College of Engineering | University of Nebraska–Lincoln



Distributed Forces

Earlier chapters have predominantly focused on problems with concentrated forces – idealized representation of forces with small areas of application.

However, neither application areas are truly infinitesimal nor forces are always applied locally. Hence, another model is needed: Distributed Force.

Examples of Distributed Forces

Dimensionality and Intensity of Distributed Forces

Distributed forces are described via intensity, direction and curve/area/volume of application.

In many problems direction is the same for the entire distributed force. Therefore, typically description consists of curve/area/volume and a corresponding intensity with units: [pic]

The most well-known example of a distributed force is a gravitational force, with intensity [pic] ������������������������������������������������ ������������������������������������.

Center of Gravity

Behavior of a rigid body, affected by gravity, can be evaluated by replacement of the distributed gravitational force with its resultant applied at a center of gravity.

Since distributed gravitational force is acting along a single direction at every point of the body, it is equivalent to a set of parallel vertical forces.

From the earlier chapter we know that two parallel forces can be replaced with a single equivalent force, parallel to original two. Hence, action of the gravity can be achieved with a single concentrated force.

Center of Gravity, Mass and Centroid

Location of a center of gravity can be determined via two methods:

* experimentally via hanging a body on a cable, with Center of Gravity being on the line of the cable (special case with no external moments applied!)

* analytically via Varignon’s theorem (moment of sum is equal sum of moments) from Ch.3, essentially replacing summation with integration.

Force Resultant of Gravity Force

[pic]

Since direction and sense of gravity is assumed to be uniform:

[pic]

Or, similarly, it can be written in component form.

Moment Resultant of Gravity Force

Lets say line of action of the resultant passes through [pic] ( [pic], [pic], [pic] ) , then:

[pic]

Using Varignon’s principle, we will continue with one component of the moment (the other components follow the same line of calculation:

[pic]

The location of the center of gravity is then:

[pic]

If ������ is constant, ������=������������ and ������������=������ ������������, center of mass:

[pic]

If ������ is constant, ������=������������ and ������������=������ ������������:

[pic]

Purely geometric! This point is called centroid.

If gravity and density are uniform, center of gravity, mass and centroid are the same point.

Topographies of Reduced Dimensionality

2D:

[pic]

1D:

[pic]

Note: Don’t mix dimensionalities of topography and embedding space! Resultants from above are both 3D.

Intelligent selection of coordinate system and taking advantage of symmetry greatly helps.

Centroids by Integration

Order of Element

Order of the element is equal to a number of infinitesimal dimensions used in its description:

First: dx, dy, dz

Second: dx*dy

Third: dx*dy*dz.

Select lowest order to have least integration passes

Whenever possible, select elements spanning through an entire area/volume to have a single integral.

Continuity

Whenever possible, select elements that can be integrated in one continuous operation to cover the figure

Discarding higher order terms

Higher order terms can be dropped in comparison to lower-order terms: dA=ydx and the second order term 1/2dxdy can be discarded. In the limit, there is no error.

Choice of Coordinate System

Generally, Coordinate system is selected to match boundaries of area/volume in the best possible way (Cartesian system would be best to describe the boundary of (a), while polar coordinates would be the best choice for (b)).

Centoridal coordinate of element

When the first or second differential element is adopted, it is important to use centroidal coordinates within it for the moment arm in setting up the the moment of the differential element.

Examples of Elements

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Centroid, 2D, Example 1

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Example 2

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Example 3

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Example 4

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Example 5

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Centroid, 3D, Example

[pic]

Centroids of Composite Bodies

Often a complex body can be split into a set of simpler bodies with known locations of centroids. In this case coordinates of centroid for the whole set can be easily computed.

[pic]

Centroids of Composite Bodies, 2D

A polygon in the figure above can be split into a sum of two triangles and a rectangle. Using Moment of Sum=Sum of Moments:

[pic]

[pic]

Where the summation is performed over the different parts. In case g and ρ are constant, the equations are reduced to geometric equations

[pic]

Similarly to the 2D, 3D case:

[pic]

Hence, evaluation of centroids can be significantly simplified by using previously computed centroids for common geometric forms.

Notes: Coordinates of centroids of the parts xi, yi, zi have a sign (can be either positive or negative).

Holes are areas/volumes to be subtracted and considered as negative areas/volumes.

Example:

Centroids of Common 1D Bodies

[pic]

Centroids of Common 2D Bodies

[pic]

[pic]

Centroids of Common 3D Bodies

[pic]

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Composite Bodies Example 1

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Composite Bodies Example 2

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Centroids of 1D Bodies, Example

[pic]

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Centroids of 2D Bodies, Example 1

[pic]

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Example 2

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Centroids of 3D Bodies, Example 1

[pic]

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Example 2

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Application of centroids to distributed forces

The concept of centroid can be used to replace a distributed force of any kind with a concentrated force

[pic]

In this case the distributed weight (or any other force such as hydrostatic force etc) can be replaced by a resultant computed from the weight distribution. The line of application is determined from determining the centroid of the “body” representing the distributed weight.

Example 1:

Example 2:

[pic]

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