The Chicago North Shore & Milwabee Railroad was an electric...

The Chicago North Shore \& Milwabee Railroad was an electric railway running between the cities in its corporate title. It had passenger cars as shorwn in the figure, which weighed $104.4$ kip, had 32-ft, 8-in truck centers, 7-ft-wheelbase tracks, and a coupled length of $55 \mathrm{ft}, 3 \frac{1}{4}$ in. Consider the case of a single car on a 100 -ft-long, simply supported deck plate ginder bridge. (a) What was the largest bending moment in the bridge? (b) Where on the bridge was the moment located? (c) What was the position of the car on the bridge? (d) Under which axle is the bending moment?

The Chicago North Shore \& Milwabee Railroad was an electric railway running between the cities in its corporate title. It had passenger cars as shorwn in the figure, which weighed $104.4$ kip, had 32-ft, 8-in truck centers, 7-ft-wheelbase tracks, and a coupled length of $55 \mathrm{ft}, 3 \frac{1}{4}$ in. Consider the case of a single car on a 100 -ft-long, simply supported deck plate ginder bridge.
(a) What was the largest bending moment in the bridge?
(b) Where on the bridge was the moment located?
(c) What was the position of the car on the bridge?
(d) Under which axle is the bending moment?

Key Concepts

Simply Supported Bridge

A simply supported bridge is one that rests on supports at both ends with no moment-resisting connections, meaning that the reaction forces at the supports are entirely vertical and any bending moment is developed solely by the applied loads. This concept is essential in understanding how a load, such as a vehicle, induces bending within the structure and how the maximum bending moment is computed especially when the load is applied at various positions along the span.

Bending Moment

The bending moment refers to the internal moment that causes a beam or structural element to bend. It is a measure of the force distribution along the length of a bridge and is calculated by summing moments about a given cross-section. In bridge design, determining the largest bending moment is critical for ensuring that the structure can safely withstand the forces induced by applied loads.

Influence Lines

Influence lines are a tool used in structural engineering to determine how moving loads affect responses such as bending moment, shear force, or reactions at specific locations along a structure. By using influence lines, engineers can identify the position of a moving load, such as a train car with multiple axles, that will produce the maximum moment, thereby ensuring the design is adequate for worst-case loading scenarios.

Vehicle Axle Configuration and Load Positioning

The configuration of a vehicle’s axles, including the spacing and the distribution of weight, plays a significant role in how loads are transferred to a bridge. Understanding the precise axle configuration and the resulting load positions is crucial for determining which axle induces the maximum bending moment on the bridge, as different positions can lead to different moment distributions along the span.

Transcript

00:01 And as a question is required to find the distance d from a to c and the width, which is w.

00:08 So i'll use the loads of signs to do so.

00:12 First of all, what do we have is the two sides with an angle that outside, it's an exterior angle to this triangle.

00:24 So i would like to find this angle here by using the complementary angle.

00:30 Because you know here it's a right triangle or a right angle.

00:38 So if this here is 25, then this will be 90 minus 25 as according to the complementary angle rule.

00:49 So a, admit this by the angle a, is 90 minus 25 equals to 65.

01:01 Then, since i know this angle, then i'll use it to be sine a over the side opposite equals to i want the angle c in order to be able next to find the angle b and hence i can find d.

01:26 So the steps like this will be sine a over 230 equal to sine c over 250.

01:38 From this i can find c.

01:40 Then i'll be able to find b since two angles of this triangle are known.

01:46 Then by using loads of signs, i'll use this angle to find d.

01:53 So, as i said, i'll use sign.

01:58 65 over 230 equals to sine c over 250.

02:12 Sign c is 0 .985 and hence c will be sine inverse 0 .985 or arc sine 0 .985.

02:32 And this gives the angle to be 80 .1.

02:38 So i got angle c, then angle b will be 180 because the summation of the interior angles of entry angle is 180.

02:49 So it would be 180 minus the angle c minus the angle a.

02:55 So 180...