?ed???d???? ?a? ???t?st??se??

1
????????S ??? ??T?????S T?????OS??S 7? ??.
???-???. ??? - ??ad. ?t?? 2007 - 08
??????? 7 ?ed???d???? ?a? ???t?st??se??
09.01.2008
2
?ed???d???? ?a? ???t?st??se??
? ?eµe???s? µp??e? ?a ???e? µe ped???d????? ?
???t?st??s? st?? pe??pt?se?? ?p?? e??a? ep???µ?t?

1. ? µe??s? t?? d?af?????? ?a????se?? µeta??
   ?e?t?????? ped????, e?te ???? p??? d?af??et????
   f??t??? e?te ???? d?af??et???? (? aß?ßa???)
   edaf???? s???????
2. ? µe??s? t?? a??a?a? p?es?? ?d?as?? t?? ped????
   st? ?daf?? (p.?. se pe??pt?se?? f??t??? µe?????
   e??e?t??t?ta? ? µe????? ??p??, ?p?? st??
   pe??pt?s? µe????? se?sµ???? f??t???)
3. ? µe??s? t?? ??????t?a? d??aµ?? p?? ??p??? p?d???
   µetaf??e? st? ?daf?? (p.?. ??a t?? ap?t??p?
   ???s??se?? t?? ped????)
4. Ge????te?a, ?p?? e??a? ep???µ?t? ? ße?t??s? t??
   s??e??as?a? µeta?? t?? ped????

• ? ?ta?
• t? p?s?st? ??????? t?? ped???? e??a? s?µa?t???
  p?s?st? t?? ep?f??e?a? ß?s?? t?? ?atas?e??? (p.?.
  gt 50),
• ? a?aµe??µe?? s??????? ?a????s? t?? ped???? e??a?
  a??et? µe???? (?p?te ?a? ? d?af????? ?a????s?
  µp??e? ?a e??a? ?????)
• ? ?atas?e?? e??a? ?d?a?te?a e?a?s??t? se
  d?af?????? ?a????se?? ? µetaf??e? s?µa?t????
  ??p?? st? ?eµe???s?

?eßa???, ? ßa??? ?eµe???s? µe pass?????
ap?te?e? µ?a ???? e?a??a?t??? ??s?
3
?ed???d????
S?µa?t??? µe??µ??e? t?se?? ?d?as?? ?a? s?µa?t???
µe??µ??e? d?af?????? ?a????se?? µeta?? t??
st???? se s?????s? µe ta µeµ???µ??a p?d??a
4
???t?st??se??
S?µa?t??? µe??µ??e? t?se?? ?d?as?? ?a? s?µa?t???
µe??µ??e? d?af?????? ?a????se?? µeta?? t??
st????, se s?????s? µe ta µeµ???µ??a p?d??a ?a?
t?? ped???d?????
5
1. ?????s? ?eµe???se?? µe ped???d?????
S??µat??? d????aµµa teµ???s?? d???µe?? ?a?
?aµpt???? ??p?? ?at? µ???? t?? ped???d????
??µ???se? d???µe??
?a?ad???? ??a t?? ?ata??µ? t?? edaf???? p?es??
(q) ?at? t? µ???? t?? ped???d????

• ??aµpte? ped???d????
• G?aµµ??? ?ata??µ?
• S?et???? e??aµpte?
• ?ata??µ? p?? a??????e? t? µ??f? t?? pa?aµ??f?s??
  t?? ped???d????

?aµpt???? ??p??
6
1. ?????s? ?eµe???se?? µe ped???d?????
1.1 ?a?ad??? ??aµµ???? ?ata??µ?? t?? edaf????
p??se?? st? ß?s?
?s??e? se ??aµpte? ped???d?????
S???st?µe?? ?ata????f? d??aµ?
???t?? ß?s?? ?
S???st?µe?? ??p? ?? p??? t?? a??? (x0)
1. ?s????p?a ?ata????f?? d???µe?? ?a? ??p??
7
1. ?????s? ?eµe???se?? µe ped???d?????
1.1 ?e pa?ad??? ??aµµ???? ?ata??µ?? t?? edaf????
p??se?? st? ß?s?
???t?? ß?s?? ?
2. ?p?????sµ?? d?a???µµat?? ?aµpt???? ??p?? ?at?
µ???? t?? d????
?p??e? ?a ???e? µe ep???s? t?? d???? ?p? ta
(???st?) ep?ßeß??µ??a f??t?a ?a? t?? a??t???
edaf???? a?t?d??se?? (sL, so)
?a?at???s? ? pa?ad??? ??aµµ???? ?ata??µ?? t??
edaf???? p??se?? a?t?st???e? st?? pa?ad???
te?e??? ??aµpt?? ped???d????. S??ep??, ? a???ße?a
t?? pa?ad???? a????e? ??a p???? ??aµpte?
ped???d?????. ??aµ??a ped???d???? ?e???? ??p?
ad?a?e?a? (?), µ???? µ???? (L), µa?a?? ?daf??
8
?a??de??µa efa?µ???? ?p???s? ped???d???? µe
pa?ad??? ??aµµ???? ?at???µ?? t?? t?se?? st? ?daf??
?ed???d???? p??t?? ? 1.20m µ????
L 12m ?daf?? de? e?d?af??e? st?? ep???s? a?t?
F??t?a
V1 V2 400 kN , V3 640 kN
G?aµµ??? ?at???µ? t?? t?se?? st? ?daf??
9
?a??de??µa efa?µ???? ?p???s? ped???d???? µe
pa?ad??? ??aµµ???? ?at???µ?? t?? t?se?? st? ?daf??
S???st?µe?? ?ata????f? d??aµ?
1440 kN
S???st?µe?? ??p? ?? p??? x0
9600 kNm
F??t?a
V1 V2 400 kN , V3 640 kN
133 kN/m2
x0
67 kN/m2
10
?a??de??µa efa?µ???? ?p???s? ped???d???? µe
pa?ad??? ??aµµ???? ?at???µ?? t?? t?se?? st? ?daf??
?????aµµa teµ???s?? d???µe??
?????aµµa ?aµpt???? ??p??
400 kN
400 kN
640 kN
400 kN
400 kN
640 kN
11
1. ?????s? ?eµe???se?? µe ped???d?????
1.2 ?e pa?ad??? a??pt???? edaf???? p??se?? ?at?
t? µ??t??? Winkler
Se s?et???? e??aµpte? ped???d?????, ? pa?ad???
??aµµ???? ?ata??µ?? t?? t?se?? st? ß?s? t?? d????
de? e??a? epa???? a???ß??. St?? pe??pt?se?? a?t??
µp??e? ?a ???s?µ?p????e? ?ata??µ? t?se??
s?µßat? µe t? µ??t??? Winkler
???t??? Winkler
?????aµµa ß???se?? (y) ?a? edaf???? p??se?? (p)
??t? ap? t?? ped???d???
p edaf??? a?t?d?as? (kPa) y ß???s? t?? d????
(m) k sta?e?? e?at????? Winkler (kN/m3) ?
?e??t?? ?d?f???
?? d????aµµa t?? edaf???? a?t?d??se?? (p) e??a?
a?????? t?? ß???se?? (y), epe?d? p k y
12
1. ?????s? ?eµe???se?? µe ped???d?????
1.2 ?e pa?ad??? a??pt???? edaf???? p??se?? ?at?
t? µ??t??? Winkler
???t??? Winkler
???S??? ? de??t?? ed?f??? k ??? e??a?
?d??t?ta t?? ed?f???. ??a?t?ta? ap? ta
?a?a?t???st??? t?? ed?f??? (µ?t?? e?ast???t?ta?
?, ????? Poisson ?) ?a? t?? ped???d???? (µ????
?a? d?s?aµ??a) ? t?? ped???? (p??t?? ?a? µ????)
?????aµµa ß???se?? (y) ?a? edaf???? p??se?? (p)
??t? ap? t?? ped???d???
13
??t?µ?s? t?? de??t? ed?f??? k ? ks (de??t??
Winkler)
1. ??t??s? t?? (ks) ap? d???µ? f??t?s?? p???a?
ks de??t?? ed?f??? ??a t? s???e???µ??? p???a
t?? d???µ??
??-??aµµ??? µ??t???
?aµp??? d???µ??
14
??t?µ?s? t?? de??t? ed?f??? k (de??t?? Winkler)
2. ??t?µ?s? t?? k µ?s? eµpe?????? s??se??
2.1 ??-s??e?t??? (aµµ?d?) ed?f?
???te???µe?e? t?µ?? t?? de??t? ed?f??? k (se ??/
m3) ?at? Terzaghi (??a tet?a?????? ? ???????
p???a e????? ?? 0.305m)
S?et??? p????t?ta ?µµ?? ?a?a?? ??s?? p????t?ta? ?????
??µ?? t?? s?et???? p????t?ta? (Dr) lt 50 50-75 gt 75
????? t?µ?? k (MN/m3) ????? ? ????? ?µµ?? 6.4 19.2 19.2 - 96 96 - 320
???te???µe?e? t?µ?? k ????? ? ????? ?µµ?? 13 MN/m3 42 MN/m3 160 MN/m3
???te???µe?e? t?µ?? k ?µµ?? ??t? ap? t?? ?d??f??? ??????ta 8 MN/m3 26 MN/m3 96 MN/m3
15
??t?µ?s? t?? de??t? ed?f??? k ? ks (de??t??
Winkler)
2.2 S??e?t??? ed?f? (???????) ????p???s? t??
s??se?? e?ast???? µ??f??
??a ??aµpta p?d??a st?? ep?f??e?a t??
ed?f??? (D0 ? ?D1)
S??se?? Steinbrenner G?a ??????? p?d??? ?s
0.79 G?a tet?a?????? p?d??? ?s ? 1 G?a ????da
(L/B?) ?s ? 2
?p?te
ks de??t?? ed?f??? ??a t? s???e???µ??? p?d???
e????? (?)
?? e?ast???? s??se?? ????? ?a??te?? efa?µ??? st??
pe??pt?se?? ast?????st?? f??t?s?? s??e?t????
(a????????) edaf?? (?0.5, ? ?u), ?p?te ??a
??????? p???a d?aµ?t??? ?? 0.305m (1 p?d?), ?
de??t?? ed?f??? ks e??a?
??d?? ??????? cu (kPa) Eu / cu Eu (MPa) ks (MN/m3)
???? µa?a?? lt 12.5 400 lt 5 lt 25
?a?a?? 12.5 25 400 5 10 25 50
S??e?t??? 25 50 350 10 17.5 50 100
St?f?? 50 -100 300 17.5 30 100 - 165
???? st?f?? 100 200 200 30 40 165 220
S????? gt 200 150 gt 35 gt 200
16
??t?µ?s? t?? de??t? ed?f??? k ? ks (de??t??
Winkler)
3. ??t?µ?s? t?? k ??a p?d??a d?af????
d?ast?se??, µe ß?s? t? µet???e?sa t?µ? k?
3.1 S??e?t??? (a???????) ed?f?
?p? t?? s??se?? e?ast???? µ??f?? p????pte? ?t?
e?? st? d???µ? f??t?s?? p???a? (µe p???a
d?ast?se?? Bo) µet???e? de??t?? ed?f??? ks ko
qo / ?? ( ?p?? qo p?es? p???a?, ?? ?a????s?
p???a?) t?te, ? de??t?? ed?f??? ??a tet?a??????
p?d??? e????? ? e??a?
ko de??t?? ed?f??? ??a p???a e????? ?? 0.305m
?a?at???s? ? de??t?? ed?f??? de? e??a? sta?e??
pa??µet??? t?? ed?f??? a??? e?a?t?ta? ap? t??
d?ast?se?? t?? ped???? (µe t?? pa?ad??? ??aµµ????
e?ast???t?ta?). S???e???µ??a, ? de??t?? k
µe???eta? s?µa?t??? µe t?? a???s? t?? ?.
?et?a?????? p?d??a
G?a ?????????? p?d??a (Lµ???? gt ?)
G?a ????da (L8)
17
??t?µ?s? t?? de??t? ed?f??? k (de??t?? Winkler)
3.2 ??-s??e?t??? (aµµ?d?) ed?f?
?e a???p???s? t?? s??se?? ?p?????sµ?? t??
?a????s?? ??aµpt?? ped????
?e??t?? ed?f??? ks q / ?i
2.1 ????d?? ?p?????sµ?? t?? ?a????s?? ped????
e????? ? ?at? Alpan
?p?te
2.2 ????d?? ?p?????sµ?? t?? ?a????s??
tet?a??????? ped???? e????? ? ?at? Terzaghi
Peck
?p?te
18
??t?µ?s? t?? de??t? ed?f??? k (de??t?? Winkler)
3.2 ??-s??e?t??? (aµµ?d?) ed?f? (s????e?a)
??a???? se p?d??a d?af???? d?ast?se??
??t?µ?s? t?? k ??a tet?a?????? p?d??a e????? ?
(?at? Terzaghi Peck )
??? ?at? t?? d???µ? f??t?s?? p???a? (d?ast?se??
Bo 0.305m) µet???e? de??t?? ed?f??? ko qo /
?? (qo p?es? p???a?, ?? ?a????s? p???a?)
t?te, ? de??t?? ed?f??? ??a tet?a?????? p?d???
e????? ? gt Bo e??a?
?p?? ?, ?? se µ?t?a
??t?µ?s? t?? k ??a ?????????? p?d??a e????? ? ?a?
µ????? L gt B
??t?µ?s? t?? k ??a ped???d????? e????? ? (L8)
19
??t?µ?s? t?? de??t? ed?f??? k (de??t?? Winkler)
S?????s? t?? µe??d?? ?p?????sµ?? t?? de??t?
ed?f??? (k) ??a tet?a?????? p?d??a
?et?a?????? p?d??a
ko de??t?? ed?f??? ??a p???a e????? ?? 0.305m
G?a aµµ?d? ed?f?
G?a a??????? ed?f?
???????? ed?f?
?µµ?d? ed?f?
20
??t?µ?s? t?? de??t? ed?f??? k (de??t?? Winkler)
4. ?e s??e?t?µ?s? t?? s?et???? d?s?aµ??a?
ped???d???? - ed?f???
????d?? Vesic (1961) ??a ?eµe???????de?
Es , ? µ?t?? e?ast???t?ta? ?a? ????? Poisson
t?? ed?f??? ?b , ? , I µ?t?? e?ast???t?ta?,
p??t?? ?a? ??p? ad?a?e?a? t?? ped???d????
?a??de??µa efa?µ???? ?ed???d???? ? 1.20m,
H0.60m, Eb 25 GPa ?daf?? S??e?t??? ???????
µe ?u 15 MPa (?u 0.5) ? ? ?3 / 12 0.0216
m4
8.5 MN/m3
?a?at???s? ?p? t?? s??se?? e?ast???? µ??f?? ??a
????da (L/B? ? Is ? 2) p????pte?
8.3 MN/m3
21
??t?µ?s? t?? de??t? ed?f??? k (de??t?? Winkler)
???te???µe?e? t?µ?? t?? de??t? ed?f??? (k) ??a
s????e?? ped???d????? se d??f????? t?p???
edaf?? (?at? Bowles)
??d?? ed?f??? k (MN/m3)
?µµ?? ?a?a??? (Dr lt 50) ??s?? p????t?ta? (Dr 50-75) ?????? (Dr gt 75) 4.8 16 9.6 80 64 128
??????de?? ?µµ?? µ?s?? p????t?ta? 32 80
????de?? ?µµ?? µ?s?? p????t?ta? 24 48
??????? ?a?a??? (qu 25-50 kPa) S??e?t???? (qu 50-100 kPa) St?f??? (qu 100-200 kPa) ???? st?f??? (qu 200-400 kPa) ???? s?????? (qu gt 800 kPa) 5 12 12 18 18 24 24 48 gt 48
???S??? ? de??t?? ed?f??? (k) e?a?t?ta? ?a? ap?
ta ?a?a?t???st??? t?? ped???d???? (L, B, I, ?b).
S??ep??, ?? a??t??? t?µ?? e??a? e?de??t????.
S?µe??s? 10 MN/m3 1 kg/cm3
22
1. ?????s? ?eµe???se?? µe ped???d?????
1.2 ?e pa?ad??? a??pt???? edaf???? p??se?? ?at?
t? µ??t??? Winkler
???t??? Winkler
p edaf??? a?t?d?as? (kPa) y ß???s? t?? d????
(m) k sta?e?? e?at????? Winkler (kN/m3)
??af????? e??s?s? t?? d????
q ?ata?eµ?µ??? f??t?s? ep? t?? d???? (kN/m) ?
p??t?? t?? d???? (m) ?b , ? µ?t?? e?ast???t?ta?
(kN/m2) ?a? ??p? ad?a?e?a? (m4) t?? d????
? ep???s? t?? a??t??? d?af?????? e??s?s?? ??a
t????ta ep?ßeß??µ??a f??t?a (V, M, q),
?a?a?t???st??? t?? d???? (?b ,? ,? ,L) ?a?
?a?a?t???st??? t?? ed?f??? (de??t?? k) µp??e? ?a
???e? µe a?a??t???? ? a???µ?t???? µe??d??? (p.?.
µe pepe?asµ??a st???e?a)
23
1. ?????s? ?eµe???se?? µe ped???d?????
1.2 ?e pa?ad??? a??pt???? edaf???? p??se?? ?at?
t? µ??t??? Winkler
??af????? e??s?s? t?? d????
Eb µ?t?? e?ast???t?ta? t?? ped???d???? ? ??p?
ad?a?e?a? t?? d?at?µ?? t?? ped???d????. G?a
?????????? d?at?µ? p??t??? ? ?a? ????? ? y
ß???s? t?? d???? (m) k sta?e?? e?at?????
Winkler ??a t?? d??? (kN/m3) ? p??t?? t?? d????
(m) q ?ata?eµ?µ??? f??t?s? ep? t?? d????
(kN/m) p p?es? edaf???? a?t?d?as?? (kPa) p
k y
?at? t?? ep???s? t?? d?af?????? e??s?s?? (se
ap??? pe??pt?se?? f??t?s??) p????pte? ? ad??stat?
pa??µet???
G?a ap??? f??t?se??, ? d???? µp??e? ?a ?e????e?
(Hetenyi, 1946) ???? ??aµpt? ? lt (p/4) ?
µp??e? ?a efa?µ?s?e? ??aµµ??? ?ata??µ? t?se??
??d??µes?? a?aµ??a? (p/4) lt ? lt p ?
a????s? µe µ??t??? Winkler ???? e??aµpt? ?
gt p ? µp??e? ?a ?e????e? ?a? ?? ape???? µ?????
24
1. ?????s? ?eµe???se?? µe ped???d?????
1.2 ?e pa?ad??? a??pt???? edaf???? p??se?? ?at?
t? µ??t??? Winkler
??a??a?t???, ? s?et??? a?aµ??a t?? ped???d???? ??
p??? t? ?daf?? µp??e? ?a e?t?µ??e? ?at? Meyerhof
µ?s? t?? ad??stat?? pa?aµ?t???
Eb µ?t?? e?ast???t?ta? t?? ped???d???? ? ??p?
ad?a?e?a? t?? d?at?µ?? t?? ped???d???? ?s µ?t??
e?ast???t?ta? t?? ed?f??? ? p??t?? t??
ped???d???? (m) L µ???? t?? ped???d???? (m)
? ped???d???? µp??e? ?a ?e????e? ?? ??aµpt? e??
? gt 0.5 St?? pe??pt?s? a?t? (? gt 0.5), µp??e? ?a
???s?µ?p????e? ? pa?ad??? ??aµµ???? ?ata??µ?? t??
t?se?? st? ß?s? t?? ped???d????. ??? ? lt 0.5,
apa?te?ta? a????s? µe µ??t??? Winkler
25
?a??de??µa efa?µ???? ?p???s? ped???d???? µe
pa?ad??? edaf???? p??se?? µ?s? e?at????? t?p??
Winkler - a???µ?t??? ep???s? (pepe?asµ??a
st???e?a)
?ed???d???? ? 1.20m, H0.60m, L 12m
S????deµa Eb 25 GPa ??a ? ? ?3 /
12 0.0216 m4 ?daf?? S??e?t??? ???????, ?u
15 MPa (?u 0.5)
F??t?a
V1 V2 400 kN , V3 640 kN
?daf???? p??se?? µ?s? e?at????? t?p?? Winkler
26
?a??de??µa efa?µ???? ?p???s? ped???d???? µe
pa?ad??? edaf???? p??se?? µ?s? e?at????? t?p??
Winkler (a???µ?t??? ep???s?)
?e??t?? ed?f??? ?at? Vesic
8.5 MN/m3
?d??stat? pa??µet??? s?et???? d?s?aµ??a? d????
ed?f???
3.146 ? p
??a ? d???? e??a? s?et???? e??aµpt?
?????aµµa ß???se?? (y) t?? ped???d????
? ep???s? t?? d?af?????? e??s?s?? t?? ped???d????
p?? f??t??eta? µe ta (???st?) ep?ßeß??µ??a f??t?a
?a? ed???eta? se s??e??? ?ata?eµ?µ??a e?at???a
Winkler ????e µe a???µ?t??? µ???d? (pepe?asµ??a
st???e?a)
27
?a??de??µa efa?µ???? ?p???s? ped???d???? µe
pa?ad??? edaf???? p??se?? µ?s? e?at????? t?p??
Winkler - a???µ?t??? ep???s? (pepe?asµ??a
st???e?a)
S?????s? µe ta ap?te??sµata t?? ep???s?? µe
?e???s? ??aµµ???? ?ata??µ?? t?? edaf????
a?t?d??se?? de???e? p??? µ????? d?af???? (a? ?a?
? ped???d???? e??a? s?et??? e??aµpt?)
?????aµµa teµ???s?? d???µe??
?????aµµa ?aµpt???? ??p??
400 kN
400 kN
640 kN
400 kN
400 kN
640 kN
28
?a??de??µa efa?µ???? ?p???s? ped???d???? µe
pa?ad??? edaf???? p??se?? µ?s? e?at????? t?p??
Winkler - a???µ?t??? ep???s? (pepe?asµ??a
st???e?a)
S?????s? µe ta ap?te??sµata t?? ep???s?? µe
?e???s? ??aµµ???? ?ata??µ?? t?? edaf????
a?t?d??se?? de???e? p??? µ????? d?af???? (a? ?a?
? ped???d???? e??a? s?et??? e??aµpt?)
?????aµµa edaf???? ß???se??
?????aµµa edaf???? a?t?d??se??
400 kN
400 kN
640 kN
400 kN
400 kN
640 kN
67 kPa
133 kPa
29
?a??de??µa efa?µ????
?p???s? t?? ped???d???? µe pa?ad??? edaf????
a?t?d??se?? µ?s? e?at????? Winkler ????
e??aµpt? ped???d????
6.066 ? 2 p
?ed???d???? ????? ?0.25m (a?t? ?0.60m)
(???p?? pa??µet??? ?? ???)
(p??? e??aµpt?)
?????aµµa ?aµpt???? ??p??
?????aµµa edaf???? ß???se??
400 kN
400 kN
640 kN
400 kN
400 kN
640 kN
S?µp??asµa ???? µ???? a??a?? st?? ?aµpt????
??p??
30
1. ?????s? ?eµe???se?? µe ped???d?????
1.2 ?e pa?ad??? a??pt???? edaf???? p??se?? ?at?
t? µ??t??? Winkler
??a??t???? s??se?? ??a p??? e??aµpte?
ped???d????? (ape???µ??e?? L ? ?)
?d??stat? pa??µet??? s?et???? d?s?aµ??a? d????
ed?f???
(Hetenyi, 1946)
1. F??t?s? µe s???e?t??µ??? f??t?? ? st? s?µe??
x0
????s?
?aµpt??? ??p?
??µ???sa d??aµ?
?1 , ?2 , ?3 s??te?est?? ep?????? (ß??pe
ep?µe?? se??da)
31
??a??t???? s??se?? ??a p??? e??aµpte?
ped???d????? (ape???µ??e??)
2. F??t?s? µe s???e?t??µ??? ??p? ?? st? s?µe??
x0
????s?
?aµpt??? ??p?
??µ???sa d??aµ?
?1 , ?3 , ?4 s??te?est?? ep?????? (ß??pe
ep?µe?? se??da)
3. F??t?s? µe f??t?a ?a? s???e?t??µ??e? ??p?? se
d??f??e? ??se??
?p??e? ?a efa?µ?s?e? ? a??? t?? epa?????a?,
??t??ta? ?? x0 t? µ?s?? t?? ped???d????
32
??a??t???? s??se?? ??a p??? e??aµpte?
ped???d????? (ape???µ??e??)
??µ?? t?? s??te?est?? ep?????? ?1 , ?2 , ?3 ,
?4 se d??f??e? ??se?? x t?? d????
33
S????? µe??d?? a????s?? ped???d????
1. ???t???a a?aµ??a? ped???d????
???t???? ??p?? ????? t?µ?? ??d?? ped???d???? / ????d?? a????s??
Hetenyi ? lt (p/4) (p/4) lt ? lt p ? gt p ??aµpt? (??aµµ??? ?ata??µ?) ???aµpt? (Winkler) ???? e??aµpt? (Winkler ? L?)
Meyerhof ? gt 0.5 ? lt 0.5 ??aµpt? (??aµµ??? ?ata??µ?) ???aµpt? (Winkler)
2. ??t?µ?s? t?? de??t? ed?f??? (k) se e??aµpte?
ped???d?????
?e ape??e?a? µ?t??s? (d???µ? f??t?s?? p???a?) ?
e?t?µ?s? (ap? p??a?e?) t?? ko ?a? st? s????e?a
e?t?µ?s? t?? (k) µe a?a???? st?? d?ast?se?? t??
ped???d???? ? µe ???s? t?? µe??d?? Vesic
3. ????d?? a????s??
??aµpte? µe pa?ad??? ??aµµ???? ?ata??µ?? t??
edaf???? p??se?? st? ß?s? ???aµpte? µe ???s?
µ??t???? Winkler ???? e??aµpte? ??
ape???µ??e??
34
?fa?µ??? t?? de??t? ed?f??? (k) se ??aµpta p?d??a
?p?????sµ?? t?? st??f?? (?) ped???? (B x L) µe
a?????? d??aµ? (V) ?a? ??p? (?)
B e???? ped???? ?at? t?? d?e????s? t??
e??e?t??t?ta?
???e?t??t?ta
??s? t?s?
????? e??e?t??t?ta e lt B/6
1. ????? e??e?t??t?ta 0 ? e ? B / 6
Winkler ymax smax / k , ymin smin / k
??a
??s?aµ??a t?? st??f???? e?at????? st? ß?s? t??
ped????
(kNm / rad)
??s?aµ??a t?? a??????? e?at????? st? ß?s? t??
ped????
(kN / m)
35
?fa?µ??? t?? de??t? ed?f??? (k) se ??aµpta p?d??a
?p?????sµ?? t?? st??f?? (?) ped???? (B x L) µe
a?????? d??aµ? (V) ?a? ??p? (?)
B e???? ped???? ?at? t?? d?e????s? t??
e??e?t??t?ta?
???e?t??t?ta
??s? t?s?
?e???? e??e?t??t?ta e gt ?/6
2. ?e???? e??e?t??t?ta ? / 6 ? e ? B / 2
Winkler ymax smax / k
??a
??s?aµ??a t?? st??f???? e?at????? st? ß?s? t??
ped????
(kNm / rad)
??s?aµ??a t?? a??????? e?at????? st? ß?s? t??
ped????
(kN / m)
36
?fa?µ??? t?? de??t? ed?f??? (k) se ??aµpta p?d??a
?e??s? t?? e??e?t??t?ta? ped???? µes?t????a? ????
d?s?aµ??a? t?? st????
St???? µes?t????a? (µ?t?? e?ast???t?ta? ?b , ??p?
ad?a?e?a? ?) µe ?e?t???? f??t?s? V ed???eta? st?
?daf?? µe p?d???. ? ?efa?? t?? st???? ?e??e?ta?
a????t? (e?e??e?a st?ept?). ??? t? ?daf?? e??a?
apa?aµ??f?t? (k? ), d??ad? t? p?d??? e??a?
ap???t?? ?st?ept?, ? a?????? d??aµ? (V) t??
st???? ?a µetafe??e? st? ?daf?? ?? ?µ???µ??f?
p?es? µe s???staµ??? V ?at? t?? ????a t?? ped????
(e0). ???t? ??e? ?? ap?t??esµa t?? a??pt???
?e????? d???µe?? S ??a t?? a?????? t?? ??p?? M
V e, ?p?te S V e / H St?? s????? pe??pt?s?
p?? t? p?d??? µp??e? ?a st?afe? (?s? t??
ep?t??pe? ? de??t?? ed?f??? k), ? a?t?d?as? t??
ed?f??? ??e? s???staµ??? V µe e??e?t??t?ta e (0 lt
e lt e). St?? pe??pt?s? a?t?, t? ?e???? d???µe??
S ??a t?? a?????? t?? ap?µ????sa? ??p?? M V (e
e) e??a? S V(e-e) / H
St???? Eb I
37
?fa?µ??? t?? de??t? ed?f??? (k) se ??aµpta p?d??a
?e??s? t?? e??e?t??t?ta? ped???? µes?t????a? ????
d?s?aµ??a? t?? st????
?p?????sµ?? t?? e??e?t??t?ta? (e)
St??f? t?? ped???? (?) ???? de??t? ed?f??? (k)
?at? t?? ep?ß??? t?? ??p?? ? V e
St??f? t?? ß?s?? t?? st???? (p?d???) ????
efa?µ???? t?? ??p?? ? V (e e) st?? st???
????? ?, ??p?? ad?a?e?a? ? µe ?????s? st?? ?efa??

St???? Eb I
???s?s? t?? a??t??? s??se?? (?? p??? ?) ?a?
ep???s? ?? p??? e d??e?
G?a ?st?ept? p?d??? k ? ? e e
38
?e??s? t?? e??e?t??t?ta? ped???? µes?t????a? ????
d?s?aµ??a? t?? st????
?fa?µ???
??d??? B 2m L 4m ??????? f??t?? V
2000 kN ?daf?? k 10 MN/m3 Eb 25 GPa
(s????deµa) ??ast?se?? st???? b 0.4m d
0.6m ???? st???? ? 4.5m ? b d3 /12 0.4 x
0.63 / 12 0.0072 m4 e B/2 d/2 1 0.3
0.7m
? e 0.13m lt B/6
St???? Eb I
S V(e-e) / H 253.3 kN
????st? ??p? st???? ? S H 1140 kN
St??f? ped???? (?a? t?? ß?s?? t?? st????)
0.00975 rad 0.56o
39
2. ?????s? ?eµe???se?? µe ???t?st??se??
????a ???t?st??s?? µp??e? ?a ?e????e? ?? e??aµpt?
e?? ?? d?ast?se?? t?? (B, L) ?pe?ßa????? t??
?a?a?t???st??? d??stas? (?c ) (Westergaard, 1939)

S?µe??s? ?? e???? Bc a?t?st???e? st?? t?µ? ?
p ??a ped???d????? ?at? Hetenyi (1946)
Eb , ?b µ?t?? e?ast???t?ta? ?a? ????? Poisson
t?? s????d?µat?? t p???? p???a? ???t?st??s?? k
de??t?? ed?f??? p?? a?t?st???e? se tet?a??????
?eµ???? e????? Bc . ? de??t?? ed?f???
?p??????eta? µe t?? µe??d??? p?? a?af?????a?
p??????µ???? (?a? d???µ?? ??a t?? p??sd????sµ?
t?? e????? Bc).

• ???aµpte? p???e? ???t?st??se?? µp????? ?a
  a?a?????? µe ???s? e?at????? Winkler ?a? t?µ? t??
  de??t? k p?? a?t?st???e? se tet?a?????? ?eµ????
  e????? Bc
• ??? ?? d?ast?se?? t?? p???a? e??a? µ????te?e? t??
  Bc/2 t?te ? p???a e??a? ??aµpt? ?a? ???eta?
  pa?ad??? ??aµµ???? ?ata??µ?? t?? edaf???? p??se??
  st? ß?s? t??. ? p??sd????sµ?? t?? edaf????
  p??se?? ???eta? µe t?? e??s?se?? ?s????p?a? (?p??
  st?? ped???d?????). St? s????e?a, ? a????s? t??
  p???a? ???t?st??s?? ???eta? µe ta ???st? f??t?a
  ?a? edaf???? a?t?d??se??.
• ??? ? lt Bc/2 ?a? L gt Bc/2 , ? p???a µp??e? ?a
  a?a???e? ?? ped???d???? e????? ?
• ??? t? e???? (?) e??a? Bc/2 lt ? lt Bc , ? p???a
  µp??e? ?a a?a???e? µe ???s? e?at????? Winkler ?a?
  de??t? k p?? a?t?st???e? st?? p?a?µat???? t??
  d?ast?se??.

40
2. ?????s? ?eµe???se?? µe ???t?st??se??
?a??de??µa efa?µ???? 1 ????a ???t?st??s?? ap?
s????deµa (Eb 25 GPa, ?b 0.30) p????? t
0.80m ep? a???????? ed?f??? µe de??t? ko 75
MN/m3 (??a p???a e????? ?? 0.305m)
G?a a??????? ?daf??
S??d?asµ?? µe t??
d??e?
31.75 m
S??ep??, e?? ? p???a ??e? d?ast?se?? (?, L) ???
31.75 / 2 15.9 m, ?e??e?ta? ?? ??aµpt?. ??? ?
p???a ??e? d?ast?se?? (B, L) gt 31.75m ?e??e?ta?
?? e??aµpt? µe de??t? ed?f???
75 / (0.305 / 31.75) 0.72 MN/m3 0.07 kg/cm3
a?e?a?t?t?? t?? p?a?µat???? d?ast?se?? t?? p???a?.
?????, e?? ? p???a ??e? d?ast?se?? µeta?? 15.9m
?a? 31.7m, ? de??t?? ed?f??? ?p??????eta? ??a t??
p?a?µat???? t?? d?ast?se??.
41
2. ?????s? ?eµe???se?? µe ???t?st??se??
?a??de??µa efa?µ???? 2 ????a ???t?st??s?? ap?
s????deµa (Eb 25 GPa, ?b 0.30) p????? t
0.80m ep? aµµ?d??? ed?f??? µe de??t? ko 50
MN/m3 (??a p???a e????? ?? 0.305m)
G?a aµµ?de? ?daf??
S??d?asµ?? µe t??
d??e?
? ?c (?c??) 242 ? Bc 15.4 m
S??ep??, e?? ? p???a ??e? d?ast?se?? ??? 15.4 / 2
7.7 m ?e??e?ta? ?? ??aµpt?. ??? ??e? d?ast?se??
gt 15.4m ?e??e?ta? ?? e??aµpt? µe de??t? ed?f???
(50/4) 1(0.305/15.4)2 13 ??/m3 1.3 kg/cm3
a?e?a?t?t?? t?? p?a?µat???? d?ast?se?? t?? p???a?.
?????, e?? ? p???a ??e? d?ast?se?? µeta?? 7.7m
?a? 15.4m, ? de??t?? ed?f??? ?p??????eta? ??a t??
p?a?µat???? t?? d?ast?se??.