10.4. Examples

Example 10-1: Steel Building Frame Analysis, (White et al. 1976)

A small, mass-produced industrial building, Fig. 10.3, is to be framed in structural steel with a typical cross section as shown below. The engineer is considering three different designs for the frame: (a) for poor or unknown soil conditions, the foundations for the frame may not be able to develop any dependable horizontal forces at its bases. In this case the idealized base conditions are a hinge at one of the bases and a roller at the other; (b) for excellent...

In this example, a structural engineer is tasked with designing a steel frame for a small industrial building. The design must account for varying soil conditions that could impact the stability and performance of the building. Three design scenarios are proposed: one where the soil conditions are poor, necessitating special support (a hinge and a roller), and two others for better soil conditions which allow for different types of support. Each choice affects the structural analysis and load distribution, thereby impacting safety and functionality.

Imagine building a treehouse. If you're attaching it to a sturdy tree, you can use simple nails (like ideal conditions), but if the tree is weak or leaning, you might need ropes and hooks (like hinge and roller supports) to ensure it stays up safely.

3. In the following discussion, the contributions to displacements due to axial strain are denoted with a single prime (') and those due to curvature by a double prime ('').

4. Consider the axial strain first. A unit length of frame member shortens as a result of the temperature decrease from 85°F to 45°F at the middepth of the member. The strain is therefore (\delta T = (0.0000055)(40) = 0.00022

5. The effect of axial strain on the relative displacements needs little analysis. The horizontal member shortens by an amount (0.00022)(20) = 0.0044 ft. The shortening of the vertical members results in no relative displacement in the vertical direction 2. No rotation occurs.

This part of the example focuses on how changes in temperature affect the frame's dimensions. When the temperature drops, the materials contract, causing a length change known as axial strain. In this case, the linear shortening of a horizontal beam is calculated based on the temperature drop. The explanation clarifies that while the horizontal member shrinks, the vertical members do not change in height, meaning there's no vertical displacement or rotation, simplifying the analysis.

Think of a balloon filled with air. When it gets cold, the air inside cools down and takes up less space, causing the balloon to shrink a bit. If you have a long balloon shaped like a dog, it might not change in height, but its length decreases. So, while the dog balloon gets shorter, it stays upright without falling over.

7. The effect of curvature must also be considered. A frame element of length dx undergoes an angular strain as a result of the temperature gradient. The change in length at an extreme fiber is $\epsilon = \delta T dx = 0.0000055(25)dx = 0.000138dx$
8. with the resulting real rotation of the cross section $d\theta = \epsilon/0.5 = 0.000138dx/0.5 = 0.000276dx$ radians.

Here, the focus shifts to how curvature (bending) in the frame affects its overall displacement. As the temperature changes, not only does the frame shorten (as previously discussed), but it also bends slightly due to thermal expansion, affecting angles between members. The calculations show how temperature changes lead to angular strains along the length of the material, which is critical for understanding the entire structural behavior under variable conditions.

Picture a metal ruler left in the sun, which can heat up. As it warms, if it gets long enough in the heat, it may start to bow in the middle. Just as the bending of the ruler affects the angles it forms with other objects around it, the curvature in frame elements needs to be accounted for in structural analysis to ensure safety and usability.

10. A virtual force Q is applied in the direction of the desired displacement and the resulting moment diagram M determined. 11. The virtual work equation $Q D = \int M d\theta$ is used to obtain each of the desired displacements D.

This section explains the virtual work method, a powerful approach in structural analysis. When analyzing the structure, a hypothetical (or virtual) force is applied to determine how the displacement changes in response to external loads. By calculating the moments created by this force, engineers can use the virtual work equation to find actual displacements in the structure. This method allows for a systematic way to assess how structures respond under different loading conditions.

Imagine you have a stretchy rubber band. If you pull on one end, the other end moves as well. If you take a friend to help pull on the rubber band, you can visualize how far it stretches by measuring the length at various points. Here, applying a virtual force is like having multiple people pulling on different parts of the rubber band—that helps you understand how each part responds to the overall force.

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