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Chapter 26: Columns Back To Chapter

26.1: Stability of structures

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Mechanical Engineering

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Stability of structures

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26.2: Euler's Formula for Pin-Ended Columns

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TRANSCRIPT

Consider two rods connected at point O by a pin and a torsional spring of constant k. The ends of both the rods are pin-connected.

If both the rods are perfectly aligned and two loads are along the same line of action, the system stays in equilibrium.

Now, if point O is moved slightly, making a small angle with respect to the vertical position, then two couples act on the rods.

The first one is due to the applied load acting at point O, causing the rod to move away from the vertical.

The second couple is due to the torsional spring constant, which attempts to bring the rod back towards the vertical position.

If the magnitude of the moments of these two couples is the same, a critical load expression is formulated.

The system's stability depends on the comparison between the applied load and the critical load.

If the applied load surpasses the critical load, the system becomes unstable. Conversely, if the applied load is less than the critical load, then the system is stable.

26.1: Stability of structures

In mechanical engineering, the stability of systems under various forces is critical for designing durable and efficient structures. One fundamental way to explore these concepts is by analyzing systems like two rods connected at a pivot point, O, with a torsional spring of spring constant k at the pivot point. This system is similar in appearance to a scissor jack used to change tires on a car. In this case, the arms of the linkage (equivalent to the rods in this system) are entirely vertical, corresponding to the case where the vehicle has been lifted as far as possible.

These rods are initially positioned vertically, and they are prevented from pivoting by the spring's action. If two external loads F and F', of equal magnitude and opposite direction, are applied to the system in such a way that they share the same line of action along the length of the rods, then the system remains in equilibrium. This alignment ensures that there is no net moment or torque acting to displace the system from its equilibrium state.

However, if the pivot point, O, is moved slightly sideways, this movement results in a small angular deviation of the rods from the vertical. This causes the ends of each rod to rotate relative to their pivot points, introducing couples into the system. The first couple results from the reaction support at point O. This force, now acting at an angle, attempts to further displace the rod from its initial vertical orientation, moving it away from equilibrium. The second couple, resulting from the torsional spring's resistance, exerts a restoring force to return the rod to its original vertical position. The system reaches a point of critical load F_cr when these two moments are equal. If the applied load exceeds the critical load, the system becomes unstable; if it is less than the critical load, the system remains stable.

This principle is universally applicable in structural systems and is extensively used in mechanical engineering designs, including the stabilization of buildings, vehicles, and machinery.

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