7 Distributed Loads and Centroids Free Online FlashCards

What distinguishes a distributed load from a point load?

A distributed load acts over a length, area, or volume rather than at a single point.

How do you find a beam load’s equivalent resultant force?

The resultant is the signed area under the intensity curve: $R=\int_a^b w(x)\,dx$.

How is the line of action of a distributed load’s resultant located?

Match the original distribution’s moment: $\bar{x}=\frac{\int_a^b xw(x)\,dx}{\int_a^b w(x)\,dx}$, provided $R\ne 0$.

What are the resultant and location for a uniform load over length $L$?

For intensity $w_0$ over length $L$, $R=w_0L$, acting at the rectangle’s centroid, $\bar{x}=\frac{L}{2}$.

What is the resultant of a triangular load rising to $w_{\max}$ over length $L$?

For a triangular load rising from zero to $w_{\max}$ over length $L$, the resultant is $R=\frac{1}{2}w_{\max}L$.

Where does a triangular load’s resultant act along its length?

It acts $\frac{L}{3}$ from the end with the larger intensity, or $\frac{2L}{3}$ from the zero-load end.

What is a plane area’s centroid, and how are its coordinates defined?

A centroid is a shape’s geometric balance point. For a plane area, $\bar{x}=\frac{1}{A}\int_A x\,dA$ and $\bar{y}=\frac{1}{A}\int_A y\,dA$.

Where is the centroid of a rectangle?

A rectangle’s centroid is at the intersection of its diagonals, halfway along each side.

Where is a triangle’s centroid relative to its base?

It is at the intersection of the medians, one-third of the altitude from the base, or two-thirds from the opposite vertex.

How do you find the centroid of a composite area?

Use area-weighted coordinates: $\bar{x}=\frac{\sum_i A_i x_i}{\sum_i A_i}$ and $\bar{y}=\frac{\sum_i A_i y_i}{\sum_i A_i}$.

How should a hole be represented in a composite-area centroid calculation?

Treat a hole as a negative area when calculating the composite shape’s centroid.

How do you combine the resultants of several load shapes?

Add the signed resultants, then use their moments: $R=\sum_i R_i$ and $\bar{x}=\frac{\sum_i R_i x_i}{\sum_i R_i}$, if $R\ne 0$.