Stability and calculation

State the two static equilibrium conditions for a rigid body in two dimensions ($\sum F = 0$ and $\sum M = 0$) and explain in your own words what each one guarantees: which motion the first prevents and which the second.

Explain the difference between a fixed support (clamped support), a pinned support, and a roller support: which movements each permits and what reactions each can provide. Give a real-life example of each in construction.

In a beam with a pinned support and a roller support, explain why the roller support provides only a vertical reaction: what horizontal movement it releases and what would happen if both supports were pinned and the beam expanded due to heat.

Define the moment of a force about a point ($M = F \cdot d$), explaining what the lever arm $d$ is, and state its SI units. When is the moment of a force equal to zero even if the force is non-zero?

To loosen a nut, a force of $150\ \text{N}$ is applied at the end of a $0{,}25\ \text{m}$ spanner, perpendicular to the spanner. Calculate the moment applied and explain why using an extension pipe of $0{,}5\ \text{m}$ enables it to be loosened with half the force.

A force of $500\ \text{N}$ pulls on an anchor at an angle of $30^\circ$ to the horizontal. Calculate its horizontal and vertical components and the moment it produces about a point located $2\ \text{m}$ below the anchor (consider only the horizontal component, which is the one causing rotation about that point). Explain why resolving into components is essential when the force is not perpendicular.

The figure shows a triangulated truss with a span of $6\ \text{m}$ with a load of $900\ \text{N}$ applied at $2\ \text{m}$ from support A. Calculate the reactions $R_A$ and $R_B$ at the supports using the equilibrium conditions, and check that the sum of vertical forces is zero.

A beam of $4\ \text{m}$ is supported at its ends A and B and carries a load of $1\,200\ \text{N}$ at its midpoint. Without doing any calculations yet, explain by symmetry what the reactions will be; then calculate them formally using $\sum M_A = 0$ and $\sum F_y = 0$ and check that they match your prediction.

A beam of $4\ \text{m}$ supported at its ends A and B is subjected to two loads: $800\ \text{N}$ at $1\ \text{m}$ from A and $400\ \text{N}$ at $3\ \text{m}$ from A. Calculate the reactions at the supports and explain why $R_A$ turns out to be greater than $R_B$ even though the total load is shared between them.

The members of a truss are modelled as pinned at the joints. Explain what this hypothesis implies (the members only transmit axial forces, not moments) and why it simplifies the calculation of the entire structure.

An industrial warehouse uses symmetrical trusses with a $12\ \text{m}$ span and a $3\ \text{m}$ depth at the ridge. In winter, it supports a downward snow load of $12\ \text{kN}$ distributed across the top joints, while during a windstorm it experiences a net upward suction force of $18\ \text{kN}$. Analyze the behavior of the central diagonals: a) Determine the sign of the stress (tension or compression) in these diagonals under both load hypotheses. b) Justify why catastrophic failure due to elastic instability (buckling) is much more likely during the windstorm than during the snowfall, even though the magnitude of the load is similar, and propose a design modification in the truss to prevent it without increasing the steel cross-section.

A technical team has designed the pin-jointed support structure for solar panels formed by four coplanar joints arranged in a quadrilateral. To stiffen it, an operator has placed two thin steel cable tensioners along both diagonals. a) Apply the static determinacy condition for planar trusses ($b = 2n - 3$) and discuss whether the structure is determinate, indeterminate, or a mechanism. b) Knowing that the cables only withstand tension and collapse in compression due to immediate buckling, predict which tensioner will become loaded and which one will slacken depending on whether wind acts from the left or the right. c) Refute a colleague's claim stating: "We can replace the two cables with a single rigid steel bar to save parts, and the structure will withstand the wind in both directions without problems." Justify what would happen to that bar under each wind direction.

Explain the relationship between a body's centre of gravity and its stability against toppling over: why a tall van topples over sooner than a low sports car on the same bend, and what role the width of the base of support plays.

A workshop gantry crane has a mass of $1\,200\ \text{kg}$ (whose center of gravity is located on its axis of symmetry), a span between supports of $4\ \text{m}$, and a height of $3{,}5\ \text{m}$. The crane has an overhang of $1{,}2\ \text{m}$ extending outward on one of its sides to load trucks. The trolley carries a suspended load of $800\ \text{kg}$ at the end of this overhang ($g = 9{,}8\ \text{m/s}^2$). a) Demonstrate whether the crane is stable against tipping over when the load is at the end of the overhang, and calculate the overturning safety factor ($C_s = M_{\text{estabilizador}} / M_{\text{vuelco}}$). b) An operator proposes two options to raise the safety factor to a minimum of $1{,}8$: adding a counterweight on the opposite leg or widening the distance between supports to $5\ \text{m}$. Determine the minimum mass of the counterweight required in the first option and argue which of the two solutions is technically and functionally more appropriate in a workshop with limited space.

In a building structure, a central column carries an axial load of $600\ \text{kN}$ and an adjacent perimeter column carries $250\ \text{kN}$. The geotechnical study determines that the allowable bearing capacity of the soil is $\sigma_{\text{adm}} = 0{,}25\ \text{MPa}$ ($250\ \text{kN/m}^2$). A contractor proposes constructing two identical square isolated footings of $1{,}8\ \text{m} \times 1{,}8\ \text{m}$ under both columns to speed up construction. a) Quantitatively evaluate whether the proposal meets the allowable soil pressure at both supports. b) Justify why the contractor's proposal, even if both pressures were below the limit, would generate a serious risk of structural failure due to differential settlement between adjacent columns. c) Design the plan dimensions ($a \times a$) of each footing so that the pressure transmitted to the soil is uniform and equal to $80\,\%$ of the allowable bearing capacity of the ground.