29.1.4. Design of a Statically Indeterminate Arch

Design a two-hinged, solid welded-steel arch rib for a hangar. The moment of inertia of the rib is to vary as necessary. The span, center to center of hinges, is to be 200 ft. Ribs are to be placed 35 ft center to center, with a rise of 35 ft. Roof deck, purlins and rib will be assumed to weight 25 lb/ft2 on roof surface, and snow will be assumed at 40 lb/ft2 of this surface. Twenty purlins will be equally spaced around the rib.

In this portion, we are tasked with designing a two-hinged arch for a hangar. The design requires consideration of the total span and the placement of ribs. The arch needs to support roof decking and purlins, which contribute to the total load it carries. By specifying the materials and dimensions, we establish a framework for analyzing the arch's structural integrity.

Imagine building a playground swing set that needs to support the weight of children playing on it. Just like the swing set's design must accommodate the weight of the kids and resist wind pressure, the arch's design must consider the roof and any additional loads like snow.

1. The center line of the rib will be taken as the segment of a circle. By computation the radius of this circle is found to be 160.357 ft, and the length of the arc AB to be 107.984 ft.

The design approach starts with defining the arch's geometric shape, which is modeled as a circular arc. The radius of the circle and the length of the arc are calculated to ensure that the arch meets the specified dimensions and looks aesthetically pleasing while also functioning effectively. Circular shapes are favored in arches for their structural efficiency.

Think of a rainbow forming an arc in the sky. Just like the curvature of a rainbow, the arch must be shaped for both visual appeal and strength, allowing it to distribute weight effectively across its structure.

2. For the analysis the arc AB will be considered to be divided into ten segments, each with a length of 10.798 ft. Thus a concentrated load is applied to the rib by the purlins framing at the center of each segment.

The arch is divided into ten segments for structural analysis. This segmentation simplifies the load calculations, allowing engineers to apply concentrated loads effectively at each segment's center. By focusing on smaller segments, we can analyze how loads affect the structure incrementally, which helps in ensuring each part can handle the stress.

Picture slicing a cake into equal pieces. Each slice corresponds to a segment of the arch, and just as you would want each slice to have an even distribution of frosting and toppings, each segment of the arch must be able to bear its share of the load without collapsing.

3. Since the total dead and snow load is 65 lb/ft2 of roof surface, the value of each concentrated force will be P = 10.798 * 35 * 65 = 24.565 k.

In this calculation step, we determine the force acting on each segment by considering the total load due to the roof and snow. The computation involves multiplying the area each segment supports by the total load per square foot, resulting in the concentrated force for proper structural assessment.

Imagine a heavy backpack in which each pocket holds a fraction of the total weight. Calculating how much weight each pocket must support is like finding the concentrated force on each segment of the arch from the roof.

4. The computations necessary to evaluate the coordinates of the centers of the various segments, referred to the hinge at A, are shown in Table 29.1. Also shown are the values of Δx, the horizontal projection of the distance between the centers of the several segments.

This step involves calculating the exact coordinates for the centers of each segment based on the arch's geometry. By evaluating the horizontal projection of distances, we can better understand how forces will redistribute across the arch and ensure that it performs as expected under load.

Consider navigating a new city. By determining your locations on a map (similar to segment coordinates), you can figure out the best route and how far apart each stop is, ensuring you maintain a clear path to your destination.

21. The sections previously designed at the centers of the segments are checked for adequacy in Table 29.8. From this table it appears that all sections of the rib are satisfactory. This cannot be definitely concluded, however, until the secondary stresses caused by the deflection of the rib are investigated.

After designing, an essential part of the process is verifying the adequacy of the rib's sections to support the calculated loads. The results indicate that the design meets requirements, but further analysis is necessary to consider any secondary stresses produced by the arch's deflection under load, which could affect its overall integrity.

Just like a doctor might conduct a final checkup after treatment to ensure a patient is healthy, engineers must ensure their designs are sound before final approval, checking for any hidden issues that could arise from stress on the materials.

--