Retaining Wall

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<p>Retaining Walls</p> <p>Function of retaining wallRetaining walls are used to hold back masses of earth or other loose material where conditions make it impossible to let those masses assume their natural slopes. Such conditions occur when the width of an excavation, cut, or embankment is restricted by conditions of ownership, use of the structure, or economy. For example, in railway or highway construction the width of the right of way is fixed, and the cut or embankment must be contained within that width. Similarly, the basement walls of the buildings must be located within the property and must retain the soil surrounding the base.</p> <p>Types of retaining wallsFree standing retaining walls, as distinct from those that form parts of structures, such as basement walls, are of various types. The gravity retaining wall retains the earth entirely by its own weight and generally contains no reinforcement. It is used up to 10 ft height. The reinforced concrete cantilever retaining wall consists of the vertical arm that retains the earth and is held in position by a footing or base slab. In this case, the weight of the fill on top of the heel, in addition to the weight of the wall, contributes to the stability of the structure. Since the arm represents a vertical cantilever, its required thickness increase rapidly, with increasing height. It is used in the range of 10 to 20 ft height.</p> <p>Types of retaining wallsIn the counterfort wall the stem and base slab are tied together by counterforts which are transverse walls spaced at intervals and act as tension ties to support the stem wall. Counterforts are of half or larger heights. Counterfort walls are economical for heights over 25 ft. Property rights or other restrictions sometimes make it necessary to place the wall at the forward edge of the base slab, i.e. to omit the toe. Whenever it is possible, toe extensions of one-third to one-fourth of the width of the base provide a more economical solution.</p> <p>Types of retaining wallsA buttress wall is similar to a counterfort wall except that the transverse support walls are located on the side of the stem opposite to the retained material and act as compression struts. Buttress, as compression elements, are more efficient than the tension counterforts and are economical in the same height range. A counterfort is more widely used than a buttress because the counterfort is hidden beneath the retained material, whereas the buttress occupies what may otherwise be usable space in front of the wall.</p> <p>Types of retaining wallsThis is an free standing wall category. A wall type bridge abutment acts similarly to a cantilever retaining wall except that the bridge deck provides an additional horizontal restraint at the top of the stem. Thus this abutment is designed as a beam fixed at the bottom and simply supported or partially restrained at the top.</p> <p>Earth Pressure For liquid P=wwh, ww is the unit weight of water. Soil retaining structure Ph=Cowh w is unit weight of the soil C0 is a constant known as the coefficient of earth pressure at rest According to Rankin, the coefficient for active and passive earth pressure are</p> <p>Cos H C a ! Cos H Cos H Cos H C p ! Cos H Cos H </p> <p>Cos 2 H Cos 2 J Cos 2 H Cos 2 JCos 2 H Cos 2 J Cos 2 H Cos 2 J</p> <p> For the case of horizontal surface H=0 si J si J C ah ! C ph ! si J si J</p> <p>Earth pressure for common condition of loadingh y ! 3 1 P ! C ah wh 2 2</p> <p>h 3 1 p ! C a wh 2 2 For H ! J C a ! Cos J y !</p> <p>Earth pressure for common condition of loading</p> <p>h 2 3h hd y ! 3 (h 2hd ) 1 P ! C ah wh ( h 2 h d ) 2</p> <p>Stability Requirement1. Individual parts should be strong enough to resist the applied forces 2. The wall as a whole should be stable against (i) Settlem ent (ii) Sliding (iii) Overtur ning</p> <p>Stability RequirementSettlement It is necessary to ensure that the pressure under the footing does not exceed the permissible bearing pressure for the particular soil. By the formula If a "</p> <p>q max !min</p> <p>N</p> <p>MC s I</p> <p>N , compression will act throughout the section 3</p> <p>Stability RequirementSettlement N N a R v R v 2 2 q 1, 2 ! s 3 N N 12 2 2 R vN R vN a R vN 2R vN a R v R v 4 2 4 q 1, 2 ! s ! s 3 3 N N N N 12 12 R v 6R va R v 3R v ?N s 3 N 6 a A q 1, 2 ! ! s 2 2 N N N N R v ?4 N 6 a A &amp; q 2 ! R 2v ?6 a 2 NA q1 ! 2 N N N R v when a ! q1 ! q2 ! N 2 Unit dimension in the direction perpendicular to the paper.</p> <p>Stability RequirementSettlement</p> <p>N N N R v v Rv 2 3 2 q 1,2 ! s 3 N N 12 N N N 3N 2N R v v 1 12 6 R v v 2 1 12 6 2 q 1,2 ! R v s ! Rv s 3 3 N N N N 1 1 q 1,2 ! R v s N N 2R v q1 ! &amp; q2 ! 0 N</p> <p>Stability RequirementSettlement</p> <p> 1 R v ! q 3 a 2 2R v q! 3a</p> <p>Stability Requirement SlidingF = QRv F u 1 .5 Ph</p> <p>OverturningStabilizing moment u2 Overturning moment</p> <p>Basis of Structural Design1. Lateral earth pressure will be considered to be live loads and a factor of 1.7 applied. 2. In general, the reactive pressure of the soil under the structure at the service load stage will be taken equal to 1.7 times the soil pressure found for service load conditions in the stability analysis. 3. For cantilever retaining walls, the calculated dead load of the toe slab, which causes moments acting in the opposite sense to those produced by the upward soil reaction, will be multiplied by a factor of 0.9.</p> <p>Basis of Structural Design</p> <p>4. For the heel slab, the required moment capacity will be based on the dead load of the heel slab itself, plus the earth directly above it, both multiplied by 1.4.</p> <p>5. Surcharge, if resent, will be treated as live load with load factor of 1.7. 6. The upward pressure of the soil under the heel slab will be taken equal to zero, recognizing that for severe over load stage a non linear pressure distribution will probably be obtained, with most of the reaction concentrated near the toe.</p> <p>Drainage Reduction in bearing capacity. Hydrostatic pressure. Ice pressure Drainage can be provided in various ways</p> <p>i.</p> <p>Weep holes, 6 to 8 in. 5 to 10 ft horizontally spaced. 1 ft3 stone at rear end at the bottom weep holes to facilitate drainage and to prevent clogging. Longitudinal drains embedded in crushed stone or gravel, along rear face of the wall. Continuous back drain consisting of a layer of gravel or crushed stone covering the entire rear face of the wall with discharge at the ends.</p> <p>ii. iii.</p> <p>Problem A gravity retaining wall consisting of plain concrete w=144 lb/ft3 is shown in fig. The bank of supported earth is assumed to weigh 110 lb/ft3 =300 and to have a coefficient of friction against sliding on soil of 0.5.</p> <p>Solution1 sin J 1 sin 30 Ca ! ! 0 .333 1 sin J 1 sin 30</p> <p>1 si J 1 si 30 Cp ! !3 1 si J 1 si 30 Pah 1 1 2 ! C a wh ! ( 0 . 333 )( 110 )( 12 ) 2 ! 2637 . 36 l 2 2 12 Overtur i g mome t ! 2637.36 v ! 10549.44 l .ft 3</p> <p>Solution Restoring moment: oment about toeComponent Force Weights W1 W2 W3 W4 W5 (6)(4)(144) (1)(11)(144) 1 (4)(144)(11) 2 1 (4)(11)(110) 2 </p> <p>oment arm 864 x 3 1584 x 11 3168 v 1 . 5 v 4 3 2 2420 v 1 . 5 v 4 3 </p> <p>oment 2592 lb.ft 1584 lb.ft 8976 lb.ft 10083.33 lb.ft 3478.75 lb.ft M=26714.08 lb.ft</p> <p>(0.5)(11)(110)</p> <p>605 x 5.75 Rv = 8641 lb</p> <p>26714 .08 ! 2.53 " 2 O.K Safety factor against overturning = 10549 .44</p> <p>Solution Distance of resultant from the toe</p> <p>26714 .08 10549 .44 a! ! 1 .87 ft 8641 Thus resultant is out of kern distance 2R v ( 2 )( 8641 ) The max. soil pressure will be q ! 3a ( 3 )( 1 . 87 )</p> <p> q = 3080.57 lb/ft2 &lt; 4000 psf</p> <p>Solution Sliding Assuming that soil above footing toe has eroded and thus the passive pressure is only due to soil depth equal to footing thickness. Pph ! 1 1 C p wh 2 ! ( 3 )(110 )(1) 2 ! 16 5 lb 2 2</p> <p> Friction force between footing concrete and soil.</p> <p>F ! Q R v ! ( 0 . 5 )( 8641 ) ! 4320 . 5 lb4320 .5 165 ! 1 .7 " 1 . 5 F.O.S. against sliding = 2637 .36</p> <p>O.K</p> <p>Estimating size of cantilever retaining wall Height of Wall The base of footing should be below frost penetration about 3 or 4.</p> <p>Stem Thickness. Stem is thickest at its base. They have thickness in the range of 8 to 12% of overall height of the retaining wall. The minimum thickness at the top is 10 but preferably 12.</p> <p>Base Thickness Preferably, total thickness of base fall between 7 and 10% of the overall wall height. inimum thickness is at least 10 to 12 used.</p> <p>Estimating size of cantilever retaining wall</p> <p>Base Length For preliminary estimates, the base length can be taken about 40 to 60% of the overall wall height. Another method refer to fig. W is assumed to be equal to weight of the material within area abcd. Take moments about toe and solve for x.</p> <p>Problem Design a cantilever retaining wall to support a bank of earth of 16 ft height above the final level of earth at the toe of the wall. The backfill is to be level, but a building is to be built on the fill. Assume that an 8 surcharge will approximate the lateral earth pressure effect. Weight of retained material = 130 lb/ft3</p> <p>Angle of internal friction = 35o Coefficient of friction b/w concrete and soil = 0.4 fc=3000 psi fy=40,000 psi aximum pressure = 5 k/ft2 soil</p> <p>Solution Height of Wall Allowing 4 for frost penetration to the bottom of footing in front of the wall, the total height becomes. </p> <p>h = 16 + 4 = 20 ft.</p> <p>Thickness of Base At this stage, it may be assumed 7 to 10% of the overall height h. Assume a uniform thickness t = 2 ( 10% of h )</p> <p>Base Length h = 20</p> <p>h = 8</p> <p>Solution1 P ! C ah wh ( h 2 h d ) 2 1 1 sin J ! 120 20 20 2 v 8 2 1 sin J ! 0 . 5 0 . 271 120 20 36 ! 11707 . 2 lb h 2 3 h h d 20 3 20 8 ! y ! 3 h 2 h d 3 20 2 v 8 ! 8 . 148 ft2</p> <p>Solution Moments about point a W = (120)(x)(20+8) = 3360 x lb ( G )( ) ! P v y 2</p> <p>G ( 3360 G )( ) ! (11707 . 2 )( 8 . 148 ) 2</p> <p> x = 7.54 ft So base length = 1.5 v x = 11.31 ft</p> <p>Use 11 ft 4 with x = 7-8 and 3-8 toe length</p> <p>Solution Stem Thickness Prior computing stability factors, a more accurate knowledge of the concrete dimensions is necessary. The thickness of the base of the stem is selected with the regard for bending and shear requirements. P for 18 height and h = 8</p> <p>1 P ! C ah wh ( h 2 h d ) 2 1 ! 0 . 271 120 18 (18 2 v 8 ) 2 ! 9951 . 12 lb h 2 3 hh ' (18 ) 2 ( 3 )( 18 )( 8 ) y ! ! ' 3 (h 2h ) 3 (18 2 v 8 ) ! 7 . 412 ft</p> <p>Solution Mu = (1.7) Py = (1.7) (9951.12) (7.412) = 125388.09 lb.ft ( 0 .85 )F 1fcd 87000 Vb ! 87000 f fy y ( 0 .85 )( 0 .85 )( 3000 ) 87000 ! 40 ,000 87000 40 ,000 ! 0 .03712</p> <p>V</p> <p>1 For adequate deflection control, choose V ! V max ! 0 . 01392 2 1 then R n ! V f y (1 V m ) 2 fy 40 ,000 ! ! 0 . 85 f cd 0 . 85 v 3000 ! 15 . 686</p> <p>ax</p> <p>! 0 .75 V b ! 0 .02784</p> <p>Solution 1 Rn ! Vfy 1 Vm 2 1 ! (0.01392)(40,000)(1 v 0.01392 v 15.686) 2 ! 496 Re quir d Mn 2 Re quired bd ! Rn Mn J 125388.09 v 12 ! ! 0.9496 Rn ! 3370.65 d d ! 16.76d Total thickness = 16.7 + 0.5 + 3 = 20.26</p> <p>Try 21 thickness of base of stem and select 12 for top of the wall</p> <p>Solutiond used now = 17.5 = 1.458</p> <p>Shear at d</p> <p> At 18 1.458 = 16.542 from top</p> <p> 1 P ! C ah wh ( h 2h d ) 2 1 120 . 542 . 542 2 v 8 16 16 ! 0 . 271 2 ! 8752 . 92 lb Vu ! 1 . 7 P ! 17879 . 96 lb bd J Vc ! J v 2 fcd ! 0 . 85 v 2 v 3000 v 12 v 16 . 542 ! 18483 . 27 lb Since J Vc " Vu , So no shear re inf orcement is required .</p> <p>Solution</p> <p>F.O.S Against Overturning</p> <p>Component Force W1 W2 W3 W4 W5 W6 Total (5.92)(18)(120)= 12787.2 1 (0.75)(18)(150)=1012.5 2 </p> <p>Arm3 . 67 21 " 5 . 92 ! 8 . 38 2</p> <p>Moment 107156.74 4981.50 11259 19255.34 4187.7 51257.62 198097.9</p> <p>3.67+1+0.25=4.92 3.67+0.5=4.1711 . 33 ! 5 . 665 2</p> <p>(18)(1)(150)=</p> <p>2700</p> <p>(11.33)(2)(150)= 3399.00 1 (18)(0.75)(120)=810 2 </p> <p>3.67+1+0.5=5.173.67 1 6.67 ! 8.005 2</p> <p>(6.67)(8)(120)=6403.2 27111.9</p> <p>Solution P = 11707.2 lb y= 8.148 ft Overturning Moment = 11707.2 x 8.148 = 95390.27 lb.ft 198097 . 9 F.O.S. against overturning ! ! 2 . 077 " 2 O.K 95390 . 27</p> <p>Location of Resultant &amp; Footing Soil Pressure Distance of the resultant from the front edge is Righting moment Overturnin g moment a! Total load ( righting ) 198097 . 9 95390 . 27 a! ! 3 . 7883 27111 . 9 Middle third = 3.7778 ft, So resultant is within the middle third.</p> <p>Rv q1 ! 2 ( 4 N 6 a ) N 27111 . 9 q1 ! 4 v 11 . 33 6 v 3 . 7883 2 11 . 33 </p> <p>Solutionq1= 4771.12 lb/ft2 &lt; 5 k/ft2q2 ! Rv ( 6 a 2 N ! 14 . 74 lb / ft 2 ) 2 N</p> <p>So O.K against bearing pressure.</p> <p>Solution F.O.S. against sliding force causing sliding = P = 11707.2 lb Frictional resistance = QR </p> <p>= (0.4) (27111.9) = 10844.76 lb</p> <p> Passive earth pressure against 2 height of footing !</p> <p> 1 sin J 1 ! (120 )( 2 ) 1 sin J 2 ! 442 . 82 lb 10844 . 76 442 F .O .S . ! ! 0 . 964 11707 . 2</p> <p>1 2 wh Caph 2</p> <p>1 . 5 So key is required .</p> <p>Solution The front of key is 4 in front of back face of the stem. This will permit anchoring the stem reinforcement in the key.. x 4756 .38 ! 6.243 11.33 x ! 2620 .84 lb / ft Total ordinate 2635 .58 lb / ft</p> <p> Frictional resistance between soil to soil = QR 4771.12 2635 .58 ! (tan J) (5.087 ) 2 ! 13191.17 lb</p> <p>Solution Frictional resistance between heel concrete to soil = QR</p> <p> 2635.58 14.74 ! (0.4) (6.243 ) 2 ! 3309.19 lb 1 2 Passive earth pressure ! wh Cph 2 2 1 ! 120 h 3.69 ! 221 .4 h 2 lb 2 F.O.S. against sliding = 1.5 13191 . 17 3309 . 19 221 . 4 h 2 1 .5 ! 11707 . 2 h ! 2 . 19 ft So use key of height = 2-3 = 2.25</p> <p>Solution Design of Heel CantileverWu = (1.7) (120) (8) + (1.4) [18 x 120 + 2 x 150 ] = 5076 lb/ft2 WN Mu ! 2 2 1 ! 5 . 76 5 . 92 2 ! 88947 . 76 lb.ft</p> <p>Vu = Factored shear a joint of stem and heel When the support reaction introduces compression into the end region, then critical shear is at a distance d from face of support. However, the support is not producing compression, therefore, critical shear is at joint of stem and heel.</p> <p>Solution Design of Heel CantileverVu = Factored shear a joint of stem and heel = (5.92) (5076) = 30049.92 lb JVc ! J2 f cd bd 3000 v 12 v 21 . 5 Vu ! 0 . 85 v 2 v ! 24023 . 11</p> <p> So depth is required to be increased. Vu 30049 . 92 d ! ! 26 . 8 9 d d ! b ( 0 . 85 )( 2 )( 3000 )( 12 ) J 2 f cd Therefore heel thickness 30,</p> <p>d = 27.5</p> <p>Solution Design of Heel CantileverNow Wu=(1.7)(120)(8)+(1.4)[(17.5)(120)+(2.5)(150)] = 5097 lb/ftMu</p> <p>1 ! ( 5097 )( 5 . 92 ) 2 ! 89315 . 75 lb . ft . 2Rn Mu 89315 v 12 ! ! ! 131 . 23 psi 2 2 J bd ( 0 . 9 )( 12 )( 27 . 5 )n</p> <p>Re quired 1 ! mmin</p> <p> 2 mR 1 1 fy 200 ! ! 0 . 005 fy</p> <p> ! 0 . 00337 </p> <p>Solution As = Vmin bd = 1.8 in2 Use # 8 @ 5 c/c (As = 1.88 in2 ) Dev. Length required = 23 top bars so 23 1.3 = 29.9 Available = 5.92-3 = 68.04 O.K</p> <p>Design of Toe Slabx 7.66 ! 4756 .38 11.33</p> <p>x = 3215.7 3215.7 + 14.74 = 3230.44 Self load=(0.9)(12150)=270 lb/ft</p> <p> 3230 .44 4771 .12 Wu ! (1.7 ) 2 (6801.33)lb/ft Wu = 3801.33 lb/ft 270 = 6531.33</p> <p>{</p> <p>Overload factor = 0.9 d=24-3.5=2...</p>