Establish ratiosbetween each end condition (taking the pinned-pinnedcondition as 1)

To establish ratios between each end condition, we will compare different support conditions of beams. The pinned-pinned condition will serve as our base condition with a ratio of 1. Here we will briefly describe common end conditions and derive their ratios.

1. Break Down the Problem

We need to consider the following common end conditions for beams:

• Pinned-Pinned
• Fixed-Fixed
• Fixed-Pinned
• Free-Free

2. Relevant Concepts

The stiffness and behavior of beams change based on their end conditions. We typically express the effective stiffness or deflection ratios relative to the pinned-pinned condition.

3. Establishing the Ratios

• Pinned-Pinned (PP): Ratio = 1 (as it is the reference condition)

• Fixed-Fixed (FF): This condition provides maximum restraint, leading to a higher stiffness. The theoretical ratio is generally considered as 4.

  $\text{Ratio (FF)} = 4$

• Fixed-Pinned (FP): The fixed end provides some restraint, while the pinned end does not. The common ratio for this case is approximately 2.

  $\text{Ratio (FP)} = 2$

• Free-Free (FF): In this case, the beam is not restrained at any end; the ratio is considered to be very low, typically 0.

  $\text{Ratio (FF)} = 0$

4. Verify and Summarize

Now that we have established the ratios, we can summarize them as follows:

• Pinned-Pinned (PP): 1
• Fixed-Fixed (FF): 4
• Fixed-Pinned (FP): 2
• Free-Free (FF): 0

Final Answer

The ratios of bending stiffness relative to the pinned-pinned condition are:

• Pinned-Pinned: 1
• Fixed-Fixed: 4
• Fixed-Pinned: 2
• Free-Free: 0