Scenario: A single-bay, single-story steel frame is constructed with pinned bases and pinned beam-to-column connections. It is subjected to a lateral wind load.

Analysis: The structure has $r = 4$ reactions (two pins, each with 2 reactions). There are $m = 3$ members and $j = 4$ joints.

Determinacy Check: Let's check the frame determinacy formula $3m + r = 3j$. $3(3) + 4 = 13$, and $3(4) = 12$. The structure appears statically indeterminate to the 1st degree ($13 > 12$).

Stability Assessment: Despite appearing indeterminate, the structure is geometrically unstable. Because all connections are pins, there is no resistance to lateral deformation. The structure will collapse laterally (sway collapse) under any horizontal force. This demonstrates that mathematical determinacy does not guarantee stability; geometric configuration is paramount.

Solution: To make the frame stable, we must provide lateral resistance. This can be done by adding a diagonal bracing member (creating a truss mechanism), changing the base supports to fixed supports, or using moment-resisting connections between the beams and columns.

Scenario: A horizontal beam is supported by three vertical links (rods) spaced equally along its length. It is subjected to a horizontal point load.

Analysis: The beam has $r = 3$ parallel vertical reactions from the links. The equilibrium equations available are $\sum F_x = 0$, $\sum F_y = 0$, and $\sum M = 0$.

Determinacy Check: $r = 3$, which equals the 3 equations of static equilibrium. It appears to be statically determinate.

Stability Assessment: The structure is unstable. All three reaction lines of action are parallel (vertical). When the horizontal load is applied, there is no horizontal reaction force available to satisfy $\sum F_x = 0$. The beam will swing laterally like a pendulum.

Solution: To stabilize the beam, at least one support must provide a horizontal reaction, meaning its line of action must intersect the other vertical lines of action, destroying the parallelism. Replacing one link with a pinned support to the ground would solve this.

Determine the stability and degree of indeterminacy of a planar truss with $m = 15$ members, $j = 9$ joints, and supported by a pin and a roller.

Determine the degree of indeterminacy of a horizontal beam that rests on 4 roller supports and 1 pinned support at the far left end. The beam is continuous over all supports.

Consider a two-story, single-bay rigid frame. The column bases are fixed to the ground. Determine the degree of indeterminacy.