3D Force Components
The addition of a third dimension alters the process of vector decomposition of an arbitrary force slightly:
F_x=F\cos\theta_x
F_y=F\cos\theta_y
F_z=F\cos\theta_z
F=\sqrt{F^2_x+F^2_y+F^2_z}
\displaystyle\bm{\overrightarrow{F}}=F_x\bm{\hat{i}}+F_y\bm{\hat{j}}+F_z\bm{\hat{k}}
\displaystyle\bm{\overrightarrow{F}}=F(\cos\theta_x\bm{\hat{i}}+\cos\theta_y\bm{\hat{j}}+\cos\theta_z\bm{\hat{k}})
The subscript for the angles denotes the axis that the angle is measured from.
We can describe the vector component of the last equation with respect to the forces unit vector:
F(\cos\theta_x\bm{\hat{i}}+\cos\theta_y\bm{\hat{j}}+\cos\theta_z\bm{\hat{k}})=F\bm{\hat{n}}
A unit vector has a magnitude of 1, and can be derived with the following definition:
\displaystyle\bm{\hat{n}}=\dfrac{(x_2-x_1)\bm{\hat{i}}+(y_2-y_1)\bm{\hat{j}}+(z_2-z_1)\bm{\hat{k}}}{\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}}=\dfrac{\bm{\overrightarrow{n}}}{\left|\bm{\overrightarrow{n}}\right|}
3D Equilibrium Equations
For a structure to be static, every point in the structure must satisfy the following:
F_x=0
F_y=0
F_z=0
M_x=0
M_y=0
M_z=0
Structure Analysis
The method of joints and frame analysis is completed using the exact same process outlined in the 2D structure analysis sections. The only difference between 2D and 3D analysis is the application of the three dimensional equilibrium equations. Because of more complex geometries, 3D analysis is typically more difficult than 2D, however taking a problem one step at a time will make the process much more manageable.