Exercises: 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.

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.

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.

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.

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.

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.

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.

A sign of $30\ \text{kg}$ hangs from the end of a horizontal arm of $1{,}5\ \text{m}$ fixed into the wall. a) Calculate the moment produced by the weight of the sign at the wall fixity (take $g = 9{,}8\ \text{m/s}^2$). b) Explain what reactions (force and moment) the wall must provide to maintain equilibrium. c) Reason why the arm is usually reinforced with an inclined tie.

A beam of $5\ \text{m}$, supported at its ends, carries a uniformly distributed load of $400\ \text{N/m}$, as shown in the figure. a) Replace the distributed load with its equivalent point resultant (magnitude and point of application). b) Calculate the reactions at the supports. c) Comment on why, by symmetry, the result could be predicted without calculation.

A see-saw of $4\ \text{m}$ is supported at its centre. A girl weighing $30\ \text{kg}$ sits at $1{,}5\ \text{m}$ from the pivot. a) Calculate the moment she produces (take $g = 9{,}8\ \text{m/s}^2$). b) At what distance from the pivot must her brother weighing $45\ \text{kg}$ sit to balance it? c) Explain what happens if the brother moves closer to the pivot.

A shelf of $1{,}2\ \text{m}$ rests on two brackets, one placed at each end, and supports a box of $20\ \text{kg}$ placed at $0{,}3\ \text{m}$ from the left bracket. Calculate the load supported by each bracket (take $g = 9{,}8\ \text{m/s}^2$) and explain which way the box must be moved to balance both brackets.

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.

A workshop jib crane has a column with a $2\ \text{m}$ arm and carries a load of up to $250\ \text{kg}$ at the end. a) Calculate the moment at the base of the column (take $g = 9{,}8\ \text{m/s}^2$). b) Explain which two systems these cranes use to resist that moment (base anchoring and counterweight or top bracket) and what check you would perform on the anchor bolts.

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.

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.

A symmetrical truss for a pedestrian bridge with an $8\ \text{m}$ span and a $2\ \text{m}$ depth is formed by four square modules of $2\ \text{m} \times 2\ \text{m}$ with $45^\circ$ diagonals. It supports a vertical point load of $12\ \text{kN}$ transmitted by a platform at the bottom joint located $2\ \text{m}$ from the left support (pin/roller support). a) Determine the reactions at the supports. b) Using the method of sections (Ritter's method), calculate the axial force in the central top chord and in the diagonal closest to the left support. c) If the top chord works in compression and the diagonal in tension, justify why the top chord requires a cross-section with a greater moment of inertia than the diagonal, even though the absolute value of the axial force is comparable, indicating the stress phenomenon that governs its sizing.

A free-standing shelving unit $3\ \text{m}$ high and $0{,}8\ \text{m}$ deep has a self-weight mass of $50\ \text{kg}$ (c.o.g. at its geometric centre). Pallets of $200\ \text{kg}$ are stored on the top shelf ($h = 2{,}8\ \text{m}$). The coefficient of static friction with the concrete floor is $\mu = 0{,}35$. A forklift accidentally hits the shelving unit at mid-height ($h = 1{,}5\ \text{m}$) with a horizontal force $F_h$. Take $g = 9{,}8\ \text{m/s}^2$. a) Analytically determine what will happen first as $F_h$ increases: will the shelving unit slide or tip over? Calculate the critical value of $F_h$ for each failure mode. b) If the rear base of the shelving unit is anchored to the floor with expansion bolts, predict how the failure mode changes and calculate the tensile force that each of the two rear bolts must withstand to resist an accidental horizontal thrust of $1\,500\ \text{N}$ at the top ($h = 3\ \text{m}$).

A $60\ \text{kg}$ advertising canopy is supported by a $2\ \text{m}$ horizontal beam hinged to the wall and a steel cable tie that connects the outer end of the beam to a point on the wall at a height $h$ above the hinge ($g = 9{,}8\ \text{m/s}^2$). a) Express the tie tension $T(h)$ and the compressive force on the beam $N(h)$ as a function of the anchoring height $h$. b) Interpret what happens asymptotically to both internal forces if, due to space limitations, the height $h$ is reduced towards zero ($h \to 0$), and explain why a small angle between the tie and the beam is structurally unfeasible. c) If the available cable withstands a maximum allowable tension of $1\,200\ \text{N}$, calculate the minimum height $h_{\text{mín}}$ at which the tie can be anchored to the wall without risk of failure.

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.

A regulation futsal goal ($3\ \text{m}$ wide, $2\ \text{m}$ high, and $1\ \text{m}$ deep at the bottom base) has a total mass of $45\ \text{kg}$, with its center of gravity located $0{,}35\ \text{m}$ behind the front plane of the goal. A $75\ \text{kg}$ player hangs from the center of the crossbar, placing their center of gravity $0{,}3\ \text{m}$ in front of the goal's front plane ($g = 9{,}8\ \text{m/s}^2$). a) Prove mathematically that the goal will tip over by calculating the moment balance with respect to the front tipping axis. b) It is decided to stabilize it using a ballast placed on the rear horizontal bar of the base (at $1\ \text{m}$ from the front). Calculate the minimum mass of the ballast to guarantee a safety factor against overturning of $C_s \ge 1{,}5$. c) A company proposes replacing the ballast with a forward horizontal extension of the base. Determine what length this extension should have to achieve the same safety factor ($C_s \ge 1{,}5$) without adding any additional weight.