De 22
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phÇn 2 - bµI to¸n tÊm ph¼ng Sè hiÖu bµI tËp: 22
§Ò bµi sè 22 Cho kÕt cÊu chÞu lùc nh trªn h×nh 01. Trong ®ã: a = 3m, t = 10cm, E = 1.2x106 N/cm2, ν= 0.18, P = 20 kN, q = 5 kN/m. Yªu cÇu: - TÝnh chuyÓn vÞ c¸c nót, - X¸c ®Þnh vÐc t¬ øng suÊt trong c¸c phÇn tö.
H×nh 01 - S¬ ®è kÕt cÊu
phÇn tÝnh to¸n bµI 22 1. C¸c sè liÖu ban ®Çu . ChiÒu dÇy tÊm t = 0.10 m . KÝch thíc cña kÕt cÊu a = 3.00 m . M« ®un ®µn håi E = 1.2E+07 kN/m2 . HÖ sè Po¸t - x«ng ν = 0.18 2. Chia kÕt cÊu thµnh c¸c phÇn tö vµ c¸c th«ng tin cho tÝnh to¸n
H×nh 02 - S¬ ®å rêi r¹c cÊu KÕt cÊu ®îc chia thµnh 16 phÇn tö tam gi¸chãa nh kÕt h×nh 02. Liªn kÕt biªn díi cña tÊm ®îc m« h×nh hãa bëi c¸c liªn kÕt gèi cè ®Þnh t¹i c¸c nót däc theo biªn nµy.
KÕt cÊu ®îc chia thµnh 16 phÇn tö tam gi¸c nh h×nh 02. Liªn kÕt biªn díi cña tÊm ®îc m« h×nh hãa bëi c¸c liªn kÕt gèi cè ®Þnh t¹i c¸c nót däc theo biªn nµy. C¸c th«ng tin vÒ nót: Sè hiÖu nót 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 To¹ ®é X 0 0 0 1.5 1.5 1.5 3 3 3 4.5 4.5### 6 6 6 To¹ ®é Y 0 1.5 3.0 0.0 1.5 3.0 0.0###3.0 0.0 1.5###0.0 1.5 3.0
TÝnh to¸n dêi t¶i träng ph©n bå vÒ c¸c nót
H×nh 03 - S¬ ®å chia t¶i träng vµ dêi t¶i träng vÒ nót C¸c th«ng tin vÒ t¶i träng: Sè hiÖu nót 1 2 3 4 Px 0 ? ? ? PY
?
0
0
?
5 0
6 0
7 ?
8 0
9 10 11 12 13 14 15 0 ? 0 0 ? ###-3.75
0
0
?
0
0
?
0
0
?
Sau khi xö lý ®iÒu kiÖn biªn ta cã vÐc t¬ chuyÓn vÞ cÇn t×m: ∆ = {'u2 v2 u14 v14
v3 u5 v5 u6 v6 u8 v8 u9 v9 u11 v11 u12 v12 u15 v15 }
0
-20
3. LËp ma trËn ®é cøng 3.1. LËp ma trËn ®é cøng phÇn tö Ma trËn ®é cøng cña phÇn tö ®îc x¸c ®Þnh bëi c«ng thøc:
[
[ k ii ] [ k ij ] [ k im ] [ k ] = [ k ji ] [ k jj ] [ k jm ] [ k mi ] [ k mj ] [ k mm ]
]
Trong ®ã:
[
1−ν br bs cr c s Et 2 [ k rs ] = 4 1− ν 2 Δ νc b 1−ν b c r s r s 2 r=i , j , m ; s=i , j , m
1−ν c r bs 2 1− ν cr c s br b s 2 νb r c s
]
Hay ma trËn [k] cã thÓ viÕt l¹i nh sau:
[
1−ν c c 2 i i 1−ν νc i b i bi ci 2 1−ν b j bi c j ci Et 2 [ k ]= 2 4 1−ν Δ νc b 1−ν b c j i j i 2 1−ν bm bi c c 2 m i 1−ν νc m b i bm ci 2 bi b i
1−ν ci bi 2 1−ν c i ci b i bi 2 1−ν νb j c i c j bi 2 1−ν c j ci b j bi 2 1−ν νb m c i cm bi 2 1−ν c m ci bm bi 2 νb i c i
1−ν c c 2 i j 1−ν νc i b j bi c j 2 1−ν b j b j cjcj 2 1−ν νc j b j bjcj 2 1−ν bm b j c c 2 m j 1−ν νc m b j bm c j 2 bi b j
1− ν cib j 2 1−ν ci c j bi b j 2 1− ν νb j c j c jbj 2 1−ν c j c j bjbj 2 1− ν νb m c j cm b j 2 1−ν cm c j bm b j 2 νb i c j
1− ν ci c m 2 1− ν νc i b m bi cm 2 1− ν b j b m c j cm 2 1− ν νc j b m b jcm 2 1− ν b m b m cm cm 2 1− ν νc m b m b m cm 2 b i b m
Ta ph©n c¸c phÇn tö tam gi¸c cña kÕt cÊu thµnh hai lo¹i: Lo¹i 1 gåm c¸c phÇn tö: 1,2,5,6,9,10,13,14 (h×nh 05) Lo¹i 2 gåm c¸c phÇn tö: 3,4,7,8,11,12,15,16 (h×nh 06) LËp ma trËn ®é cøng cña phÇn tö lo¹i 1 y m
i
Tªn nót To¹ ®é X To¹ ®é Y
j
x
H×nh 05 - PhÇn tö lo¹i 1
i 0 0
TÝnh c¸c hÖ sè: ai= xjym-xmyj = 2.3 bi = yj-ym = 0.0 ci = xm-xj = -1.5 aj = xmyi-xiym = 0.0 bj = ym-yi = 1.5 cj = xi-xm =0.0 am = xiyj-xjyi = 0.0 bm = yi-yj = -1.5 cm = xj-xi =1.5
DiÖn tÝch tam gi¸c ijm: 1
xi
yi
j 1.5 1.5
m 0 1.5
1−ν c b 2 i m 1−ν ci c i bi b m 2 1−ν νb j c m c j bm 2 1− ν c j c m b j bm 2 1−ν νb m c m c b 2 m m 1− ν c m c m b m bm 2 νb i c m
]
∆ = 0.5*
∆ = 0.5*
1 1
xj xm
yj ym
1 1 1
0 1.5 0
0 1.5 1.5
=1.1
i 0 0
j 1.5 0
LËp ma trËn ®é cøng cña phÇn tö lo¹i 2 y
Tªn nót To¹ ®é X To¹ ®é Y
m
j i
x
H×nh 06 - PhÇn tö lo¹i 2
TÝnh c¸c hÖ sè: ai = xjym-xmyj = 2.3 bi = yj-ym =-1.5 ci = xm-xj =0.0 aj = xmyi-xiym = 0.0 bj = ym-yi = 1.5 cj = xi-xm = -1.5 am = xiyj-xjyi = 0.0 bm = yi-yj =0.0 cm = xj-xi =1.5
DiÖn tÝch tam gi¸c ijm:
∆ = 0.5*
1 1 1
0 1.5 1.5
0 0 1.5
=1.1
m 1.5 1.5
Ma trËn ®é cøng phÇn tö tam gi¸c lo¹i 1
(14) (13) (10) (9) (6) (5) (2) (1)
[k]1 =
u11
v11
0
u15 u14 u12 u11 u9 u8 u6 u5
v15 v14 v12 v11 v9 v8 v6 v5
u12 u11 u9 u8 u6 u5 u3 u2
v12 v11 v9 v8 v6 v5 v3 v2
0 u8
0 v8
0 u5
0 v5
0 u2
0 v2
0 254237
0
0
-254237
-254237
254237
0
620091
-111616
0
111616
-620091
0
-111616
620091
0
-620091
111616
-254237
0
0
254237
254237
-254237
-254237
111616
-620091
254237
874328
-365854
254237
-620091
111616
-254237
-365854
874328
0 0 u5 v5 u2 v2
u2 v2 u6 v6 u3 v3
0 0 u8 v8 u5 v5
u5 v5 u9 v9 u6 v6
0 0 u11 v11 u8 v8
u8 v8 u12 v12 u9 v9
0 0 u14 v14 u11 v11
u11 v11 u15 v15 u12 v12
(1)
(2)
(5)
(6)
(9)
(10)
(13)
(14)
Ma trËn ®é cøng phÇn tö tam gi¸c lo¹i 2 (16) (15) (12) (11) (8) (7) (4) (3)
[k]2 =
u11
v11
u14
v14
0
u15 u14 u12 u11 u9 u8 u6 u5
v15 v14 v12 v11 v9 v8 v6 v5
0 u8
0 v8
0 u11
0 v11
0 u5
0 v5
0 u8
0 v8
0 u2
0 v2
0 u5
0 v5
0
0
0
620091
0
-620091
111616
0
-111616
0
254237
254237
-254237
-254237
0
-620091
254237
874328
-365854
-254237
111616
111616
-254237
-365854
874328
254237
-620091
0
-254237
-254237
254237
254237
0
0 0 0 0 u5
u2 v2 u5 v5 u6
0 0 0 0 u8
u5 v5 u8 v8 u9
0 0 0 0 u11
u8 v8 u11 v11 u12
0 0 0 0 u14
u11 v11 u14 v14 u15
-111616
0
111616
-620091
0
620091
v5
v6
v8
v9
v11
v12
v14
v15
(3)
(4)
(7)
(8)
(11)
(12)
(15)
(16)
3.2.LËp ma trËn ®é cøng cña kÕt cÊu
u2
v2
u3
v3
u5
v5
u6
v6
u8
v8
u9
v9
u11
v11
u12
1240182
0
620091
-111616
0
0
0
-223233
0
0
0
-223233
0
0
0
0
508475
-254237
254237
0
0
-508475
0
0
0
-508475
0
0
0
-508475
620091
-254237
874328
-365854
0
0
254237
-111616
0
0
254237
-111616
0
0
254237
-111616
254237
-365854
874328
0
0
-254237
620091
0
0
-254237
620091
0
0
-254237
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-508475
254237
-254237
0
0
508475
0
0
0
508475
0
0
0
508475
-223233
0
-111616
620091
0
0
0
1240182
0
0
0
1240182
0
-254237
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-254237
0
0
508475
0
0
0
508475
0
0
0
508475
[K]=0
-508475
-223233
0
-111616
620091
0
0
0
1240182
0
0
0
1240182
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-254237
0
0
0
0
0
0
0
-254237
0
0
508475
0
0
0
508475
0
0
0
508475
-111616
620091
0
0
0
1240182
0
0
0
0
0
0
0
0 -620091
111616
0
0
0
223233
0
0
0
223233
0
0
0
254237
-254237
0
0
508475
0
0
0
508475
0
0
0
508475
-620091
-254237 -620091
111616
0
0
254237
111616
0
0
254237
111616
0
0
254237
-111616
-254237 254237
-254237
0
0
254237
620091
0
0
254237
620091
0
0
254237
0 -223233 -1240182 0
0
0
-508475 254237 0 -508475
v12
u14
-223233 -1240182
v14 0
u15
v15
-620091 -111616
0
0
-508475 -254237 -254237
-111616
-620091
254237
-620091
254237
620091
111616
-254237
111616
-254237
0
0
0
0
0
0
0
0
0
0
0
0
508475
254237
254237
1240182
223233
0
111616
620091
0
0
0
0
0
0
0
0
0
0
0
0
508475
254237
254237
0
223233
0
111616
620091
0
0
0
0
0
u2 v2 u3 v3 u5 v5 u6 v6 u8 v8 u9 v9 u11
0
0
0
0
0
0
0
508475
254237
254237
1240182
223233
0
111616
620091
223233
1240182
0
620091
111616
0
0
508475
254237
254237
111616
620091
254237
874328
0
620091
111616
254237
0
874328
v11 u12 v12 u14 v14 u15 v15
4. LËp ph¬ng tr×nh c©n b»ng 1240182
0
0
508475
620091 -111616
0
0
0
-223233
0
0
0
-223233
0
0
0
-223233
###
254237
0
0
-508475
0
0
0
-508475
0
0
0
###
0
0
620091 -254237 874328 -365854
0
0
254237
-111616
0
0
254237 -111616
0
0
-111616 254237
-254237 620091
0
0
###
620091
111616
###
###
874328
0
0
-254237
620091
0
0
254237 -111616 -620091
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
508475
0
0
0
508475
0
0
0
508475
0
0
0
-508475 254237 -254237
-223233
0
###
620091
0
0
0
1240182
0
0
0
###
0
-254237
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-508475
0
-254237
0
0
508475
0
0
0
508475
0
0
0
508475
0
0
-223233
0
###
620091
0
0
0
1240182
0
0
0
###
0
0
0
0
223233
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-254237
0
0
0
0
0
0
0
0
0
0
0
508475
0
0
0
508475
0
0
0
508475
0
0
0
-508475 254237 -254237
-223233
0
###
620091
0
0
0
1240182
0
0
0
0
0
0
0
###
0
###
111616
0
0
0
223233
0
0
0
223233
0
0
0
-508475 254237 -254237
0
0
508475
0
0
0
508475
0
0
0
508475
0
1240182 223233
1240182 223233 223233 1240182 0
0
-620091 -254237
###
111616
0
0
254237
111616
0
0
254237 111616
0
0
254237 111616
620091
-111616 -254237 254237 -254237
0
0
254237
620091
0
0
254237 620091
0
0
254237 620091
111616
5
6
7
8
9
10
11
12
13
14
15
5. C¸c chuyÓn vÞ nót tªn nót
1
2
3
4
u
0
###
###
0
### #VALUE!
0
###
###
0
###
###
0
###
###
v
0
###
###
0
### #VALUE!
0
###
###
0
###
###
0
###
###
###
###
u2
0
u2
#VALUE!
###
###
###
v2
0
v2
#VALUE!
254237
###
254237
u3
0
u3
#VALUE!
###
111616
###
v3
0
v3
#VALUE!
0
0
0
u5
0
u5
#VALUE!
0
0
0
v5
0
v5
#VALUE!
u6
0
u6
#VALUE!
v6
0
v6
#VALUE!
u8
0
u8
#VALUE!
0
v8
508475 254237 254237 0
111616 620091
0
0
0
0
0
0
508475 254237 254237 0
111616 620091
x
v8
=
↑
0
=
#VALUE!
u9
0
u9
#VALUE!
v9
0
v9
#VALUE!
0
0
0
u11
0
u11
#VALUE!
0
0
0
v11
0
v11
#VALUE!
u12
0
u12
#VALUE!
508475 254237 254237 0
111616 620091
v12
0
v12
#VALUE!
0
620091 111616
u14
-7.5
u14
#VALUE!
508475 254237 254237
v14
0
v14
#VALUE!
254237 874328 254237
0
0
u15
-3.75
u15
#VALUE!
874328
v15
-20
v15
#VALUE!
6. X¸c ®Þnh c¸c vÐc t¬ øng suÊt trong c¸c phÇn tö VÐc t¬ øng suÊt trong c¸c phÇn tö ®îc x¸c ®Þnh th«ng qua vÐc t¬ chuyÓn vÞ nót cña theo c«ng thøc sau:
{}
σx { σ } = σ y = [ D ][ B ] { δ }e τ xy e
[
1 0 [ D ] [ B ]= E 2 ν 0 1−ν 0 1− ν 2
0 0 1− ν 2
ν −1 1 −ν 1− ν 0 − 2
−ν −1 1−ν − 2
]
{}
ui vi u { δ }e = j vj um vm
[D][B]
=
### 2232327 0
0 0 ###
0 0 5084746
0 1 0
-1.00 -0.18 -0.41
-0.18 -1 -0.41
TÝnh cho phÇn tö thø nhÊt: 0.00E+00 0.00E+00 #VALUE!
σx {σ}
e
1
=
σy
=
τxy 1
[D][B] x
#VALUE! #VALUE! #VALUE!
#VALUE! =
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø hai: #VALUE! #VALUE! #VALUE!
σx {σ}
e
2
=
σy
=
[D][B] x
τxy
#VALUE!
#VALUE! =
#VALUE! #VALUE!
2
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø ba: 0.00E+00 0.00E+00 0.00E+00
σx {σ}
e
3
=
σy
=
[D][B] x
τxy
0.00E+00
#VALUE! =
#VALUE! #VALUE!
3
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø t: #VALUE! #VALUE! #VALUE!
σx {σ}
e
4
=
σy
=
[D][B] x
τxy
#VALUE!
#VALUE! =
#VALUE! #VALUE!
4
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø n¨m 0.00E+00 0.00E+00 #VALUE!
σx {σ}
e
1
=
σy τxy
=
[D][B] x
#VALUE! #VALUE!
#VALUE! =
#VALUE! (kN/m2) #VALUE!
5
#VALUE!
TÝnh cho phÇn tö thø s¸u: #VALUE! #VALUE! #VALUE!
σx {σ}
e
6
=
σy
=
[D][B] x
τxy
#VALUE!
#VALUE! =
#VALUE! #VALUE!
6
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø b¶y: 0.00E+00 0.00E+00 0.00E+00
σx {σ}
e
7
=
σy
=
[D][B] x
τxy
0.00E+00
#VALUE! =
#VALUE! #VALUE!
7
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø t¸m: #VALUE! #VALUE! #VALUE!
σx {σ}
e
8
=
σy
=
[D][B] x
τxy
#VALUE!
#VALUE! =
#VALUE! #VALUE!
8
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø chÝn: 0.00E+00 0.00E+00 #VALUE!
σx {σ}
e
9
=
σy
=
[D][B] x
#VALUE!
#VALUE! =
#VALUE! (kN/m2)
τxy
#VALUE! #VALUE!
9
#VALUE!
TÝnh cho phÇn tö thø mêi: #VALUE! #VALUE! #VALUE!
σx {σ}
e
10
=
σy
=
[D][B] x
τxy
#VALUE!
#VALUE! =
#VALUE! #VALUE!
10
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø mêi mét: 0.00E+00 0.00E+00 0.00E+00
σx {σ}
e
11
=
σy
=
[D][B] x
τxy
0.00E+00
#VALUE! =
#VALUE! #VALUE!
11
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø mêi hai: #VALUE! #VALUE! #VALUE!
σx {σ}
e
12
=
σy
=
[D][B] x
τxy 12
#VALUE! #VALUE! #VALUE!
#VALUE! =
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø mêi ba:
σx
0.00E+00 0.00E+00 #VALUE!
#VALUE!
{σ}e
13
=
σy
=
[D][B] x
τxy
#VALUE!
=
#VALUE! #VALUE!
13
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø mêi bèn: #VALUE! #VALUE! #VALUE!
σx {σ}
e
14
=
σy
=
[D][B] x
τxy
#VALUE!
#VALUE! =
#VALUE! #VALUE!
14
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø mêi n¨m: 0.00E+00 0.00E+00 0.00E+00
σx {σ}
e
15
=
σy
=
[D][B] x
τxy
0.00E+00
#VALUE! =
#VALUE! #VALUE!
15
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø mêi s¸u: #VALUE! #VALUE! #VALUE!
σx {σ}e
16
=
σy
=
τxy 16
[D][B] x
#VALUE! #VALUE! #VALUE!
#VALUE! =
#VALUE! (kN/m2) #VALUE!
chuyÓn vÞ nót cña phÇn tö
§Ò bµi sè 22 Cho kÕt cÊu chÞu lùc nh trªn h×nh 01. Trong ®ã: a = 3m, t = 10cm, E = 1.2x106 N/cm2, ν= 0.18, P = 20 kN, q = 5 kN/m. Yªu cÇu: - TÝnh chuyÓn vÞ c¸c nót, - X¸c ®Þnh vÐc t¬ øng suÊt trong c¸c phÇn tö.
H×nh 01 - S¬ ®è kÕt cÊu
phÇn tÝnh to¸n bµI 22 1. C¸c sè liÖu ban ®Çu . ChiÒu dÇy tÊm t = 0.10 m . KÝch thíc cña kÕt cÊu a = 3.00 m . M« ®un ®µn håi E = 1.2E+07 kN/m2 . HÖ sè Po¸t - x«ng ν = 0.18 2. Chia kÕt cÊu thµnh c¸c phÇn tö vµ c¸c th«ng tin cho tÝnh to¸n
H×nh 02 - S¬ ®å rêi r¹c cÊu KÕt cÊu ®îc chia thµnh 16 phÇn tö tam gi¸chãa nh kÕt h×nh 02. Liªn kÕt biªn díi cña tÊm ®îc m« h×nh hãa bëi c¸c liªn kÕt gèi cè ®Þnh t¹i c¸c nót däc theo biªn nµy.
KÕt cÊu ®îc chia thµnh 16 phÇn tö tam gi¸c nh h×nh 02. Liªn kÕt biªn díi cña tÊm ®îc m« h×nh hãa bëi c¸c liªn kÕt gèi cè ®Þnh t¹i c¸c nót däc theo biªn nµy. C¸c th«ng tin vÒ nót: Sè hiÖu nót 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 To¹ ®é X 0 0 0 1.5 1.5 1.5 3 3 3 4.5 4.5### 6 6 6 To¹ ®é Y 0 1.5 3.0 0.0 1.5 3.0 0.0###3.0 0.0 1.5###0.0 1.5 3.0
TÝnh to¸n dêi t¶i träng ph©n bå vÒ c¸c nót
H×nh 03 - S¬ ®å chia t¶i träng vµ dêi t¶i träng vÒ nót C¸c th«ng tin vÒ t¶i träng: Sè hiÖu nót 1 2 3 4 Px 0 ? ? ? PY
?
0
0
?
5 0
6 0
7 ?
8 0
9 10 11 12 13 14 15 0 ? 0 0 ? ###-3.75
0
0
?
0
0
?
0
0
?
Sau khi xö lý ®iÒu kiÖn biªn ta cã vÐc t¬ chuyÓn vÞ cÇn t×m: ∆ = {'u2 v2 u14 v14
v3 u5 v5 u6 v6 u8 v8 u9 v9 u11 v11 u12 v12 u15 v15 }
0
-20
3. LËp ma trËn ®é cøng 3.1. LËp ma trËn ®é cøng phÇn tö Ma trËn ®é cøng cña phÇn tö ®îc x¸c ®Þnh bëi c«ng thøc:
[
[ k ii ] [ k ij ] [ k im ] [ k ] = [ k ji ] [ k jj ] [ k jm ] [ k mi ] [ k mj ] [ k mm ]
]
Trong ®ã:
[
1−ν br bs cr c s Et 2 [ k rs ] = 4 1− ν 2 Δ νc b 1−ν b c r s r s 2 r=i , j , m ; s=i , j , m
1−ν c r bs 2 1− ν cr c s br b s 2 νb r c s
]
Hay ma trËn [k] cã thÓ viÕt l¹i nh sau:
[
1−ν c c 2 i i 1−ν νc i b i bi ci 2 1−ν b j bi c j ci Et 2 [ k ]= 2 4 1−ν Δ νc b 1−ν b c j i j i 2 1−ν bm bi c c 2 m i 1−ν νc m b i bm ci 2 bi b i
1−ν ci bi 2 1−ν c i ci b i bi 2 1−ν νb j c i c j bi 2 1−ν c j ci b j bi 2 1−ν νb m c i cm bi 2 1−ν c m ci bm bi 2 νb i c i
1−ν c c 2 i j 1−ν νc i b j bi c j 2 1−ν b j b j cjcj 2 1−ν νc j b j bjcj 2 1−ν bm b j c c 2 m j 1−ν νc m b j bm c j 2 bi b j
1− ν cib j 2 1−ν ci c j bi b j 2 1− ν νb j c j c jbj 2 1−ν c j c j bjbj 2 1− ν νb m c j cm b j 2 1−ν cm c j bm b j 2 νb i c j
1− ν ci c m 2 1− ν νc i b m bi cm 2 1− ν b j b m c j cm 2 1− ν νc j b m b jcm 2 1− ν b m b m cm cm 2 1− ν νc m b m b m cm 2 b i b m
Ta ph©n c¸c phÇn tö tam gi¸c cña kÕt cÊu thµnh hai lo¹i: Lo¹i 1 gåm c¸c phÇn tö: 1,2,5,6,9,10,13,14 (h×nh 05) Lo¹i 2 gåm c¸c phÇn tö: 3,4,7,8,11,12,15,16 (h×nh 06) LËp ma trËn ®é cøng cña phÇn tö lo¹i 1 y m
i
Tªn nót To¹ ®é X To¹ ®é Y
j
x
H×nh 05 - PhÇn tö lo¹i 1
i 0 0
TÝnh c¸c hÖ sè: ai= xjym-xmyj = 2.3 bi = yj-ym = 0.0 ci = xm-xj = -1.5 aj = xmyi-xiym = 0.0 bj = ym-yi = 1.5 cj = xi-xm =0.0 am = xiyj-xjyi = 0.0 bm = yi-yj = -1.5 cm = xj-xi =1.5
DiÖn tÝch tam gi¸c ijm: 1
xi
yi
j 1.5 1.5
m 0 1.5
1−ν c b 2 i m 1−ν ci c i bi b m 2 1−ν νb j c m c j bm 2 1− ν c j c m b j bm 2 1−ν νb m c m c b 2 m m 1− ν c m c m b m bm 2 νb i c m
]
∆ = 0.5*
∆ = 0.5*
1 1
xj xm
yj ym
1 1 1
0 1.5 0
0 1.5 1.5
=1.1
i 0 0
j 1.5 0
LËp ma trËn ®é cøng cña phÇn tö lo¹i 2 y
Tªn nót To¹ ®é X To¹ ®é Y
m
j i
x
H×nh 06 - PhÇn tö lo¹i 2
TÝnh c¸c hÖ sè: ai = xjym-xmyj = 2.3 bi = yj-ym =-1.5 ci = xm-xj =0.0 aj = xmyi-xiym = 0.0 bj = ym-yi = 1.5 cj = xi-xm = -1.5 am = xiyj-xjyi = 0.0 bm = yi-yj =0.0 cm = xj-xi =1.5
DiÖn tÝch tam gi¸c ijm:
∆ = 0.5*
1 1 1
0 1.5 1.5
0 0 1.5
=1.1
m 1.5 1.5
Ma trËn ®é cøng phÇn tö tam gi¸c lo¹i 1
(14) (13) (10) (9) (6) (5) (2) (1)
[k]1 =
u11
v11
0
u15 u14 u12 u11 u9 u8 u6 u5
v15 v14 v12 v11 v9 v8 v6 v5
u12 u11 u9 u8 u6 u5 u3 u2
v12 v11 v9 v8 v6 v5 v3 v2
0 u8
0 v8
0 u5
0 v5
0 u2
0 v2
0 254237
0
0
-254237
-254237
254237
0
620091
-111616
0
111616
-620091
0
-111616
620091
0
-620091
111616
-254237
0
0
254237
254237
-254237
-254237
111616
-620091
254237
874328
-365854
254237
-620091
111616
-254237
-365854
874328
0 0 u5 v5 u2 v2
u2 v2 u6 v6 u3 v3
0 0 u8 v8 u5 v5
u5 v5 u9 v9 u6 v6
0 0 u11 v11 u8 v8
u8 v8 u12 v12 u9 v9
0 0 u14 v14 u11 v11
u11 v11 u15 v15 u12 v12
(1)
(2)
(5)
(6)
(9)
(10)
(13)
(14)
Ma trËn ®é cøng phÇn tö tam gi¸c lo¹i 2 (16) (15) (12) (11) (8) (7) (4) (3)
[k]2 =
u11
v11
u14
v14
0
u15 u14 u12 u11 u9 u8 u6 u5
v15 v14 v12 v11 v9 v8 v6 v5
0 u8
0 v8
0 u11
0 v11
0 u5
0 v5
0 u8
0 v8
0 u2
0 v2
0 u5
0 v5
0
0
0
620091
0
-620091
111616
0
-111616
0
254237
254237
-254237
-254237
0
-620091
254237
874328
-365854
-254237
111616
111616
-254237
-365854
874328
254237
-620091
0
-254237
-254237
254237
254237
0
0 0 0 0 u5
u2 v2 u5 v5 u6
0 0 0 0 u8
u5 v5 u8 v8 u9
0 0 0 0 u11
u8 v8 u11 v11 u12
0 0 0 0 u14
u11 v11 u14 v14 u15
-111616
0
111616
-620091
0
620091
v5
v6
v8
v9
v11
v12
v14
v15
(3)
(4)
(7)
(8)
(11)
(12)
(15)
(16)
3.2.LËp ma trËn ®é cøng cña kÕt cÊu
u2
v2
u3
v3
u5
v5
u6
v6
u8
v8
u9
v9
u11
v11
u12
1240182
0
620091
-111616
0
0
0
-223233
0
0
0
-223233
0
0
0
0
508475
-254237
254237
0
0
-508475
0
0
0
-508475
0
0
0
-508475
620091
-254237
874328
-365854
0
0
254237
-111616
0
0
254237
-111616
0
0
254237
-111616
254237
-365854
874328
0
0
-254237
620091
0
0
-254237
620091
0
0
-254237
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-508475
254237
-254237
0
0
508475
0
0
0
508475
0
0
0
508475
-223233
0
-111616
620091
0
0
0
1240182
0
0
0
1240182
0
-254237
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-254237
0
0
508475
0
0
0
508475
0
0
0
508475
[K]=0
-508475
-223233
0
-111616
620091
0
0
0
1240182
0
0
0
1240182
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-254237
0
0
0
0
0
0
0
-254237
0
0
508475
0
0
0
508475
0
0
0
508475
-111616
620091
0
0
0
1240182
0
0
0
0
0
0
0
0 -620091
111616
0
0
0
223233
0
0
0
223233
0
0
0
254237
-254237
0
0
508475
0
0
0
508475
0
0
0
508475
-620091
-254237 -620091
111616
0
0
254237
111616
0
0
254237
111616
0
0
254237
-111616
-254237 254237
-254237
0
0
254237
620091
0
0
254237
620091
0
0
254237
0 -223233 -1240182 0
0
0
-508475 254237 0 -508475
v12
u14
-223233 -1240182
v14 0
u15
v15
-620091 -111616
0
0
-508475 -254237 -254237
-111616
-620091
254237
-620091
254237
620091
111616
-254237
111616
-254237
0
0
0
0
0
0
0
0
0
0
0
0
508475
254237
254237
1240182
223233
0
111616
620091
0
0
0
0
0
0
0
0
0
0
0
0
508475
254237
254237
0
223233
0
111616
620091
0
0
0
0
0
u2 v2 u3 v3 u5 v5 u6 v6 u8 v8 u9 v9 u11
0
0
0
0
0
0
0
508475
254237
254237
1240182
223233
0
111616
620091
223233
1240182
0
620091
111616
0
0
508475
254237
254237
111616
620091
254237
874328
0
620091
111616
254237
0
874328
v11 u12 v12 u14 v14 u15 v15
4. LËp ph¬ng tr×nh c©n b»ng 1240182
0
0
508475
620091 -111616
0
0
0
-223233
0
0
0
-223233
0
0
0
-223233
###
254237
0
0
-508475
0
0
0
-508475
0
0
0
###
0
0
620091 -254237 874328 -365854
0
0
254237
-111616
0
0
254237 -111616
0
0
-111616 254237
-254237 620091
0
0
###
620091
111616
###
###
874328
0
0
-254237
620091
0
0
254237 -111616 -620091
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
508475
0
0
0
508475
0
0
0
508475
0
0
0
-508475 254237 -254237
-223233
0
###
620091
0
0
0
1240182
0
0
0
###
0
-254237
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-508475
0
-254237
0
0
508475
0
0
0
508475
0
0
0
508475
0
0
-223233
0
###
620091
0
0
0
1240182
0
0
0
###
0
0
0
0
223233
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-254237
0
0
0
0
0
0
0
0
0
0
0
508475
0
0
0
508475
0
0
0
508475
0
0
0
-508475 254237 -254237
-223233
0
###
620091
0
0
0
1240182
0
0
0
0
0
0
0
###
0
###
111616
0
0
0
223233
0
0
0
223233
0
0
0
-508475 254237 -254237
0
0
508475
0
0
0
508475
0
0
0
508475
0
1240182 223233
1240182 223233 223233 1240182 0
0
-620091 -254237
###
111616
0
0
254237
111616
0
0
254237 111616
0
0
254237 111616
620091
-111616 -254237 254237 -254237
0
0
254237
620091
0
0
254237 620091
0
0
254237 620091
111616
5
6
7
8
9
10
11
12
13
14
15
5. C¸c chuyÓn vÞ nót tªn nót
1
2
3
4
u
0
###
###
0
### #VALUE!
0
###
###
0
###
###
0
###
###
v
0
###
###
0
### #VALUE!
0
###
###
0
###
###
0
###
###
###
###
u2
0
u2
#VALUE!
###
###
###
v2
0
v2
#VALUE!
254237
###
254237
u3
0
u3
#VALUE!
###
111616
###
v3
0
v3
#VALUE!
0
0
0
u5
0
u5
#VALUE!
0
0
0
v5
0
v5
#VALUE!
u6
0
u6
#VALUE!
v6
0
v6
#VALUE!
u8
0
u8
#VALUE!
0
v8
508475 254237 254237 0
111616 620091
0
0
0
0
0
0
508475 254237 254237 0
111616 620091
x
v8
=
↑
0
=
#VALUE!
u9
0
u9
#VALUE!
v9
0
v9
#VALUE!
0
0
0
u11
0
u11
#VALUE!
0
0
0
v11
0
v11
#VALUE!
u12
0
u12
#VALUE!
508475 254237 254237 0
111616 620091
v12
0
v12
#VALUE!
0
620091 111616
u14
-7.5
u14
#VALUE!
508475 254237 254237
v14
0
v14
#VALUE!
254237 874328 254237
0
0
u15
-3.75
u15
#VALUE!
874328
v15
-20
v15
#VALUE!
6. X¸c ®Þnh c¸c vÐc t¬ øng suÊt trong c¸c phÇn tö VÐc t¬ øng suÊt trong c¸c phÇn tö ®îc x¸c ®Þnh th«ng qua vÐc t¬ chuyÓn vÞ nót cña theo c«ng thøc sau:
{}
σx { σ } = σ y = [ D ][ B ] { δ }e τ xy e
[
1 0 [ D ] [ B ]= E 2 ν 0 1−ν 0 1− ν 2
0 0 1− ν 2
ν −1 1 −ν 1− ν 0 − 2
−ν −1 1−ν − 2
]
{}
ui vi u { δ }e = j vj um vm
[D][B]
=
### 2232327 0
0 0 ###
0 0 5084746
0 1 0
-1.00 -0.18 -0.41
-0.18 -1 -0.41
TÝnh cho phÇn tö thø nhÊt: 0.00E+00 0.00E+00 #VALUE!
σx {σ}
e
1
=
σy
=
τxy 1
[D][B] x
#VALUE! #VALUE! #VALUE!
#VALUE! =
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø hai: #VALUE! #VALUE! #VALUE!
σx {σ}
e
2
=
σy
=
[D][B] x
τxy
#VALUE!
#VALUE! =
#VALUE! #VALUE!
2
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø ba: 0.00E+00 0.00E+00 0.00E+00
σx {σ}
e
3
=
σy
=
[D][B] x
τxy
0.00E+00
#VALUE! =
#VALUE! #VALUE!
3
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø t: #VALUE! #VALUE! #VALUE!
σx {σ}
e
4
=
σy
=
[D][B] x
τxy
#VALUE!
#VALUE! =
#VALUE! #VALUE!
4
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø n¨m 0.00E+00 0.00E+00 #VALUE!
σx {σ}
e
1
=
σy τxy
=
[D][B] x
#VALUE! #VALUE!
#VALUE! =
#VALUE! (kN/m2) #VALUE!
5
#VALUE!
TÝnh cho phÇn tö thø s¸u: #VALUE! #VALUE! #VALUE!
σx {σ}
e
6
=
σy
=
[D][B] x
τxy
#VALUE!
#VALUE! =
#VALUE! #VALUE!
6
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø b¶y: 0.00E+00 0.00E+00 0.00E+00
σx {σ}
e
7
=
σy
=
[D][B] x
τxy
0.00E+00
#VALUE! =
#VALUE! #VALUE!
7
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø t¸m: #VALUE! #VALUE! #VALUE!
σx {σ}
e
8
=
σy
=
[D][B] x
τxy
#VALUE!
#VALUE! =
#VALUE! #VALUE!
8
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø chÝn: 0.00E+00 0.00E+00 #VALUE!
σx {σ}
e
9
=
σy
=
[D][B] x
#VALUE!
#VALUE! =
#VALUE! (kN/m2)
τxy
#VALUE! #VALUE!
9
#VALUE!
TÝnh cho phÇn tö thø mêi: #VALUE! #VALUE! #VALUE!
σx {σ}
e
10
=
σy
=
[D][B] x
τxy
#VALUE!
#VALUE! =
#VALUE! #VALUE!
10
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø mêi mét: 0.00E+00 0.00E+00 0.00E+00
σx {σ}
e
11
=
σy
=
[D][B] x
τxy
0.00E+00
#VALUE! =
#VALUE! #VALUE!
11
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø mêi hai: #VALUE! #VALUE! #VALUE!
σx {σ}
e
12
=
σy
=
[D][B] x
τxy 12
#VALUE! #VALUE! #VALUE!
#VALUE! =
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø mêi ba:
σx
0.00E+00 0.00E+00 #VALUE!
#VALUE!
{σ}e
13
=
σy
=
[D][B] x
τxy
#VALUE!
=
#VALUE! #VALUE!
13
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø mêi bèn: #VALUE! #VALUE! #VALUE!
σx {σ}
e
14
=
σy
=
[D][B] x
τxy
#VALUE!
#VALUE! =
#VALUE! #VALUE!
14
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø mêi n¨m: 0.00E+00 0.00E+00 0.00E+00
σx {σ}
e
15
=
σy
=
[D][B] x
τxy
0.00E+00
#VALUE! =
#VALUE! #VALUE!
15
#VALUE! (kN/m2) #VALUE!
TÝnh cho phÇn tö thø mêi s¸u: #VALUE! #VALUE! #VALUE!
σx {σ}e
16
=
σy
=
τxy 16
[D][B] x
#VALUE! #VALUE! #VALUE!
#VALUE! =
#VALUE! (kN/m2) #VALUE!
chuyÓn vÞ nót cña phÇn tö
