Beam Deflection A simply supported beam is 64 feet long and...

Beam Deflection A simply supported beam is 64 feet long and has a load at the center (see figure). The deflection (bending) of the beam at its center is 1 inch. The shape of the deflected beam is parabolic. (a) Find an equation of the parabola. (Assume that the origin is at the center of the beam.) (b) How far from the center of the beam is the deflection equal to $\frac{1}{2}$ inch?

Beam Deflection A simply supported beam is 64 feet long and has a load at the center (see figure). The deflection (bending) of the beam at its center is 1 inch. The shape of the deflected beam is parabolic.
(a) Find an equation of the parabola. (Assume that the origin is at the center of the beam.)
(b) How far from the center of the beam is the deflection equal to $\frac{1}{2}$ inch?

Key Concepts

Parabolic Functions

A parabolic function is a quadratic relationship that can be represented by an equation of the form y = ax^2 + bx + c. This form is particularly useful in modeling the deflection behavior of structures under load, where the shape of the deflected beam is parabolic, and the vertex represents a point of maximum or minimum deflection.

Beam Deflection

Beam deflection refers to the bending or displacement of a beam when subjected to external loads. Understanding beam deflection is vital in engineering as it helps assess whether a beam will perform safely under applied loads by ensuring that the deflections remain within acceptable limits.

Simply Supported Beams

A simply supported beam is a basic structural element that is supported at both ends, allowing it to rotate but not translate at those supports. This configuration provides clear boundary conditions, such as zero deflection at the supports, which simplify the analysis of the beam's behavior under load.

Symmetry in Structural Analysis

Symmetry plays a crucial role in simplifying structural problems. When a load is applied symmetrically, such as at the center of a beam, the deflection curve is symmetric about the midpoint. This allows for easier determination of the equation of the deflected shape by aligning the coordinate system with the center of symmetry.

Transcript

00:01 So i have a picture drawn for us in our figure there, and i'm making the x -axis and the y -axis be scaled very, very differently than what it really should be.

00:10 But in any case, we know that that width from here to here is supposed to be 64 feet, and then that height is only supposed to be one inch.

00:23 Now, we either have to have both measurements in feet or both measurements in inches.

00:28 And let's go in feet.

00:29 So i'm going to say that this is 112th of a foot.

00:33 So this point on the parabola would end up being the point 32 this way.

00:41 This would be negative 32 that way.

00:43 So this point is going to be 32 and then the y coordinate will be 112th.

00:48 And again, these are both in terms of feet.

00:51 So we can't have them mixed.

00:53 And i'm going to write the parabola in the good old fashion, y equals a x squared.

00:58 I looked at the text and the text.

01:00 Has something about 4p here, but we don't need to worry about the focus.