Exercises: Strong, stable and rigid structures

A bookcase can fail in three ways: the shelf breaks, the entire bookcase topples forward, or the structure twists into a rhombus shape. Match each failure with the condition that is broken — strength, stability, or rigidity — and explain in your own words what each of the three conditions means.

The figure shows a bridge structure made of joined triangles, with a load in the centre.

Explain why this structure is made of triangles and not rectangles: what happens to the bars of an articulated rectangle when pushed from the side, and why a triangle, on the other hand, cannot deform without breaking a bar.

There are four basic types of structure: frame (beams and columns), triangulated (truss/triangular), suspension (tensioned cables), and shell/laminar (a curved sheet that supports itself). Classify these examples: the skeleton of a building under construction, a harbour crane, a suspension bridge, and a car bonnet.

Build a hinged square using four toothpicks and sticker dots (gomets), and a triangle using three toothpicks. Push each one from the side and answer:

1. a) Describe what happens: which one deforms without breaking and which one does not?
2. b) What piece would you add to the square to make it rigid?

The centre of gravity is the point where the weight of an object is "concentrated": if you hold it right at that point, it stays balanced. Explain the idea with a simple example (balancing a ruler on a finger) and reason why that point is higher in a full glass than in the same empty glass.

An object is more stable the wider its base is and the lower its centre of gravity is. Apply this rule by ordering from least to most stable: a pencil standing upright on the table, a full water bottle, and an Egyptian pyramid, and explain in each case the role played by the base and the height of the weight.

Classify these structures into frame, triangulated, suspension, or shell/laminar, justifying your answer: a bookcase made of stacked cubes, the structure of a Ferris wheel, a washing line with taut ropes between two walls, and a plastic bottle supporting the weight of a book on top.

Look at the next construction crane you see (or find a photo): you will see that its jib and tower are full of triangles. Explain why those triangles are there, what would happen if the jib were a simple rectangular frame of bars, and why this solution, as well as being rigid, uses little material.

The figure shows two objects with their centre of gravity marked and the vertical line dropping from it: one with a wide base and the other narrow and tilted.

Explain, looking at where the vertical line from the centre of gravity falls relative to the base of support in each object, why the first stays upright and the second topples over, and state the general rule of stability in your own words.

The Leaning Tower of Pisa does not fall because, even though it is tilted, the vertical line from its centre of gravity still falls within its base. Explain this rule in your own words and apply it: reason how far a tower could tilt before toppling over and why counterweights or widened foundations help.

Match each situation with the idea of stability that explains it: (a) a bicycle supported on two wheels falls over with a light gust, but does not with a kickstand, (b) floor lamps have a heavy, wide base, and (c) a lorry loaded very high is more dangerous around bends than one loaded low. Explain what changes in each case: the base of support or the height of the centre of gravity.

An empty plastic bottle can support a person's weight without being crushed, even though the plastic sheet is extremely thin.

1. a) Explain what type of structure it belongs to based on its morphology.
2. b) Justify why its curved and closed shape gives it a resistance that the same flat sheet would not have.
3. c) Give another example of this type of structure that you can find at home.

In a suspension bridge, the deck hangs from tensioned cables between two towers. Analyse how its elements work: what stress the cables support, what stress the towers support, and where all the loads end up in the end. Explain why this type of structure makes it possible to span huge distances without intermediate supports.

The frame of a building is a framework of vertical columns and horizontal beams. Explain what loads the beams support (the floors and whatever is on top) and where they pass them to, what stress the columns support, and why, if you remove a column, the beam that previously rested on it works much worse.

A cube shelving unit without a back panel shifts to one side, taking on a rhomboidal shape when pushed sideways. However, if a back panel is screwed on (or two cross braces are placed behind it), the structure does not deform.

Based on this situation, answer the following questions:

1. a) Of the three basic conditions a structure must satisfy (strength, stability, and rigidity), which one failed in the initial shelving unit? Justify your answer.
2. b) Why does the back panel solve this problem, despite being a very thin material?
3. c) How is this solution related to the triangulation technique used in structures?

Choose the most suitable type of structure (frame, triangulated, suspension, or shell/laminar) for each case and justify it: (a) a bus stop shelter, (b) a $60\ \text{m}$ footbridge over a river with no supports in the middle, (c) a tent with tensioned poles, and (d) the village water tank.

The figure shows four schematic structural typologies (from a to d):

1. A bridge is planned to span a 300 m wide gorge without intermediate supports, as well as the roof for a large, open-plan sports arena. Justify which of the four structures you would choose for each case and why you would rule out scheme a) in both situations.

2. In scheme b), a worker removes the interior diagonal bars to open up space for a window. Predict what will happen when the structure supports a horizontal load and justify which basic requirement (strength, stability, or rigidity) will be compromised.

3. In scheme d), justify why that curved shape withstands loads better than a flat plate made of the same material and thickness.

A tower crane is erected on a construction site to lift loads of up to 2,5 tonnes:

1. Analyze the mechanical stresses that develop in its main components when it holds a suspended load at the end of the jib: determine which one works in bending, which in compression, and which in tension, justifying each case based on how the element tends to deform.

2. A novice worker suggests removing the concrete counterweight at the end of the workday, arguing that «with no load to lift, removing that weight will prevent fatiguing the tower at night». Rebut his proposal technically by explaining the function of the counterweight and what would happen to the crane if it were removed.

3. Justify why the base is anchored to a large underground concrete footing and which basic structural requirements (strength, rigidity, or stability) this anchorage ensures against side winds.

Two identical lorries carry the same weight, but one has it stacked very high and the other distributed across the bottom of the trailer. Using the centre of gravity, explain which is more dangerous around bends and why, and deduce the loading rule that lorries (and hiking backpacks) should follow.

A 20 m long pedestrian footbridge must be built to cross a river, and two options are compared: a solid concrete beam or a steel lattice (triangulated truss):

1. When the truss bends under the weight of pedestrians, deduce with reasoning which chord of bars (top or bottom) works in tension and which in compression.

2. Knowing that slender, long elements can undergo buckling under compression, predict in which part of the truss this danger will exist and what design solution you would propose to prevent it without adding excessive weight.

3. Justify why the truss achieves strength and rigidity comparable to those of the solid beam while consuming much less material, explaining which areas inside a solid beam barely withstand any stresses.

In an office, a vertical four-drawer filing cabinet rests perfectly on the floor. However, upon opening both upper drawers filled with folders at the same time, the filing cabinet suddenly tips forward:

1. Explain the technical cause of the tipping by analyzing the position of the center of gravity relative to the base of support, and identify which basic structural requirement has failed.

2. Design three independent technical solutions to prevent this accident: one that modifies the base or the mass distribution of the furniture itself, another based on a safety mechanism in the drawer slides, and another using external anchoring. Justify the physical or structural principle of each proposal.

3.

Analyze the mechanical behavior of a folding scissor chair (with «X» legs) compared to a conventional fixed four-legged chair:

1. An «X» frame hinged at its crossing point is not inherently undeformable. Explain what element the design adds so that the chair becomes undeformable when unfolded, and what basic geometric shape is created to ensure its rigidity.

2. When a person sits down, deduce what mechanical stress (tension or compression) the strap or tie connecting the lower part of the legs is subjected to, and justify your deduction by describing where the supports would tend to shift if that element broke.

3. If someone sitting down rocks and leans sharply to one side, and assuming that, as is usual, the folding chair is laterally narrower than the fixed one and both have their seats at a similar height, predict which of the two chairs will tip over more easily and justify your prediction based on its base of support and its center of gravity.

Classroom engineering challenge: using $20$ sheets of paper and a metre of sticky tape, your team must build a $1\ \text{m}$ tower that can support a book on top. Design your proposal: overall shape of the tower and its base, where you place triangles, how you ensure the centre of gravity stays low, and what tests you would carry out before the final test. Justify each decision using the topic's vocabulary.

During a storm with gusty winds, the behavior of a vertical metal streetlight and a young tree on the same sidewalk is observed:

1. Deduce what main mechanical stress the wind subjects both elements to, and identify with reasoning at what point of the streetlight the highest risk of failure is concentrated.

2. Compare the two mechanical solutions they represent: absolute rigidity versus elastic flexibility. Predict what mechanical failure mode each will undergo if the storm exceeds its strength limit.

3. When designing skyscrapers to withstand hurricane-force winds and earthquakes, engineers mimic the tree's strategy rather than the streetlight's. Justify why they are designed to sway slightly instead of being built completely rigid.

Topic challenge. Is the bookshelf in your room safe? You are going to audit and optimize a real structure in your home.

1. Data collection: Choose a bookshelf shelf or a table. Measure its length with a tape measure (e.g. $0,8\text{ m}$) and weigh the objects it supports using a kitchen or bathroom scale to calculate the load in newtons ($P = m \cdot 9,8\text{ N/kg}$).

2. Structural analysis: Identify which elements undergo bending, compression, or tension. Evaluate stability (position of the center of gravity and base of support) and check its rigidity: does it deform or wobble if you apply a slight horizontal force?

3. Proposal and redesign: Write a short report using LibreOffice (or model the solution in Tinkercad) proposing a technical improvement to increase the strength and rigidity of the furniture piece against buckling or tipping over, applying triangulation, brackets, or tie rods.

Assessment criteria:

• Rigorous collection of real data and correct calculation of the load in newtons.

• Accurate identification of mechanical stresses and analysis of rigidity and stability.

• Technical feasibility, creativity, and justification of the reinforcement proposal in the report.