Instruction to project nr 1
Statical scheme
d = a2 + c2
a
sin a ć
d
sinÄ… = a Ä™ d
a
d
c
cos a ć
d
cosÄ… = c Ä™ d
c
d
a
tan a ć
c
2 Instruction to project 1 mma7.nb
tanÄ… = a Ä™ c
a
c
|
Instruction to project 1 mma7.nb 3
Base structure
|
4 Instruction to project 1 mma7.nb
Equilibrium of nodes
ü Node 2
Pix ć 0; N"12" - T"2A" cosHaL - N"2A" sinHaL - r"2i" ć 0

Piy ć 0 ; -N"2A" cosHaL - T"21" + T"2A" sinHaL ć 0

ü Node 1
Piy ć 0;
 -N"1B" + T"12" - HPL ć 0
Pix ć 0 ;
 -N"12" + T"1C" ć 0
Solve@8N12 - T2A cosÄ… - N2A sinÄ… - r2i 0,
- N2A cosÄ… - T21 + T2A sinÄ… 0,
- N1B + T12 - P 0,
- N12 + T1C 0<, 8N12, N1B, N2A, r2i-c d T1C - a d T21 + a2 T2A + c2 T2A d T21 - a T2A
::r2i - , N1B -P + T12, N12 T1C, N2A - >>
c d c
FullSimplify@%D
d Hc T1C + a T21L - Ia2 + c2M T2A
-d T21 + a T2A
::r2i , N1B -P + T12, N12 T1C, N2A >>
c d c
% Ä™. 8a ^2 + c ^2 d ^ 2<
d Hc T1C + a T21L - d2 T2A -d T21 + a T2A
::r2i , N1B - P + T12, N12 T1C, N2A >>
c d c
Instruction to project 1 mma7.nb 5
FullSimplify@%D
c T1C + a T21 - d T2A -d T21 + a T2A
::r2i , N1B -P + T12, N12 T1C, N2A >>
c c
ü Solution
a T"21" d T"2A"
r"2i" := T"1C" + - ,
c c
N"1B" := T"12" - HPL,
a T"2A" d T"21"
N"2A" := - ,
c c
N"12" := T"1C"
|
6 Instruction to project 1 mma7.nb
Base scheme
Equilibrium of node 1
q b2
r"1P" := -P a +
8
Instruction to project 1 mma7.nb 7
a T"21" d T"2A"
r"2i" := T"1C" + -
c c
3 q b
T"1C" = 0, T"21" = , T"2A" = 0
8
3 q b
a
d 0 3 q a b
8
r"2P" := 0 + - =
c c 8 c
|
8 Instruction to project 1 mma7.nb
Auxliar system 1
Instruction to project 1 mma7.nb 9
3 EJ2 4 EJ1
r"11" := +
b c
a T"21" d T"2A"
r"2i" := T"1C" + -
c c
3 EJ1 3 EJ2
T"1C" = - , T"21" = - , T"2A" = 0
c2 b2
3 EJ2
3 EJ1 - a d 0 3 EJ1 3 a EJ2
b2
r"21" := - + - = - -
c c
c2 c2 c b2
|
10 Instruction to project 1 mma7.nb
Auxliar system 2
d a
D1 = , D2 =
c c
Instruction to project 1 mma7.nb 11
3 EJ1 3 EJ2 3 EJ1 3 a EJ2
r"12" := - - D2 = - -
c2 b2 c2 c b2
a T"21" d T"2A"
r"2i" := T"1C" + -
c c
3 EJ1 3 EJ2 3 a EJ2 3 EJ1 3 EJ1
T"1C" = , T"21" = D2 = , T"2A" = - D1 = -
c3 b3 c b3 d3 c d2
3 EJ2 3 EJ1
3 EJ1 b3 a J- N d 3 Hc + dL EJ1 3 a EJ2
c d2
r"22" := + - = +
c c
c3 c3 d b3 c
|
12 Instruction to project 1 mma7.nb
Sulution of the canonical system
q b2
r"1P" := -P a +
8
3 q a b
r"2P" :=
8 c
3 EJ2 4 EJ1
r"11" := +
b c
3 EJ1 3 a EJ2
r"21" := - -
c2 c b2
3 EJ1 3 a EJ2
r"12" := - -
c2 c b2
3 Hc + dL EJ1 3 a EJ2
r"22" := +
c3 d b2 c
EJ2 = n EI; EJ1 = EI
EI
Solve@8
r"11" Z1 + r"12" Z2 + r"1P" 0,
r"21" Z1 + r"22" Z2 + r"2P" 0<, 8 Z1, Z2::Z1 I8 a b4 c2 P + 8 a b4 c d P + 8 a2 b2 c3 d n P -
b6 c2 q - 3 a b5 c d q - b6 c d q - 3 a2 b3 c2 d n q - a b4 c3 d n qM ë
I8 EI I4 b4 c + b4 d + 3 b3 c2 n - 6 a b2 c d n + 3 b3 c d n + 4 a b2 c2 d n - 3 a2 c2 d n2 + 3 a b c3 d n2MM,
4 EI 3 EI n
3 a b J + N q
b2 q
3 EI 3 a EI n
c b
- + J- - N J- a P + N
8 c c2 b2 c 8
Z2 - >>
2
3 EI 3 a EI n 4 EI 3 EI n 3 Hc+dL EI 3 a EI n
J- - N - J + N J + N
c2 b2 c c b c3 d b2 c
FullSimplify@%D
99Z1 Ø Ib2 c I8 a P Ia c2 d n + b2 Hc + dLM - b q Ib2 H3 a d + b Hc + dLL + a c d n H3 a + b cLMMMë
I8 EI Ib2 c n H2 a H2 c - 3L d + 3 b Hc + dLL - 3 a c2 d n2 Ha - b cL + b4 H4 c + dLMM,
Z2 Ø Ib2 c2 d I8 a P Ia c n + b2M - b2 q I4 a Hb + c nL + b2MMM ë
I8 EI Ib2 c n H2 a H2 c - 3L d + 3 b Hc + dLL - 3 a c2 d n2 Ha - b cL + b4 H4 c + dLMM==
|
Instruction to project 1 mma7.nb 13
Final diagrams
3
0
c d
M"A2" 0
3 n
3 a n
M"12 -q b2 b
b2 c
8
= + a1 + a2
1
M"1B" 0 -
0
c
M"1C" 0 3
3
-
c
c2
3
0
0 c d
3 n
1 3 a n
H-qL b2 b
8 b2 c
+ a1 + a2
1
-
0
c 0
3
3
0
-
c
c2
3 a2
c d
q b2 3 a n a2 3 n a1
- + +
8 b2 c b
a1
-
c
3 a1 3 a2
-
c c2
FullSimplify@%D
3 a2
c d
b2 q
3 n Hb c a1+a a2L
-
b2 c 8
a1
-
c
3 c a1-3 a2
c2
% Ä™. 99Ä…1 Ib2 c I8 a IHc + dL b2 + a c2 d nM P - b IH3 a d + b Hc + dLL b2 + a c H3 a + b cL d nM qMM ë
I8 IH4 c + dL b4 + c H2 a H2 c - 3L d + 3 b Hc + dLL n b2 - 3 a c2 Ha - b cL d n2MM,
Ä…2 Ib2 c2 d I8 a Ib2 + a c nM P - b2 Ib2 + 4 a Hb + c nLM qMM ë
I8 IH4 c + dL b4 + c H2 a H2 c - 3L d + 3 b Hc + dLL n b2 - 3 a c2 Ha - b cL d n2MM==
::9I3 b2 c I8 a Ib2 + a c nM P - b2 Ib2 + 4 a Hb + c nLM qMM ë
b2 q
I8 Ib4 H4 c + dL + b2 c H2 a H- 3 + 2 cL d + 3 b Hc + dLL n - 3 a c2 Ha - b cL d n2MM=, :- +
8
1
I3 n IIb3 c2 I8 a Ib2 Hc + dL + a c2 d nM P - b Ib2 H3 a d + b Hc + dLL + a c H3 a + b cL d nM qMM ë
b2 c
I8 Ib4 H4 c + dL + b2 c H2 a H- 3 + 2 cL d + 3 b Hc + dLL n - 3 a c2 Ha - b cL d n2MM +
Ia b2 c2 d I8 a Ib2 + a c nM P - b2 Ib2 + 4 a Hb + c nLM qMM ë
I8 Ib4 H4 c + dL + b2 c H2 a H-3 + 2 cL d + 3 b Hc + dLL n - 3 a c2 Ha - b cL d n2MMMM>,
9-Ib2 I8 a Ib2 Hc + dL + a c2 d nM P - b Ib2 H3 a d + b Hc + dLL + a c H3 a + b cL d nM qMM ë
I8 Ib4 H4 c + dL + b2 c H2 a H-3 + 2 cL d + 3 b Hc + dLL n - 3 a c2 Ha - b cL d n2MM=,
1
: II3 b2 c2 I8 a Ib2 Hc + dL + a c2 d nM P - b Ib2 H3 a d + b Hc + dLL + a c H3 a + b cL d nM qMM ë
c2
I8 Ib4 H4 c + dL + b2 c H2 a H-3 + 2 cL d + 3 b Hc + dLL n - 3 a c2 Ha - b cL d n2MM -
I3 b2 c2 d I8 a Ib2 + a c nM P - b2 Ib2 + 4 a Hb + c nLM qMM ë
I8 Ib4 H4 c + dL + b2 c H2 a H-3 + 2 cL d + 3 b Hc + dLL n - 3 a c2 Ha - b cL d n2MMM>>>
14 Instruction to project 1 mma7.nb
FullSimplify@%D
3 b2 c I8 a Ib2+a c nM P-b2 Ib2+4 a Hb+c nLM qM 24 a c n IHa d+b Hc+dLL b2+a c Ha+b cL d nM P-b2 IH4 c+dL b4+2 c I6 d a2+b H2 c+3L d a+3 b2 Hc+dLM n b+6 a c2 H3 a+
K : > :
8 IH4 c+dL b4+c H2 a H2 c-3L d+3 b Hc+dLL n b2-3 a c2 Ha-b cL d n2M 8 IH4 c+dL b4+c H2 a H2 c-3L d+3 b Hc+dLL n b2-3 a c2 Ha-b cL d n2M
MatrixForm@%D
b2 c I8 a Ib2 Hc+dL+a c2 d nM P-b Ib2 H3 a d+b Hc+dLL+a c H3 a+b cL d nM qM b2 c2 d I8 a Ib2+a c nM P-b2 Ib2+4 a Hb+c nLM
K Z1 Z2
8 EI Ib4 H4 c+dL+b2 c H2 a H-3+2 cL d+3 b Hc+dLL n-3 a c2 Ha-b cL d n2M 8 EI Ib4 H4 c+dL+b2 c H2 a H-3+2 cL d+3 b Hc+dLL n-3 a c2