printed on June 24, 2003 LECTURE 4 - LOADS II 4.1 OBJECTIVE OF THE LESSON

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1 LECTURE 4 - LOADS II 4.1 OBJECTIVE OF THE LESSON The purpose of this lesson is to continue the developments of the loads for bridge design required by the LRFD Specification by providing information on ice loads, earth pressures. 4.2 ICE LOADS General The Specification requires consideration of the following types of ice action. Dynamic pressure due to moving sheets or floes of ice being carried by stream flow, wind or currents and striking a pier. Static pressure due to thermal movements of ice sheets. Static forces may be caused by the thermal expansion of ice in which a pier is embedded, or by irregular growth of the ice field. Pressure resulting from hanging dams or jams of ice. Hanging dams are the phenomenon of frazil ice passing under the surface layer of ice and accumulating under the surface ice at the bridge site. The frazil ice comes typically from rapids or waterfalls upstream. The hanging dam can cause a back-up of water, which exerts pressure on the pier, and can also cause scour around or under the piers as the water flows at an increased velocity. Static uplift or vertical load resulting from adhering ice in waters of fluctuating level. The behavior of ice and the forces that it generates is a very complex issue and is not yet fully understood. When undergoing long-term changes in temperature and sustained loadings, ice can behave in a relatively plastic manner. Ice which is moving with a current can either create a substantial impact load on a pier or breakup when it hits the pier reducing the load. For purposes of classification, the commentary of the Specification uses the types of ice failure developed by Montgomery in The reference to Montgomery is given in the Specification. Thus, the following types of ice failure are considered: Lecture - 4-1

2 crushing - the ice fails by local crushing across the width of the pier. The crushed ice is continually cleared from a zone around the pier as the floe moves past, bending - for piers with inclined noses, a vertical reaction component acts on the impinging ice floe. This reaction causes the floe to rise up the pier nose and fail as flexural cracks form, splitting - where a comparatively small floe strikes a pier, stress cracks propagating from the pier into the floe split the floe into smaller parts, impact - if the floe is small, it is brought to a halt when impinging on the nose of the pier before it has failed by crushing over the full-width of the pier, by bending or by splitting, and buckling - for very wide piers, where a large floe cannot clear the pier as it fails, compressive forces cause the floe to fail by buckling in front of the pier nose Design for Ice The design for ice typically starts with the determination of the effective crushing strength of the ice (p). The following values are specified when cite specific information is not available MPa where break-up occurs at melting temperatures and the ice is substantially disintegrated in its structure, 0.77 MPa where break-up occurs at melting temperatures and the ice is somewhat disintegrated in its structure, 1.15 MPa where break-up or major ice movement occurs at melting temperatures, but the ice moves in large pieces and is internally sound, and 1.53 MPa where break-up or major ice movement occurs with the ice temperature, averaged over its depth, measurably below the melting point. As indicated in the commentary to the Specification, the values identified above are considerably less than ice values which may be measured under laboratory conditions. The lowest value is appropriate for piers where experience with similar sites indicates that ice forces are very low, but ice should be considered in the design at that location. The maximum value of 1.53 MPa is based on an observed history of bridges that have survived significant Lecture - 4-2

3 icing conditions, and these are detailed in the reference list in Section 3 of the Specification. Once the effective ice strength, P, has been determined for the given site, breakup conditions, the next step is to determine the horizontal force, F, resulting from the pressure of moving ice. This horizontal force is related to the effective strength by a series of expressions in S , which are also a function of the pier geometry. A distinction is made as to whether the horizontal force is caused by an ice flow failing by compression over the full-width of the pier, or whether the ice flow fails by flexure as it rides up an inclined ice breaking pier nose. The design expressions require an estimate of the ice thickness. The preferred method to determine the ice thickness is historical records of actual ice thickness at a potential bridge site over some number of years. When this data is not available, an empirical expression is provided in the commentary to S , which provides a means of estimating ice thickness, depending on the accumulation of freezing days at the site, per year, and the particular conditions of wind and snow apt to occur at the site. Snow cover is found to be an important factor in determining ice thickness. The snow cover in excess of 150 mm in thickness has been shown to reduce ice thickness by almost one-half. The specification permits the reduction in ice forces on small streams not conducive to the formation of large ice flows. The reduction is limited to not more than 50% of the design force. Once the ice force, F, has been determined, it is necessary to consider combinations of longitudinal and transverse forces acting on a pier. It would be unrealistic to expect ice to move exactly parallel to a pier, so that a minimum lateral component of 15% of the longitudinal force is specified. Piers are usually aligned in the direction of stream flow, usually assumed to be the direction of ice movement. Two design cases are investigated as follows: a longitudinal force equal to F shall be combined with a transverse force of 0.15F, or a longitudinal force of 0.5F shall be combined with a transverse force of F t. The transverse force, F t, shall be taken as: F t ' F 2 tan(β/2% θ f ) ( ) Lecture - 4-3

4 where: β = nose angle in a horizontal plane, for a round nose taken as 100 (DEG) θ f = friction angle between ice and pier nose (DEG) Both the longitudinal and transverse forces shall be assumed to act at the pier nose. Where the longitudinal axis of a pier is not parallel to the principal direction of ice action, or where the direction of ice action may shift, the total force on the pier shall be determined on the basis of the projected pier width and resolved into components. Under such conditions, forces transverse to the longitudinal axis of the pier is taken to be at least 20% of the total force Static Ice Loads on Piers Ice pressures on piers frozen into ice sheets shall be investigated where the ice sheets are subject to significant thermal movements relative to the pier where the growth of shore ice is on one side only, or other situations which may produce substantial unbalanced forces on the pier. Unfortunately, little guidance is available for predicting static ice loads on piers. Under normal circumstances, the effects of static ice forces on piers may be strain limited, but expert advice should be sought if there is reason for concern Hanging Dams and Ice Jams The frazil accumulation in a hanging dam may be taken to exert a pressure of to MPa as it moves by the pier. An ice jam may be taken to exert a pressure of 0.96x10-3 to 9.6x10-3 MPa. The wide spread of pressures quoted reflects both the variability of the ice and the lack of firm information on the subject Vertical Forces due to Ice Adhesion The vertical force on a bridge pier due to rapid water level fluctuation is given by: for a circular pier, in N: F v ' 0.3t 2 % Rt 1.25 ( ) Lecture - 4-4

5 for an oblong pier, in N/mm of pier perimeter: F v ' 2.3x10 &3 t 1.25 ( ) where: t = ice thickness (mm) R = radius of circular pier (mm) Equations and neglect creep and are, therefore, conservative for fluctuations occurring over more than a few minutes, but they are also based on the assumption that failure occurs on the formation of the first crack, which is non-conservative Ice Accretion and Snow Loads on Superstructures No specific ice accretion or snow loads are specified in the LRFD Specification. However, Owners in areas where unique accumulations of snow and/or ice are possible should specify appropriate loads for that condition. The following discussion of snow loads is taken from Ritter (1991). Snow loads should be considered where a bridge is located in an area of potentially heavy snowfall. This can occur at high elevations in mountainous areas with large seasonal accumulations. Snow loads are normally negligible in areas of the United States that are below mm elevation and east of longitude 105 W, or below mm elevation and west of longitude 105 W. In other areas of the country, snow loads as large as MPa may be encountered in mountainous locations. The effects of snow are assumed to be offset by an accompanying decrease in vehicle live load. This assumption is valid for most structures, but is not realistic in areas where snowfall is significant. When prolonged winter closure of a road makes snow removal impossible, the magnitude of snow loads may exceed those from vehicular live loads. Loads also may be notable where plowed snow is stockpiled or otherwise allowed to accumulate. The applicability and magnitude of snow loads are left to Designer's judgment. Snow loads vary from year to year and depend on the depth and density of snow pack. The depth used for design should be based on a mean recurrence interval or the maximum recorded depth. Density is based on the degree of compaction. The lightest accumulation is produced by fresh snow falling at cold temperatures. Density increases when the snow pack is subjected Lecture - 4-5

6 to freeze-thaw cycles or rain. Probable densities for several snow pack conditions are as follows, ASCE (1980): Table Snow Density CONDITION OF SNOW PACK PROBABLE DENSITY (kg/m 3 ) Freshly Fallen 96 Accumulated 300 Compacted EARTH LOADS General The Specification defines several broad classifications of walls which are also referred to herein. For reference, these definitions are repeated below: Abutment - A structure that supports the end of a bridge span, and provides lateral support for fill material on which the roadway rests immediately adjacent to the bridge. Anchored Wall - An earth retaining system typically composed of the same elements as non-gravity cantilevered walls, and which derive additional lateral resistance from one or more tiers of anchors. Mechanically Stabilized Earth Wall - A soil retaining system, employing either strip or grid-type, metallic or polymeric tensile reinforcements in the soil mass, and a discrete modular precast concrete facing which is either vertical or nearly vertical. Non-Gravity Cantilever Wall - A soil retaining system which derives lateral resistance through embedment of vertical wall elements and support retained soil with facing elements. Vertical wall elements may consist of discrete elements, e.g., piles, caissons, drilled shafts or auger-cast piles spanned by a structural facing, e.g., lagging, panels or shotcrete. Alternatively, the vertical wall elements and facing may be continuous, e.g., diaphragm wall panels, tangent piles or tangent drilled shafts. Prefabricated Modular Wall - A soil retaining system employing interlocking soil-filled timber, reinforced concrete or steel modules Lecture - 4-6

7 or bins to resist earth pressures by acting as gravity retaining walls. Prefabricated modular walls consist of individual structural units assembled at the site into a series of hollow bottomless cells known as cribs. The cribs are filled with soil, and their stability depends not only on the weight of the units and their filling, but also on the strength of the soil used for the filling. The units themselves may consist of reinforced concrete, fabricated metal, or timber. Rigid Gravity, Semi-Gravity and Cantilever Retaining Walls - A structure that provides lateral support for a mass of soil and that owes its stability primarily to its own weight and to the weight of any soil located directly above its base. This classification of walls includes: A gravity wall depends entirely on the weight of the stone or concrete masonry and of any soil resting on the masonry for its stability. Only a nominal amount of steel is placed near the exposed faces to prevent surface cracking due to temperature changes. A semi-gravity wall is somewhat more slender than a gravity wall and requires reinforcement consisting of vertical bars along the inner face and dowels continuing into the footing. It does not rely on the weight of the overlying soil for stability. It is provided with temperature steel near the exposed face. A cantilever wall consists of a concrete stem and a concrete base slab, both of which are relatively thin and fully reinforced to resist the moments and shears to which they are subjected. A counterfort wall consists of a thin concrete face slab, usually vertical, supported at intervals on the inner side by vertical slabs or counterforts that meet the face slab at right angles. Both the face slab and the counterforts are connected to a base slab, and the space above the base slab and between the counterforts is backfilled with soil. All the slabs are fully reinforced. 1. Several of these types of walls are illustrated in Figure Lecture - 4-7

8 Figure Illustration of Several Wall Types (from Das, B. M., Principles of Foundation Engineering, Brooks/Cole Engineering Division, 1984) Retained earth exerts lateral pressure on retaining walls and abutments. In general, the magnitude and distribution of the lateral earth pressure on such structures is a function of the composition and consistency of the retained earth and the magnitude of external loads applied to the retained soil mass. Typically, development of a design earth pressure considers the following: The type, unit weight, shear strength and creep characteristics of the retained earth; The anticipated or permissible magnitude and direction of lateral deflection at the top of the wall or abutment; The degree to which backfill soil retained by the wall is to be compacted; Lecture - 4-8

9 The location of the groundwater table within the retained soil; The magnitude and location of surcharge loads on the retained earth mass; and The effects of horizontal acceleration of the retained earth mass during an earthquake. The degree to which a wall (or abutment) is permitted to deflect laterally, and the characteristics of the retained earth are the two most significant factors in the development of lateral earth pressure distributions. Walls which are permitted to tilt or move laterally away from the retained soil permit the development of an active state of stress in the retained soil mass and should be designed for the active earth pressure. Walls which are restrained against movement (e.g., integral abutments) or walls for which lateral deflection and associated ground movements may adversely impact adjacent facilities (typically within a distance behind the wall less than about one-half the wall height) should be designed to resist the at-rest earth pressure, which may be 50% greater than the magnitude of the active pressure. Walls which may deflect laterally into the retained soil should be designed to resist the passive earth pressure, which can be 10 to 20 times greater than the active pressure. (The passive state of stress is limited, for all practical purposes, to lateral deflection of the embedded portions of flexible cantilever retaining walls into the supporting soil.) The lateral wall movement required to permit development of the minimum active earth pressure or maximum passive earth pressure is affected by the type of soil retained, as shown in Table where: = lateral movement at top of wall (achieved through rotation or translation) required for development of active or passive earth pressure (mm) H = wall height (mm) Lecture - 4-9

10 Table Approximate Values of Relative Movements Required to Reach Minimum Active or Maximum Passive Earth Pressure Conditions, Clough (1991) Values of /H Type of Backfill Active Passive Dense sand Medium dense sand Loose sand Compacted silt* Compacted lean clay* Compacted fat clay* *Not typically used to backfill highway structures Nearly all conventional retaining walls of typical proportions, except very short walls, deflect sufficiently to permit development of active earth pressures. Gravity and semi-gravity walls designed with a sufficient mass to support only active earth pressures will deflect (tilt or translate) in response to more severe loading conditions (e.g., at-rest earth pressures) until stresses in the retained soil mass are relieved sufficiently to permit development of an active state of stress in the retained soil. The most significant potential for development of at-rest earth pressures on such walls is on the stems of cantilevered retaining walls, where a rigid stemto-base connection may prevent lateral deflection of the stem with respect to the base in response to the lateral pressure of backfill soil retained above the base. For such a condition, excessive lateral earth pressures on the stem could conceivably lead to a structural failure of the stem or stem-to-base connection. A comparison of estimated actual lateral deflections to deflections required to mobilize active earth pressure on the stem of a cantilevered retaining wall backfilled with dense sand is provided in Tables and Table assumes that the full section modulus of the stem is effective in resisting bending, and that the base of the stem is fixed to the foundation slab. Table assumes that a reduced section modulus is effective in resisting bending to account for cracking and creep of the concrete. Neither Table nor Table includes lateral deflections that would occur due to differential settlement of the wall base slab. Lecture

11 Table Cantilever Wall Stem Deflections Using Full Section Modulus Estimated Stem Deflection Wall Height H (mm) Average Wall Stem Thickness t (mm) Under Active Earth Pressure /H (dim) Under At- Rest Earth Pressure /H (dim) Deflection Required to Mobilize Active Earth Pressure /H (dim) Note: Assumes Backfill φ = 37, k a = 0.25, k o = 0.40 and f' c = 27.6 MPa Table Cantilever Wall Stem Deflections Using Reduced Section Modulus Estimated Stem Deflection Wall Height H (mm) Average Wall Stem Thickness t (mm) Under Active Earth Pressure /H (dim) Under At- Rest Earth Pressure /H (dim) Deflection Required to Mobilize Active Earth Pressure /H (dim) Note: Assumes Backfill φ = 37, k a = 0.25, k o = 0.40 and f' c = 27.6 MPa Tables and indicate that, with the exception of very short walls, cantilever wall stems will generally deflect, crack Lecture

12 and creep sufficiently to permit mobilization of active earth pressures in backfill soils composed of relatively dense sand, which is the typical backfill material for such walls. For walls bearing on yielding (soil) foundation materials, tilting of the base slab due to settlement of the foundation soils will result in even greater deflections, such that active earth pressures will be mobilized even on the stems of very short walls. It is likely that, unless walls are otherwise restrained against rotation or translation or have a massive cross-section, active earth pressure conditions will be achieved on the stems of nearly all cantilever retaining walls backfilled with compacted granular soil, with the exception of very short walls (less than 1.5 m tall) bearing directly on rock. For walls retaining cohesive soils, the effects of soil creep may prevent the permanent establishment of active and passive earth pressure. Under stress conditions producing the minimum active or maximum passive earth pressure, cohesive soils continually creep, such that shear stresses within the soil mass are partially received. As a result, the movements indicated in Table are produced only temporarily. Without further movement, the lateral earth pressure exerted by a cohesive soil initially in the active stress state will increase eventually to a value approaching the at-rest earth pressure. Likewise, the lateral earth pressure exerted by a cohesive soil initially in the passive stress state will decrease eventually to a value approaching approximately 40% of the passive earth pressure Compaction When mechanical compaction equipment is operated within a distance behind a retaining wall equal to about one-half of the wall height, additional lateral earth pressures are induced on the wall due to the compaction effort. Excessive backfill compaction can increase lateral earth pressures to values significantly greater than the active or even at-rest lateral earth pressure. Such compactioninduced pressures continue to act, even after the compaction equipment has been removed due to the inelastic behavior of the soil. A typical lateral earth pressure distribution for an unyielding wall, including compaction-induced residual pressures, is shown in Figure Lecture

13 Figure Residual Earth Pressure after Compaction of Backfill Behind an Unyielding Wall (after Clough and Duncan, 1991) The induced residual pressures would be somewhat less on a flexible or unrestrained wall subjected to the same compaction loading conditions since lateral deflection or movement of the wall would permit partial relief of the stress in the retained soil. The heavier the compaction equipment and the closer it operates to the wall, the greater are the compaction-induced pressures. Therefore, the use of soils which are difficult to compact (e.g., fine-grained, moisture sensitive soils) and heavy compaction equipment immediately behind earth retaining structures is likely to cause unacceptably large lateral soil pressures and should be avoided.use of free-draining granular soils and light compaction equipment within a distance of H/2 behind retaining walls is usually specified to preclude development of excessive compactioninduced lateral earth pressures. If the use of heavy static or vibratory compaction equipment behind a retaining wall cannot be avoided, the residual compaction-induced lateral earth pressures should be estimated by available procedures (e.g., Clough and Duncan, 1991). Lecture

14 4.3.3 Earth Pressure As described in Article 4.3.1, the magnitude and distribution of lateral earth pressure on a retaining structure is primarily a function of the retained soil characteristics and the degree to which the wall tilts or translates in response to the loading. For most abutments and conventional retaining walls, the earth pressure distribution is assumed to increase linearly with depth in accordance with the following: p = k h γ g z 10-9 ( ) where: γ = unit density of soil (kg/m 3 ) p = lateral earth pressure (MPa) k h = lateral earth pressure coefficient taken as k a or k o, depending on the magnitude of lateral deflection (dim) (see Article 4.3.1) z = depth below backfill surface (mm) g = gravitational constant (m/sec 2 ) Although the lateral earth pressure due to the retained soil is assumed to increase linearly with depth, the resultant lateral load due to the earth pressure is assumed to act at a height of 0.4H above the base of the wall for conventional gravity retaining walls (where H is the total wall height measured from the top of the backfill to the base of the footing) rather than 0.33H, as would be expected for a linearly proportional (triangular) distribution. As a conventional gravity wall deflects laterally (translates) in response to lateral earth loading, the backfill behind the wall must slide down along the back of the wall for the retained soil mass to achieve the active state of stress. Experimental results indicate that the backfill arches against the upper portion of the wall as the wall translates, causing an upward shift in the location at which the resultant of the lateral earth load is transferred to the wall (Terzaghi, 1934; Clausen, 1972, and Sherif, 1982) For non-gravity cantilever retaining walls or other flexible walls which tilt or deform laterally in response to lateral loading, significant arching of the backfill against the wall does not occur, and the resultant lateral load due to earth pressure is assumed to act at a height of 0.33H above the base of the wall. Lecture

15 AT-REST PRESSURE COEFFICIENT, k o When a retaining wall is restrained against lateral movement or lateral movement of the wall is unacceptable, the lateral earth pressure coefficient (k h ) in Equation is taken as k o. For a normally consolidated soil, the at-rest lateral earth pressure coefficient, k o, can be computed by the following: k o = 1 - sin φ f ( ) where: φ f = effective stress angle of internal friction of the drained soil For overconsolidated soils, the at-rest lateral earth pressure coefficient is generally considered to vary as a function of the stress history, the value of k o increasing with increasing degree of overconsolidation in accordance with the following: k o = (1 - sin φ f )(OCR) sin φ f ( ) where: OCR = overconsolidation ratio (dim) Values of k o for various soil types and degrees of overconsolidation are presented in Table Lecture

16 Table Typical Coefficients of At-Rest Lateral Earth Pressure Coefficient of Lateral Earth Pressure, k o Soil Type OCR = 1 OCR = 2 OCR = 5 OCR = 10 Loose Sand Medium Sand Dense Sand Silt (ML) Lean Clay (CL) Highly Plastic Clay (CH) ACTIVE PRESSURE COEFFICIENT, k a When a retaining wall deflects laterally in response to loading by the retained earth, a wedge of the retained soil moves laterally and downward along the back of the wall, as shown in Figure Lecture

17 Figure Active Failure Wedge for Conventional Gravity and Cantilever Retaining Walls, Coulomb Analysis As the soil wedge moves, the shear strength of the soil is gradually mobilized along the failure plane shown in Figure When the full shear strength of the soil is mobilized, the additional force required to maintain the stability of the wedge (and the corresponding force acting on the wall) reaches a minimum value equal to the active earth pressure, P a. Prior to wall movement and mobilization of shear strength in the soil, the wall must support the at-rest earth pressure (Article ). One of two theories is generally used to estimate the active earth pressure on retaining walls. Rankine earth pressure theory neglects the vertical friction force applied to the surface of the wall by the retained soil wedge as it moves downward along the back of the wall. For the Rankine earth pressure theory, the active earth pressure resultant is assumed to have a line of action parallel to the backfill surface. Coulomb earth pressure theory accounts for the friction force exerted on the wall by the retained earth, which results in an inclination of the earth pressure resultant of δ with respect to the back face of the wall (for a gravity wall) or to the vertical pressure surface extending up from the heel of the wall (for a cantilever wall), as shown in Figure Typical values of δ, Lecture

18 the friction angle between the wall and backfill, are presented in Table Table Friction Angles Between Dissimilar Materials Interface Materials Friction Angle, δ (deg) Mass concrete on the following foundation materials: Clean sound rock Clean gravel, gravel-sand mixtures, coarse sand Clean fine to medium sand, silty medium to coarse sand, silty or clayey gravel Clean fine sand, silt or clayey fine to medium sand Fine sandy silty, non-plastic silt Very stiff and hard residual or preconsolidated clay Medium stiff and stiff clay and silty clay to to to to to to 19 Masonry on foundation materials has same friction factors Steel sheet piles against the following soils: Clean gravel, gravel-sand mixtures, well-graded rock fill with spalls Clean sand, silty sand-gravel mixtures, single-size hard rock fill Silty, sand, gravel, or sand mixed with silty or clay Fine sandy silt, non-plastic silt Lecture

19 Interface Materials Friction Angle, δ (deg) Formed or precast concrete or concrete sheet piling against the following soils: Clean gravel, gravel-sand mixtures, well-graded rock fill with spalls Clean sand, silty sand-gravel mixtures, single-size hard rock fill Silty sand, gravel or sand mixture with silt or clay Find sandy silt, non-plastic silt Various structural materials: Masonry on masonry, igneous and metamorphic rocks: dressed soft rock on dressed soft rock dressed hard rock on dressed soft rock dressed hard rock on dressed hard rock Masonry on wood in direction of cross grain Steel on steel at sheet pile interlocks 22 to to Both AASHTO Standard Specification (AASHTO 1992) and the LRFD Specification (AASHTO 1993) employ the Coulomb earth pressure theory. For the typical case when a retaining wall is permitted to deflect sufficiently to develop the active state of stress in the retained soil, the lateral earth pressure coefficient (k h ) in Equation is, therefore, taken as the Coulomb active earth pressure coefficient, k a. For the case of a vertical retaining wall and a horizontal backfill surface, the value of k a can be obtained from Figure For theoretical solutions like those shown in Table 1 and Figure 1, the angle of internal friction is denoted simply as φ. The value of φ, shown in these solutions, is to be interpreted as the effective stress friction angle, φ f, determined from a drained shear test, when analyses are performed using effective stresses, and the total stress friction angle φ, determined from an undrained shear test, when analyses are performed using total stresses. For longterm conditions, the earth pressures should be calculated using effective stresses, and adding water pressures as appropriate. Lecture

20 Figure Active and Passive Pressure Coefficients for Vertical Wall and Horizontal Backfill - Based on Log Spiral Failure Surfaces For the more general case of an inclined wall face and sloping backfill surface, the value of the k a can be obtained from Table Lecture

21 Table Value of k a for Log Spiral Failure Surface δ (DEG) i (DEG) β (DEG) φ (DEG) φ Studies have shown that the failure surface defining the soil wedge loading the wall is approximated more closely by a log spiral curve than a straight line. The values of k a, provided in Figure and Table , were, therefore, obtained from analyses using log spiral surfaces (Caquot and Kerisel, 1948). Both the AASHTO Standard and LRFD Specifications also provide guidance for estimating earth pressures on special types of earth retaining structures (e.g., non-gravity cantilevered walls, anchored walls and mechanically stabilized earth walls) and walls subjected to unusual loading conditions (e.g., passive earth pressures). Where unusual backfill geometries or surcharge conditions exist, the active pressure may be estimated using a graphical trial wedge procedure. Lecture

22 EQUIVALENT FLUID PRESSURE For simplicity, lateral earth pressure is often estimated as an equivalent fluid pressure, wherein the resultant of the earth pressure is equivalent to the resultant of a fictitious fluid exerting hydrostatic pressure on the wall. Where equivalent fluid pressure is used, the unit earth pressure (in MPa) at any depth is taken as: p = γ eq g Z 10-9 ( ) where: γ eq = equivalent fluid unit density of soil, not less than 480 (kg/m 3 ) Typical values of γ eq for design of walls up to 6.5 m in height are shown in Table Table Typical Values for Equivalent Fluid Unit Densities of Soil Type of Soil At-Rest γ eq (kg/m 3 ) Level Backfill Active ( /H = 1/240) γ eq (kg/m 3 ) Backfill with i = 25 At-Rest γ eq (kg/m 3 ) Active ( /H = 1/240) γ eq (kg/m 3 ) Loose sand or gravel Medium dense sand or gravel Dense sand or gravel Compacted silt (ML) Compacted lean clay (CL) Compacted fat clay (CH) The resultant lateral earth load due to the equivalent fluid pressure is assumed to act at a height of 0.4H above the base of Lecture

23 the wall footing for conventional gravity and semi-gravity retaining walls Presence of Water If soil mass retained by a retaining wall or abutment contains groundwater and the groundwater is not eliminated through drainage, a water table will develop behind and exert lateral pressure on the structure above the water table, the horizontal pressure is given as: P = k h γ g z 10-9 ( ) The presence of the water table behind the wall has two additional effects, as indicated below and in Figure The unit weight of the retained soil is reduced to its submerged or buoyant value: γ s = γ s -γ w ( ) As a result, the lateral earth pressure below the water table is reduced to: P = (k h γ s (z-z w ) + k h γ' s z) g 10-9 ( ) The retained water exerts a horizontal hydrostatic water pressure equal to: where: P w = γ w g z w 10-9 ( ) γ s = submerged unit density of soil (kg/m 3 ) γ s = total unit density of soil (kg/m 3 ) γ w = unit density of water (kg/m 3 ) k h = horizontal earth pressure coefficient (dim) z = depth below backfill surface (mm) z w = depth below water table (mm) P = horizontal earth pressure (MPa) P w = hydrostatic water pressure (MPa) Lecture

24 Figure Effect of Groundwater Table on Earth Pressure With some algebra, the equations for pressure below the groundwater surface can be rearranged for ease of computation as: P = [k h γ z + γ w z w - k h γ w z w ] g x 10-9 ( ) The location of the resultant corresponding terms containing "k h " is taken at 0.4 z or 0.4 z w above the design section defined by z and z w. The location of the resultant of the term γ w z w is taken as z w /3 above the design section. Where possible, the development of hydrostatic water pressures should be prevented through the use of free-draining granular backfill and/or by providing a positive means of backfill drainage, such as weep holes, perforated and solid pipe drains, gravel drains or geofabric drains. When groundwater levels differ on opposite sides of a retaining wall, seepage occurs beneath the wall. The effect of seepage forces is to increase the load on the back of the wall (and decrease any passive resistance in front of the wall). Pore pressures in the backfill soil can be approximated through development of a flow net or using various analytical methods, and must be added to the effective horizontal earth pressures to determine the total lateral pressures on the wall Surcharge Surcharge loads on the retained earth surface produce additional lateral earth pressure on retaining walls. Where the surcharge is uniform over the retained earth surface, the additional lateral earth pressure due to the surcharge is assumed to remain constant with depth and has a magnitude, p, of: p = k s q s ( ) where: Lecture

25 p = constant horizontal earth pressure due to uniform surcharge (MPa) k s = coefficient of earth pressure due to surcharge q s = uniform surcharge applied to the upper surface of the active earth wedge (MPa) For active earth pressure conditions, k s is taken as k a, and for at-rest conditions, k s is taken as k o. Where vehicular traffic is anticipated within a distance behind a wall equal to about the wall height, a live load surcharge is assumed to act on the retained earth surface. The uniform increase in horizontal earth pressure due to live load surcharge is typically estimated as: p = k s γ s g h eq 10-9 ( ) where: p = constant horizontal earth pressure due to uniform surcharge (MPa) γ s = unit density of soil (kg/m 3 ) k = coefficient of earth pressure h eq = equivalent height of soil for the design live load (mm) Equivalent heights of soil, h eq, for highway loading on retaining walls of various heights can be taken from Table Table Equivalent Height of Soil for Vehicular Loading Wall Height (mm) h eq (mm) # The tabulated values of h eq were determined based on evaluation of horizontal pressure distributions produced on retaining walls by the vehicular live loads specified in the LRFD Specification (AASHTO 1993). The pressure distributions were obtained from Lecture

26 elastic half-space solutions with Poisson's ratio of 0.5, doubled to account for the non-deflecting wall. Alternatively, the increase in horizontal earth pressure, p, on a retaining wall resulting from a live load surcharge, p, can be taken as: p ' 2p π (α&sinαcos(α % 2δ)) ( ) where: p = live load intensity (MPa) α = angle illustrated in Figure (RAD) δ = angle illustrated in Figure (RAD) Figure Horizontal Pressure on Wall Caused by Uniformly Loaded Strip Effect of Earthquake Lateral earth pressures on retaining structures are amplified during an earthquake due to horizontal acceleration of the retained earth mass. The Mononobe-Okabe method of analysis is a pseudo-static method which is commonly used to develop an Lecture

27 equivalent static fluid pressure to model seismic earth pressure on retaining walls. The Mononobe-Okabe method is contingent upon the following assumptions (Barker, et al, 1991): The wall is unrestrained and capable of deflecting sufficiently to mobilize the active earth pressure condition in the backfill; The backfill is cohesionless and unsaturated; The failure surface defining the active wedge of soil loading the wall is planar; and Accelerations are uniform throughout the soil mass. As indicated above, the Mononobe-Okabe method assumes that backfill soils are unsaturated and, as such, not susceptible to liquefaction. The potential for liquefaction should be evaluated where saturated soils may be subjected to earthquake or other cyclical or instantaneous dynamic loadings. The Mononobe-Okabe method (AASHTO 1992 and AASHTO 1993) considers equilibrium of the soil wedge retained by the wall as shown in Figure Figure Definition Sketch for Seismic Loading (after Barker, 1991) The result of the combined active static and seismic earth pressures is taken as: P AE = 0.5 x K AE γ g (1-k v ) H ( ) where the seismic active earth pressure coefficient, K AE, is: Lecture

28 s 2 (φ&θ&β) 2 βcos(δ%β%θ) 1% sin(φ%δ)si cos(δ%β%θ) ( ) where: γ = unit density of soil (kg/m 3 ) H = height of soil face (mm) φ = angle of internal of soil (deg) θ = arc tan (k h /(1-k v )) δ = angle of friction between soil and wall (deg) (refer to Article ) k h = horizontal acceleration coefficient (dim) k v = vertical acceleration coefficient (dim) ι = slope of backfill surface (deg) β = slope of wall back face (deg) g = gravitational constant (m/sec 2 ) For estimation of lateral earth pressures under earthquake conditions, the vertical acceleration coefficient, k v, is commonly assumed equal to zero, and the horizontal acceleration coefficient, k h, is taken as: k h = 0.5α ( ) for walls designed to move horizontally 250 α (mm), and k h = 1.5 ( ) for walls designed for zero horizontal displacement where: α = A/100 A = horizontal earthquake acceleration (percent of g) g = acceleration of gravity (m/sec 2 ) Lecture

29 Values of the earthquake acceleration coefficient, A, are shown in Figures S , S and S Although the Mononobe-Okabe method of analysis does not specify the point of application of the horizontal seismic earth pressure resultant, the resultant of the dynamic component of the earth pressure ( P AE ) is significantly higher on the wall than the static active earth pressure resultant (P a ), as indicated in Figure Both the AASHTO Standard Specifications (AASHTO 1992) and the LRFD Specifications (AASHTO 1993) indicate that the combined static and seismic lateral earth pressure can be assumed to be uniformly distributed with a resultant (P AE ) acting at the mid-height of the wall. The AASHTO Standard Specification and the LRFD Specification also provide guidelines for determination of passive seismic earth pressure on retaining structures which are being forced horizontally into the backfill, and methods for design of retaining structures for a limited tolerable displacement under earthquake loading rather than for zero permanent displacement as assumed in the Mononobe-Okabe method. As with many current methods of seismic analysis, the Mononobe-Okabe method neglects inertial forces due to the mass of the retaining structure, concentrating only on the inertia of the retained soils mass. For gravity structures which rely solely on their mass for stability, this assumption is unconservative. The effects of wall inertia on the behavior of gravity retaining walls is discussed further by Richards and Elms (1979), who conclude that the structure inertia forces should not be neglected. Richards and Elms suggest a design approach based on limiting wall displacements to tolerable levels, rather than designing for no movement, and computing the weight of wall required to prevent movements greater than specified. The work of Richards and Elms is incorporated into the tolerable displacement procedure presented in the AASHTO Specifications Reduction due to Earth Pressure In some cases, lateral earth pressures may reduce the effects of other loads and forces on culverts, bridges and their components. One such case is that of the top slab of a box culvert, for which the maximum bending moment in the top slab is reduced due to the effect of lateral earth pressure on the side walls. Such reductions in loading should be limited to the extent that the applied earth pressures can be expected to be permanent. In lieu of more precise methods of estimating the lateral earth pressure force effects, the AASHTO Standard Specification (AASHTO 1992) and the LRFD Specification (AASHTO 1993) permit a load effect reduction of 50% where earth pressures are present. Lecture

30 4.3.8 Downdrag When a point bearing pile or drilled shaft penetrates a soft layer subject to settlement (e.g., where an overlying embankment may cause settlement of the layer), the soil settling around the shaft exerts a frictional force, or downdrag, around the perimeter of the pile or shaft. This frictional force acts as an additional axial load on the pile or shaft and, if sufficient in magnitude, could cause a structural failure of the foundation element or a bearing capacity failure at the tip. The methods used to estimate downdrag loads are the same as those used to estimate the side resistance of shafts and piles due to skin friction, as the same load transfer mechanism is responsible for both Design of a Cantilever Retaining Wall The purpose of this example is to illustrate the application of the various load factors and load combinations. In order to fully develop the example, references to provisions for wall and footings, and references to geotechnical textbooks are required and so noted. Some rounding of numbers has been done so it may not be possible to exactly duplicate all values to the full precision shown. The cantilever retaining wall below has been proposed to support an embankment. Lecture

31 Figure Schematic of Example During the subsurface exploration, it was determined that the foundation soils are predominantly clay to a depth of 6 m below the proposed bottom of footing, and, therefore, a 150 mm blanket of compacted granular material was placed below the footing. Dense sand and gravel underlies the clayey foundation soils. Assume elastic settlement of the dense sand and gravel to be negligible. The proposed wall backfill will consist of a free draining granular fill. Assume the seasonal high water table to be at the bottom of the footing. Apply the vehicular live load surcharge (LS) on the backfill as indicated in the figure above. Determine the lateral pressure distribution on the wall and estimate the bearing capacity for the proposed design. Check the design for adequate protection against sliding and estimate the consolidation settlement of the underlying clay. Solution: Step 1: Calculate the Unfactored Loads with η = 1.0 (A) Dead Load of Structural Components and Nonstructural Attachments (DC) Unit Density of Concrete = 2,400 kg/m 3 Lecture

32 Figure Retaining Wall Area Designation Weight of Concrete W1 = (0.3)(4.5)(2,400)g = 31,784 N per m of length W2 = (0.5)(4.5)(0.2)(2,400)g = 10,595 N per m of length W3 = (0.5)(3.0)(2,400)g = 35,316 N per m of length DC = Sum of W1 - W3 = 77,695 N per m of length (B) Vertical Earth Pressure (EV) Weight of Soil P EV = W = (2.0)(4.5)(1,920)g = 169,517 N per m of length (C) Live Load Surcharge (LS) The live load surcharge shall be applied where vehicular load is expected to act on the backfill within a distance equal to the wall height behind the wall (S ). An equivalent height of soil for the design vehicular loading (heq) is estimated using Table , repeated below, and a wall height of 5 m. Lecture

33 Table Equivalent Height of Soil for Vehicular Loading Wall Height (mm) h eq (mm) # h eq = 907 mm Use soil density of the backfill = 1920 kg/m 3 Vertical Component of LS = (1920)(907)(9.81) x 10-9 m 3 /mm 3 = MPa For a width of 2m, P LSV = 34,167 N per m of length Assume an active earth pressure coefficient k a using Figure with a wall friction angle, δ = φ, repeated below. Lecture

34 Figure Active and Passive Pressure Coefficients for Vertical Wall and Horizontal Backfill - Based on Log Spiral Failure Surfaces k = k a = 0.29 using Equation S : Lecture

35 p = (k) (γ' 1 ) (g) (h eq ) (10-9 ) p = MPa Using a rectangular distribution, the live load horizontal earth pressure acting over the entire wall will be: P LSH = (0.0050)(5)(10 6 ) = 24,771 N per m of length (D) Horizontal Earth Pressure (EH) The basic earth pressure should be assumed to be linearly proportional to the depth of earth and is given by Equation S : p = k(γ' 1 )gz(10-9 ) Use k = k a (as above) At the base of the footing: p = (0.29)(1,920)(9.81)(5000)(10-9 ) p = MPa The horizontal earth pressure acting over the entire wall will be: P EH = (0.5)(0.0273)(5)(10 6 ) = 68,278 N per m of length Note triangular pressure distribution (E) Summary of Unfactored Loads Table Vertical Loads and Resisting Moments - Unfactored Item V N/m Arm about 0 m Moment about 0 N-m/m W1 31, ,017 W2 10, ,710 W3 35, ,974 P EV 169, ,034 P LSV 34, ,334 TOTAL 281,379 Lecture

36 Table Horizontal Loads and Overturning Moments Moment - Unfactored Item H N/m Arm about 0 m Moment about 0 N-m/m P LSH 24, ,928 P EH 68, ,555 For location of resultant for lateral earth pressure see S Step 2: Determine the Appropriate Load Factors Using Tables S and S given in this manual as Tables and , respectively: Table Load Factors Group DC EV LSv P LSH (active) EH Probable Use Strength I-a Strength I-b Strength IV Bearing Capacity (eccent.)&sliding Bearing Capacity (max. value) Bearing Capacity (max. value) Service I Settlement By inspection, the following conclusions can be drawn for the case of bearing capacity (maximum) Strength I-b will probably govern, however, the factored loads will also be checked for Strength IV. Lecture

37 Step 3: Calculate the Factored Loads Table Factored Vertical Loads Group/ Item Units W1 N/m W2 N/m W3 N/m P EV N/m P LSV N/m Total N/m V (Unf.) 31,784 10,595 35, ,517 34, ,379 Strength I-a Strength I-b Strength IV 28,606 9,535 31, ,517 59, ,235 39,731 13,244 44, ,848 59, ,759 47,677 15,892 52, , ,390 Service I 31,784 10,595 35, ,517 34, ,379 Table Factored Moments M v Group/ Item Units W1 N-m/m W2 N-m/m W3 N-m/m P EV N-m/m P LSV N-m/m Total N-m/m M v (Unf.) Strength I-a Strength I-b Strength IV 27,017 6,710 52, ,034 68, ,068 24,315 6,039 47, , , ,649 33,771 8,388 66, , , ,656 40,525 10,065 79, , ,747 Service I 27,017 6,710 52, ,034 68, ,068 Lecture