Examples |
The following example represents truss with 9 joints and 16 members under forces applied to different joints. |
Input number of joints: |
Number of joints n>2: | 9 |
Input coordinates of joints: |
x_{1} : | 0 | y_{1} : | 0 |
x_{2} ≥0 : | 10 | y_{2} ≥0 : | 7.5 |
x_{3} ≥0 : | 20 | y_{3} ≥0 : | 15 |
x_{4} ≥0 : | 20 | y_{4} ≥0 : | 0 |
x_{5} ≥0 : | 30 | y_{5} ≥0 : | 22.5 |
x_{6} ≥0 : | 40 | y_{6} ≥0 : | 15 |
x_{7} ≥0 : | 50 | y_{7} ≥0 : | 7.5 |
x_{8} ≥0 : | 60 | y_{8} ≥0 : | 0 |
x_{9} ≥0 : | 40 | y_{9} ≥0 : | 0 |
Input truss members: |
First end (joint) of member # 1: | 1 | Second end (joint) of member # 1: | 2 |
First end (joint) of member # 2: | 1 | Second end (joint) of member # 2: | 4 |
First end (joint) of member # 3: | 2 | Second end (joint) of member # 3: | 3 |
First end (joint) of member # 4: | 2 | Second end (joint) of member # 4: | 4 |
First end (joint) of member # 5: | 3 | Second end (joint) of member # 5: | 4 |
First end (joint) of member # 6: | 3 | Second end (joint) of member # 6: | 5 |
First end (joint) of member # 7: | 4 | Second end (joint) of member # 7: | 5 |
First end (joint) of member # 8: | 4 | Second end (joint) of member # 8: | 9 |
First end (joint) of member # 9: | 5 | Second end (joint) of member # 9: | 6 |
First end (joint) of member # 10: | 5 | Second end (joint) of member # 10: | 9 |
First end (joint) of member # 11: | 6 | Second end (joint) of member # 11: | 7 |
First end (joint) of member # 12: | 6 | Second end (joint) of member # 12: | 9 |
First end (joint) of member # 13: | 7 | Second end (joint) of member # 13: | 8 |
First end (joint) of member # 14: | 7 | Second end (joint) of member # 14: | 9 |
First end (joint) of member # 15: | 8 | Second end (joint) of member # 15: | 9 |
Input constraints: |
Number of joint of horizontal constraint # 1: | 1 | Number of joint of vertical constraint # 1: | 1 |
Number of joint of horizontal constraint # 2: | Number of joint of vertical constraint # 2: | 8 |
Input vertical and horizontal components of external forces applied to joints: |
X_{1} : | Reaction | Y_{1} : | Reaction |
X_{2} : | Y_{2} : | -6 | |
X_{3} : | Y_{3} : | -6 | |
X_{4} : | -4 | Y_{4} : | |
X_{5} : | Y_{5} : | -6 | |
X_{6} : | Y_{6} : | -11 | |
X_{7} : | Y_{7} : | -8 | |
X_{8} : | Y_{8} : | Reaction | |
X_{9} : | Y_{9} : |
Output: |
The horizontal reaction is: H1=4 . The vertical reactions are: V1=17, V8=20 .
The computed internal forces are: N_{12}=-28.333; N_{14}=18.667; N_{23}=-23.333; N_{24}=-5; N_{34}=-6; N_{35}=-23.333; N_{45}=9.849; N_{49}=14.667; N_{56}=-26.667; N_{59}=16.415; N_{67}=-26.667; N_{69}=-11; N_{78}=-33.333; N_{79}=-6.667; N_{89}=26.667;
Solution |
Equilibrium equation of truss (the sum of moments about joint with number 8):
V1=(+Y_{2}(x_{2}-x_{8})+Y_{3}(x_{3}-x_{8})+Y_{5}(x_{5}-x_{8})+Y_{6}(x_{6}-x_{8})+Y_{7}(x_{7}-x_{8})-X_{4}(y_{4}-y_{8})-H1(y_{1}-y_{8}))/(x_{8}-x_{1})=(-6(10-60)-6(20-60)-6(30-60)-11(40-60)-8(50-60)+4(0-0)-4(0-0))/(60-0)=17;
Equilibrium equation of truss with respect to y-axis:
V8=-Y_{2}-Y_{3}-Y_{5}-Y_{6}-Y_{7}-V1=+6+6+6+11+8-17=20;
Internal forces of the truss:
Equilibrium equations of joint with number 1 with respect to y and x axes:
N_{12}v_{12}+N_{14}v_{14}+V1=0;
N_{12}h_{12}+N_{14}h_{14}+H1=0;
After substitution of known values we have the system of linear equations
0.6N_{12}+-0N_{14}+17=0;
0.8N_{12}+1N_{14}+4=0;
The solution of the system: N_{12}=-28.333; N_{14}=18.667
Equilibrium equations of joint with number 2 with respect to y and x axes:
N_{23}v_{23}+N_{24}v_{24}+N_{21}v_{21}+Y_{2}=0;
N_{23}h_{23}+N_{24}h_{24}+N_{21}h_{21}=0;
After substitution of known values we have the system of linear equations
0.6N_{23}-0.6N_{24}+11=0;
0.8N_{23}+0.8N_{24}+22.667=0;
The solution of the system: N_{23}=-23.333; N_{24}=-5
Equilibrium equations of joint with number 3 with respect to y and x axes:
N_{34}v_{34}+N_{35}v_{35}+N_{32}v_{32}+Y_{3}=0;
N_{34}h_{34}+N_{35}h_{35}+N_{32}h_{32}=0;
After substitution of known values we have the system of linear equations
-1N_{34}+0.6N_{35}+8=0;
-0N_{34}+0.8N_{35}+18.667=0;
The solution of the system: N_{34}=-6; N_{35}=-23.333
Equilibrium equations of joint with number 4 with respect to y and x axes:
N_{45}v_{45}+N_{49}v_{49}+N_{41}v_{41}+N_{42}v_{42}+N_{43}v_{43}=0;
N_{45}h_{45}+N_{49}h_{49}+N_{41}h_{41}+N_{42}h_{42}+N_{43}h_{43}+X_{4}=0;
After substitution of known values we have the system of linear equations
0.914N_{45}+-0N_{49}-9=0;
0.406N_{45}+1N_{49}-18.667=0;
The solution of the system: N_{45}=9.849; N_{49}=14.667
Equilibrium equations of joint with number 5 with respect to y and x axes:
N_{56}v_{56}+N_{59}v_{59}+N_{53}v_{53}+N_{54}v_{54}+Y_{5}=0;
N_{56}h_{56}+N_{59}h_{59}+N_{53}h_{53}+N_{54}h_{54}=0;
After substitution of known values we have the system of linear equations
-0.6N_{56}-0.914N_{59}-1=0;
0.8N_{56}+0.406N_{59}+14.667=0;
The solution of the system: N_{56}=-26.667; N_{59}=16.415
Equilibrium equations of joint with number 6 with respect to y and x axes:
N_{67}v_{67}+N_{69}v_{69}+N_{65}v_{65}+Y_{6}=0;
N_{67}h_{67}+N_{69}h_{69}+N_{65}h_{65}=0;
After substitution of known values we have the system of linear equations
-0.6N_{67}-1N_{69}-27=0;
0.8N_{67}+-0N_{69}+21.333=0;
The solution of the system: N_{67}=-26.667; N_{69}=-11
Equilibrium equations of joint with number 7 with respect to y and x axes:
N_{78}v_{78}+N_{79}v_{79}+N_{76}v_{76}+Y_{7}=0;
N_{78}h_{78}+N_{79}h_{79}+N_{76}h_{76}=0;
After substitution of known values we have the system of linear equations
-0.6N_{78}-0.6N_{79}-24=0;
0.8N_{78}-0.8N_{79}+21.333=0;
The solution of the system: N_{78}=-33.333; N_{79}=-6.667
Equilibrium state of joint with number 8:
N_{89}h_{89}+N_{87}h_{87}=0;
After substitution of known values we have
-1N_{89}+26.667=0;
The solution is N_{89}=26.667;
Here h_{ij} and v_{ij} are cosine and sine of the angle between x-axis and truss member connecting joints i and j .
They are calculated using formulas: h_{ij}=(x_{j}-x_{i})/l_{ij}; v_{ij}=(y_{j}-y_{i})/l_{ij} ,
l_{ij} is the distance between joints i and j (the length of truss member).
For the considered truss: l_{12}=12.5; l_{14}=20; l_{23}=12.5; l_{24}=12.5; l_{34}=15; l_{35}=12.5; l_{45}=24.622; l_{49}=20; l_{56}=12.5; l_{59}=24.622; l_{67}=12.5; l_{69}=15; l_{78}=12.5; l_{79}=12.5; l_{89}=20;
h_{12}=0.8; h_{14}=1; h_{23}=0.8; h_{24}=0.8; h_{34}=-0; h_{35}=0.8; h_{45}=0.406; h_{49}=1; h_{56}=0.8; h_{59}=0.406; h_{67}=0.8; h_{69}=-0; h_{78}=0.8; h_{79}=-0.8; h_{89}=-1;
v_{12}=0.6; v_{14}=-0; v_{23}=0.6; v_{24}=-0.6; v_{34}=-1; v_{35}=0.6; v_{45}=0.914; v_{49}=-0; v_{56}=-0.6; v_{59}=-0.914; v_{67}=-0.6; v_{69}=-1; v_{78}=-0.6; v_{79}=-0.6; v_{89}=-0;