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Page 1 PRESSURE ISOBARS Definition An isobar i s a lin e whic h connect s al l point s o f equa l stres s belo w th e groun d surface . I n othe r words, a n isoba r i s a stres s contour . W e ma y dra w an y numbe r o f isobar s a s show n i n Fig . 6.1 9 fo r any give n loa d system . Eac h isoba r represent s a fractio n o f th e loa d applie d a t th e surface . Sinc e these isobar s for m close d figure s an d resembl e th e for m o f a bulb , the y ar e als o terme d bulb o f pressure o r simpl y th e pressure bulb. Normall y isobars ar e draw n fo r vertical , horizonta l an d shea r stresses. Th e on e tha t i s mos t importan t i n th e calculatio n o f settlement s o f footing s i s th e vertica l pressure isobar . Page 2 PRESSURE ISOBARS Definition An isobar i s a lin e whic h connect s al l point s o f equa l stres s belo w th e groun d surface . I n othe r words, a n isoba r i s a stres s contour . W e ma y dra w an y numbe r o f isobar s a s show n i n Fig . 6.1 9 fo r any give n loa d system . Eac h isoba r represent s a fractio n o f th e loa d applie d a t th e surface . Sinc e these isobar s for m close d figure s an d resembl e th e for m o f a bulb , the y ar e als o terme d bulb o f pressure o r simpl y th e pressure bulb. Normall y isobars ar e draw n fo r vertical , horizonta l an d shea r stresses. Th e on e tha t i s mos t importan t i n th e calculatio n o f settlement s o f footing s i s th e vertica l pressure isobar . Lines o f equal vertica l pressure o r isobars Figure 6.19 Bulb of pressure Significant Depth In hi s openin g discussio n o n settlement o f structure s a t th e First Internationa l Conferenc e on Soi l Mechanic s an d Foundation Engineerin g (hel d i n 1936 a t Harvar d Universit y i n Cambridge, Mass , USA) , Terzaghi stresse d th e importance o f th e bul b o f pressure an d it s relationshi p with th e sea t o f settlement . A s stated earlie r w e ma y dra w an y number o f isobar s fo r an y give n load system , bu t th e on e tha t i s of practica l significanc e i s th e one whic h enclose s a soi l mas s whic h i s responsibl e fo r th e settlemen t o f th e structure . Th e dept h o f this stresse d zon e ma y b e terme d a s th e significant depth D S whic h i s responsibl e fo r th e settlemen t of th e structure . Terzagh i recommende d tha t fo r al l practica l purpose s on e ca n tak e a stress contour which represent s 2 0 pe r cen t o f th e foundatio n contac t pressur e q , i.e , equa l t o Q.2q. Th e dept h o f such a n isoba r ca n b e take n a s th e significant depth D s whic h represent s th e sea t o f settlemen t fo r the foundation . Terzaghi' s recommendatio n wa s base d o n hi s observatio n tha t direc t stresse s ar e considered o f negligibl e magnitud e whe n the y ar e smalle r tha n 2 0 pe r cen t o f th e intensit y o f th e applied stres s fro m structura l loading , an d tha t mos t o f th e settlement , approximatel y 8 0 pe r cen t o f the total , take s plac e a t a dept h les s tha n D s . Th e dept h D s i s approximatel y equa l t o 1. 5 time s th e width o f squar e o r circula r footing s [Fig . 6.20(a)] . If severa l loade d footing s ar e space d closel y enough , th e individua l isobar s o f eac h footin g in questio n woul d combin e an d merg e int o on e larg e isoba r o f the_intensit y a s show n i n [Fig. 6.20(b)] . Th e combine d significan t dept h D i s equa l t o abou t 1. 5 B . a z = Q.2q D<=\.5B\ Stresse d zon e Isobar (a) Significan t dept h o f stresse d zon e for singl e footin g Isobar Combined stresse d zon e (b) Effec t o f closel y place d footing s Page 3 PRESSURE ISOBARS Definition An isobar i s a lin e whic h connect s al l point s o f equa l stres s belo w th e groun d surface . I n othe r words, a n isoba r i s a stres s contour . W e ma y dra w an y numbe r o f isobar s a s show n i n Fig . 6.1 9 fo r any give n loa d system . Eac h isoba r represent s a fractio n o f th e loa d applie d a t th e surface . Sinc e these isobar s for m close d figure s an d resembl e th e for m o f a bulb , the y ar e als o terme d bulb o f pressure o r simpl y th e pressure bulb. Normall y isobars ar e draw n fo r vertical , horizonta l an d shea r stresses. Th e on e tha t i s mos t importan t i n th e calculatio n o f settlement s o f footing s i s th e vertica l pressure isobar . Lines o f equal vertica l pressure o r isobars Figure 6.19 Bulb of pressure Significant Depth In hi s openin g discussio n o n settlement o f structure s a t th e First Internationa l Conferenc e on Soi l Mechanic s an d Foundation Engineerin g (hel d i n 1936 a t Harvar d Universit y i n Cambridge, Mass , USA) , Terzaghi stresse d th e importance o f th e bul b o f pressure an d it s relationshi p with th e sea t o f settlement . A s stated earlie r w e ma y dra w an y number o f isobar s fo r an y give n load system , bu t th e on e tha t i s of practica l significanc e i s th e one whic h enclose s a soi l mas s whic h i s responsibl e fo r th e settlemen t o f th e structure . Th e dept h o f this stresse d zon e ma y b e terme d a s th e significant depth D S whic h i s responsibl e fo r th e settlemen t of th e structure . Terzagh i recommende d tha t fo r al l practica l purpose s on e ca n tak e a stress contour which represent s 2 0 pe r cen t o f th e foundatio n contac t pressur e q , i.e , equa l t o Q.2q. Th e dept h o f such a n isoba r ca n b e take n a s th e significant depth D s whic h represent s th e sea t o f settlemen t fo r the foundation . Terzaghi' s recommendatio n wa s base d o n hi s observatio n tha t direc t stresse s ar e considered o f negligibl e magnitud e whe n the y ar e smalle r tha n 2 0 pe r cen t o f th e intensit y o f th e applied stres s fro m structura l loading , an d tha t mos t o f th e settlement , approximatel y 8 0 pe r cen t o f the total , take s plac e a t a dept h les s tha n D s . Th e dept h D s i s approximatel y equa l t o 1. 5 time s th e width o f squar e o r circula r footing s [Fig . 6.20(a)] . If severa l loade d footing s ar e space d closel y enough , th e individua l isobar s o f eac h footin g in questio n woul d combin e an d merg e int o on e larg e isoba r o f the_intensit y a s show n i n [Fig. 6.20(b)] . Th e combine d significan t dept h D i s equa l t o abou t 1. 5 B . a z = Q.2q D<=\.5B\ Stresse d zon e Isobar (a) Significan t dept h o f stresse d zon e for singl e footin g Isobar Combined stresse d zon e (b) Effec t o f closel y place d footing s Pressure Isobars for Footings Pressure isobar s o f square , rectangula r an d circula r footing s ma y convenientl y b e use d fo r determining vertica l pressure , (J z , a t an y depth , z , belo w th e bas e o f th e footings . Th e depth s z fro m the groun d surface , an d th e distanc e r (o r jc ) fro m th e cente r o f th e footin g ar e expresse d a s a function o f th e widt h o f th e footin g B . I n th e cas e o f circula r footin g B represent s th e diameter . The followin g pressur e isobar s ar e give n base d o n eithe r Boussines q o r Westergaard' s equations 1. Boussines q isobar s fo r squar e an d continuou s footings , Fig . 6.21 . 2. Boussines q isoba r fo r circula r footings , Fig . 6.22 . 3. Westergaar d isobar s fo r squar e an d continuou s footings , Fig . 6.23 . B/2=b BI2=b Continuous 25 Figure 6.21 Pressure isobars based on Boussinesq equation for square and continuous footings Page 4 PRESSURE ISOBARS Definition An isobar i s a lin e whic h connect s al l point s o f equa l stres s belo w th e groun d surface . I n othe r words, a n isoba r i s a stres s contour . W e ma y dra w an y numbe r o f isobar s a s show n i n Fig . 6.1 9 fo r any give n loa d system . Eac h isoba r represent s a fractio n o f th e loa d applie d a t th e surface . Sinc e these isobar s for m close d figure s an d resembl e th e for m o f a bulb , the y ar e als o terme d bulb o f pressure o r simpl y th e pressure bulb. Normall y isobars ar e draw n fo r vertical , horizonta l an d shea r stresses. Th e on e tha t i s mos t importan t i n th e calculatio n o f settlement s o f footing s i s th e vertica l pressure isobar . Lines o f equal vertica l pressure o r isobars Figure 6.19 Bulb of pressure Significant Depth In hi s openin g discussio n o n settlement o f structure s a t th e First Internationa l Conferenc e on Soi l Mechanic s an d Foundation Engineerin g (hel d i n 1936 a t Harvar d Universit y i n Cambridge, Mass , USA) , Terzaghi stresse d th e importance o f th e bul b o f pressure an d it s relationshi p with th e sea t o f settlement . A s stated earlie r w e ma y dra w an y number o f isobar s fo r an y give n load system , bu t th e on e tha t i s of practica l significanc e i s th e one whic h enclose s a soi l mas s whic h i s responsibl e fo r th e settlemen t o f th e structure . Th e dept h o f this stresse d zon e ma y b e terme d a s th e significant depth D S whic h i s responsibl e fo r th e settlemen t of th e structure . Terzagh i recommende d tha t fo r al l practica l purpose s on e ca n tak e a stress contour which represent s 2 0 pe r cen t o f th e foundatio n contac t pressur e q , i.e , equa l t o Q.2q. Th e dept h o f such a n isoba r ca n b e take n a s th e significant depth D s whic h represent s th e sea t o f settlemen t fo r the foundation . Terzaghi' s recommendatio n wa s base d o n hi s observatio n tha t direc t stresse s ar e considered o f negligibl e magnitud e whe n the y ar e smalle r tha n 2 0 pe r cen t o f th e intensit y o f th e applied stres s fro m structura l loading , an d tha t mos t o f th e settlement , approximatel y 8 0 pe r cen t o f the total , take s plac e a t a dept h les s tha n D s . Th e dept h D s i s approximatel y equa l t o 1. 5 time s th e width o f squar e o r circula r footing s [Fig . 6.20(a)] . If severa l loade d footing s ar e space d closel y enough , th e individua l isobar s o f eac h footin g in questio n woul d combin e an d merg e int o on e larg e isoba r o f the_intensit y a s show n i n [Fig. 6.20(b)] . Th e combine d significan t dept h D i s equa l t o abou t 1. 5 B . a z = Q.2q D<=\.5B\ Stresse d zon e Isobar (a) Significan t dept h o f stresse d zon e for singl e footin g Isobar Combined stresse d zon e (b) Effec t o f closel y place d footing s Pressure Isobars for Footings Pressure isobar s o f square , rectangula r an d circula r footing s ma y convenientl y b e use d fo r determining vertica l pressure , (J z , a t an y depth , z , belo w th e bas e o f th e footings . Th e depth s z fro m the groun d surface , an d th e distanc e r (o r jc ) fro m th e cente r o f th e footin g ar e expresse d a s a function o f th e widt h o f th e footin g B . I n th e cas e o f circula r footin g B represent s th e diameter . The followin g pressur e isobar s ar e give n base d o n eithe r Boussines q o r Westergaard' s equations 1. Boussines q isobar s fo r squar e an d continuou s footings , Fig . 6.21 . 2. Boussines q isoba r fo r circula r footings , Fig . 6.22 . 3. Westergaar d isobar s fo r squar e an d continuou s footings , Fig . 6.23 . B/2=b BI2=b Continuous 25 Figure 6.21 Pressure isobars based on Boussinesq equation for square and continuous footings Figure 6.22 Pressure isobars based on Boussinesq equation for uniformly loaded circular footings B/2=b B/2=b 5b 6b Continuous IB 2 B 3 5 Figure 6.23 Pressure isobars based on Westergaard equation for square and continuous footing Page 5 PRESSURE ISOBARS Definition An isobar i s a lin e whic h connect s al l point s o f equa l stres s belo w th e groun d surface . I n othe r words, a n isoba r i s a stres s contour . W e ma y dra w an y numbe r o f isobar s a s show n i n Fig . 6.1 9 fo r any give n loa d system . Eac h isoba r represent s a fractio n o f th e loa d applie d a t th e surface . Sinc e these isobar s for m close d figure s an d resembl e th e for m o f a bulb , the y ar e als o terme d bulb o f pressure o r simpl y th e pressure bulb. Normall y isobars ar e draw n fo r vertical , horizonta l an d shea r stresses. Th e on e tha t i s mos t importan t i n th e calculatio n o f settlement s o f footing s i s th e vertica l pressure isobar . Lines o f equal vertica l pressure o r isobars Figure 6.19 Bulb of pressure Significant Depth In hi s openin g discussio n o n settlement o f structure s a t th e First Internationa l Conferenc e on Soi l Mechanic s an d Foundation Engineerin g (hel d i n 1936 a t Harvar d Universit y i n Cambridge, Mass , USA) , Terzaghi stresse d th e importance o f th e bul b o f pressure an d it s relationshi p with th e sea t o f settlement . A s stated earlie r w e ma y dra w an y number o f isobar s fo r an y give n load system , bu t th e on e tha t i s of practica l significanc e i s th e one whic h enclose s a soi l mas s whic h i s responsibl e fo r th e settlemen t o f th e structure . Th e dept h o f this stresse d zon e ma y b e terme d a s th e significant depth D S whic h i s responsibl e fo r th e settlemen t of th e structure . Terzagh i recommende d tha t fo r al l practica l purpose s on e ca n tak e a stress contour which represent s 2 0 pe r cen t o f th e foundatio n contac t pressur e q , i.e , equa l t o Q.2q. Th e dept h o f such a n isoba r ca n b e take n a s th e significant depth D s whic h represent s th e sea t o f settlemen t fo r the foundation . Terzaghi' s recommendatio n wa s base d o n hi s observatio n tha t direc t stresse s ar e considered o f negligibl e magnitud e whe n the y ar e smalle r tha n 2 0 pe r cen t o f th e intensit y o f th e applied stres s fro m structura l loading , an d tha t mos t o f th e settlement , approximatel y 8 0 pe r cen t o f the total , take s plac e a t a dept h les s tha n D s . Th e dept h D s i s approximatel y equa l t o 1. 5 time s th e width o f squar e o r circula r footing s [Fig . 6.20(a)] . If severa l loade d footing s ar e space d closel y enough , th e individua l isobar s o f eac h footin g in questio n woul d combin e an d merg e int o on e larg e isoba r o f the_intensit y a s show n i n [Fig. 6.20(b)] . Th e combine d significan t dept h D i s equa l t o abou t 1. 5 B . a z = Q.2q D<=\.5B\ Stresse d zon e Isobar (a) Significan t dept h o f stresse d zon e for singl e footin g Isobar Combined stresse d zon e (b) Effec t o f closel y place d footing s Pressure Isobars for Footings Pressure isobar s o f square , rectangula r an d circula r footing s ma y convenientl y b e use d fo r determining vertica l pressure , (J z , a t an y depth , z , belo w th e bas e o f th e footings . Th e depth s z fro m the groun d surface , an d th e distanc e r (o r jc ) fro m th e cente r o f th e footin g ar e expresse d a s a function o f th e widt h o f th e footin g B . I n th e cas e o f circula r footin g B represent s th e diameter . The followin g pressur e isobar s ar e give n base d o n eithe r Boussines q o r Westergaard' s equations 1. Boussines q isobar s fo r squar e an d continuou s footings , Fig . 6.21 . 2. Boussines q isoba r fo r circula r footings , Fig . 6.22 . 3. Westergaar d isobar s fo r squar e an d continuou s footings , Fig . 6.23 . B/2=b BI2=b Continuous 25 Figure 6.21 Pressure isobars based on Boussinesq equation for square and continuous footings Figure 6.22 Pressure isobars based on Boussinesq equation for uniformly loaded circular footings B/2=b B/2=b 5b 6b Continuous IB 2 B 3 5 Figure 6.23 Pressure isobars based on Westergaard equation for square and continuous footing Example 6.12 A singl e concentrate d loa d o f 100 0 k N act s a t th e groun d surface . Construc t a n isoba r fo r <7 = 4 0 kN/m 2 b y makin g us e o f th e Boussines q equation . Solution From Eq . (6.la ) w e hav e 3(2 1 We ma y no w writ e b y rearrangin g a n equatio n fo r th e radia l distanc e r a s 1 Now fo r Q = 100 0 kN , cr , = 4 0 kN/m 2 , w e obtai n th e value s o f r p r 2 , r y etc . fo r differen t depths z, , z 2 , z v etc . Th e value s s o obtaine d ar e z(m) 0.25 0.50 1.0 2.0 3.0 3.455 r(m) 1.34 1.36 1.30 1.04 0.60 0.00 g=1000kN a, = 4 0 kN/m J Isobar 3.455 Figure Ex. 6.12Read More
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