Question

Project: The bending moment for a simply supported beam of length \(l\) carrying a distributed load \(w\) is \(M=\frac{1}{2} w l x-\frac{1}{2} w x^{2} .\) The bending moment curve is therefore a parabola. Plot the curve of \(M\) vs. \(x\) for a 10 -ft-long beam carrying a load of 1000 Ib/ft. Take 1 -ft intervals along the beam. Graphically locate (a) the points of zero bending moment and (b) the point of maximum bending moment.

Short answer

The zero bending moments are at 0 ft and 10 ft, and the maximum bending moment of 12500 lb-ft occurs at 5 ft (midspan) along the beam.

Step-by-step solution

Define the Bending Moment Equation

The bending moment equation for a simply supported beam subjected to a uniformly distributed load is given by \(M = \frac{1}{2} w l x - \frac{1}{2} w x^2\), where \(w\) is the distributed load per unit length, \(l\) is the length of the beam, and \(x\) is the distance from one end of the beam.

Substitute Known Values

Substitute the known values (beam length \(l = 10\) ft and distributed load \(w = 1000\) lb/ft) into the equation to get \(M = \frac{1}{2} \times 1000 \times 10 \times x - \frac{1}{2} \times 1000 \times x^2 = 5000x - 500x^2\).

Calculate Bending Moment at 1-ft Intervals

To plot the curve, calculate the bending moment \(M\) at 1-ft intervals along the beam (i.e., at \(x = 0, 1, 2, ..., 10\) ft).

Plot the Bending Moment Curve

Using the values obtained, plot the bending moment curve on a graph with \(x\) (distance along the beam) on the horizontal axis and \(M\) (bending moment) on the vertical axis. The curve will be a parabolic shape opening downward.

Locate Points of Zero Bending Moment

The points of zero bending moment will occur at the ends of the beam where \(x=0\) ft and \(x=l\) ft (i.e., at \(x=10\) ft for this beam). This is where the bending moment curve intersects the \(x\)-axis.

Determine the Point of Maximum Bending Moment

The maximum bending moment will occur at the vertex of the parabola. Since the equation is in standard form, the vertex can be found using the formula \(x_v = -\frac{b}{2a}\), where \(a = -500\) and \(b = 5000\) from the simplified bending moment equation. Thus, \(x_v = -\frac{5000}{2\times(-500)} = 5\) ft.

Calculate Maximum Bending Moment

Substitute \(x_v = 5\) ft back into the bending moment equation to find the maximum bending moment \(M_{max} = 5000 \times 5 - 500 \times 5^2 = 25000 - 12500 = 12500\) lb-ft.

Key concepts

Simply Supported Beam

Imagine you're holding a ruler on the tips of your fingers, one at each end – this resembles a simply supported beam. In structural engineering, a simply supported beam is one of the simplest forms of a beam, commonly used in bridges and buildings. It's supported at both ends but is free to rotate and has no resistance to warping. When you place a load on it, like books on the ruler, the beam will bend, creating a bending moment which is essentially the force the beam must resist to avoid collapsing.

The bending moment at any point along the beam is the result of the forces acting on it. It depends not only on the load but also on the distance from the supports. At the ends of a simply supported beam, the bending moment is zero because that's where it rests on the supports. Picture the ruler bending under the weight of the books – the highest bend, or the highest bending moment, is somewhere in the middle, not at the ends where your fingers are.

Uniformly Distributed Load

A uniformly distributed load (UDL) is a load spread evenly across a beam, much like evenly sprinkling sugar across a stick of toast. In our beam scenario, this could be akin to snow falling evenly across a beam bridge or a shelf holding a row of identical books. The importance of a UDL is in its predictability: it makes calculating stresses and bending moments simpler because the load is consistent along the entire length of the beam.

When you hear 'uniformly distributed load', think of it as balanced; there's no heavier concentration of weight in one spot than another. In the ruler analogy, instead of books, imagine a well-dispersed line of marbles causing the ruler to bend. This even distribution allows us to use formulas, like the one in our exercise, to calculate the exact bending moment at any position along the beam for scenarios like this, which are frequent in construction and engineering calculations.

Parabolic Bending Moment

The beauty of mathematics meets the physical world in the parabolic bending moment curve. Like a symmetrical bow when shooting an arrow, the parabolic curve describes how the bending moment changes along the length of a simply supported beam under a uniformly distributed load.

The parabola is not just a pretty shape; it signifies that at the ends of the beam, where there is no bending moment, and the greatest bending moment is right in the middle of the beam – this is the apex or the vertex of the parabola. In other words, the bending moment forms this 'u' shape because the beam tries harder to resist bending where it feels the most burden, which is generally in its center. So, plotting this curve is not just an academic exercise; it's crucial for engineers to understand where a beam will experience the most stress and design it to be strong enough in the right places to hold up to the load it's carrying.