Exercises: Forces, loads, and stresses

Explain in your own words the difference between a dead load (static load) and a live load (variable load), and classify these loads on a bridge: the weight of the deck itself, passing cars, the weight of the fixed streetlights, winter snow, and storm winds.

The figure shows five members subjected to the five basic stresses, labelled with the letters a to e, with arrows representing the forces.

Match each letter with its stress — tension, compression, bending, torsion, or shear — and justify your answer by looking at the direction of the arrows: whether they stretch, squash, bend, twist, or shear the member.

Match each component with the stress it mainly withstands: (a) a zip line cable, (b) a porch pillar, (c) the shaft of a screwdriver when tightening, (d) a bookshelf full of books, and (e) a sheet of paper when cut with scissors. Explain your choice in two of the five cases.

A shelf loaded with books gradually sags in the middle. Explain what stress it is experiencing, why it sags downwards in the middle and not at the ends, and what two simple changes you would make so that it sags less (without emptying the shelf).

When you stretch elastic bands with your hands, the band stretches and returns to its original shape. Explain what stress the rubber band undergoes while you pull it, what happens if you stretch it too much, and why rubber bands are used for that exact purpose (think about what material you would choose for a zip line cable).

When wringing out a mop or cloth, you twist it with both hands rotating in opposite directions. State what stress you are applying, describe precisely how the forces act, and name two objects or components in your home that usually work under that same stress.

Scissors cut paper because their two blades push very close together in opposite directions, one upwards and the other downwards. Explain what stress the paper experiences at that point and why scissors cut much better near the pivot screw than at the tip (think about the force applied by your hand).

The self-weight of a roof is a load that is always present, but snow only appears a few days a year. Explain why these two types of loads are treated differently when designing, and reason what would happen if a roof were designed considering only its dead weight, without accounting for snow.

The figure shows a bridge with three components labelled with the letters a, b, and c: a pillar, one of the cables supporting the deck, and the deck itself where a lorry travels.

State which stress each labelled component mainly withstands (tension, compression, or bending), reasoning what the loads do to each one: which is stretched, which is squashed, and which is bent.

A coat hanger carrying a heavy coat seems simple, but its parts work differently: the wire hook is stretched and the horizontal bar bends slightly. Analyse what stress each part of the hanger withstands and explain why the same plastic hanger holds the coat but breaks if you also hang a school bag from the bar.

The drive shaft of a blender transmits the motor's rotation to the blades. Explain what stress that shaft experiences while blending, what would happen if it were made of a soft material like rubber, and name two other drive shafts in household appliances that work in the same way.

On a table, the legs withstand the weight of everything on top without bending, but the tabletop sags if something very heavy is placed in the centre. Explain what stress each part experiences and why the shape of each element (a vertical leg versus a horizontal tabletop) makes one resist its stress better than the other.

A construction crane has these parts: the cable lifting the bag of cement, the protruding horizontal jib, the concrete counterweight, and the vertical tower. State which stress or stresses each part withstands (tension, compression, or bending) and reason what the counterweight is for by looking at where the bag 'pulls'.

You bend a plastic ruler and it returns to its shape; you bend it further and it remains deformed; you bend it even further and it breaks. Explain what happens to materials at these three stages (elastic deformation, plastic deformation, and fracture) and why structures are designed to operate strictly within the first stage, with a large margin of safety.

A sports hall must withstand its own weight, but also variable loads: snow on the roof, wind against the walls, and thousands of spectators on a match day. Explain why the designer must consider the worst imaginable day rather than an ordinary day, and what could happen if loads were estimated 'by eye' instead of using safety margins.

A stone bench and a plastic chair support the same person sitting down, but they break in different places if overloaded. Compare the stresses experienced by each (legs, seat) and reason why stone is good under compression (squashed) but not under bending, whereas the chair's plastic withstands seat bending well.

The figure shows a playground swing set with three labeled elements: the seat chains (a), the horizontal top bar (b), and the frame legs (c), with a person seated.

During a technical inspection, it is discovered that the horizontal bar (b) has sagged permanently downward. A maintenance worker proposes two possible solutions:

1st. Replace the chains (a) with rigid steel bars of the same thickness to "help push the bar upward."

2nd. Move the lower legs (c) inward toward the center so that they support closer to the seat.

Answer with reasoned explanations:

1. a) Following the path of the weight from the seated person to the ground, justify what stress each labeled element supports (a, b, and c).
2. b) Refute the 1st proposal technically: predict what will actually happen if rigid bars are used and explain why it does not correct the bending of the bar (b).
3. c) Analyze what would happen to the stability and tipping risk of the swing set if the 2nd proposal were implemented.
4. d) Design a valid structural modification to the frame to reduce the bending of bar (b) without changing its material or impeding swinging.

On a recreational sailboat, a student claims that the tensioned steel cables (shrouds and stays) that secure the mast to the deck are an unnecessary expense, arguing that the mast is a very thick aluminum tube capable of resisting tons of compression without being crushed.

1. a) Identify what main stress appears in the mast when the wind pushes strongly against the sail from the side, and what stress is absorbed by the cables located on the side opposite the wind.
2. b) Refute the student's argument: predict what will happen to the mast without cables if it faces strong gusts of wind, justifying your answer based on the combined effects of bending and buckling.
3. c) Explain the technical principle of the assembly: how is it possible that flexible cables, which only withstand tension, can prevent a rigid element like the mast from failing under lateral loads?

A team of students is building a load-bearing structure using rolled steel I-beams (with an "I"-shaped cross-section). When resting one of the beams on two pillars, a group member proposes placing it on its side (with a wide "H"-shaped cross-section instead of a vertical "I"), arguing that "by having a wider bearing surface at the base, it will support more vertical load without deforming."

1. a) Explain in detail what mechanical stresses (tension and compression) develop on the top face, on the bottom face, and along the centerline (neutral axis) of a beam when it supports a centered vertical load.
2. b) Refute the student's proposal: prove in which of the two positions (vertical "I" or lying flat "H") the beam offers greater resistance to bending, justifying it based on the distance of the material relative to the centerline.
3. c) If the structure had to simultaneously withstand heavy vertical loads and strong lateral wind gusts, predict what limitation the I-beam presents compared to a hollow square tube of the same weight.

A shop needs to install a heavy sign (40 kg) that cantilevers over the sidewalk. Two mounting systems are compared:

• System 1: A horizontal bar anchored to the wall, with an angled steel cable (tie rod/guy wire) connecting the end of the sign to a point on the wall located 1,5 m higher.

• System 2: The same horizontal bar embedded into the wall, but removing the angled cable so that the facade looks cleaner visually.

1. a) Analyze System 1: determine with justification which mechanical stress the angled cable experiences and which one the horizontal bar experiences while supporting the sign.
2. b) Predict what radical change in stresses the horizontal bar will undergo if System 2 is adopted. Justify at what exact point on the bar the greatest risk of failure will exist.
3. c) In System 1, if by mistake the cable is anchored to the wall only 5 cm above the bar (remaining almost horizontal), predict what will happen to the tension that the cable must withstand and to the wall anchor in order to support the same sign.

A particleboard shelf 1,2 m long, supported only at its two ends, has become permanently deformed, sagging into a pronounced downward curve after supporting several toolboxes piled in the center for months.

1. a) Diagnose the failure: name the responsible stress and describe what has happened microscopically to the wood fibers on the upper surface and on the lower surface in the central area of the board.
2. b) Analyze the effect of load distribution: if those same boxes had been divided into two groups placed close to the end supports, would the same deformation have occurred? Justify your answer mechanically.
3. c) Design two different technical solutions so that the shelf can support the heavy load without bending: one by modifying the shelf's geometry or reinforcements without changing the material, and another by altering the support structure. Justify the mechanical operation of each one.

A student claims: "Paper is a material that cannot be used to build load-bearing structures because it lacks rigidity." To test their statement, you perform two tests with identical 80 g/m² sheets:

• Test 1: You take a flat vertical sheet, hold it by the top edge, and pull firmly downward from the bottom edge with a wide clip. Then, you try to make that same flat sheet stand upright vertically on the table and push downward from the top edge.

• Test 2: You place a flat horizontal sheet across two blocks spaced 15 cm apart; the sheet sags under its own weight. You fold that same sheet into continuous zigzag folds (accordion shape) and place it across the blocks again: now it supports several coins and a pencil sharpener without sagging.

1. a) Justify what stress the paper successfully resists in Test 1 and which one causes its instant failure when pushed, explaining why this occurs.
2. b) In Test 2, explain technically why zigzag folding allows the sheet to overcome bending without having added more material or changed the paper.
3. c) Transfer this principle to technical applications: mention a real packaging product or construction element that takes advantage of this same geometric principle to support loads, and explain how it works.

An urban furniture company presents a prototype of a public children's swing. The design consists of two straight vertical wooden posts embedded into the ground without lateral triangulation, a cylindrical horizontal PVC top bar attached between them, and two seats suspended by nylon ropes.

1. a) As the safety officer, identify and justify three serious structural design and material selection errors in the prototype, indicating which mechanical stress will cause failure in each defective element during normal use.
2. b) Redesign the frame: define the geometry that the lateral supports must have to prevent deformation and overturning during back-and-forth swinging, indicating how each bar works.
3. c) Justify why the foundation footings anchoring the legs to the ground must be designed to withstand alternating tensile and compressive stresses when users swing forcefully.

To cross a 6 m wide canal, two options using wooden planks of identical cross-section are compared:

• Option A: A single continuous 6 m plank supported only at both banks.

• Option B: Two 3 m planks supported on the banks and on a rigid intermediate pier erected in the center of the channel.

1. a) Identify the main stress experienced by the wood when a person crosses the plank and explain how tension and compression are distributed across the thickness of the plank.
2. b) In bending, the maximum deflection at the center of a beam increases proportionally to the cube of the clear span between supports ($L^3$). If a 60 kg person crossing in Option A causes the plank to sag 8 cm, calculate and justify with sound reasoning the maximum sag in Option B when that same person passes through the center of one of the two 3 m spans.
3. c) If the canal bed were made of soft mud and did not allow founding a central pier, propose and schematically sketch two different structural solutions (one using top triangulation and another using stay cables or arches) that prevent excessive bending of the plank without relying on supports in the water.