unfavorable partial coefficients γ G = 1.35 and γ Q = 1.50 . Hence, H_Ed = 91.44855 KN/m

The stabilizing actions are P A v γ and W. They are all permanent and weighted by favourable partial coefficients γ G , f a v = 1.00 and the partial resistance coefficient is equal to γ R h = 1.00 . So

H R d = 16.00 + 37.825 B   KN / m (50)

The stability condition

H R d ≥ H E d (51)

Hence B ≥ 1.994   m .

Checking rollover stability

The destabilizing actions are M A h γ and M A h q . They are weighted by unfavorable partial coefficients γ G = 1.35 and γ Q = 1.50 . Hence M E d = 173.786 KNm/m.

The stabilizing actions are M A v γ and M_W. They are all permanent and weighted by favourable partial coefficient γ G , f a v = 1.00 . So

M R d = 34.657 B + 49.7952 B 2   KNm / m (52)

The stability condition requires that

M R d ≥ M E d (53)

The roots of this inequation are B₁ = -2.24 m and B₂ = 1.55 m.

The solution to this inequation is B ≥ 1.55 m.

Verification of punching stability

N = 46.786 + 110.565 B (54)

N R d = 46.786 + 110.565 B   KN / m (55)

The resultant of the moments is

Δ M = 49.7952 B 2 + 34.657 B − 173.7860 (56)

Load eccentricity e_B with respect to the axis passing through the center of the footplate:

e B = B 2 − Δ M N (57)

e B = 0.5 × B − Δ M N (58)

In this case, check that the eccentricity is in the middle third

| e B | ≤ B 6 (59)

The fictitious cross-section of the wall base for a unit length is:

B ′ = B − 2 e B (60)

and

L ′ = 1   m (61)

or

A ′ = B ′ × L ′ (62)

so

A ′ = B − 2 e B (63)

The average stress exerted by the wall under the various loads is:

q E d = N E d A ′ (64)

this implies

q E d = 46.786 + 110.565 B ( B − 2 ) e B (65)

The load-bearing capacity (ultimate stress) is calculated as follows:

➢ Determination of load-bearing capacity factors in drained condition N_q, N_c and N_γ

The formulas below are proposed by Eurocode 7, but other relationships are also accepted: δ d , f d n = 24.79 ∘ = φ ′ d , f d n ; N q = 10.428 ; N c = 20.143 ; N γ = 8.708 .

➢ Determination of resultant inclination factors in drained condition i_q, i_c and i_γ

Eurocode 7 defines three factors that take into account the inclination of the load resultant R with respect to the vertical. For very long walls: L ′ = ∞   m ; so m B = 2 .

H E d = 91.44588   KN / m and

N E d = 34.786 + 110.565 B (66)

A ′ = B ′ × 1 (67)

so

A ′ = ( B − 2 e B ) ⋅ m 2 (68)

After replacing the numerical values, the coefficients respectively take on the following expressions:

i q = [ 1 − 91.44855 46.786 + 183.748 B − 146.366 e B ] 2 (69)

i C = 1.106 ⋅ i q − 0.1060 (70)

i γ = [ 1 − 91.44855 46.786 + 183.748 B − 146.366 e B ] 3 (71)

It goes without saying that the coefficients i q ( B ) , i C ( B ) and i γ ( B ) are a function of and cannot be calculated directly. The ultimate load-bearing capacity can then be deduced from the relationship:

q u l t = ( N γ × i γ × γ f d n × B ′ 2 ) + ( N q × i q × γ f d n × t ) + ( N c × i c × C ′ d , f d n ) (72)

N γ = 8.708 ; N C = 20.413 ; N q = 10.428 ; γ f d n = 14.3   KN / m 3 ; C ′ d , f d n = 33 .8   KN / m 2 and t = 0.6 m.

Expression becomes:

q u l t = ( 79.6782 × i γ × B ′ ) + ( 89.47224 × i q ) + ( 689.9594 × i c ) (73)

With the partial resistance coefficient γ R v = 1.0 ; we find:

q R d = q u l t γ R v (74)

q R d = q u l t (75)

Punching stability means ensuring that:

q E d ≤ q R d (76)

The solution to this inequality is more complex than previous inequalities, given the dependence of several parameters on the variable B. It is therefore solved by iteration. The results are shown in Table 5, where the variable is the base width B. The objective is to determine the value of B that satisfies the stability condition

q E d ≤ q R d (77)

and the condition of the central third

| e B | ≤ B 6 (78)

From the results in Table 5 and Figure 7, it can be seen that punching stability is already achieved for B ≥ 3.1 m where B_min = 3.1 m where:

q E d ≤ q E d (79)

q E d q R d = 29.147 % (80)

e B ≤ e l i m = B 6 (81)

e B e l i m = 95.145 % (82)

➢ Interpretation of results

After verification of the three main stability criteria, the minimum dimensions found for the width of the footing (base) are:

1) Sliding stability: B ≥ 1.9945 m;

2) Rollover stability: B ≥ 1.55 m;

3) Punching stability: B ≥ 3.1 m.

Hence, the static dimensioning gives the following geometric dimensions for the wall:

Sole width: B = 3.1 m;

Width of small base: b = 0.775 m and Actual wall weight: W = 253.89 KN/m.

Checking stability criteria: Unlike static design, seismic design is based on accidental (non-permanent) project situations. Consequently, the partial coefficients of actions and their effects are taken to be equal to 1.0 (EN 1997-1 2.4.7.1(3)).

Verification of sliding stability

v Destabilizing actions are:

The horizontal component of static thrust: P A h = 66.142   KN / m ;

The horizontal component of dynamic thrust: Δ P A E h = 29.407   KN / m ;

The horizontal inertia force due to the weight of the wall:

F h = 14.612 B (83)

So

H E d = P A h + Δ P A E h + F h (84)

this implies

H E d = 95.549 + 14.612 B (85)

v The stabilizing actions are:

The vertical component of static thrust: P A v = 34.657   KN / m ;

The vertical component of dynamic thrust: Δ P A E v = 29.307   KN / m ;

The resultant of the wall’s weight and vertical inertia force:

W + F v = ( 1 + k v ) w (86)

So

H R d = γ G , f a v × ( P A v + Δ P A E v + W + F v ) × tan ( δ d , f d n ) γ R h (87)

γ G , f a v = γ R h = 1.0 and δ d , f d n = 24.79 ˚

This implies

H R d = 29.541 + 41.200 B (88)

The stability condition requires that:

H R d ≥ H E d (89)

The solution to the inequality is: B ≥ 2.482 m.

Checking rollover stability

Rollover stability is verified by the balance of moments around point o.

The moment of static thrust P A h : M A h = 118.923   KNm .

The moment of dynamic thrust Δ P A E h : Δ M A E h = 95.161   KNm .

The moment of inertia of the wall F h :

M F h = F h × y G (90)

hence

M F h = 34.875 × B (91)

So

M E d = M A h + Δ M A E h + M F h

M E d = 214.084 + 34.875 × B (92)

The stabilizing actions are:

The moment of vertical static thrust P A v :

M A v = P A v × B (93)

M A v = 34.657 × B (94)

The moment of dynamic thrust Δ P A E v :

Δ M A E v = Δ P A E v × B (95)

Δ M A E v = 29.307 × B (96)

The moment of resultant weight and vertical force of inertia:

M R w = 54.237 B 2 (97)

So

M R d = M A v + M R W

M R d = 63.964 × B + 54.237 × B 2 (98)

The stability condition requires that

M R d ≥ M E d (99)

The roots of this inequation are B₁ = −2.2719 m and B₂ = 1.7366 m.

The solution is B ≥ 1.7366 m.

Verification of punching stability

The resultant of the vertical loads acts unfavorably with respect to punching stability, and is therefore equal to:

N E d = 63.964 + 89.206 × B (100)

The resultant of the moments is:

Δ M = 54.237 × B 2 + 29.089 × B − 214.084 (101)

The eccentricity of load e_B with respect to the axis passing through the center of the footing:

e B = 0.5 B − Δ M N (102)

In this case, check that the eccentricity is in the middle third:

| e B | ≤ B 6 (103)

The fictitious cross-section of the wall base for a unit length is:

B ′ = B − 2 e B (104)

L ′ = 1   m

so

A ′ = B ′ × L ′ (105)

And

A ′ = B − 2 × e B (106)

The average stress exerted by the wall under the various loads is:

q E d = N E d A ′ (107)

The load-bearing capacity (ultimate stress) is calculated as follows:

➢ Determination of load-bearing capacity factors in drained condition N_q, N_c, N_γ

Bearing capacity factors in drained condition: N_q = 10.428; N_c = 20.413; N_γ = 8.708.

➢ Determination of resultant inclination factors in drained condition i_q, i_c and i_γ

The load inclination factors become: m_B = 2;

H E d = 95.549 + 14.612 × B (108)

N E d = 63.961 + 89.206 × B (109)

A ′ = B − 2 × e B (110)

c ′ d , f d n = 7.142   KN / m 2

φ d , f d n = 24.79 ˚

i q = [ 1 − 95.549 + 14.612 B 63.961 + 104.669 B − 30.927 e B ] 2 (111)

i C = 1.106 i q − 0.106 (112)

i γ = [ 1 − 164.009 + 14.612 B 109.794 + 104.669 B − 30.927 e B ] 3 (113)

The ultimate load-bearing capacity can then be deduced from the relationship:

q u l t = ( 62.2622 × i γ × B ′ ) + ( 89.472 × i q ) + ( 145.789 × i c ) (114)

With the partial resistance coefficient γ R v = 1.0 ; we find:

q R d = q u l t γ R v (115)

q R d = q u l t (116)

Punching stability means ensuring that:

q E d ≤ q R d (117)

Using the same procedure as for the static calculation, the results are presented in Table 6, where the only variable is the width of the footing (base).

From the results in Table 6 and Figure 8, it can be seen that punching stability is already achieved for B ≥ 3.40 m where B_min = 3.40 m.

q E d ≤ q R d (118)

q E d q R d = 87.6788 % (119)

and

e B ≤ e l i m = B 6 (120)

e B e l i m = 92.3133 % (121)

Assessing the impact of seismic action: Interpreting the results

1) Sliding stability: B ≥ 2.482 m;

2) Rollover stability: B ≥ 1.7366 m;

3) Punching stability: B ≥ 3.4 m.

Hence, the static dimensioning gives the following geometric dimensions for the wall:

➢ Width of sole (large base): B = 3.4 m;

➢ Width of small base: b = 0.85 m;

Actual wall weight (per meter length): W = 278.46 KN/m².

A comparison of the minimum dimensions for the two sizing approaches is shown in Table 7.

The two design methods show a difference of 24.47% in slip stability, 12.038% in tilt stability and 9.677% in punching stability. This difference is substantial and demonstrates the impact of earthquakes on gravity wall design in this study. The tilting stability condition is the highest and most dangerous. The article by P. A. Yadav et al. (2018) [6] presents the results of evaluation of retaining walls stability under static and seismic loads. Studies have shown that for slender walls with low backfill heights, the pseudo-static Mononobe-Okabe method provides results close to more rigorous dynamic analyses.

In this work, it is assumed that the natural climatic conditions corresponding to the exposure environment of these materials generally vary and thus produce certain thermal effects that may be characterized by a longer stabilization time.