Geotechnical Analysis of Stepped Gravity Walls
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1 Geotechnical Analysis of Stepped Gravity Walls Baleshwar Singh 1 * and Birjukumar Mistri 2 1 Associate Professor, Civil Engineering Department, IIT Guwahati, India 2 Former Post-Graduate Student, Civil Engineering Department, IIT Guwahati, India ABSTRACT Retaining walls are structures that help to maintain different elevations of the ground surface on either side of it. When the height of cut, backfill or slope is beyond a certain limit, a single gravity retaining wall becomes unfeasible and uneconomical. In such situations, stepped gravity walls can be an attractive option. The wall at each level of the retaining system should be checked for safety against sliding, overturning and bearing capacity failure. Apart from these, a global stability check is necessary. The paper outlines the analysis approach for such stepped retaining walls that can be used for various purposes in site development. Keywords: Gravity wall, Stepped retaining wall, Global stability *Author for Correspondence baleshwar@iitg.ernet.in 1. INTRODUCTION Ground surface is stable so long as it is generally horizontal or is characterized by mild slopes. From the available topography and geotechnical data, the stability of the sloping terrain can be evaluated. If the existing slope is unstable, either the gradient can be suitably reduced or retaining walls can be adopted. Structures built to maintain different levels of earth surface on either side of it are called retaining walls. In the absence of such a structure, the retained soil would move downwards till it achieves a stable slope. They are also required in several projects involving site development in sloping terrains. They are classified in terms of their relative mass, flexibility and anchorage conditions. They can be made from different materials such as reinforced concrete, masonry, gabions, etc. Depending on their structural features, retaining walls can be of several types such as gravity walls, cantilever walls and counterfort walls. Gravity walls are the oldest and simplest type of retaining walls, in which the stability is by virtue of its own self-weight. It can be constructed easily by using stone/brick masonry or mass lean concrete. It may be sloped either on a single face or on both the faces. They are thick and stiff enough that they do not bend. Their movement occurs by rigid body translation and rotation. Cantilever walls bend, translate and rotate, and they rely on their flexural strength to resist lateral pressures. Counterfort retaining walls are similar to cantilever walls except that there are bracings which are built at regular intervals along the wall. These bracings tie the wall and base together and then reduce bending moments and shear ISSN: STM Journals All Rights Reserved Page 1
2 stresses in the wall. Both cantilever and counterfort walls are made of reinforced concrete. earth pressure develops as the wall moves towards the soil producing compressive strain in the soil. Site investigation is necessary to understand the soil and drainage characteristics of the existing slopes at the location of the proposed walls. If the height of slope or backfill is excessive, a single gravity wall becomes inefficient and uneconomical. Instead, stepped gravity walls constructed at different elevations can be used to retain the soil effectively. The walls are to be constructed from bottom upwards one over the other by ensuring that the top of a lower wall and the bottom of a next upper wall have a minimum or required clearance for a walkway. Even more than three steps can be incorporated based on site conditions and requirements. This paper covers the analysis of stepped gravity retaining walls that can be adopted in site development. 2. GEOTECHNICAL ASPECTS The analysis and design of gravity retaining walls requires knowledge of the static earth pressure acting on the back of the wall from the retained soil backfill. The magnitude of earth pressure itself is a function of the magnitude and nature of the absolute and relative movements of the wall and the soil. Active earth pressure develops as the wall moves away from the soil thereby inducing extensional lateral strain in the soil. Passive If H is the height of the wall, deflections of the order of H and H are required to develop active pressure in cohesionless and cohesive soils, respectively. For the development of passive pressure, a deflection of about 0.05 H is required in cohesionless soils, whereas deflections needed in cohesive soils are not clearly defined. In addition to the earth pressure, there may be uniform surcharge load, live load or vehicle load acting over the retained soil. In earthquake-prone areas, the walls are also required to withstand seismic forces. Gravity walls can fail by rigid-body mechanisms such as sliding, overturning or bearing failure. Sliding occurs when horizontal force equilibrium is not satisfied. In this, the lateral pressure on the back of the wall produces a thrust that exceeds the available passive pressure acting in the front and sliding resistance on the base of the wall. Overturning occurs when moment equilibrium is not maintained, in which bearing failure at the base is often involved. The bearing pressure is obtained from the resultant of forces at the base and is then compared with the safe bearing capacity of the ground. The walls can also be damaged by gross instability of the soils behind and ISSN: STM Journals All Rights Reserved Page 2
3 beneath them as a global slope stability failure that encompasses the wall. Prediction of deformation is a complicated soil-structure interaction problem. Hence, deformations are rarely considered in analysis and design. The usual analysis approach is to estimate the forces acting on the wall and design the wall to resist them with a safety factor high enough to produce acceptably small deformations. The walls should be checked for external stability, which comprises of factors of safety against sliding, overturning, bearing and global slip failure. The external stability is determined by limit equilibrium analysis, in which the restoring forces are compared with disturbing forces and their ratio is expressed as the safety factor. Internal stability refers to the ability of individual parts to act as a single unit. This is ensured by adequate structural strength against compressive stress, shear stress and bending moment. Generally gravity retaining walls are made up of stones, brick masonry or mass concrete. To achieve strength and durability, good quality of stone and bricks should be used. The masonry used in the gravity wall cannot resist appreciable tension. The analysis is based on the assumption that wall can move sufficiently to keep the backfill in an active state. The first step is to fix basic dimensions of the wall and to estimate the various forces acting on the wall. The driving forces include the lateral pressure from backfill, any surcharge load, live load and earthquake load. The resisting forces are weight of the structure, passive pressure at the toe, and frictional resistance force at the bottom. Figure 1 shows the various forces acting on a gravity retaining wall. The passive earth pressure against the toe is generally ignored due to possibility of scour and erosion. Fig. 1: Forces Acting on Gravity Retaining Wall. ISSN: STM Journals All Rights Reserved Page 3
4 If there is a choice for the backfill, granular backfill should be preferred as there is less likelihood of the build-up of hydrostatic pressure under adequate provision of drainage. Clay backfills should be avoided as far as possible as they are susceptible to volume changes during rainy and dry seasons. When a change takes place from a simple wet condition to a fully saturated condition, the lateral pressure may increase by even up to 30%. Swelling leads to wall movements whereas shrinkage causes tensions cracks in the soil, which may subsequently get filled with water and cause considerable increase in the lateral pressure. Some form of filter material should be provided behind the gravity wall to prevent development of pore water pressure. Weepholes should be provided at regular intervals within the wall section to drain away the water collected in the filter Determination of Active Earth Pressure For calculating active earth pressure, Coulomb s theory can be used. The soil properties required for computation of the theoretical active earth pressure coefficient are the unit weight, angle of internal friction and cohesion intercept. Coulomb s equation for coefficient of active earth pressure, K A, is as follows: where β = slope of backfill, Ø = internal friction angle of soil, θ = inclination angle of back of wall with the vertical, and δ = soilwall interface friction angle. Vibration of ground caused by earthquake will increase the lateral earth pressure against the gravity wall. It decreases the stability of a slope by increasing shear stresses and pore pressures and by decreasing shear strength. As the actual loading on the gravity walls during earthquakes is complicated, seismic pressures on the walls are usually estimated using simplified methods. The dynamic pressure acting on the wall can be computed by using the pseudostatic method of Mononobe-Okabe, which is a direct extension of the static Coulomb theory. In this, the effects of the earthquake are represented by constant horizontal and vertical accelerations that produce inertial forces, which act through the centroid of the failure mass. The coefficient of dynamic active earth pressure, K AE, is given by ISSN: STM Journals All Rights Reserved Page 4
5 where ψ = tan -1 [k h /(1 k v )], k h = horizontal pseudostatic coefficient, and k v = vertical pseudostatic coefficient. Earthquake effect need not be considered if the wall is located in an area of low seismic activity Effect of Surcharge If a uniformly distributed surcharge load of intensity q per unit area acts on the backfill surface, the effective vertical pressure at any depth is increased by the same magnitude q. This causes a uniform increase of active.. (2) earth pressure throughout the back of the wall and is equal in magnitude to K A.q. For other types of surcharge loading such as line loading and strip loading, the theory of elasticity is used to determine the active earth pressure. According to this approach, the horizontal stress at any depth Z on a retaining wall of height H caused by a line load of intensity q per unit length acting at a distance X from the wall is given by for a > (3) for a (4) where a = X/H and b = Z/H. Each line load gives rise to a curved lateral pressure diagram which varies from zero value at the top of the wall to a maximum value and then decreases gradually towards the base of the wall. 3. ANALYSIS OF TOP WALL The analysis starts from the top wall and forces are transmitted to lower walls step by step. The top wall of stepped gravity walls can be analyzed in a similar way to that of a single-level wall. The analysis is based on the assumption that the wall can move far enough to reduce backfill pressure to the active state. This will be the most economical design. In the first step, the dimensions of the wall are selected tentatively so as to estimate the forces that act on it. The forces include lateral earth pressure, weight of the structure, weight of any backfill portion that acts directly over the structure, and effects of surcharge and earthquake. Once the forces acting on the wall have been determined, calculation of the safety factors against sliding, ISSN: STM Journals All Rights Reserved Page 5
6 overturning, and bearing capacity failure can be carried out. wall itself. The minimum safety factor against overturning should be at least Check against Sliding Sliding failure involves outward translation of the wall along the base or along the soil below the base. The active earth pressure is resisted by friction between the wall base and soil below it plus any passive resistance of embedded toe of the wall. The friction between the wall base and soil is equal to the effective normal pressure on the base times the tangent of the friction angle between the base and soil. Site-specific data should be used. The unit weight may range up to 17 kn/m 3 for cohesionless soils and up to 19 kn/m 3 for cohesive soils. The friction angle can be taken not less than 30 for a coarse-grained soil containing no silt or clay and as 24 for a coarse-grained soil containing silt. Shear key can be provided at the base to improve the sliding stability of wall. The safety factor against sliding must not be less than 1.5. The passive resistance against the toe of the wall can be ignored where there is possibility of scour and settlement Check against Overturning The rotation of the wall about its toe due to lateral pressure is termed as overturning. The safety factor against overturning is the ratio of moment of driving forces to the moment of resisting forces. The driving forces are lateral back pressure and surcharge pressure. The resisting forces are self-weight of the 3.3. Check against Bearing Failure The wall should be dimensioned in such a way so that no tensile forces are allowed to develop in the base. The pressure must be compressive over the entire width of the base. To achieve this, the resultant of the forces should fall within the middle third width of the base. Further, the maximum base pressure which occurs at the toe of the wall must not exceed the safe bearing capacity of the underlying soil. Since the load reaching the foundation is inclined, appropriate inclination factors should be used for evaluating the ultimate bearing capacity. 4. ANALYSIS OF LOWER WALLS For the analysis of any lower wall below the top wall, in addition to the forces of earth pressure and uniform surcharge, additional horizontal force to be considered is due to the base pressure of the upper wall. The base pressure diagram from the upper wall is first divided into a number of strips parallel to the wall. Each strip load is then expressed as a line load. For each line load, a curved lateral pressure diagram is obtained. Thereafter, at equal vertical intervals along the height of the lower wall, add up the lateral pressures on account of all line loads so as to determine the total horizontal pressure distribution due to the base pressure of the upper wall. ISSN: STM Journals All Rights Reserved Page 6
7 4.1. Check for Global Stability The overall stability refers to stability of the soil below and behind the wall. If stepped gravity walls are to be used to retain a slope, the possibility should also be considered that a failure may occur by sliding along a curved surface that passes beneath the walls and behind the backfill. Hence, a check for global stability must be included in the analysis. Various solution techniques with the method of slices can be employed for this [1]. The primary difference among all these methods lies in the equations of statics, consideration of normal and shear forces, and the assumed relationship between the interslice forces. In this study, Bishop s simplified method is used which is based on the effective stress approach. In this method, a circular slip surface of sliding is first defined, and the soil mass above the assumed slip surface is divided into a number of vertical slices of equal width. Each slice is acted upon by its total weight, interslice normal forces, shear force acting along the base, and normal force acting on the base. Equilibrium equations are written and solved for each slice. This method satisfies overall moment equilibrium but not overall horizontal force equilibrium. The total resisting moment and the total driving moment about the center of rotation of the slip circle are then computed to determine the safety factor. As the determination of the minimum safety factor is very crucial for design, it is important to locate the most critical slip circle. It should also be ensured that no slip circle intersects any of the stepped gravity walls. This is a trial-and-error procedure in which the centre of the slip circle, the radius, and the intercept distance in front of the toe of the lowest wall are varied. A minimum safety factor of 1.3 should be used for walls designed for only static loads. If earthquake loading is also considered, the minimum safety factor must be ANALYSIS EXAMPLE A sloping terrain of 5.0 m elevation is to be developed into a park with gardens and lawns at various elevations. For this purpose, it is decided to construct three steps of gravity retaining walls. Clearances are to be provided between any two tiers of the walls for provision of walkways. Firstly, determine the natural backfill soil parameters, surcharge loading and the drainage arrangements to be used. There is a surcharge load of 4 kn/m 2 acting at the top of the slope. The angle of internal friction of the backfill soil is 30 and cohesion is 12.5 kn/m 2. The unit weight of the soil is 16 kn/m 3, and soil-wall friction angle is 28. The safe bearing capacity of soil is 100 kn/m 2. The unit weight of the stone masonry used is 24 kn/m 3. ISSN: STM Journals All Rights Reserved Page 7
8 Secondly, select tentative dimensions of the walls and then compute forces and stresses. As always, the analysis process is an iterative one using repeated trials until the desired safety factors are achieved. For global stability analysis, limit equilibrium software [2] has been used. The final layout of the walls and the composite loading diagram are illustrated in Figures 2 and 3, with dimensions shown in mm. The loading details on the top, middle and lower walls are depicted separately in Figures 4 6, respectively. Each step of the gravity walls is safe. The critical slip circle for global stability is shown in Figure 7. The details of each wall along with corresponding safety factors are Top Wall Depth in ground = 0.3 m Height above ground = 1.5 m Width at top of wall = 0.45 m Width at bottom of wall = 0.9 m Safety factor (sliding) = 2.25 > 1.5 (OK) Safety factor (overturning) = 2.2 > 2 (OK) Safety factor (bearing) = 2.92 > 1.0 (OK) Middle Wall Depth in ground = 0.3 m Height above ground = 1.5 m Width at top of wall = 0.45 m Width at bottom of wall = 0.9 m Safety factor (sliding) = 1.89 > 1.5 (OK) Safety factor (overturning) = 2.01 > 2.0 (OK) Safety factor (bearing) = 2.92 > 1 (OK) Bottom Wall Depth in ground = 0.45 m Height above ground = 2.0 m Width at top of wall = 1.2 m Width at bottom of wall = 2.4 m Safety factor (sliding) = 2.13 >1.5 (OK) Safety factor (overturning) = 3.55 > 2.0 (OK) Safety factor (bearing) = 2.1 >1.0 (OK) Overall Stability Global safety factor = 2.24 > 1.3 (OK) Fig. 2: Layout of Stepped Gravity Retaining Walls. ISSN: STM Journals All Rights Reserved Page 8
9 Fig. 3: Composite Loading on Stepped Gravity Retaining Walls. Fig. 4: Loading on Top Wall. Fig. 5: Loading on Middle Wall. Fig. 6: Loading on Bottom Wall. ISSN: STM Journals All Rights Reserved Page 9
10 Fig. 7: Check for Global Stability. 6. CONCLUSIONS The analysis of stepped gravity walls is an iterative procedure, and it is started with some tentative dimensions. The total height of the sloping terrain should be divided into vertical divisions and for each division, a separate wall should be constructed. Proper profiling of the walls is a must to economize the design. Adequate amount of weepholes are required to drain out any backfill moisture and avoid pore water pressure build-up. The analysis proceeds from the top wall and downwards to the bottom wall. A geotechnical assessment is necessary for determining correct total lateral active earth pressure diagrams. Global stability check should be carried out after designing individual walls. Dimensions of the walls are to be revised if necessary. REFERENCES 1. Duncan J. M. and Wright S. G. Soil Strength and Slope Stability. New Jersey. John Wiley & Sons SLOPE/W. A software package for slope stability analysis. Version 7. Geo-Slope International, Calgary, Alberta ISSN: STM Journals All Rights Reserved Page 10
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