Normal Stress
Imagine pulling on the end of a long, fixed rod, applying tension to the member. According to Newton’s 3rd law, by applying this force the rod will necessarily react by applying an equal and opposite force internally. This resulting reaction force will create what is known as a normal stress along the rod. Normal stress acts axially along an object, the acting force is parallel to the normal of the cross-section of interest.
Stress is a measure of the local intensity of a force on an object. Localized stress has the following definition:
\displaystyle\sigma=\lim_{\Delta A\to 0}\dfrac{\Delta F}{\Delta A}
Where \Delta A signifies a portion of the objects cross-sectional area and \Delta F is a portion of the overall force applied to that area. Commonly the average stress is used to get a quick understanding of an object under load.
\sigma_{avg}=\dfrac{F}{A}
This average normal stress will generally be referred to as \sigma going forward.
Shearing Stress
Shearing stress occurs when an acting force is perpendicular to the normal of the cross-section of interest.
The average shear stress for a given force is as follows:
\tau_{avg}=\dfrac{F}{A}
This average shear stress will generally be referred to as \tau going forward.
Bearing Stress
Bearing stress is a stress that occurs at a bearing surface, such as bolts or pins. For a simple pin the bearing stress is as follows:
\sigma_b=\dfrac{F}{A}=\dfrac{F}{td}
Oblique Planes
Imagine cutting a member under tension or compression at an angle. The resulting normal and shear stress relative to the cut will be different from the normal stress applied to the member.
\sigma=\dfrac{F\cos\theta}{A/\cos\theta}=\dfrac{F\cos^2\theta}{A}
\tau=\dfrac{F\sin\theta}{A/\cos\theta}=\dfrac{F\sin\theta\cos\theta}{A}
General Loading
The stresses shown in the following diagram can occur for a particle under general loading conditions. The first subscript denotes the direction of the face, whereas the second subscript denotes the direction of applied force.
The general loading condition of a particle can be summarized using a matrix known as a tensor. Tensors provide a useful mathematical framework which is compatible with vectors, a staple of physics.
\sigma_{ij}=\begin{bmatrix} \sigma_x & \tau_{xy} & \tau_{xz}\\ \tau_{yx} & \sigma_{y} & \tau_{yz}\\ \tau_{zx} & \tau_{zy} & \sigma_z \end{bmatrix}
To ensure that the particle is static the following relations must be true:
\tau_{xy}=\tau_{yx}
\tau_{xz}=\tau_{zx}
\tau_{yz}=\tau_{zy}
As such, there are only six independent stressors for any static particle.
Safety Factor
Factors of safety are used to ensure designs meet safety standards. The safety factor can be thought of as a scalar applied to the theoretically allowable load which will ensure the design can exceed required loading expectations.
FS=\dfrac{\sigma_u}{\sigma_{all}}=\dfrac{\text{ultimate stress}}{\text{allowable stress}}