Exercises: Types of loads and stresses

Define each of the five basic stresses (tension, compression, bending, torsion and shear) in one sentence and give an example of an everyday element subject primarily to each one.

Explain the difference between a dead load (or permanent load) and a live load (or variable load) on a structure, and classify these loads on a footbridge: the weight of the deck itself, people walking across, accumulated snow and the handrail.

The figure shows five members of a structure subjected to different stresses; four are labelled and the fifth is covered. Indicate which stress the member marked with a question mark supports, and justify your answer based on how the forces act.

The pillars of a porch support the roof. Indicate what dominant stress they experience, what would happen to them if the roof became loaded with snow, and why the central pillar of a large porch works harder than the outer ones.

Name three everyday objects whose main component works in torsion in normal use, and explain for each where the torque is applied and which cross-section resists it.

Explain what buckling is: why a long, slender rod bows laterally under compression long before squashing, and which two geometric characteristics (length and cross-sectional shape) most influence the buckling load.

Distinguish between a point load and a distributed load using examples on a beam: a person standing at its centre versus a row of books resting along its entire length. How is each represented in calculation diagrams?

Explain in your own words what bending moment is in a beam and why it is maximum at the centre of a simply supported beam loaded in the middle (and zero at the supports). Relate this to where a ruler breaks if you bend it while resting it on two books.

In a construction tower crane, identify the dominant stress in each member and justify it: the cable supporting the load, the vertical mast, the horizontal cantilever jib and the tie rods connecting the end of the jib to the top of the mast.

A garage shelving unit was sagging in the central shelf but not in the side ones. The owner "fixed" it by turning the central shelf upside down. Explain, using the concepts of bending and deflection, what stress the shelf was experiencing, why it always sagged in the same direction, and evaluate whether the fix is a durable solution or just a temporary patch.

Explain why scissors cut (shear stress), why a screwdriver transmits torsion, and why a bicycle chain works under tension. In each case, explain what would happen if the element were subjected to the opposite stress or one not intended.

Compare a simply supported beam with the same cantilever beam (fixed at one end) under the same load applied at the centre or at the free end: which deflects more, where the maximum moment is located in each case, and why balconies are calculated as cantilevers.

Explain how a stone arch works: into what stress it converts vertical loads, why masonry/stone materials (which have low tensile strength) are suitable for arches, and what role the lateral abutments play.

In a suspension bridge, the main cables and vertical hangers work almost exclusively in tension. Explain why it is advantageous for the main structural elements to work in tension (they utilise the entire cross-section and do not suffer from buckling) and which elements of the bridge do work in compression (towers, deck).

Analyse a dining table under different loads: the tabletop when someone sits on the edge, the legs under a central load, and when the loaded table is dragged. Indicate the dominant stress in each element and where reinforcement should be added.

Distinguish between a static load and a dynamic load (impact or repeated load): why a swing that supports a person sitting still might break if someone drops down onto the seat, and how safety factors account for this in design calculations.

During emergency braking with the front brake of a bicycle, the deceleration generates an inertial force on the centre of mass of the rider-bicycle system that almost completely unloads the rear wheel and overloads the front wheel.

1. a) Analyse the mechanical internal forces: identify and justify which dominant internal forces (tension, compression, bending, or torsion) occur in the front fork blades, in the down tube of the frame, and in the drive chain during that moment.
2. b) A manufacturer is hesitating between building the frame with $12\ \text{mm}$ diameter solid rods or with $30\ \text{mm}$ outer diameter hollow tubes that have exactly the same mass per linear metre. Demonstrate qualitatively, using Navier's normal bending stress distribution ($\sigma = \frac{M \cdot y}{I}$), why the hollow tube withstands much higher bending and torsional moments without deforming.
3. c) What new mechanical failure mode due to elastic instability would appear if the manufacturer decided to further increase the outer diameter while reducing the wall thickness to tenths of a millimetre to save more weight?

An inexperienced worker installs the steel reinforcement of a cantilever balcony by placing all the rebars at the bottom face of the concrete slab, arguing that "in interior span beams they are always placed at the bottom, so here it must be done the same way".

1. a) Draw the deflection diagram of the cantilever under its self-weight and live load, and represent qualitatively the bending moment diagram. On which face (top or bottom) do tensile stresses appear, and on which do compressive stresses appear?
2. b) Predict what will happen the moment the formwork props of the balcony are removed if the worker's reinforcement is kept: describe the cracking and collapse process, justifying it based on the characteristic mechanical behaviour of plain concrete in tension.
3. c) Design the correct cross-section of the balcony, indicating the exact position of the main longitudinal tensile rebars and explaining what function the minimum concrete cover over the steel bars serves.

You must design the load-bearing structure of a folding wall-mounted cantilever clothesline with a projection of $0,80\ \text{m}$ that will support a distributed wet laundry load of $20\ \text{kg}$ ($200\ \text{N}$). The system is anchored to a vertical wall via two points fixed at different heights.

1. a) Compare two triangulated configurations: option A (lower horizontal arm hinged to the wall and upper oblique tie at $45^\circ$ working in tension) versus option B (upper horizontal arm and lower oblique strut at $45^\circ$ working in compression). Using static equilibrium, determine the value of the axial force in the inclined member and in the horizontal arm for option A.
2. b) Justify which option is structurally safer against the risk of buckling if very slender profiles or cables are used.
3. c) Reasonably select the cross-section (hollow circular tube, solid L-angle profile, or braided cable) for each member of option A, justifying your choice based on the stress supported by each member and the folding assembly requirements.

A shelf is fixed to the wall using identical triangular brackets, each secured with two through bolts or expansion anchors, as shown in the diagram. A worker claims that, since the bracket is at rest, the weight of the books is divided equally between both bolts ($50\ \%$ shear and tension each).

1. a) Refute the worker's claim by performing a static analysis of the assembly: considering that the lower end of the vertical plate acts as a fulcrum or bearing point in compression against the wall, analytically demonstrate what stress (axial tension or compression, and shear) the upper bolt actually bears and what the lower one bears.
2. b) Using the lever principle, justify why the pull-out force on the upper bolt increases when the books are moved toward the outer edge of the shelf, deriving the literal expression for said tension as a function of the load $P$, the horizontal distance $d$ to the wall, and the vertical spacing $h$ between the upper bolt and the lower support point.
3. c) If the lower diagonal member (strut) were replaced with a diagonal tension tie located above the shelf, reason how the axial forces in the bracket members and in the anchor bolts would change.

To shore a floor slab located at a height of $4,0\ \text{m}$, a contractor proposes to splice end-to-end two $2,0\ \text{m}$ steel telescopic props (identical hollow tubular section), claiming that "since the resisting cross-section and the yield strength of the steel are the same, the double prop of $4,0\ \text{m}$ will withstand exactly the same compressive load before failing".

1. a) Quantitatively refute the contractor's claim using Euler's critical buckling load formula ($P_k = \frac{\pi^2 E I}{L_k^2}$): calculate by what factor the maximum allowable load of the $4,0\ \text{m}$ prop is reduced compared to the $2,0\ \text{m}$ one, assuming pinned-end boundary conditions at both ends.
2. b) Explain the physical mechanism of buckling and why it is a failure due to geometric instability rather than an exhaustion of the material's simple compressive strength.
3. c) Design a modification to the jobsite setup that would allow the $4,0\ \text{m}$ props to be used while achieving the same load capacity as the $2,0\ \text{m}$ ones, without changing the metal profile of the prop or increasing the tube thickness.

In the packaging industry, there is an ongoing effort to reduce the grammage of PET plastic bottles to lower costs and minimize waste. By reducing the wall thickness from $0,35\ \text{mm}$ to $0,18\ \text{mm}$ in a smooth-surfaced cylindrical bottle, it is observed to collapse easily by crushing when stacking pallets or simply when gripping it with fingers. To solve this without increasing the mass of plastic, the bottle is molded with horizontal annular ribs and perimeter corrugations.

1. a) Justify from the theory of bending and section strength how the addition of ribs modifies the moment of inertia of the wall ($I$) with respect to its neutral axis, relating it to the stiffness against lateral crushing (ovalization) without the need to increase the net area of material.
2. b) Analyze the behavior under axial compression produced by stacking: why does a thin smooth sheet or wall fail at very low loads due to crippling or local buckling, and how do the corrugations act to prevent such instability?
3. c) Compare this case with the load-bearing behavior of a corrugated roof sheet versus a flat sheet of identical thickness and material supported on two beams spaced $3,0\ \text{m}$ apart: explain, using the stress distribution, why the flat sheet deflects excessively while the corrugated sheet is self-supporting.

A user intends to attach a fixed pulley to their garage ceiling to vertically hoist a $100\ \text{kg}$ ($1\ \text{kN}$) engine by pulling vertically downwards from the floor with a rope. The floor system consists of reinforced concrete joists with a $4,0\ \text{m}$ span spaced $0,70\ \text{m}$ apart, with intermediate hollow ceramic filler blocks.

1. a) Determine the total downward vertical force transmitted by the pulley to the ceiling anchor during hoisting at a uniform speed, and compare it with the load that would be generated if the load were hoisted using a motorized hoist (electric winch) fixed directly at the same point.
2. b) Qualitatively draw the shear force and bending moment diagrams for a simply supported joist when placing the concentrated load in two alternative positions: right at the midspan ($x = 2,0\ \text{m}$) or a few centimeters from the support ($x = 0,20\ \text{m}$). Justify in which position the failure risk is governed by bending and in which by shear.
3. c) Explain why it would be a critical mistake to anchor the fixture into the ceramic filler blocks, and design a simple technical solution to distribute the load between two adjacent joists instead of overloading a single one.

A novice designer sizes the steel profiles of a grandstand for an outdoor music festival and concludes: "The yield strength of the structural steel is $f_y = 275\ \text{MPa}$. After adding the self-weight and the average weight of the seated attendees, the maximum calculated stress in the beams is $260\ \text{MPa}$. Since $260 < 275$, the structure is completely safe, will not yield, and we also avoid the waste of over-dimensioning with arbitrary safety factors".

1. a) Technically refute the designer's argument by identifying and justifying four real sources of uncertainty (two associated with actions or loads and two associated with material resistance and assembly) that make designing with a $5\ \%$ margin almost certainly lead to disaster.
2. b) In modern building codes (such as the CTE or Eurocodes), a single global safety factor is not used; instead, partial safety factors are applied: actions amplification ($\gamma_F > 1$) and material resistance reduction ($\gamma_M > 1$). Justify what technical advantage is provided by disaggregating safety into independent factors for permanent actions, variable actions, and materials.
3. c) Argue why the required safety factor and Ultimate Limit State verification criteria are much stricter for this spectator grandstand than for an auxiliary structure without permanent human occupancy (such as a tool shed), evaluating the consequences of collapse and the sudden or progressive nature of structural failure.