29.1.4 - Design of a Statically Indeterminate Arch
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Interactive Audio Lesson
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Introduction to Statically Indeterminate Structures
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Today's lesson focuses on statically indeterminate structures, particularly the two-hinged arch. Can anyone tell me what a statically indeterminate structure is?
I think it's a structure where the supports and connections provide more reactions than necessary to maintain equilibrium.
Exactly right! Because of this extra support, such structures require more complex calculations. What do we know about arches and their design?
Arches are curved structures that can effectively carry loads through compression.
Correct! Now, remember the acronym 'ARCH' for how we analyze them: A - Assumptions, R - Reactions, C - Calculations, H - Hinge locations. Let’s dive into designing a two-hinged arch.
Loading Conditions
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In the design of our arch, we mentioned specific loading conditions. Can anyone recap the types of loads that need to be considered?
The roof itself adds a dead load, and then we also have a snow load.
Right! The total load becomes the combination of the roof load and snow load, which is calculated at 65 lb/ft². Now, how do we find the concentrated load on our segments?
We can use the length of the segments multiplied by the total loading.
Exactly! Each segment applies a concentrated load of 24.565 k, which is crucial in our calculations.
Calculating Forces in the Arch
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So, we've distributed our loads. Now let’s look at how we calculate the forces. What equations or methods should we use?
I remember we need to create shear and moment diagrams for the arch.
Absolutely! We can evaluate the shear and moments at segments across the arch, which provides insights into axial thrust and bending moments. How do we represent these computations?
We can tabulate the results for each segment, summarizing the shear and moment values.
Perfect! This tabulation helps verify if each segment of our arch rib meets the structural requirements. Don't forget to check the adequacy of our sections!
Finalizing the Arch Design
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As we finalize our design, what are the critical aspects we need to check for adequacy?
We must ensure each segment is adequate for the loads it carries and investigate secondary stresses due to deflections.
That's correct! These checks are vital to ensure the safety and performance of our arch under service conditions. Remember, adequate design leads to better resilience against unforeseen loads!
What do we do if a section is found inadequate?
We would need to adjust either the material or the geometry until all sections are satisfactory. Remember to document these changes!
Introduction & Overview
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Quick Overview
Standard
The section describes how to design a two-hinged, solid welded-steel arch rib for a hangar, detailing crucial parameters such as span, loading conditions, and the market calculations required for effective structural design. It emphasizes understanding how to compute emergent forces and moments along the arch's rib.
Detailed
Design of a Statically Indeterminate Arch
This section focuses on the design of a two-hinged arch rib made from solid welded steel, specifically intended for use as a hangar. Key features of the design include:
- Span and Rise: The arch spans 200 ft with a rise of 35 ft and a spacing of ribs set at 35 ft.
- Loading: The roof and purlins exert a uniform load of 25 lb/ft², combined with an additional snow load of 40 lb/ft², leading to a total load of 65 lb/ft² across the roof surface.
- Arch Geometry: The rib's center line follows a circular arc with a calculated radius of 160.357 ft and an arc length of approximately 107.984 ft, split into ten segments for analysis.
- Load Calculation: Each segment receives a concentrated load due to the purlins, computed to be 24.565 k.
- Structural Analysis: The section provides detailed calculations on shear and moments at various segments of the arch, leading to a thorough investigation of axial forces and the adequacy of structural cross-sections, ensuring each segment can sustain the required loads.
- Final Validation: Comprehensive checks of the design are performed to confirm that all sections of the arch rib meet the requisite safety and stability criteria before finalizing the overall design.
Audio Book
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Introduction to Arch Design
Chapter 1 of 6
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Chapter Content
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.
Detailed Explanation
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.
Examples & Analogies
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.
Shape of the Arch
Chapter 2 of 6
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Chapter Content
- 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.
Detailed Explanation
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.
Examples & Analogies
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.
Segmenting the Arch
Chapter 3 of 6
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Chapter Content
- 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.
Detailed Explanation
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.
Examples & Analogies
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.
Calculating Concentrated Loads
Chapter 4 of 6
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Chapter Content
- 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.
Detailed Explanation
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.
Examples & Analogies
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.
Determining Segment Coordinates
Chapter 5 of 6
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Chapter Content
- 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.
Detailed Explanation
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.
Examples & Analogies
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.
Assessing Sectional Adequacy
Chapter 6 of 6
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Chapter Content
- 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.
Detailed Explanation
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.
Examples & Analogies
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.
Key Concepts
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Arch Geometry: The spanned length and rise of the arch are critical to determining the forces, which are influenced by the curvature.
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Loading Calculations: Accurate assessment of loads, including dead weights and snow, is essential for structural design.
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Moment Redistribution: Understanding how moments are shared across segments ensures each section is designed adequately.
Examples & Applications
Designing a two-hinged arch for a hangar with specific loading conditions and determining the concentrated loads at each arch segment.
Calculating shear forces and moments for various segments to validate the adequacy of structural components.
Memory Aids
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Rhymes
Arch, arch, structural art; carrying loads, it plays its part.
Stories
Imagine an arch standing proudly over a river, with two hinges at its base dancing with the load of the world above it, feeling strong and united.
Memory Tools
To remember the steps in arch design: S-P-L-M (Span, Purlin loads, Loads calculation, Moments at segments).
Acronyms
LOAD
- Live load
- Other induced loads
- Arch geometry
- Design checks.
Flash Cards
Glossary
- Twohinged Arch
An arch structure supported at two points (hinges) that allows for rotation, thus the moments are not restrained at the supports.
- Load
The forces or weight applied to a structure, frequently categorized as dead load (permanent) and live load (temporary, e.g., snow).
- Moment of Inertia
A measure of an object's resistance to changes in its rotation, varying along the arch's length in this context.
- Shear Force
The internal force that acts parallel to the cross section of a member, crucial for analyzing structural integrity.
- Axial Thrust
The longitudinal force in the arch due to bending moments and external loads that must be accounted for in design.
Reference links
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