Verification Analysis of the Gravity Wall

Verification Manal no. Update 0/06 Verification Analysis of the Gravity Wall Progra File Gravity Wall Deo_v_en_0.gtz In this verification anal yo will find hand-ade verification analysis calclations of gravity wall in peranent and seisic design sitations. The reslts of the hand-ade calclations are copared with the reslts fro the GEO5 Gravity Wall progra. Ters of Reference In Figre, an exaple of a gravity wall with inclined footing botto in 0 inclination is shown. The earth body is coprised of two soil layers and the terrain is adjsted in 0 inclination. The top layer of the earth body (depth.5 ) is fored of sandy silt (MS). The lower layer of the earth body is fored of clayey sand (SC), which is at the front face of the wall too. The grondwater table is in the depth of.5 behind the wall and 3.7 in front of the wall. The properties of soils (effective vales) are in Table. The gravity wall is ade fro plain concrete C0/5 with nit weight = 3 kn/ 3. A verification analysis of the wall is perfored with the help of the theory of liit states. Figre Constrction of gravity wall diensions

Soil Unit weight [kn/ 3 ] Satrated nit weight [kn/ 3 ] sat Angle of internal friction [ ] ef Cohesion of soil c [kpa] ef Angle of friction strc.-soil [ ] Poisson s ratio [-] MS 8.00 0.00 6.50.00 5.00 0.35 SC 8.50 0.50 7.00 8.00 5.00 0.35 Table Soil properties characteristic effective vales The angle of friction and cohesion enter the first phase of the calclation as design vales and that s why the soil paraeters fro Table are redced with coefficients. and. 4. The design vales sed in calclation are in Table. c Soil Angle of internal friction [ ] ef,d Cohesion of soil [kpa] c ef, d Angle of friction strc.-soil [ ] d MS 4.09 8.57 3.636 SC 4.545 5.74 3.636 Table Soil properties design vales. The First Stage - Peranent Design Sitation Verification of the Whole Wall Calclation of the weight force and the centroid of the wall. The wall is divided into 5 parts, which are shown in Figre. Parts 4 and 5 are nder the grondwater table, therefore the nit weight of concrete is redced by the nit weight of water = 0 kn/ 3. Table 3 shows the diensions of the parts of the wall, their weight forces and centroids. Part height h i [] width b i [] Area A i [ ] Part weight i [kn/ 3 ] w Weight force W i [kn/] Point of action x i [] z i [] G i xi G i zi 3.500 0.700.450 3 56.350.950 -.550 09.883-43.693 3.500 0.700.5 3 8.75.367 -.967 38.506-55.4 3 0.00.300 0.460 3 0.580.50-0.700.67-7.406 4 0.600.300.380 3 7.940.50-0.300 0.63-5.38 5 0.30.300 0.65 3 3.439.533 0.077 5.73 0.64 Total 6.484 - - 86.460 -.68 Table 3 Diensions, weight force and centroids of the individal blocks

Centroid of the constrction 5 Wi xi 86.460 x t. 60 5 6.484 Wi 5 Wi zi.68 z t. 87 5 6.484 Wi Calclation of the front face resistance. The depth of soil in front of the wall is 0.6. Pressre at rest is considered. Hydralic gradient ( hw - water tables difference, d d - seepage path downwards, d - seepage path pwards) i d d h w d 3.7.5 0.606 3.03 0.6 Coefficient of earth pressre at rest (For cohesive soils the Terzaghi forla for copting of the coefficient of earth pressre at rest K is sed) K r r 0.35 0.538 0.35 Unit weight of soil in the area of ascending flow sat w w i 0.5 0.0 0 0.606 4.439 kn/ 3 Vertical noral effective stress z in the footing botto z h 4.439 0.6. 663 kpa Pressre at rest in the footing botto r z K r.663 0.538. 433 kpa Resltant force of stress at rest S r (Resltant force S acts only in horizontal direction, therefore S S and S 0 ) r S r r h.433 0.6 0.430 kn/ r rx rz 3

Point of action of the resltant force S r x 0. 000 z h 0.6 0. 00 3 3 Calclation of the active pressre. There are two layers of soils in the area, where we solve the active pressre. The strctre is therefore divided into two sections, in both of which the geostatic pressre z, the active earth pressre a and the resltant forces S a and S a are calclated. The active earth pressre is calclated sing Colob s theory. Figre Geostatic pressre z and active pressre a Coefficients of active earth pressre in both sections ( 0 - back face inclination of the strctre, 0 -inclination of the terrain; design vales of soils fro Table are sed in the calclation) K a - coefficient of active earth pressre K a cos ( ) sin( ) sin( ) cos ( ) cos( ) cos( ) cos( ) 4

K ac - coefficient of active earth pressre de to cohesion K ac cos( ) cos( ) cos( ) tg( ) tg( ) sin( ) cos( ) Calclation for the first section arctg 5. 7 0 K cos (4.09 0) sin( 4.09 3.636) sin( 4.09 5.7) cos (0) cos(0 3.636) cos(0 3.636) cos(0 5.7) a cos(4.09) cos(5.7) cos(3.636 0) tg( 0) tg(5.7) sin( 4.09 3.636 0 5.7) 0.4097 cos(3.636 0) K ac Calclation for the second section tg( ) 8.0 tg(5.7) arctg arctg 5. 557 i 8.5 K a cos (4.545 0) sin( 4.545 3.636) sin( 4.545 5.557) cos (0) cos(0 3.636) cos(0 3.636) cos(0 5.557) 0.406 0.5936 cos(4.545) cos(5.557) cos(3.636 0) tg( 0) tg(5.557) sin(4.545 3.636 0 5.557) cos(3.636 0) K ac Unit weight of soil SC in the area of descending flow sat w w i.5 0.0 0 0.606 6.56 kn/ 0 3 0.588 Vertical geostatic pressre z in two sections z h 8.0.5 7. 000 kpa z h h 8.0.5 6.56 3.03 77. 80 kpa Calclation of high in first layer of soil MS, where the active earth pressre is netral h cef, d K K 8.57 0.5936 ac 0 a 8.0 0.4097.380 Active earth pressre a in two sections a z K a cef, d K ac 7.00 0.4097 8.57 0.5936 0. 886 kpa 5

aa z K a cef, d K ac 7.00 0.406 5.74 0.588 4. kpa ab z K a cef, d K ac 77.8 0.406 5.74 0.588 4. 74 kpa Resltant forces of active earth pressre S a, S a and vertical and horizontal coponents Sa a ( h h0 ) 0.886 (.500.380) 0.053 kn/ Sa, x Sa cos( ) 0.053 cos(3.636) 0.05 kn/ Sa, z Sa sin( ) 0.053 sin(3.636) 0.03 kn/ Sa ( ab aa ) h aa h (4.74 4.) 3.03 4. 3.03 43.08 kn/ Sa, x Sa cos( ) 43.08 cos(3.636) 4.806 kn/ Sa, z Sa sin( ) 43.08 sin(3.636) 0.4 kn/ Points of action of resltant forces S a and S a x. 300.5.38 z 0.8.0. 840 3 x. 300 3.03 3.03 3.03 4. 3.03 0.3 (4.74 4.) 0.3 3 z 0. 97 3.03 4. 3.03 (4.74 4.) Total resltant force of active earth pressre S a Sax Sa, x Sa, x 0.05 4.806 4.858 kn/ Saz Sa, z Sa, z 0.03 0.4 0.55 kn/ S a S ax S az 4.858 0.55 43.07 kn/ Point of action of total resltant force 6

x a. 300 S ai, z zi 0.03 (.840) 0.4 ( 0.97) za 0. 99 0.55 S ai, z Calclation of the water pressre. The heel of the strctre is snken in a pereable sbsoil, which allows free water flow below the strctre. Therefore, the hydrodynaic pressre st be considered and its resltant force is calclated as shown in Figre 3. The area of the hydrodynaic pressre is divided into two sections. Figre 3 Hydrodynaic pressre w Horizontal water pressre w at interface of section and section (depth 3.7 ) w w ( 3.7.5) 0.. 000 kpa Resltant force of water pressre S w in two sections Sw w hw.000. 4.00 kn/ 7

Sw w hw.000 0.83 9.30 kn/ Points of action of resltant forces x. 300. z 0.6. 333 3 x. 300 0.6 0.3 z 0.6 0. 33 3 Total resltant force of water pressre S w S w S w, i 4.00 9.30 33.330 kn/ Point of action of resltant force S w x w. 300 S wi zi 4. (.333) 9.3 ( 0.33) z w. 056 33.33 S wi Checking for overtrning stability. The oents calclated in the analysis rotate abot the origin of the coordinate syste (left botto corner of the strctre). Resisting oent M and overtrning oent M are calclated for verification. ovr res Calclation of resisting oent M res and its redction by coefficient S. M res W r S az r 6.484.60 0.55.3 09.847 kn / M res S 09.847 90.770 kn/. Reslt fro the GEO5 Gravity Wall progra M res 90.74 kn / Calclation of overtrning oent M ovr M ovr 0.430 0. 4.858 0.99 33.33.056 73.997 kn / Reslt fro the GEO5 Gravity Wall progra M ovr 74.0 kn / 8

Usage M V M ovr res 73.997 00 00 38.8 %, SATISFACTORY 90.770 Reslt fro the GEO5 Gravity Wall progra V 38.8 %, SATISFACTORY Checking for slip. Slip in the inclined footing botto (Figre 4). Figre 4 Forces acting in the footing botto Total vertical and horizontal forces Fver and F hor F ver 6.484 0.55 6.639 kn / F hor 0.43 4.858 33.33 74.758 kn / Noral force N b 5. 7 N F ver cos( ) F b hor sin( ) 6.639 coc (5.7) 74.758 sin(5.7) 33.450 kn / b Shear force T T F ver sin( ) F b hor cos( ) 6.639 sin(5.7) 74.758 cos(5.7) 6.785 kn / b Eccentricity of noral force d - inclined width of footing botto 9

ealw - axial allowable eccentricity d.3 cos( ) b.3 cos(5,7).3 M e ovr M res N N d 33.450.3 73.997 09.847 0.38 33.450 In the progra, eccentricity is calclated as a ratio. e ratio e alw e 0.38 0.060 d.3 0.333 e 0.060, SATISFACTORY ratio Resisting horizontal force H res and its redction by coefficient. - redction coefficient of contact base - soil.0 (withot redction) S Fres - resisting force F res 0 kn H res cd ( d e) 5.74 (.3 0.38) N tg d Fres 33.450 tg(4.545) +0.0 H res 7.57 kn / H res S 7.57 65.974 kn/. Reslt fro the GEO5 Gravity Wall progra H res 65.98 kn / Acting horizontal force H act H act T 6.785 kn / Reslt fro the GEO5 Gravity Wall progra H act 6.79 kn / Usage H V H act res 6.785 00 00 93.7 %, SATISFACTORY 65.974 Reslt fro the GEO5 Gravity Wall progra V 93.6 %, SATISFACTORY 0

Bearing Capacity of the Fondation Soil The bearing capacity of the fondation sol is set to stress in the inclined footing botto. R d 00 kpa, and is copared with the Usage eccentricity V e e alw 0,060 00 00 8,0 %, SATISFACTORY 0,333 Reslt fro the GEO5 Gravity Wall progra V 8.0 %, SATISFACTORY Stress in the footing botto N 33.450 65.577 kpa d e.3 0.38 Reslt fro the GEO5 Gravity Wall progra 65.57 kpa Usage V 65.577 00 00 65.6 %, SATISFACTORY R 00 d Reslt fro the GEO5 Gravity Wall progra V 65.6 %, SATISFACTORY Diensioning Wall Ste Check In this exaple, a cross-section in the level of x-axis in Figre 5 is verified. The verified crosssection is ade fro plain concrete C 0/5 (characteristic cylindrical strength of concrete in copression with height f ck 0000 kpa, characteristic strength of concrete in tension f ct 00 kpa ) h. 40 and widthb. 00. The verification of a cross-section ade fro plain concrete is realized in accordance with EN 99--.

Figre 5 Diensioning wall ste check, cross-section Calclation of the weight force and the centroid of the wall W 3 (0.7 3.5 0.7 3.5) 84.55 kn/ 0.7 0.7 3 0.7 3.5 0.7 0.7 3.5 3 x t 0. 856 84.55 3.5 3.5 3 0.7 3.5 0.7 3.5 3 z t. 556 84.55 Calclation of the active earth pressre. The area behind the evalated part of the constrction is divided into two sections. In the first section, the active pressre is the sae as in the analysis of the whole wall. The centroids of all forces st be recalclated. Vertical geostatic stress z at the end of the second section z z h 7.0 6.56.0 60. kpa Active earth pressre ab at the end of the second section ab 0.406 60. 5.74 0.588 7. 43 kpa

Resltant force of active earth pressre S a and vertical and horizontal coponent (Resltant force S a at the beginning is the sae) S a (7.43 4.).0 4..0.544 kn/ Sa, x Sa cos( ).544 cos(3.636) 0.937 kn/ Sa, z Sa sin( ).544 sin(3.636) 5.079 kn/ Calclation of points of action x. 400 z (.50.38). 040 3 x. 400.00.00.00 4..00 (7.43 4.) 3 z 0. 794,00 4..00 (7.43 4.) Total resltant force of active earth pressre S a and horizontal and vertical coponent Sax Sa, x Sa, x 0.05 0.937 0.989 kn/ Saz Sa, z Sa, z 0.03 5.079 5.09 kn/ S a S ax S az 0.989 5.09.598 kn/ Point of action of resltant force S a x a. 400 0.05 (.840 0.800) 0.937 ( 0.794) z a 0. 797 0.05 0.937 Calclation of the water pressre. Water pressre becoes higher with the increasing depth. Horizontal water pressre w in depth of 3.5 nder the srface of the adjsted terrain w w ( 3.5.5) 0.0 0. 000 kpa 3

Resltant force of water pressre S w Sw w hw 0.000.0 0.000 kn/ Point of action of resltant force S w x. 400 z,0 0. 667 3 Verification of the shear strength. The design shear force, design noral force, design bending oent and shear strength of the cross-section are calclated. The design bending oent rotates abot the iddle of the verified cross-section. Design shear force V V S w S ax 0.000 0.989 40.989 kn / Reslt fro the GEO5 Gravity Wall progra V 40.94 kn / Design noral force N N S az W 5.09 84.55 89.67 kn / Reslt fro the GEO5 Gravity Wall progra N 89.57 kn / Design bending oent M M W r S az r S ax r3 S w r4 M 84.55 (0.856 0.700) 5.09 0.700 0.989 0.797 0.000 0.667 3.3 kn / Reslt fro the GEO5 Gravity Wall progra M 3.3 kn / Calclation of the area of copressed concrete A cc Deterining the copressed area of concrete is necessary in order to deterine the stresses on the front and the rear edges of the verified cross-section. The noral force N akes copressive stress and therefore is considered to be a negative force. Stress fro the design noral force N N A N A 89.67 64.0 kpa.00.40 Stress fro the design bending oent M 4

M W M W y 3.3.00.40 6 40.748 kpa Figre 6 Corse of tension on a cross-section of a wall ste Fro Figre 6 it can be seen, that the whole area of the cross-section is copressed. A cc b. h.00.40.40 c Stress in cross-section area cp N 89.67 cp 0. 064 MPa A 000.400 000 cc Design strength of concrete in copression f cd f ck 0.00 f cd cc, pl 0.8 0. 667 MPa.5 c Design strength of concrete in tension f ctd f ctk,005 - lower vale of characteristic strength of concrete in tension f f 0.7 f 0.7.0 ctk,005 ct ctd ct, pl ct, pl 0.8 0. 8 c c.5 MPa Liit stress c, li f cd f ctd ( f cd f ctd ) 0.667 0.8 (0.667 0.8) 4. 55 MPa Shear strength f cvd f cvd f ctd cp f ctd ax(0, cp c,li 0.8 ax(0;0.064 4.55 0.064 0.8 f cvd 0. 85 MPa 5

Design shear strength V Rd k.5 fcvd Acc 0.85.4 VRd 000 000 795.00 kn/ k.5 Reslt fro the GEO5 Gravity Wall progra V Rd 795.74 kn / Usage V V V Rd 40.989 00 00 5. %, SATISFACTORY 795.00 Reslt fro the GEO5 Gravity Wall progra V 5. %, SATISFACTORY Verification of a cross-section loaded by bending oent and noral force. Calclation of eccentricity e M e Max abs N e 0. 49 h 3.3.400 ; ;0.0 Max abs ; ;0.0 Max 30 89.67 30 0.49; 0.047; 0.0 Effective height of cross-section h e.400 0.49. 0 Design noral strength N Rd Max( f ; 50) 50 Max(0; 50) 50 50 50.0 ck.0.0.0 00 00 00 N Rd ( b f cd ) 000 (.0.0.00.667) 000 754.667 kn / Reslt fro the GEO5 Gravity Wall progra N Rd 758.60 kn / Usage N V N Rd 89.67 00 00 0.8 %, SATISFACTORY 754.667 Reslt fro the GEO5 Gravity Wall progra V 0.8 %, SATISFACTORY 6

. The Second Stage Seisic Design Sitation Verification of the Whole Wall The second stage of calclation ses the sae wall inflenced by an earthqake. The calclation of earthqake effects is ade according to the Mononobe-Okabe theory. The factor of horizontal acceleration is k 0. 05 (inertial force acts horizontally in an nfavorable direction) and h the factor of vertical acceleration is k 0. 04 (inertial force acts downwards). The coefficients of v redction of soil paraeters and the coefficients of overall stability of constrction are eqal to one. Therefore, the design vales of soil properties are the sae as the characteristic vales in Table. Calclation of the weight force of the wall. To deterine the horizontal and vertical coponents of a force fro an earthqake, it is necessary to calclate the weight force of wall withot the boyancy exerted on the wall by the grondwater. The calclation is shown in Table 4. Block Height h i [] Width b i [] Area A i [ ] Block weight i [kn/ 3 ] Weight force W i [kn/] Point of action x i [] z i [] G i xi G i zi 3.500 0.700.450 3 56.350.950 -.550 09.883-43.693 3.500 0.700.5 3 8.75.367 -.967 38.506-55.4 3 0.00.300 0.460 3 0.580.50-0.700.67-7.406 4 0.600.300.380 3 3.740.50-0.300 36.50-9.5 5 0.30.300 0.65 3 6.095.533 0.077 9.344 0.469 Total 3.940 - - 06.407-5.563 Table 4 Diensions, weight force and centroids of the individal blocks Centroid of the strctre 5 Wi xi 06.407 x t. 553 5 3.940 Wi 5 Wi zi 5.563 z t. 6 5 3.940 Wi Horizontal and vertical coponent of the force fro the earthqake W k W 0.053.940 6.647 kn eq, x h / Weq, z kv W ( 0.04) 3.940 5.38 kn/ 7

Calclation of the front face resistance. The pressre at rest on the front face of the wall is the sae as in the first stage of the calclation. The resltant force of the pressre at rest is S r 0.430 kn /. Calclation of the active earth pressre. The corse of the geostatic pressre is the sae as in the first stage of calclation. In the calclation coefficients of active earth pressre K a and K ac are sed as characteristic vales of soil properties. The active earth pressre a and the resltant force of active earth pressre S are calclated. a Corse of geostatic stress z 7. 000 kpa z 77. 80 kpa Coefficients of active earth pressre in both sections ( 0 - back face inclination of the strctre, 0 -inclination of terrain; characteristic vales of soils fro Table are sed in calclation) Calclation for the first layer arctg 5. 7 0 K cos (6.5 0) sin( 6.5 5.0) sin( 6.5 5.7) cos (0) cos(0 5.0) cos(0 5.0) cos(0 5.7) a 0.37 cos(6.5) cos(5.7) cos(5.0 0) tg( 0) tg(5.7) sin(6.5 5.0 0 5.7) cos(5.0 0) K ac 0.569 Calclation for the second layer tg( ) 8.0 tg(5.7) arctg arctg 5. 557 i 8.5 K a cos (7.0 0) sin( 7.0 5.0) sin( 7.0 5.557) cos (0) cos(0 5.0) cos(0 5.0) cos(0 5.557) cos(7.0) cos(5.557) cos(5.0 0) tg( 0) tg(5.557) sin(7.0 5.0 0 5.557) 0.363 cos(5.0 0) K ac 0.5563 8

Calclation of height in the first layer of soil MS, where the active earth pressre is netral h cef, K K 0.569 ac 0 a 8.0 0.37.09 >.50 Active earth pressre a is calclated only for the second section (in the first section it s eqal to zero) aa z K a cef, d K ac 7.00 0.363 8.0 0.5563 0. 903kPa ab z K a cef, d K ac 77.8 0.363 8.0 0.5563 9. 3 kpa Resltant force of the active earth pressre S a and its horizontal and vertical coponents S a ( ab aa ) h aa h (9.3 0.903) 3.03 0.903 3.03 30.339 kn/ S ax S a cos( ) 30.339 cos(5.0) 9.306 kn / S az S a sin( ) 30.339 sin(5.0) 7.85 kn / Point of action of the resltant force x, 300 3.03 3.03 3.03 0.903 3.03 0.3 (9.3 0.903) 0.3 3 z 0. 86 3.03 0.903 3.03 (9.3 0.903) Increase of the active earth pressre cased by an earthqake. An earthqake increases the effect of active earth pressre. Calclation of the seisic inertia angle in the first layer (withot restricted water inflence) kh 0.05 arctg arctg. k ( 0.04) 75 v Calclation of the seisic inertia angle in the second layer (with restricted water inflence) sat, kh sat, kh 0.5 0.05 arctg arctg 5. 36 s, ( kv ) ( sat, w ) ( kv ) (0.5 0) 0.04 9

Coefficient K ae for the active earth pressre in both sections K ae cos cos ( ) cos( ) cos ( ) sin( ) sin( ) cos( ) cos( ) K ae K ae 0.40 K ae cos(.75) cos K ae 0.449 cos(5.36) cos (0) cos(.75 0 5.0) cos (6.5.75 0) (0) cos(.75 0 5.0) cos (7.0 5.36 0) sin( 6.5 5.0) sin(6.5.75 5.7) cos(0 5.0.75) cos(0 5.7) sin( 7.0 5.0) sin( 7.0 5.36 5.557) cos(0 5.0 5.36) cos(0 5.557) Calclation of noral stress d fro the earthqake effects. The noral stress is calclated fro the botto of the wall d 0. 000 kpa - noral stress in the footing botto ( 0.04) 5. kpa d h ( kv ) 6.56 3.03 87 ( 0.04) 80. kpa d 0 d h ( kv ) 5.87 8.0.50 67 Increase of the active earth pressre cased by the earthqake in both sections ae, a d0 ( Kae Ka) 80.67 (0.40 0.37) 3. 38 kpa ae, b d ( Kae Ka) 5.87 (0.40 0.37). 04 kpa ae, a d ( Kae Ka) 5.87 (0.449 0.363) 4. 65 kpa Resltant forces of the increase of the active earth pressre S ae in both sections S S ae ( ae,a ae,b ) h ae,b h (3.38.04).50.04.50 3.884 kn/ S cos( ) 3.884 cos(5.0) 3.75 kn ae x ae / S ae z S ae sin( ) 3.884 sin(5.0).005 kn / Sae ae,a h 4.65 3.03 6.30 kn/ 0

Sae x Sae cos( ) 6.30 cos(5.0) 6.095 kn / S ae z S ae sin( ) 6.30 sin(5.0).633 kn / Points of action of the resltant forces x. 300 z.50.50.04.50 (3.03 0.3) (3.38.04) (.50) (3.03 0.3) 3.50.04.50 (3.38.04) z 3. 603 x. 300 z ( 3.03) 0.3. 790 3 Total resltant force of the increase of active earth pressre S ae and its horizontal and vertical coponent Sae, x Saex Saex 3.75 6.095 9.847 kn/ Sae, z Saez Saez.005.633.638 kn/ Sae Sae, x Sae, z 9.847.638 0.94 kn/ Point of action of the resltant force S ae x ae. 300 3.75 ( 3.603) 6.095 (.790) z ae. 48 3.75 6.095 Calclation of water pressre. The water pressre is the sae as in the verification of the whole wall in the first stage. The resltant force of the water pressre is S w 33.330 kn / and has the sae point of action as in the first stage. Calclation of the hydrodynaic pressre acting on the front face of the wall. The action of the hydrodynaic pressre cased by the earthqake is calclated fro the grondwater table to the botto of the wall. The direction of the force is the sae as the direction of the horizontal acceleration.

Calclation of the resltant force of the hydrodynaic pressre Pwd H 0.6 0.3 0. 83 cased by the earthqake P 7 7 wd kh w H 0.05 0.0 0.83 0.0 kn/ Point of action of the resltant force P wd x. 300 z ywd 0.3 (0.4 H ) 0.3 (0.4 0.83) 0.3 0. 0 Checking for overtrning stability. The oents calclated in the analysis rotate abot the origin of the coordinate syste (left botto corner of the strctre). Resisting oent M and overtrning oent M are calclated for verification. ovr res Calclation of the resisting oent M res M res M res W r Weq, z r Saz r3 Sae, z r4 6.484.60 5.38.553 7.85.300.638.300 6.484.60 5.38.553 7.85.300.638.300 M res 8.877 kn / Reslt fro the GEO5 Gravity Wall progra M res 8.86 kn / Calclation of the overtrning oent M ovr M ovr S r r Weq, z r Sax r3 Sae, x r4 Sw r5 Pwd r6 M ovr 0.43 0.00 6.647.6 9.306 0.86 9.847.48 33.330.056 0.0 0.0 M ovr 94.550 kn / Reslt fro the GEO5 Gravity Wall progra M ovr 94.59 kn /

Usage M V M ovr res 94.550 00 00 43. %, SATISFACTORY 8.877 Reslt fro the GEO5 Gravity Wall progra V 43. %, SATISFACTORY 4.). Checking for slip. The slip in the inclined footing botto in 0 inclination is checked (Figre Total vertical and horizontal forces Fver a F hor F ver 6.484 5.38 7.85.638 3.9 kn / F hor 0.43 6.647 9.606 9.847 33.33 79.000 kn / Noral force in the footing botto N b 5. 7 N F ver cos( ) F b hor sin( ) 3.9 coc (5.7) 79.000 sin(5.7) 39.496 kn / b Shear force in the footing botto T T F ver sin( ) F b hor cos( ) 3.9 sin(5.7) 79.000 cos(5.7) 65.444 kn / b Eccentricity of the noral force d - inclined width of the footing botto ealw - axial allowable eccentricity d M e.3 cos( ovr ) b M res N.3 cos(5.7) N d.3 39.496.3 94.550 8.877 39.496 In the progra, the eccentricity is calclated as a ratio. e ratio e 0.64 0.4 d.3 e 0,333 e 0.4, SATISFACTORY alw ratio 0.64 Resisting horizontal force H res and its redction by coefficient. - redction coefficient of contact base - soil.0 (withot redction) S 3

Fres - resisting force F res 0 kn H res cef, ( d e) 8.00 (.3 0.64) N tg ef, Fres 39.496 tg(7.0) +0.0 H res 85.34 kn / Reslt fro the GEO5 Gravity Wall progra H res 85.33 kn / Acting horizontal force H act H act T 65.444 kn / Reslt fro the GEO5 Gravity Wall progra H act 65.37 kn / Usage V H H act res 65.444 00 00 76.7 %, SATISFACTORY 85.34 Reslt fro the GEO5 Gravity Wall progra V 76.6 %, SATISFACTORY Bearing Capacity of the Fondation Soil The bearing capacity of the fondation soil is set to stress in the inclined footing botto. R d 00 kpa, and is copared with the Usage eccentricity V e e alw 0.4 00 00 34. %, VYHOVUJE 0.333 Reslt fro the GEO5 Gravity Wall progra V 34.6 %, SATISFACTORY Stress in the footing botto N 39.496 78.37 kpa d e.3 0.64 Reslt fro the GEO5 Gravity Wall progra 78.9 kpa 4

Usage V 78.37 00 00 78. %, SATISFACTORY R 00 d Reslt fro the GEO5 Gravity Wall progra V 78.3 %, SATISFACTORY Diensioning Wall Ste Check In this exaple, the cross-section on the level of the x-axis in Figre 5 is verified. The crosssection is ade fro plain concrete C 0/5 (characteristic cylindrical strength of concrete in copression with height f ck 0000 kpa, characteristic strength of concrete in tension f ct 00 kpa ) h. 40 and widthb. 00. The verification of the cross-section ade fro plain concrete is realized in accordance with EN 99--. Calclation of the weight force and the centroid is the sae as in the first stage W 84.55 kn/ x t 0. 856 z t. 556 Horizontal and vertical coponent of the force cased by the earthqake (the centroid is the sae as the centroid of the weight force) Weq, x kh W 0.0584.55 4.6 kn/ Weq, z kv W ( 0.04) 84.55 3.38 kn/ Calclation of the active earth pressre. The area behind the evalated part of the constrction is divided into two sections. The centroids of all forces st be recalclated. Vertical geostatic stress z and z in both sections z h 8.0.5 7. 000 kpa z z h 7.0 6.56.0 60. kpa Active earth pressre in the second section aa and ab first section is eqal to zero) aa 0.363 7.000 8.0 0.5563 0. 903 kpa (the active earth pressre in the ab 0.363 60. 8.0 0.5563. 99 kpa 5

Total resltant force of the active earth pressre S a and its horizontal and vertical coponents S a (.99 0.903).0 0.903.0 3.83 kn/ Sa, x Sa cos( ) 3.83 cos(5.0) 3.360 kn/ Sa, z Sa sin( ) 3.83 sin(5.0) 3.580 kn/ Point of action of the resltant force S a x. 400.00.00.00 0.903.00 (.99 0.903) 3 z 0. 70.00 0.903.00 (.99 0.903) Increase of the active earth pressre cased by an earthqake. An earthqake increases the effect of the active earth pressre. Calclation of the noral stress d fro the earthqake effects. The vertical pressre is calclated fro the lower part of the ste d 0. 000 kpa - noral stress at the level of the lower part of the ste ( 0.04) 34. kpa d ( h ) ( kv ) 6.56 (.00) 447 ( 0.04) 6. kpa d 0 d h ( kv ) 34.447 8.0.50 57 Increase of the active earth pressre cased by the earthqake effects in both sections ae, a d0 ( Kae Ka) 6.57 (0.40 0.37). 445 kpa ae, b d ( Kae Ka) 34.447 (0.40 0.37). 347 kpa ae, a d ( Kae Ka) 34.447 (0.449 0.363). 749 kpa Resltant forces of the increase of the active earth pressre S ae in both sections S S ae ( ae,a ae,b ) h ae,b h (.445.347).50.347.50 S cos( ).844 cos(5.0).747 kn ae x ae /.844 kn/ 6

Sae z S ae sin( ).844 sin(5.0) 0.736 kn / Sae ae,a h.749.00.749 kn/ S ae x S ae cos( ).749 cos(5.0).655kn / S ae z S ae sin( ).749 sin(5.0) 0.7 kn / Points of action of the resltant forces x. 400.50.50.347.50 00 (.445.347) (.50).00 3 z. 84.50.347.50 (.445.347) x. 400 z (.0). 333 3 Total resltant force S aeand its horizontal and vertical coponents Sae, x Saex Saex.747.655 5.40 kn/ Sae, z Saez Saez 0.736 0.7.447 kn/ Sae Sae, x Sae, z 5.40.447 5.59 kn/ Point of action of the resltant force S ae x ae. 400.747 (.84).655 (.333) z ae. 09.747.655 Calclation of the water pressre. Water pressre becoes higher with the increasing depth. Horizontal water pressre w in depth of 3.5 nder the srface of the adjsted terrain w w ( 3.5.5) 0.0 0. 000 kpa Resltant force of the water pressre S w 7

Sw w hw 0.000.0 0.000 kn/ Point of action of the resltant force S w x. 400 z,0 0. 667 3 Verification of shear strength. The design shear force, design noral force, design bending oent and shear strength of the cross-section are calclated. The design bending oent rotates abot the iddle of the verified cross-section. Design shear force V V S w Sa, x Sae, x Weq, x 0.000 3.360 5.40 4.6 4.988 kn/ Reslt fro the GEO5 Gravity Wall progra V 4.95 kn / Design noral force N N Sa, z Sae, z W Weq, z 3.580.447 84.55 3.38 9.933 kn/ Reslt fro the GEO5 Gravity Wall progra N 9.89 kn / Design bending oent M M W r Saz r Sax r3 Sw r4 Weq, z r 5Sae, z r6 Weq, x r7 Sae, x r8 M 84.55 (0.856 0.700) 3.580 0.700 3.360 0.70.,0 0.667 3.38 (0.856 0.700).447 0.700 4.6.556 5.40.09 M 3.458 kn / Reslt fro the GEO5 Gravity Wall progra M 3.46 kn / Calclation of the area of copressed concrete A cc Deterining the copressed area of concrete is necessary in order to deterine the stresses on the front and the rear edges of the verified cross-section. The noral force N akes copressive stress and therefore is considered to be a negative force. Stress fro design noral force N N N A A 9.933 66.38 kpa.00.40 8

Stress fro design bending oent M M W M W y 3.458.00.40 6 7.80 kpa Figre 7 Corse of tension on the cross-section of wall ste Fro Figre 7 it can be seen, that not the whole cross section is copressed, only a part, h c 38.9 h c. 347 38.9 5.49.400 A cc b. h.00.347.347 c Stress on the cross-section area cp N 9.933 cp 0. 06899 MPa A 000.347 000 cc Design strength of concrete in copression f cd f ck 0.00 f cd cc, pl 0.8 0. 667 MPa,5 c Design strength of concrete in tension f ctd f ctk,005 - lower vale of characteristic strength of concrete in tension f f 0.7 f 0.7.0 ctk,005 ct ctd ct, pl ct, pl 0.8 0. 8 c c.5 MPa 9

Liit stress c, li f cd f ctd ( f cd f ctd ) 0.667 0.8 (0.667 0.8) 4. 55 MPa Shear strength f cvd f cvd f ctd f cvd 0. 855 cp MPa f ctd ax(0, cp c,li 0.8 ax(0;0.066 4.55 0.06899 0.8 Design shear strength V Rd k.5 fcvd Acc 0.855.347 VRd 000 000 767.790 kn/ k.5 Reslt fro the GEO5 Gravity Wall progra V Rd 767.58 kn / Usage V V V Rd 4.988 00 00 5.6 %, SATISFACTORY 767.790 Reslt fro the GEO5 Gravity Wall progra V 5.6 %, SATISFACTORY Verification of a cross-section loaded by bending oent and noral force. Calclation of eccentricity e M e Max abs N e 0. 5 h 3.458.400 ; ;0.0 Max abs ; ;0.0 Max 30 9.933 30 0.5; 0.047; 0.0 Effective high of cross-section h e.400 0.5 0. 896 Design noral strength N Rd Max( f ; 50) 50 Max(0; 50) 50 50 50,0 ck,0,0.0 00 00 00 N Rd ( b f cd ) 000 (.0 0.896.00.667) 000 9557.63 kn / Reslt fro the GEO5 Gravity Wall progra N Rd 9543. kn / Usage 30

V N N Rd 9.933 00 00.0 %, SATISFACTORY 9557.63 Reslt fro the GEO5 Gravity Wall progra V.0 %, SATISFACTORY 3