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4.4: Distributed Forces

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    A distributed force is any force where the point of application of the force is an area or a volume. This means that the "point of application" is not really a point at all. Though distributed forces are more difficult to analyze than point forces, distributed forces are quite common in real world systems so it is important to understand how to model them.

    Distributed forces can be broken down into surface forces and body forces. Surface forces are distributed forces where the point of application is an area (a surface on the body). Body forces are forces where the point of application is a volume (the force is exerted on all molecules throughout the body). Below are some examples of surface and body forces.

    A dam in a river, with an extremely large drop in the water level downstream of the dam compared to upstream.
    Figure \(\PageIndex{1}\): The water pressure pushing on the surface of this dam is an example of a surface force. Image by Curimedia CC-BY-SA 2.0.
    An apple falling from a tree, with a blurring effect applied to the apple to indicate the progress of its fall.
    Figure \(\PageIndex{2}\): The gravitational force on this apple is distributed over the entire volume of the fruit. Gravitational forces are an example of body forces. Image by Zátonyi Sándor CC-BY 3.0.

    Representing Distributed Forces:

    Distributed forces are represented as a field of vectors. This is drawn as a number of discrete vectors along a line, over a surface, or over a volume, that are connected with a line or a surface as shown below.

    A long, thin, horizontal rectangular body experiences a surface force against its top edge, represented as a line of closely set downwards force vectors whose magnitude increases from 5 kN/m at the left end of the rectangle to 10 kN/m at the right end, with a line connecting the tails of all the vectors.
    Figure \(\PageIndex{3}\): This is a representation of a surface force in a 2D problem (a force distributed over a line). The magnitude is given in units of force per unit distance.
    A rectangular plane experiences a surface force against its top, represented as a grid of closely set downwards force vectors with the outermost ones connected by a rectangle. The magnitude of the surface force is uniform across the plane, labeled as 60 lbs per square inch.
    Figure \(\PageIndex{4}\): This is a representation of a surface force in a 3D problem (a force distributed over an area). The magnitude is given in units of force per unit area (also called a pressure).

    Though these representations show a discrete number of individual vectors, there is actually a magnitude and direction at all points along the line, surface, or body. The individual vectors represent a sampling of these magnitudes and directions.

    It is also important to realize that the magnitudes of distributed forces are given in force per unit distance, area, or volume. We must integrate the distributed force over its entire range to convert the force into the usual units of force.

    Analyzing Distributed Forces:

    For analysis purposes in statics and dynamics, we will usually substitute in a single point force that is statically equivalent to the distributed force in the problem. This single point force is called the equivalent point load, and it will cause the same accelerations or reaction forces as the distributed force while simplifying the math. However, in analysis that focuses on the strength of materials where the bodies are not rigid, this substitution will not work as the distributed forces will not cause the same deformations and stresses as the point force.

    Video lecture covering this section, delivered by Dr. Jacob Moore. YouTube source:

    This page titled 4.4: Distributed Forces is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Jacob Moore & Contributors (Mechanics Map) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.