AMME1802 Chap.6 Centroids and Distributed Loads
Centroids and Distributed Loads
Centroids and distributed resultants share one idea: a total magnitude acts at a weighted average location. Composite regions use signed areas so holes subtract, while load diagrams use area for resultant force and the diagram centroid for line of action. Symmetry and a sense check should be applied before any table is calculated.
The weighting quantity must match the problem: area for a uniform plate, volume or mass for a nonuniform body, and load intensity for a distributed action. Removed pieces receive negative size while retaining their ordinary coordinates from a common origin. A uniform load acts at its midpoint; a triangular load acts closer to its larger-intensity end.
Replacing a distribution by a resultant preserves external force and moment but does not preserve every internal shear or bending value along the beam.
What this chapter covers
- 01
Centroids and centres of mass
- 02
Signed composite areas
- 03
Symmetry checks
- 04
Distributed load resultants
- 05
Lines of action
- 06
Piecewise and trapezoidal load decomposition
- 07
Reference-origin shifts and invariance of the physical centroid location
- 08
A final weighted-location check against symmetry and geometric bounds
Worked example · free
Plate with an off-centre hole
- +1List the positive rectangle area and centroid.
- +1List the circular hole as negative area at its actual coordinates.
- +1Sum signed areas.
- +1Calculate the signed first moments and divide by total area.
- +1Confirm the centroid shifts away from the removed region.
Key terms
- Centroid
- The geometric weighted-average location of a line, area or volume.
- First moment
- A size multiplied by its centroid coordinate about a chosen reference axis.
- Distributed load
- A force spread continuously along a length, area or volume.
- Signed area
- A positive area for material or negative area for a removed region.
- Load intensity
- Distributed force per unit length, area or volume.
- Resultant location
- The point whose force moment matches the distributed loading moment.
- Symmetry axis
- A line dividing geometry into mirrored portions and containing its centroid.
- Trapezoidal load
- A linearly varying distribution that can be decomposed into rectangular and triangular parts.
Centroids and Distributed Loads FAQ
How are holes handled in centroid tables?
Treat each removed region as a negative area or mass while retaining its ordinary positive or negative coordinate from the common origin. Making both the area and coordinate negative would reverse the intended subtraction.
Where does a triangular load act?
Its resultant equals the triangular area and acts at the triangle centroid, one-third of the base length from the larger-intensity end. A quick sketch prevents the common reversal toward the zero-intensity end.
Can a distributed load always be replaced by one force?
The equivalent force and its centroidal line of action preserve external force and moment effects for support reactions. They do not reproduce the internal shear and bending distribution at every section, so retain the distribution when those internal functions are required.
Why does the centroid shift away from a hole?
A removed region enters the weighted average with negative size at its location. Subtracting material on one side reduces that side of the first moment, so the remaining centroid moves toward the material that is still present.
Assessment move
Begin every composite problem with symmetry and a labelled decomposition. Use a table of signed sizes, centroid coordinates and first moments. For load diagrams, calculate area and centroid separately, then check that the line of action sits closer to the heavier part of the distribution. Practise the same composite shape with two valid decompositions and verify that both give the same location.
Include off-centre holes so the direction of centroid shift can be predicted before arithmetic. Move the reference origin and confirm the physical point shifts by the coordinate transformation rather than changing on the body. For distributed actions, sketch rectangles, triangles and trapezoids with their component centroids, then combine their forces through a moment balance.
Write force units in the resultant column and force-length units in the first-moment column. When an internal beam diagram is requested, retain the original load function instead of replacing it too early. End each calculation with two checks: the centroid should lie toward positive material or heavier load, and the final length should fit within the geometry unless negative regions justify otherwise.
Create a reference sheet of standard component centroids, but always redraw the local reference axis because tables may measure from a different face. Practise holes that cross an axis so coordinate sign and area sign remain separate. For load diagrams, convert a trapezoid into a rectangle plus triangle in both possible ways and confirm the same resultant location.
Add one piecewise distribution and calculate component resultants before combining them. Finally, compare the equivalent resultant with the original load when finding support reactions, then explain why that replacement cannot generate the full internal shear function without returning to w(x). This distinction connects the geometric method to its engineering purpose.
