Axial stress describes the amount of force per unit of cross-sectional area that acts in the lengthwise direction of a beam or axle. Axial stress can cause a member to compress, buckle, elongate or fail. Some parts that might experience axial force are building joists, studs and various types of shafts. The simplest formula for axial stress is force divided by cross-sectional area. The force acting on that cross section, however, may not be immediately obvious.
Determine the magnitude of force that acts directly normal (perpendicular) to the cross section. For example, if a linear force meets the cross section at a 60-degree angle, only a portion of that force directly causes axial stress. Use the trigonometric function sine to gauge how perpendicular the force is to the face; the axial force equals the magnitude of the force times the sine of the incident angle. If the force enters at 90-degrees to the face, 100 percent of the force is axial force.
Choose a specific point at which to analyze the axial stress. Calculate the cross-sectional area at that point.
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Calculate the axial stress due to linear force. This is equal to the component of linear force perpendicular to the face divided by the cross-sectional area.
Calculate the total moment acting on the cross section of interest. For a static beam, this moment will be equal and opposite to the sum of moments acting on either side of the cross section. There are two types of moments: direct moments, as applied by a cantilever support, and moments created about the cross section by vertical forces. The moment due to a vertical force equals its magnitude times its distance from the point of interest. Use the cosine function to calculate the vertical component of any linear forces applied to the ends of the axle.
Calculate the axial stress due to moments. When a moment acts on an axle, it creates tension in either the top or bottom half of it, and compression in the other. The stress is zero along the line that runs through the center of the axle (called the neutral axis), and increases linearly toward both its top and bottom edge. The formula for stress due to bending is (M * y) / I, where M = moment, y = the height above or below the neutral axis, and I = the moment of inertia at the axle's centroid. You can think of moment of inertia as a beam's ability to resist bending. This number is easiest to obtain from tables of previous calculations for common cross-sectional shapes.
Add the stresses caused by linear forces and moments to obtain the total axial stress for the point analyzed.