Instruction to project nr 1
Statical scheme
d =
a
2
+ c
2
sin a ã
a
d
sinα = a
ê d
a
d
cos a ã
c
d
cosα = c
ê d
c
d
tan a ã
a
c
Instruction to project nr 1
Statical scheme
d =
a
2
+ c
2
sin a ã
a
d
sinα = a
ê d
a
d
cos a ã
c
d
cosα = c
ê d
c
d
tan a ã
a
c
tanα = a
ê c
a
c
|
2
Instruction to project 1 mma7.nb
Base structure
|
Instruction to project 1 mma7.nb
3
Equilibrium of nodes
ü Node 2
‚P
ix
ã 0; N
"12"
- T
"2A"
cos
HaL - N
"2A"
sin
HaL - r
"2i"
ã 0
‚P
iy
ã 0 ; -N
"2A"
cos
HaL - T
"21"
+ T
"2A"
sin
HaL ã 0
ü Node 1
‚P
iy
ã 0; -N
"1B"
+ T
"12"
-
HPL ã 0
‚P
ix
ã 0 ; -N
"12"
+ T
"1C"
ã 0
Solve
@8N12 − T2A cosα − N2A sinα − r2i 0,
− N2A cosα − T21 + T2A sinα 0,
− N1B + T12 − P 0,
− N12 + T1C 0
<, 8N12, N1B, N2A, r2i<D
::r2i → −
− c d T1C − a d T21 + a
2
T2A
+ c
2
T2A
c d
, N1B
→ − P + T12, N12 → T1C, N2A → −
d T21
− a T2A
c
>>
FullSimplify
@%D
::r2i →
d
Hc T1C + a T21L − Ia
2
+ c
2
M T2A
c d
, N1B
→ − P + T12, N12 → T1C, N2A →
− d T21 + a T2A
c
>>
%
ê. 8a ^2 + c ^2 → d ^ 2<
::r2i →
d
Hc T1C + a T21L − d
2
T2A
c d
, N1B
→ − P + T12, N12 → T1C, N2A →
− d T21 + a T2A
c
>>
4
Instruction to project 1 mma7.nb
FullSimplify
@%D
::r2i →
c T1C
+ a T21 − d T2A
c
, N1B
→ − P + T12, N12 → T1C, N2A →
− d T21 + a T2A
c
>>
ü Solution
r
"2i"
:= T
"1C"
+
a T
"21"
c
-
d T
"2A"
c
,
N
"1B"
:= T
"12"
-
HPL,
N
"2A"
:=
a T
"2A"
c
-
d T
"21"
c
,
N
"12"
:= T
"1C"
|
Instruction to project 1 mma7.nb
5
Base scheme
Equilibrium of node 1
r
"1P"
:= -P a +
q b
2
8
6
Instruction to project 1 mma7.nb
r
"2i"
:= T
"1C"
+
a T
"21"
c
-
d T
"2A"
c
T
"1C"
= 0, T
"21"
=
3 q b
8
, T
"2A"
= 0
r
"2P"
:= 0 +
a
3 q b
8
c
-
d 0
c
=
3 q a b
8 c
|
Instruction to project 1 mma7.nb
7
Auxliar system 1
8
Instruction to project 1 mma7.nb
r
"11"
:=
3 EJ
2
b
+
4 EJ
1
c
r
"2i"
:= T
"1C"
+
a T
"21"
c
-
d T
"2A"
c
T
"1C"
= -
3 EJ
1
c
2
, T
"21"
= -
3 EJ
2
b
2
, T
"2A"
= 0
r
"21"
:= -
3 EJ
1
c
2
+
-
3 EJ
2
b
2
a
c
-
d 0
c
= -
3 EJ
1
c
2
-
3 a EJ
2
c b
2
|
Instruction to project 1 mma7.nb
9
Auxliar system 2
D
1
=
d
c
, D
2
=
a
c
10
Instruction to project 1 mma7.nb
r
"12"
:= -
3 EJ
1
c
2
-
3 EJ
2
b
2
D
2
= -
3 EJ
1
c
2
-
3 a EJ
2
c b
2
r
"2i"
:= T
"1C"
+
a T
"21"
c
-
d T
"2A"
c
T
"1C"
=
3 EJ
1
c
3
, T
"21"
=
3 EJ
2
b
3
D
2
=
3 a EJ
2
c b
3
, T
"2A"
= -
3 EJ
1
d
3
D
1
= -
3 EJ
1
c d
2
r
"22"
:=
3 EJ
1
c
3
+
3 EJ
2
b
3
a
c
-
J-
3 EJ
1
c d
2
N d
c
=
3
Hc + dL EJ
1
c
3
d
+
3 a EJ
2
b
3
c
|
Instruction to project 1 mma7.nb
11
Sulution of the canonical system
r
"1P"
:= − P a +
q b
2
8
r
"2P"
:=
3 q a b
8 c
r
"11"
:=
3 EJ
2
b
+
4 EJ
1
c
r
"21"
:= −
3 EJ
1
c
2
−
3 a EJ
2
c b
2
r
"12"
:= −
3 EJ
1
c
2
−
3 a EJ
2
c b
2
r
"22"
:=
3
Hc + dL EJ
1
c
3
d
+
3 a EJ
2
b
2
c
EJ
2
= n EI; EJ
1
= EI
EI
Solve
@8
r
"11"
Z
1
+ r
"12"
Z
2
+ r
"1P"
0,
r
"21"
Z
1
+ r
"22"
Z
2
+ r
"2P"
0
<, 8 Z
1
, Z
2
<D
::Z
1
→
I8 a b
4
c
2
P
+ 8 a b
4
c d P
+ 8 a
2
b
2
c
3
d n P
−
b
6
c
2
q
− 3 a b
5
c d q
− b
6
c d q
− 3 a
2
b
3
c
2
d n q
− a b
4
c
3
d n q
M ë
I8 EI I4 b
4
c
+ b
4
d
+ 3 b
3
c
2
n
− 6 a b
2
c d n
+ 3 b
3
c d n
+ 4 a b
2
c
2
d n
− 3 a
2
c
2
d n
2
+ 3 a b c
3
d n
2
MM,
Z
2
→ −
−
3 a b
J
4 EI
c
+
3 EI n
b
N q
8 c
+
J−
3 EI
c
2
−
3 a EI n
b
2
c
N J−a P +
b
2
q
8
N
J−
3 EI
c
2
−
3 a EI n
b
2
c
N
2
−
J
4 EI
c
+
3 EI n
b
N J
3
Hc+dL EI
c
3
d
+
3 a EI n
b
2
c
N
>>
FullSimplify
@%D
99Z
1
Ø
Ib
2
c
I8 a P Ia c
2
d n + b
2
Hc + dLM - b q Ib
2
H3 a d + b Hc + dLL + a c d n H3 a + b cLMMMë
I8 EI Ib
2
c n
H2 a H2 c - 3L d + 3 b Hc + dLL - 3 a c
2
d n
2
Ha - b cL + b
4
H4 c + dLMM,
Z
2
Ø
Ib
2
c
2
d
I8 a P Ia c n + b
2
M - b
2
q
I4 a Hb + c nL + b
2
MMMë
I8 EI Ib
2
c n
H2 a H2 c - 3L d + 3 b Hc + dLL - 3 a c
2
d n
2
Ha - b cL + b
4
H4 c + dLMM==
|
12
Instruction to project 1 mma7.nb
Final diagrams
M
"A2"
M
"12
M
"1B"
M
"1C"
=
0
-q b
2
8
0
0
+
0
3 n
b
-
1
c
3
c
a
1
+
3
c d
3 a n
b
2
c
0
-
3
c
2
a
2
0
1
8
H-qL b
2
0
0
+
0
3 n
b
-
1
c
3
c
a
1
+
3
c d
3 a n
b
2
c
0
-
3
c
2
a
2
3 a
2
c d
-
q b
2
8
+
3 a n a
2
b
2
c
+
3 n a
1
b
-
a
1
c
3 a
1
c
-
3 a
2
c
2
FullSimplify
@%D
3 a
2
c d
3 n
Hb c a
1
+a a
2
L
b
2
c
-
b
2
q
8
-
a
1
c
3 c a
1
-3 a
2
c
2
%
ê. 99α
1
→
Ib
2
c
I8 a IHc + dL b
2
+ a c
2
d n
M P − b IH3 a d + b Hc + dLL b
2
+ a c
H3 a + b cL d nM qMM ë
I8 IH4 c + dL b
4
+ c
H2 a H2 c − 3L d + 3 b Hc + dLL n b
2
− 3 a c
2
Ha − b cL d n
2
MM,
α
2
→
Ib
2
c
2
d
I8 a Ib
2
+ a c n
M P − b
2
Ib
2
+ 4 a
Hb + c nLM qMM ë
I8 IH4 c + dL b
4
+ c
H2 a H2 c − 3L d + 3 b Hc + dLL n b
2
− 3 a c
2
Ha − b cL d n
2
MM==
::9I3 b
2
c
I8 a Ib
2
+ a c n
M P − b
2
Ib
2
+ 4 a
Hb + c nLM qMM ë
I8 Ib
4
H4 c + dL + b
2
c
H2 a H−3 + 2 cL d + 3 b Hc + dLL n − 3 a c
2
Ha − b cL d n
2
MM=, :−
b
2
q
8
+
1
b
2
c
I3 n IIb
3
c
2
I8 a Ib
2
Hc + dL + a c
2
d n
M P − b Ib
2
H3 a d + b Hc + dLL + a c H3 a + b cL d nM qMM ë
I8 Ib
4
H4 c + dL + b
2
c
H2 a H−3 + 2 cL d + 3 b Hc + dLL n − 3 a c
2
Ha − b cL d n
2
MM +
Ia b
2
c
2
d
I8 a Ib
2
+ a c n
M P − b
2
Ib
2
+ 4 a
Hb + c nLM qMM ë
I8 Ib
4
H4 c + dL + b
2
c
H2 a H−3 + 2 cL d + 3 b Hc + dLL n − 3 a c
2
Ha − b cL d n
2
MMMM>,
9−Ib
2
I8 a Ib
2
Hc + dL + a c
2
d n
M P − b Ib
2
H3 a d + b Hc + dLL + a c H3 a + b cL d nM qMM ë
I8 Ib
4
H4 c + dL + b
2
c
H2 a H−3 + 2 cL d + 3 b Hc + dLL n − 3 a c
2
Ha − b cL d n
2
MM=,
:
1
c
2
II3 b
2
c
2
I8 a Ib
2
Hc + dL + a c
2
d n
M P − b Ib
2
H3 a d + b Hc + dLL + a c H3 a + b cL d nM qMM ë
I8 Ib
4
H4 c + dL + b
2
c
H2 a H−3 + 2 cL d + 3 b Hc + dLL n − 3 a c
2
Ha − b cL d n
2
MM −
I3 b
2
c
2
d
I8 a Ib
2
+ a c n
M P − b
2
Ib
2
+ 4 a
Hb + c nLM qMM ë
I8 Ib
4
H4 c + dL + b
2
c
H2 a H−3 + 2 cL d + 3 b Hc + dLL n − 3 a c
2
Ha − b cL d n
2
MMM>>>
Instruction to project 1 mma7.nb
13
FullSimplify
@%D
K :
3 b
2
c
I8 a Ib
2
+a c n
M P-b
2
Ib
2
+4 a
Hb+c nLM qM
8
IH4 c+dL b
4
+c
H2 a H2 c-3L d+3 b Hc+dLL n b
2
-3 a c
2
Ha-b cL d n
2
M
> :
24 a c n
IHa d+b Hc+dLL b
2
+a c
Ha+b cL d nM P-b
2
IH4 c+dL b
4
+2 c
I6 d a
2
+b
H2 c+3L d a+3 b
2
Hc+dLM n b+6 a c
2
H3 a+
8
IH4 c+dL b
4
+c
H2 a H2 c-3L d+3 b Hc+dLL n b
2
-3 a c
2
Ha-b cL d n
2
M
MatrixForm
@%D
K Z
1
→
b
2
c
I8 a Ib
2
Hc+dL+a c
2
d n
M P−b Ib
2
H3 a d+b Hc+dLL+a c H3 a+b cL d nM qM
8 EI
Ib
4
H4 c+dL+b
2
c
H2 a H−3+2 cL d+3 b Hc+dLL n−3 a c
2
Ha−b cL d n
2
M
Z
2
→
b
2
c
2
d
I8 a Ib
2
+a c n
M P−b
2
Ib
2
+4 a
Hb+c nLM
8 EI
Ib
4
H4 c+dL+b
2
c
H2 a H−3+2 cL d+3 b Hc+dLL n−3 a c
2
14
Instruction to project 1 mma7.nb