Mathcad Obliczeniaa


1. Charakterystyki geometryczne:
2
( ( ) ( ))
A T" 2 " (0.8 " (10 + 10 + 10 ) + 0.7 " (20 + 10 )) = 90
3
( ( ) ( ))
Sy1 T" 2 " (0.7 " (20 " 10 + 0.5 " 20 " 20 ) - 0.8 " (0.5 " 10 " 10 + 10 " 10 )) = 320
Sy1
z0 T" = 3.556
 
A
( ( ) (
Jz T" 2 " (0.7 " (0.5 " 10 " 10 " 13.3333 + 0.5 " 20 " 10 " 16.6666 + 20 " 10 " 10 ) + 0.8 " (0.5 " 10 " 10 " 13.3333 + 0.5 " 20 " 10 " 16.6666 + 0.5 " 10 " 10 " 6.6666 + 10 " 10
4
( ( ) ( ))
Jy1 T" 2 " (0.7 " (10 " 20 " 20 + 0.5 " 20 " 20 " 0.6666 " 20 ) + 0.8 " (0.5 " 10 " 10 " 6.6666 + 10 " 10 " 10 )) = 11466.288
2 4
Jy T" Jy1 - z0 " A = 10328.51
Sprawdzenie
4
( ( ) ( ))
Jy T" 2 " (0.7 " (16.44 " 10 " 16.44 + 0.5 " 16.44 " 16.44 " 0.6666 " 16.44 ) + 0.8 " (0.5 " 13.56 " 13.56 " 0.6666 " 13.56 + 13.56 " 10 " 13.56 + 3.56 " 10 " 3.56 )) = 10331.558
3
( ( ) ( ))
Sy T" 2 " (0.7 " (16.44 " 10 + 0.5 " 16.44 " 16.44 ) - 0.8 " (0.5 " 13.56 " 13.56 + 13.56 " 10 + 3.56 " 10 )) = -1.667
Obranie bieguna B i punktu K
1 T" 0
2
2 T" 1 - 10 " 10 = -100
2
3 T" 2 + 10 " 10 = 0
2
4 T" 1 + 10 " 20 = 200
2
5 T" 4 - 10 " 20 = 0
zB T" -3.56
2 2 5
#0.7 " #0.5 " 200 2 10 " 13.3333 + 0.5 " 200 2 20 " 10 ś# #0.5 ś#ś#
J.y T" 2 " # # " " # - 0.8 " # " 100 " 10 " 10 + 0.5 " 100 " 10 " 13.3333 # # = 27999.98
J.y
zA T" = 2.346
 
Jz
Obranie bieguna A i punktu K0
2
1 T" -zA " 10 = -23.464
2
2 T" 1 - 10 " 10 = -123.464
176.54 + 46.93 23.46 + 176.54
176.54 -          
" 3.3333 = 102.051 -23.46 + " 0.3333 " 20 = 43.2
2 100
10 20
# - zA # = -46.927
ś#
3 T" 2 + 10 " " 3.3333 + 23.46 = 56.793
 
#10
10
2
10 20 176 536
2
4 T" 1 + 10 " 20 = 176.536
176.54 + 46.93 23.46 + 176.54
176.54 -          
" 6.6666 = 27.561 -23.46 + " 0.6666 " 20 = 109.86 10 " 6.6666 + 23.46 = 90.126
2
10 20
# ś#
5 T" 4 - 10 " + 20 = -46.927
#z #
A
2 2
#0.7 " #0.5 " -46.93 2 10 " 27.56 2 0.5 " 176.54 2 10 " 102.05 2 0.5 " 176.54 2 20 " 109.86 2 0.5 " 23.46 2 20 " 43.2 2 ś# #0.5
J T" 2 " # # " + " + " - " # + 0.8 " # " 23.46 " 10 " 56.79 + 0.5 " 123.46
2. Belka zginana
q T" 15 P T" 50
 
Belka zginana w płaszczyznie XZ
Metoda sił - belka statycznie niewyznaczalna
RB T" 0 RA T" 0
7
" 6 = 4.2

10
4.2 m 3
( )
q " 4.2 m " - RB " 6 m0 q " 4.2 m " (2.1 m + 1.8 m) - RA " 6 m0 " 6 = 1.8
   
2 10
# ś# # ś#
T" = 22.05 T" = 40.95
#R # #R #
B A
+ - q " 4.2 = 0
" 1.8 = 39.69 "
# ś#
2
4.2 " 2

ś# ź#
# ś# ( ) # ś#
4.2 + 0.6 3 2 q " (4.2 ) 2.1
10 T" 0.5 " 39.69 " " 1.8 " + " "
ź#
ś#     + 0.5 " 39.69 " " 4.2 " ś#         4.2 " ś#  
ź# ź#
# 6 # # 6 # 3 8 # 6 #
2
11 T" 0.5 " 1 " 6 "

3
10
X T" = 49.943 "
 
11
Wyznaczenie reakcji
RA T" 0
RB T" 0
Constraints
Guess Values
Constraints
Guess Values
Solver
Solver
alues
ues
RA 0
4.2 m
q " 4.2 m " - RB " 6 m + X0
  
( )
q " 4.2 m " (2.1 m + 1.8 m) - RA " 6 m - X0
2
# ś#
T" = 30.374
#R #
B
# ś#
T" = 32.626
#R #
A
Siły przekrojowe
9
t1 T" 0 ,0.01 % 4.2 t2 T" 4.2 ,4.21 % 6 " 6 = 5.4

10
# ś# # ś#
T1 #t1 # T" - q " t1 T2 #t2 # T" - q " 4.2
t1
# ś# # ś# # ś#
M1 #t1 # T" " t1 - q " t1 " M2 #t2 # T" " t2 - q " 4.2 " - 2.1

#t #
2
2
( ) ( )
M2 (5.4 ) = -31.719 " T2 (5.4 ) = -30.37388
39 54
32.5
45
26
36
19.5
27
13
18
6.5
9
0
# ś# ( )
# ś# ( ) -M1 #t1 # ( " )
T1 #t1 # ( )
0 0.6 1.2 1.8 2.4 3 3.6 4.2 4.8 5.4 6
0
-6.5 0 0.6 1.2 1.8 2.4 3 3.6 4.2 4.8 5.4 6
-9
-13
# ś# ( )
# ś# ( ) -M2 #t2 # ( " )
T2 #t2 # ( )
-18
-19.5
-27
-26
-36
-32.5
( )
t1 ( )
( )
t1 ( )
( )
( ) t2 ( )
t2 ( )
Belka zginana w płaszczyznie XY
6
RA T" 50 M T" 50 " " 6 = 180 " H T" 0

6
10
" 6 = 3.6

10
t1 T" 0 ,0.01 % 2.4 t2 T" 2.4 ,2.41 % 6
# ś# # ś#
T t 0 T t R
Constraints
Guess V
Constraints
Guess Valu
Solver
Solver
# ś# # ś#
T1 #t1 # T" 0 T2 #t2 # T" RA
# ś# # ś# # ś#
M1 #t1 # T" 0 M2 #t2 # T" RA " - 2.4
#t #
2
( ) ( )
M2 (5.4 ) = 150 " T2 (5.4 ) = 50
50 0
0 0.6 1.2 1.8 2.4 3 3.6 4.2 4.8 5.4 6
45
-20
40
-40
35
-60
30
-80
25
# ś# ( ) -100 # ś# ( )
T1 #t1 # ( ) -M1 #t1 # ( )
20
-120
15
# ś# ( ) # ś# ( )
T2 #t2 # ( ) -M2 #t2 # ( " )
-140
10
-160
5
0 -180
0 0.6 1.2 1.8 2.4 3 3.6 4.2 4.8 5.4 6
( ) ( )
t1 ( ) t1 ( )
( ) ( )
t2 ( ) t2 ( )
3. Belka skręcana
3
1 #( )3 4
( ) ( ) ( )ś#
Ks T" " #(0.7 ) " (20 + 40 ) + (0.8 ) " (20 + 2 " 20 ) # = 17.1

3
E T" 210000  T" 0.3
E
E1 T" = 230769.231
  
2
1 - 
E
G T" = 80769.231
  
( )
2 (1 + )
mx T" -q " 10 = -1.5
Mx T" P " 10 = 5 "
> > > > > >
G " Ks
1
ą T" = 0.316
   
E1 " J
Układ równań
1 2
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
Y1 () T" cosh () Y2 () T" sinh () Y3 () T" 1 - cosh () Y4 () T"  - sinh () Y5 () T" "  - cosh ()

2
 6 1 894
 T" ą " 6 = 1.894
1
Mx0 T" 5 " Ś' T"

0
# ś# # # ś# ś#
1 1 1 4 1 7
( ) ( ) ( )
0Ś' " " Y2 () + Mx0 " " Y4 () " Y4 ś# "
    - Mx "
   - ą " " 6 mź# + mx "
     - ą " " 6 mź# - Y5 ()ź#
0
ś#Y5 ś# 
2
ą G " Ks " ą G " Ks " ą # 10 # # # 10 # #
G " Ks " ą
# ś# # # ś# ś#
1 1 4 1 7
( ) ( ) ( )
0Ś' " Y1 () + Mx0 " " Y3 () - Mx " Y3 "
    - ą " " 6 mź# + mx "
    - ą " " 6 mź# - Y4 ()ź#
0
ś#Y4 ś# 
G " Ks G " Ks ś# 10 # G " Ks " ą # # 10 # #
#
Ą# -1094.474
ń#
Ą#
1
#
T"
#M , Ś' ś# = ó# 0.001  Ą#
#
x0 0
ó#
Ł# Ś#
Mx0 T" = -1.094 "
0
1
Ś' T" = 0.001

0
1
x T" 0 ,0.05 % 6
Siły przekrojowe
 |
|
4
( )
Ś () T" if 0 d"  < ą " 6 "

 |
|
10
 |
|

1 1 1
 |
( ) ( ) # ( ) ( )ś#
Ś' " " Y2 () + Mx0 " " Y4 () + mx " " (0)
        - Y5 () # |
 #Y
0 5
2
 |
ą G " Ks " ą
G " Ks " ą

 |
 |
|
 4 7 |
if ą " 6 " d"  < ą " 6 "
 
|
 |
10 10
|
 |

# ś#
1 1 1 1 4
|
 |
( ) ( ) # ( ) ( )ś#
Ś' " " Y2 () + Mx0 " " Y4 () + mx " " (0) " Y4 ś#
        - Y5 () # - Mx "
   - ą " " 6

 #Y
0 5
ź#
2
ą G " Ks " ą G " Ks " ą # 10 #
 |
G " Ks " ą

 |
 |
|
7

|
if ą " 6 " d"  < ą " 6

|
 10
|
|


|
# # ś# ś# # ś#
1 1 1 7 1 4
|
( ) ( ) ( )
Ś' " " Y2 () + Mx0 " " Y4 () + mx " " " Y4 ś#
        - ą " " 6 - Y5 ()ź# - Mx "
   - ą " " 6


 0
|
ś#Y5 ś#  ź# ź#
2
ą G " Ks " ą # # 10 # # G " Ks " ą # 10 #
G " Ks " ą


 |
 |
 |
|
4
( )
B () T" if 0 d"  < ą " 6 "

 |
|
10
 |
Constraints
Guess Values
Solver
B () if 0 d"  < ą 6
 |
|
10
 |
|

G " Ks
 |
1 1
( ) ( ) # ( ) ( )ś#
 -Ś' " " Y2 () + Mx0 " " Y2 () - mx " " (0) - Y3 () # |
    
#Y
0 3
 |
2
ą ą
 ą
 |
 |
|
 4 7 |
if ą " 6 " d"  < ą " 6 "
 
|
 |
2
10 10
(ą
)
B ( " 5.4 ) = -0.12084 "
|
 |

G " Ks
# ś#
1 1 1 4
|
 |
( ) ( ) # ( ) ( )ś#
 -Ś' " " Y2 () + Mx0 " " Y2 () - mx " " (0) - Y3 () # - Mx " " Y2 ś# - ą " " 6
      
#Y
0 3
ź#
2
 ą ą ą # 10 # |
 ą
 |
 |
|
7

|
if ą " 6 " d"  < ą " 6

|
 10
|
|


|
G " Ks
# # ś# ś# # ś#
1 1 7 1 4
|
( ) ( ) ( )

 -Ś' " " Y2 () + Mx0 " " Y2 () - mx " " - ą " " 6 - Y3 ()ź# - Mx " " Y2 ś# - ą " " 6
      
|
0
ś#Y3 ś#  ź# ź#
2
ą ą # # 10 # # ą # 10 #
 ą

 |
 |
 |
|
4
( )
Mx. () T" if 0 d"  < ą " 6 "
 |
 |
10
 |
|


 |
|
Mx0 - mx "


 |
ą
 |
 |
|
4 7

if ą " 6 " d"  < ą " 6 "
  | |
10 10
 |
|

 |

|
Mx0 - mx " - Mx


 |
ą
 |
 |
|
7
 |
if ą " 6 " d"  < ą " 6

|

|
10
|

|

7
 - mx " " 6 - Mx | |
Mx0


10
  |
 |
 |
|
4
( )
Ms () T" if 0 d"  < ą " 6 "

 |
|
10
 |
|

1
 |
( ) ( )
Ś' " G " Ks " Y1 () + Mx0 " Y3 () + mx " " (0) - Y4 () # |
 # 4 ( ) ( )ś#
 #Y
0
 |
ą
 |
 |
(ą
)
Ms ( " 5.4 ) = -0.003506 "
|
4 7
 |
if ą " 6 " d"  < ą " 6 "
 
|
10 10
 |
|

 |
# ś#
1 4
|
( ) ( )
Ś' " G " Ks " Y1 () + Mx0 " Y3 () + mx " " (0) - Y4 () # - Mx " Y3 ś# - ą " " 6
 # 4 ( ) ( )ś# 
 #Y
0
 |
ź#
ą # 10 #
 |
 |
|
7
 |
if ą " 6 " d"  < ą " 6

|

|
10
|

|

# # ś# ś# # ś#
1 7 4
|
( ) ( ) ( )
 Ś' " G " Ks " Y1 () + Mx0 " Y3 () + mx " " Y4 ś# - ą " " 6 - Y4 ()ź# - Mx " Y3 ś# - ą " " 6 |
 ś#  

0
ź# ź#
ą # # 10 # # # 10 #
  | |

|

 |
 |
|
4
( )
M () T" if 0 d"  < ą " 6 "

 |
|
10
 |
M () if 0 d"  < ą 6
 |
|
10
 |
|

1
 |
( ) ( )
-Ś' " G " Ks " Y1 () + Mx0 " Y1 () + mx " " (0) - Y2 () # |
 # 2 ( ) ( )ś#
 #Y
0
 |
(ą
)
ą M ( " 5.4 ) = 0.20903 "
 |
 |
|
4 7
 |
if ą " 6 " d"  < ą " 6 "
 
|
10 10
 |
|

 |
# ś#
1 4
|
( ) ( )
-Ś' " G " Ks " Y1 () + Mx0 " Y1 () + mx " " (0) - Y2 () # - Mx " Y1 ś# - ą " " 6
 # 2 ( ) ( )ś# 
 #Y
0
 |
ź#
ą # 10 #
 |
 |
|
7
 |
if ą " 6 " d"  < ą " 6

|

|
10
|

|

# # ś# ś# # ś#
1 7 4
|
( ) ( ) ( )
 -Ś' " G " Ks " Y1 () + Mx0 " Y1 () + mx " " Y2 ś# - ą " " 6 - Y2 ()ź# - Mx " Y1 ś# - ą " " 6 |
 ś#  

0
ź# ź#
ą # # 10 # # # 10 #
  | |

|

 |
Wykresy funkcji
2
0.01
1.75
0.009
1.5
0.008
1.25
0.007
1
0.006
0.75
2
# ś#
(
B (x " ą)
) # " #
0.005
(
Ś (x " ą)
)
0.5
0.004
0.25
0.003
0
0 0.6 1.2 1.8 2.4 3 3.6 4.2 4.8 5.4 6
0.002
-0.25
0.001
-0.5
0
0 0.6 1.2 1.8 2.4 3 3.6 4.2 4.8 5.4 6
( )
x ( )
( )
x ( )
2.5
250
2
200
1.5
150
1
100
100
0.5
50
0
0 0.6 1.2 1.8 2.4 3 3.6 4.2 4.8 5.4 6
(
Mx. (x " ą) ( )
) ( " )
0
0 0.6 1.2 1.8 2.4 3 3.6 4.2 4.8 5.4 6
-0.5 ( )
M (x " ą) (
) ( " )
-50
-1
-100
-1.5
-150
-2
-200
-2.5
-250
( )
x ( )
( )
x ( )
7.5
2.5
6
2
4.5
1.5
3
1
1.5
0.5
0
0 0 0.6 1.2 1.8 2.4 3 3.6 4.2 4.8 5.4 6
(
Ms (x " ą) ( )
) ( " )
0 0.6 1.2 1.8 2.4 3 3.6 4.2 4.8 5.4 6
(
M (x " ą) ( ą) ( )
) + Ms (x " ) ( " )
-1.5
-0.5
-3
-1
-4.5
-1.5
-6
-2
-7.5
-2.5
( )
x ( )
( )
x ( )
4. Naprężenia normalne
2
My T" -31.718925 " Mz T" 150 " B T" -0.12084 "
4 4 6
Jy T" 10331.5582 Jz T" 11933.3030 J T" 600906.7902
Punkt 1
y T" 10
2
z T" -3.56  T" -23.46
y 10
My Mz
B
" z = 10.93 -  -125.7 "  = 0.47
" y =
  
Jy Jz J
2
My Mz B explicit , ALL
-31.718925 " 150 kN " m -0.12084 kN " m 2
x T" " z " y + "      " -3.56 cm -             -23.46 cm = -114.297
" 10 cm + "
  -          
4 4 6
Jy Jz J
10331.5582 cm 11933.3030 cm 600906.7902 cm
Punkt 2
2
z T" -13.56 y T" 10  T" -123.46
My Mz
B
" z = 41.63 -  -125.7 "  = 2.48
" y =
  
Jy Jz J
2
My Mz B explicit , ALL
-31.718925 " 150 kN " m -0.12084 kN " m 2
x T" " z " y + "      " -13.56 cm -             -123.46 cm = -81.585
" 10 cm + "
  -          
4 4 6
Jy Jz J
10331.5582 cm 11933.3030 cm 600906.7902 cm
Punkt 3
2
z T" -13.56 y T" 20  T" -46.93
My Mz
B
" z = 41.63 -  -251.4 "  = 0.94
" y =
  
Jy Jz J
2
My Mz B explicit , ALL
-31.718925 " 150 kN " m -0.12084 kN " m 2
x T" " z " y + "      " -13.56 cm -             -46.93 cm = -208.823
" 20 cm + "
  -          
4 4 6
Jy Jz J
10331.5582 cm 11933.3030 cm 600906.7902 cm
Punkt 4
2
z T" 16.44 y T" 10  T" 176.54
My Mz
B
" z = " y =
  -50.47 -  -125.7 "  =
 -3.55
Jy Jz J
2
My Mz B explicit , ALL
-31.718925 " 150 kN " m -0.12084 kN " m 2
x T" " z " y + "      " 16.44 cm -            
" 10 cm + " 176.54 cm = -179.721
  -          
4 4 6
Jy Jz J
10331.5582 cm 11933.3030 cm 600906.7902 cm
Punkt 5
2
z T" 16.44 y T" 20  T" -46.93
My Mz
B
50 47 251 4 0 94
My Mz
B
" z = " y =
  -50.47 -  -251.4 "  = 0.94

Jy Jz J
2
My Mz B explicit , ALL
-31.718925 " 150 kN " m -0.12084 kN " m 2
x T" " z " y + "      " 16.44 cm -             -46.93 cm = -300.926
" 20 cm + "
  -          
4 4 6
Jy Jz J
10331.5582 cm 11933.3030 cm 600906.7902 cm
Punkt 6
2
z T" -3.56 y T" -10  T" 23.46
My Mz
B
" z = 10.93 - 
" y = 125.7 "  =
   -0.47
Jy Jz J
2
My Mz B explicit , ALL
-31.718925 " 150 kN " m -0.12084 kN " m 2
x T" " z " y + "      " -3.56 cm -            
" -10 cm + " 23.46 cm = 136.156
  -          
4 4 6
Jy Jz J
10331.5582 cm 11933.3030 cm 600906.7902 cm
Punkt 7
2
z T" -13.56 y T" -10  T" 123.46
My Mz
B
" z = 41.63 - 
" y = 125.7 "  =
   -2.48
Jy Jz J
2
My Mz B explicit , ALL
-31.718925 " 150 kN " m -0.12084 kN " m 2
x T" " z " y + "      " -13.56 cm -            
" -10 cm + " 123.46 cm = 164.846
  -          
4 4 6
Jy Jz J
10331.5582 cm 11933.3030 cm 600906.7902 cm
Punkt 8
2
z T" -13.56 y T" -20  T" 46.93
My Mz
B
" z = 41.63 - 
" y = 251.4 "  =
   -0.94
Jy Jz J
2
My Mz B explicit , ALL
-31.718925 " 150 kN " m -0.12084 kN " m 2
x T" " z " y + "      " -13.56 cm -            
" -20 cm + " 46.93 cm = 292.084
  -          
4 4 6
Jy Jz J
10331.5582 cm 11933.3030 cm 600906.7902 cm
Punkt 9
2
z T" 16.44 y T" -10  T" -176.54
My Mz
B
" z = " y = 125.7 "  = 3.55
  -50.47 -  
Jy Jz J
2
My Mz B explicit , ALL
-31.718925 " 150 kN " m -0.12084 kN " m 2
x T" " z " y + "      " 16.44 cm -             -176.54 cm = 78.776
" -10 cm + "
  -          
4 4 6
Jy Jz J
10331.5582 cm 11933.3030 cm 600906.7902 cm
Punkt 10
2
z T" 16.44 y T" -20  T" 46.93
My Mz
B
" z = " y = 251.4 "  =
  -50.47 -   -0.94
Jy Jz J
2
My Mz B explicit , ALL
-31.718925 " 150 kN " m -0.12084 kN " m 2
x T" " z " y + "      " 16.44 cm -            
" -20 cm + " 46.93 cm = 199.981
  -          
4 4 6
Jy Jz J
10331.5582 cm 11933.3030 cm 600906.7902 cm
5. Naprężenia styczne
Momenty statyczne, Sy:
Sy5 T" 0 Sy3 T" 0
3
Sy4 T" 0.7 " 16.44 " 10 = 115.08
3
Sy2 T" -0.8 " 13.56 " 10 = -108.48
13.56 + 3.56 3
Sy12 T" Sy2 - 0.8 " " 10 = -176.96
      
2
3
Sy14 T" Sy4 + 0.7 " 0.5 " 16.44 " 16.44 - 0.7 " 0.5 " 3.56 " 3.56 = 205.24
3
Sy1 T" Sy14 + Sy12 = 28.28
Momenty statyczne, Sz:
Sz5 T" 0 Sy3 T" 0
3
( )
Sz4 T" 0.7 " 0.5 " (20 + 10 ) " 10 = 105
3
( )
Sz2 T" 0.8 " 0.5 " (20 + 10 ) " 10 = 120
3
Sz12 T" Sz2 + 0.8 " 10 " 10 = 200
3
Sz14 T" Sz4 + 0.7 " 10 " 20 = 245
3
Sz1 T" Sz14 + Sz12 = 445
3
Sz0 T" Sz1 + 0.8 " 0.5 " 10 " 10 = 485
Momenty statyczne, Sw:
S5 T" 0 S3 T" 0
4
#176.54 2 46.93 2 ś#
S4 T" 0.7 " 0.5 " 10 # - # = 453.64
2 2
46.93 + 123.46 4
S2 T" -0.8 " " 10 = -681.56
       
2
2 2
123.46 + 23.46 4
S12 T" S2 - 0.8 " " 10 = -1269.24
       
2
4
#176.54 2 23.46 2 ś#
S14 T" S4 + 0.5 " 0.7 " 20 " # - # = 1525.2
4
S1 T" S14 + S12 = 255.96
2 4
S0 T" S1 - 0.5 " 0.8 " 23.46 " 10 = 162.12
Naprężenia styczne:
Tz T" -30.37388 Ty T" 50 M T" 0.20903 " Ms T" -0.003506 "
-Tz " Sy2 -Ty " Sz2 -M " S2
y2 T" = z2 T" = 2 T" = 0.3
    -3.99
    -6.28
   
Jy " 0.8 Jz " 0.8 J " 0.8
-Tz " Sy12 -Ty " Sz12 -M " S12
y12 T" = z12 T" = 12 T" = 0.55
    -6.5
    -10.47
   
Jy " 0.8 Jz " 0.8 J " 0.8
-Tz " Sy4 -Ty " Sz4 -M " S4
y4 T" = 4.83 z4 T" = 4 T" =
        -6.28
    -0.23
Jy " 0.7 Jz " 0.7 J " 0.7
-Tz " Sy14 -Ty " Sz14 -M " S14
y14 T" = 8.62 z14 T" = 14 T" =
        -14.66
    -0.76
Jy " 0.7 Jz " 0.7 J " 0.7
-Tz " Sy1 -Ty " Sz1 -M " S1
y1 T" = 1.04 z1 T" = 1 T" =
        -23.31
    -0.11
Jy " 0.8 Jz " 0.8 J " 0.8
-Ty " Sz0 -M " S0
z0 T" = 0 T" =
    -25.4
    -0.07
Jz " 0.8 J " 0.8
Lewa strona Prawa strona
2 T" y2 + z2 + 2 = -9.98 2 T" -y2 + z2 + 2 = -2
12 T" y12 + z12 + 12 = -16.43 12 T" -y12 + z12 + 12 = -3.42
4 T" y4 + z4 + 4 = -1.68 4 T" -y4 + z4 + 4 = -11.34
14 T" y14 + z14 + 14 = -6.8 14 T" -y14 + z14 + 14 = -24.04
22 38 24 46
1 T" y1 + z1 + 1 = -22.38 1 T" -y1 + z1 + 1 = -24.46
0 T" z0 + 0 = -25.47 0 T" z0 + 0 = -25.47
Naprężenia w punkcie L
3
SyL T" -0.8 " 13.56 " 5 = -54.24
20 + 15 3
SzL T" 0.8 " " 5 = 70
    
2
2 2
# ś#
123.46 - 46.93 2
+ 46.93
       
ś# ź#
2 4
ś#            " 5 ź# -170.39
SL T" -0.8 " =
# 2 #
-Tz " SyL
yL T" =
    -1.99
Jy " 0.8
-Ty " SzL
zL T" =
    -3.67
Jz " 0.8
-M " SL
L T" = 0.07
   
J " 0.8
L T" yL + zL + L = -5.59
2 2
123.46 + 46.93 2
z T" -13.56 y T" 15  T" -        -85.2
=
2
My Mz
B
" z = 41.63 -  -188.55 "  = 1.71
" y =
  
Jy Jz J
2
My Mz B explicit , ALL
-31.718925 " 150 kN " m -0.12084 kN " m 2
xL T" " z " y + "      " -13.56 cm -             -85.195 " = -145.204
" 15 cm + "
...
  -          
4 4 6
Jy Jz J
10331.5582 cm 11933.3030 cm 600906.7902 cm
2 2
> > > > > > > > > >
zastL T" xL + 4 L = 145.63
Naprężenia w punkcie K
3
SyK T" -0.7 " 16.44 " 5 = -57.54 x T" 0
20 + 15 3
SzK T" 0.7 " " 5 = 61.25
    
2
4
#46.93 2 2.1 - 2.9 " 64.81 2 ś#
SK T" -0.7 " 0.5 " # " # = 31.29
#46.93 cm2 176.54 cm2 ś#
2 # + #
-176.54 cm + " x0
        
traints
Guess Values
10 cm
-Tz " SyK
yK T" =
    -2.42
Jy " 0.7
-Ty " SzK ( )
(x) = 7.9
zK T" =
    -3.67
Jz " 0.7
-M " SK
K T" =
    -0.02
#46.93 2 176.54 2 ś#
2 # + # 2
J " 0.7
-176.54 + " 5 =
         -64.81
10
K T" yK + zK + K = -6.1
2
z T" 16.44 y T" -15  T" -64.81
My Mz
" z = " y = 188.55
  -50.47 - 
Jy Jz
2
My Mz B explicit , ALL
-31.718925 " 150 kN " m -0.12084 kN " m 2
xK T" " z " y + "      " 16.44 cm -             -64.81 = 139.379 ...
" -15 cm + "
  -          
4 4 6
Jy Jz J
10331.5582 cm 11933.3030 cm 600906.7902 cm
2 2
> > > > > > > > > >
zastK T" xK + 4 K = 139.91
Solver
Con


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