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Chapter 5 Distributed Force

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All of the forces we have used up until now have been forces applied at a single point. Most forces are not applied at a single point. They are distributed over a geometrical region, and called distributed forces. Distributed forces are more difficult to work with, therefore it is often more convenient to transform them into a single force called a concentrated force. Once this is done we can analyze the forces using the techniques previously discussed. Distributed over a very small contact area, concentrated force. Distributed over a larger contact area, distributed force. A force is considered distributed if the geometrical region is comparable to the dimensions of the object.

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Force Distributions Linear Distribution – Forces are distributed over the length of an object (1D) w – Load intensity F – Applied force L – Length over which the force is applied Examples - Cables, ropes, rods or beams Area Distribution – Forces are distributed over an area (2D) P – Pressure (or stress) F – Applied force A – Contact area for applied force Examples – Water against a dam, air on the table surface, a block resting on a table Volume Distribution – Forces are distributed over the volume of an object (3D) –Weight density – Density g – Gravitational acceleration V – Volume over which the force is applied Examples – Gravitational force, electric force, magnetic force

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Center of Gravity (COG), Center of Mass (COM) and Centroids In order to transform a distributed force to a concentrated force it is necessary to determine the location at which the concentrated force must be applied. This can be done by determining the center of the distribution, which is analogous to the determination of the Center of Gravity, Center of Mass or Centroid of an object. Center of Gravity – “Average” location of the weight of an object. Center of Mass – “Average” location of the mass of an object. Centroid – Geometric center of an object. The center of gravity, the center of mass and the centroid are not necessarily in the same location! The center of gravity can be experimentally determined by hanging an object from different attachment points on the object. A vertical line, concurrent with the attachment point, is drawn for each orientation and the intersection point for all of the lines is the location of the center of gravity. If the gravitational field of the planet you are on is uniform the center of mass should be at the same location as the center of gravity.

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The center of gravity of an object can be determined mathematically by comparing the moment caused by the concentrated weight at the center of gravity to the net moment from all of the infinitesimal weight elements, dW. The two moments must be equal since they represent the same situation. Where, W is the total weight of the object. - Discrete particles - Continuous objects or Separating into x, y and z components or

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Using these expressions allows us to determine the x, y and z components of the center of gravity. We include the z component for completeness. If we can assume the gravitational acceleration is constant, which is an appropriate assumption close to the surface of the Earth, we can easily determine the location of the center of mass. Where M is the total mass of the object. Center of Gravity Center of Mass

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If we assume that the density of the object is constant we can obtain expressions for the location of the centroid of the object. If the thickness, h, is constant: If the width, w, is constant: Centroid (Volume) Centroid (Area) Centroid (Length)

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Considerations during analysis: Choose appropriate reference axis. Look for symmetry, the center typically lies along an axis of symmetry. Cartesian coordinates may not be the most appropriate system. Consider cylindrical and spherical coordinate systems. Centers do not have to be inside the object, especially for lines and areas (consider a bent line and a curved area!).

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Choosing Element of Integration Integration is an key part of determining the centroid of a shape. There will be two integration methods shown, both of which are acceptable. The author uses a single integration technique where he defines a function such that only a single integral must be performed. I will also show you a multi-integration technique where you will integrate several times. You are always integrating over a volume (3D), but if one dimension is constant it is possible to simplify to an area (2D) or length (1D) integration. It is also possible to choose your integration element such that you are only integrating over one of those dimensions. Integration over a volume requires three differential elements dx, dy and dz, for example, and therefore requires solving a triple integral. This same situation can be simplified into a single integration by choosing an appropriate function that describes the area and integrating only over dy. Similarly, an object with a constant thickness can use a double integral over the elements dx and dy to define the area. This situation can also be simplified by choosing an appropriate length function such that only a single integration is required.

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1) Continuity – Choose integration element such that integration can be performed over entire object. Integration over the element dx requires a second function after x 1 (and hence a second integration), while integration over dy would only require a single integral for the entire object. dx dy Additional considerations when using author’s method 2) Disregard higher order terms – when choosing a single integration element the shape of the integration region is a rectangle, which may not completely encompass the region of interest. Inclusion of this gap means that you need to add an additional smaller region, which it is necessary to integrate over. dA 1 dA 2 Single differential quantity – 1 st order differential Two differential quantities – 2 nd order differential The second order term is considered a higher order term and may be dropped. Inclusion of higher order terms increases the precision of the result.

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3) Choose appropriate coordinate system – Cartesian coordinate is not always the most appropriate coordinate system. It is often more convenient to use a cylindrical (dr, d , dz) or spherical (dR, d , d ) coordinate system. Polar coordinates (dr, d ), which is just a 2D version of cylindrical and spherical, is also acceptable. Align reference axis with planes of symmetry if possible, since the centroid usually lies along one of these planes.

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