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