User:Eml4500.f08.delta 6.ramirez/staticsolutions

Statics Solution Method


P = 7 L(1) = 4 L(2) = 2 θ(1) = 30° θ(2) = 45°



Observation:

By using trigonometry and the angles given the unknown lengths can be determined. These lengths will be needed when analyzing the summation of the moments of each node.

$$\sin {30^{\circ} } = \frac{L_c}{4} \Rightarrow L_c = 4 \sin {30^{\circ} } = 2$$

$$\cos {30^{\circ} } = \frac{L_x}{4} \Rightarrow L_x = 4 \cos {30^{\circ} } = 3.46$$

$$\sin {45^{\circ} } = \frac{L_b}{2} \Rightarrow L_b = 2 \sin {45^{\circ} } = 1.4$$

$$\cos {45^{\circ} } = \frac{L_y}{2} \Rightarrow L_y = 2 \cos {45^{\circ} } = 1.4$$

Lz = Lx + Ly = 3.46 + 1.4 = 4.86

La = Lc - Lb = 2 - 1.4 = 0.6

Moment equations can now be analyzed since all lengths are known.

(1) M1: -(P)(Lx) + (F4)(Lz) + (F3)(La) = -(7)(3.46) + (F4)(4.86) + (F3)(0.6)

(2) M2: (P)(Ly) + (F1)(La) - (F2)(Lz) = (7)(1.4) + (F1)(0.6) - (F2)(4.86)

(3) M3: (F1)(Lc) - (F2)(Lx) - (F3)(Lb) + (F4)(Ly) = (F1)(2) - (F2)(3.46) - (F3)(1.4) + (F4)(1.4)

Note: There are 4 unknowns with only 3 equations. However, by inspection it is known that the summation of the forces in the x direction are equal to zero, giving us 4 equations with 4 unknowns. (4) $$\sum{F_x} = 0: F_1 - F_3 = 0 $$

All forces can now be determined by using the system of equations. Rearrange the equations solving for a common variable in order to calculate the first unknown force. Next, using the rearrange equations all unknown forces can be calculated.

(1) -24.22 + 4.86 F4 + 0.6 F3 = 0

(2) 9.8 + 0.6 F1 - 4.86 F2 = 0

(3) 2 F1 - 3.46 F2 - 1.4 F3 + 1.4 F4 = 0

(4) F1 = F3

Since F3 is already in terms of F1, F2 and F4 will also solved in terms of F1

(1) $$F_4 = \frac{24.22 - 0.6 F_3}{4.86} = \frac{24.22 - 0.6 F_1}{4.86}$$ (2) $$F_2 = \frac{9.8 + 0.6 F_1}{4.86}$$ (4) F3 = F1

Plugging in these relationships into equation (3), F1 and the rest of the forces can be solved for as follows:

(3) $$2 F_1 - 3.46 (\frac{9.8 + 0.6 F_1}{4.86}) - 1.4 F_1 + 1.4 (\frac{24.22 - 0.6 F_1}{4.86}) = 0 \Rightarrow $$

Combining like terms,

(3) $$2 F_1 - 6.976 - 0.427 F_1 - 1.4 F_1 + 6.976 - 0.173 F_1 = 0 \Rightarrow 0 = 0$$

Thus, F1 = F3 = 0

(2) $$F_2 = \frac{9.8 + 0.6(0)}{4.86} = 2.01$$ (1) $$F_4 = \frac{24.22 + 0.6(0)}{4.86} = 4.98$$

Results

F1 = 0 F2 = 2.01 F3 = 0 F4 = 4.98