PROJEKT: KRATOWNICA

Gr. Nr 3 ( ŚR. TP 7:30-9:00)

Obliczenia wg. punktu B:

$\sum_{}^{}{Fx = 0}\ \sum_{}^{}{Fy = 0}$ $\sum_{}^{}{MB = 0}$

Dane	Obliczenia	Wynik
F1 = 1500 F2 = 2000 F4 = 2500 F5 = 3000 F6 = 3500	$$\sum_{}^{}\text{Fx}$$ RBx −  F1 • cos30 + F6 • cos60 = 0 RBx =  F1 • cos30 + F6 • cos60 $$RB_{x} = \ 1500\ \bullet \frac{\sqrt{}3}{2} + 3500 \bullet 0.5 = 3049.04$$	RBx =3049.04
	$$\sum_{}^{}{Fy = 0}$$ F1 • sin30 − F2 − RA − RBy − F5 − F4 − F6sin60 = 0 RA + RBy = F1 • sin30 − F2 − F5 − F4 − F6sin60 $$R_{A} + R_{\text{By}} = 1500 \bullet 0.5 - 2000 - 3000 - 2500 - 3500 \bullet \frac{\sqrt{3}}{2}$$	RA + RBy = − 11281.09
	$$\sum_{}^{}{MB = 0}$$ $F_{5} \bullet a\sqrt{2} + F_{6} \bullet \sin{60}\left( a\sqrt{2} - \frac{a\sqrt{2}}{2} \right) + F_{6} \bullet cos60\left( a\frac{\sqrt{3}}{2} \right) + F_{4} \bullet \frac{a\sqrt{2}}{2} - F_{2}\left( \frac{a\sqrt{2}}{2} + a\sqrt{2} \right)$- +$F_{1}\sin 30\left( 2a\sqrt{2} \right) - R_{A}a\sqrt{2} = 0\ \ $ a=1 $3000\sqrt{2} + 3500 \bullet \frac{\sqrt{3}}{2} \bullet \left( \sqrt{2} + \frac{\sqrt{2}}{2} \right) + F_{6} \bullet 3500 \bullet 0.5 \bullet \frac{\sqrt{2}}{2} + 2500 \bullet \frac{\sqrt{2}}{2} - 2000 \bullet \left( \frac{\sqrt{2}}{2} + \sqrt{2} \right)$+ $$1500 \bullet 0.5 \bullet 2\sqrt{2} - R_{1}\sqrt{2} = 0$$ $$R_{A}\sqrt{2} = 7313.79$$	RA = 5171.63
	RBy = −11281.09 − RA RBy = −11281.09 − 5171.63	RBy = −16452.72

Liczenie relacji w węzłach po kolei:

Węzeł	Obliczenia	Wynik
1.	$$\sum_{}^{}F_{x} = 0 = - F_{1}\cos 30 + S_{1} + S_{2}cos45$$ $$\sum_{}^{}{F_{y} = 0 =} - F_{1}\sin 30 - S_{2}sin45$$ S2sin45 = −F1sin30 $$S_{2} = \frac{{- F}_{1}\sin 30}{sin45} = \frac{- 1500 \bullet 0.5}{\frac{\sqrt{}2}{2}}$$ S1 = F1cos30 − S2cos45 $$S_{1} = 1500 \bullet \frac{\sqrt{}3}{2} + 1060.66 \bullet \frac{\sqrt{}2}{2}$$	S2 = −1060.66 S1 = 2049.04
2.	$$\sum_{}^{}F_{x} = 0 = - S_{2}cos45 + S_{4}cos45 + S_{3}$$ $$\sum_{}^{}{F_{y} = 0 = S_{2}sin45 + S_{4}sin45 - F_{2}}$$ $$S_{4} = \frac{F_{2} - S_{2}sin45}{sin45}$$ $$S_{4} = \frac{2000 + 1060.66 \bullet \frac{\sqrt{}2}{2}}{\frac{\sqrt{}2}{2}}$$ S3 = −S4cos45 + S2cos45 $$S_{3} = - 3889.09 \bullet \frac{\sqrt{}2}{2} - 1060.66 \bullet \frac{\sqrt{}2}{2}$$	S4 = 3889.09 S3 = 3500.00
3.	$$\sum_{}^{}F_{x} = 0 = S_{1} + S_{5} - S_{4}cos45 + S_{6}cos45$$ $${\sum_{}^{}{F_{y} = 0 = - R_{A} -}S}_{4}sin45 - S_{6}sin45$$ $$S_{6} = \frac{{- S}_{4}sin45 - R_{A}}{sin45}$$ $$S_{6} = \frac{- 3889.09 \bullet \frac{\sqrt{}2}{2} - 5171.63}{\frac{\sqrt{}2}{2}}$$ S5 = S1+S4cos45 − S6cos45 $$S_{5} = 2049.04 + 3889.09 \bullet \frac{\sqrt{}2}{2} + 11202.88 \bullet \frac{\sqrt{}2}{2}$$	S6 = −11202, 88 S5 = 12720.67
4.	$$\sum_{}^{}F_{x} = 0 = {- S}_{3} - S_{6}cos45 + S_{7}cos45 + S_{8}$$ $$\sum_{}^{}{F_{y} = 0 =}S_{6}sin45 + S_{7}sin45$$ $$S_{7} = \frac{{- S}_{6}cos45}{cos45} = {- S}_{6}$$ S8 = S3 + S6cos45 − S7cos45 $$S_{8} = - 3500 - 11202.88 \bullet \frac{\sqrt{}2}{2} - 11202.88 \bullet \frac{\sqrt{}2}{2}$$	S7 = 11202.88 S8 = −19343.26
5.	$$\sum_{}^{}F_{x} = 0 = - S_{5} + R_{\text{Bx}} + S_{10} - S_{7}cos45 + S_{9}cos45$$ $$\sum_{}^{}{F_{y} = 0 =} - R_{\text{By}} - S_{7}sin45 - S_{9}sin45$$ $$S_{9} = \frac{- R_{\text{By}} - S_{7}sin45}{sin45}$$ $$S_{9} = \frac{16457.72 - 11202.88 \bullet \frac{\sqrt{}2}{2}}{\frac{\sqrt{}2}{2}}$$ S10 = S5 − RBx + S7cos45 − S9cos45 $$S_{10} = 12780.67 - 3049.04 + 11202.88 \bullet \frac{\sqrt{}2}{2} - 12064.78 \bullet \frac{\sqrt{}2}{2}$$	S9 = 12064.78 S10 = 9068.17
6.	$$\sum_{}^{}F_{x} = 0 = {- S}_{8} - S_{9}cos45{+ S}_{11}cos45 + S_{12}$$ $$\sum_{}^{}{F_{y} = 0 =}S_{9}sin45 + S_{11}sin45 - F_{4}$$ $$S_{11} = \frac{F_{4} - S_{9}sin45}{sin45}$$ $$S_{11} = \frac{2500 - 12064.78 \bullet \frac{\sqrt{}2}{2}}{\frac{\sqrt{}2}{2}}$$ S12 = S8 + S9cos45 − S11cos45 $$S_{12} = - 19343.26 + 12064.78 \bullet \frac{\sqrt{}2}{2} + 8529.29 \bullet \frac{\sqrt{}2}{2}$$	S11 = −8529.25 S12 = −4781.08
7.	$$\sum_{}^{}F_{x} = 0 = {- S}_{10} - S_{11}cos45 + S_{13}cos45$$ $$\sum_{}^{}{F_{y} = 0 =} - F_{5} - S_{11}sin45 - S_{13}sin45$$ $$S_{13} = \frac{- S_{11}sin45 - F_{5}}{sin45}$$ $$S_{13} = \frac{8529.25 \bullet \frac{\sqrt{}2}{2} - 3000}{\frac{\sqrt{}2}{2}}$$	S13 = 4286.61
8.	$$\sum_{}^{}F_{x} = 0 = - S_{12} - S_{13}cos45 - F_{6}cos60$$ $$\sum_{}^{}{F_{y} = 0 = S_{13}sin45 - F_{6}sin60}$$ $$S_{13} = \frac{F_{6}sin60}{sin45} = 4286.61$$ S12 = S13cos45 − F6cos60 = −4781.08	Węzeł policzony dla sprawdzenia: takie same wyniki jak wyżej, czyli policzono dobrze

Metoda Rittera:

Dane	Obliczenia	Wynik
F1 = 1500 F2 = 2000 F4 = 2500 F5 = 3000 F6 = 3500 a=1	$$\sum_{}^{}F_{x} = 0 = - S_{10} - S_{11}cos45 - S_{12} - F_{6}cos60$$ $$\sum_{}^{}{F_{y} = 0 = {- S}_{11}sin45} - F_{5} - F_{6}sin60$$ $$S_{11} = \frac{- F_{5} - F_{6}sin60}{sin45}$$ $$S_{11} = \frac{- 3000 - 3500 \bullet \frac{\sqrt{}3}{2}}{\frac{\sqrt{}2}{2}}$$	S11 = −8529.25
	$$\sum_{}^{}{M = 0 = S_{12} \bullet}\frac{a\sqrt{2}}{2} + F_{6}sin60 \bullet \frac{a\sqrt{2}}{2} + F_{6}\ cos60 \bullet \frac{a\sqrt{2}}{2}$$ $$S_{12} \bullet \frac{\sqrt{2}}{2} + F_{6}sin60 \bullet \frac{\sqrt{2}}{2} + F_{6}\ cos60 \bullet \frac{\sqrt{2}}{2} = 0$$ $$S_{12} = \frac{- 3500 \bullet \frac{\sqrt{2}}{2} \bullet \frac{\sqrt{}3}{2} - 3500 \bullet \frac{\sqrt{2}}{2} \bullet \frac{1}{2}}{\frac{\sqrt{}2}{2}}$$	S12 = −4781.09
	S10 = −S11cos45 − S12 − F6cos60 $$S_{10} = 8529.25 \bullet \frac{\sqrt{2}}{2} + 4782.08 - 3500 \bullet 0.5$$	S10 = 9062.18