Global Stability, Stiffness & Torsion in Building Structures

Goal: achieve overall (global) stability without resorting to "ginormous piled foundations" that are prohibitively expensive.
Jenga / masonry‐block analogy
- Start with a stable stack → resultant vertical force (R) sits near the centre of the base contact.
- Keep adding blocks on one side → R shifts rightward; once it leaves an allowable zone, the stack topples.
- Remedy: add counter-weight on the opposite side to pull R back.
Engineering translation
- Every structure has an overturning moment M_{OT} and a stabilising gravity load W.
- Eccentricity of R from the centroid of the base is e = \frac{M_{OT}}{W}.
- Classical limit for gravity-only systems on soil:
- If e \le \frac{B}{6} (B = footing width) ⇒ compressive stress remains everywhere (no tension), positive soil pressure throughout → safe.
- If e > \frac{B}{6} ⇒ tensile zone develops, uplift starts, global overturning possible.
- Practical rule of thumb: “Keep the resultant inside the middle third of the base” → guarantees pure compression, zero tension, and therefore global stability.
Visual/physical surrogate: thrust line or isostatic stress trajectory should stay inside the kern (middle third) of the footing / wall / arch.
Ways to bring R back into the middle third
- Add mass (ballast slabs, heavy roofs, concrete cores, etc.).
- Increase plan dimensions (larger footings, struts).
- Introduce tie-downs if tension is permitted (rare for unreinforced masonry; common in anchored foundations).

Same “middle-third” reasoning applies to a single pad or pile cap:
- Positive soil pressure only if e \le \frac{B}{6};
- Beyond that, part of the base lifts, wasting concrete and risking cracking.
No derivation shown, but accepted as standard soil–structure truth (Triangular pressure distribution flops to zero at one edge when e=B/6).

Even stable buildings deflect under load; columns behave like vertical/horizontal springs.
Generic spring picture
- Force–displacement law \delta = \frac{F}{k} (k = spring stiffness).
Vertical spring model of entire building mass
- Mass m atop global vertical spring of stiffness kv → vertical shortening \deltav = \frac{W}{k_v}.
- Mostly secondary issue except for very long‐span floors/slender columns.
Horizontal (lateral) spring / rotational spring model
- Building treated as cantilever of height H with rotational base stiffness k_\theta = E I.
- Small-angle assumption \theta \approx \frac{\Delta}{H}.
- Moment from lateral load M = F H.
- Rotation \theta = \frac{F H}{k_\theta} = \frac{F H}{E I} ⇒ lateral sway \Delta = \theta H = \frac{F H^2}{E I} (proportional trends highlighted).
Key sensitivities (cantilever point-load case)
- \Delta \propto F (double the force, double the drift).
- \Delta \propto H^{3} (double the height → 8 × drift).
- \Delta \propto \frac{1}{E I} (stiffer material or fatter section → smaller drift).

Structural steel: E \approx 200\,\text{GPa}.
Normal concrete (28–37 MPa compressive strength): E \approx 28\text{–}37\,\text{GPa} (compression only).
Timber: E \approx 7.8\text{–}16\,\text{GPa} (species dependent, tension value quoted).
Steeper \sigma\text{–}\varepsilon slope ⇒ stiffer material ⇒ less strain for same stress.

For rectangles bent about a centroidal axis: I = \frac{b d^{3}}{12} (b = breadth parallel to neutral axis; d = depth perpendicular).
Orientation matters dramatically:
- Example 1×2 rectangle
- Upright (2 deep): I = \frac{1\times 2^{3}}{12} = \frac{8}{12} = \frac{2}{3}.
- Laid flat (1 deep): I = \frac{2\times 1^{3}}{12} = \frac{2}{12} = \frac{1}{6}.
- Upright configuration ≈ 4 × stiffer in bending.
Rubber ruler demo: wide‐flat orientation much floppier than tall‐narrow.
Strong vs Weak axis
- Strong axis = axis about which I is largest; weak = smallest.
- Example steel section 203 × 203 × 46 UC:
- I_x = 45.6\times10^{6}\,\text{mm}^{4} (strong).
- I_y = 15.4\times10^{6}\,\text{mm}^{4} (weak) ⇒ ~3:1 stiffness ratio.
- Hence I‐beams installed “web vertical”, not flat.

Cantilever, point load at tip: \delta_{max}=\frac{P L^{3}}{3 E I}.
Simply supported, uniformly distributed load w: \delta_{max}=\frac{5 w L^{4}}{384 E I} (speaker simplified to \frac{w L^{4}}{384 E I} for emphasis).
Doubling span length multiplies deflection by L^{3} or L^{4} ⇒ span control is a cost saver.

Shear walls, braced bays, cores supply horizontal stiffness.
Demonstration models
1. Two opposing walls + X-bracing → building translates nearly rigidly when pushed.
2. Remove bracing (keep only end walls) → load applied away from stiff wall plane causes twist; columns on soft side fail first.
Plan examples
- Rectangular building with identical walls on both ends faces wind along its width:
- All engaged about strong axis ⇒ equal stiffness each side ⇒ pure translation, zero torsion.
- Same plan, wind across length (weak axis for those walls): central core may be sole stiff element ⇒ torsional rotation develops.
- L-shaped or cantilevered podium forms exacerbate eccentricity and torsion.

Cause: line of action of lateral load does not coincide with the centre of stiffness (C.o.S.).
Coordinate of C.o.S.
xs = \frac{\sum\left( ki xi \right)}{\sum ki},\qquad ys = \frac{\sum\left( ki yi \right)}{\sum ki}
where ki = E Ii of each wall/brace, and xi, yi are its centroidal coordinates.
If all walls share same material E, the E cancels; stiffness weighting depends only on I.
Illustrative cases (width = B)
1. Four equal walls → x_s = \frac{B}{2}, so e = 0, no torsion.
2. Remove one wall → x_s = \frac{B}{3}, load still at B/2 ⇒ e = \frac{B}{6}.
- Torsional moment M_t = F e = F \frac{B}{6}.
1. Make one wall twice as thick ⇒ its I doubles ⇒ C.o.S. shifts toward that wall; twist direction and magnitude follow.
Earthquake anecdote: Kobe, Japan – stiffness concentrated in stair‐core corner, torsional vibration snapped peripheral columns.

Strive for plan symmetry: mirror walls/cores about both principal axes.
Distribute stiffness: avoid putting all walls in one corner; add partner walls or braces on opposite side.
For complex shapes (L, U, C, etc.) introduce movement/expansion joints so each segment acts as a simpler rectangle.
Consider diagonal wind/quake directions: treat oblique load as orthogonal components; actual design often controlled by orthogonal envelopes, but must still check torsion.
Use ballast, outriggers, or belt trusses to drag additional resisting elements into action when needed.

Over-conservatism (e.g.
massive pile caps) wastes resources; good engineering seeks efficient stability.
Neglecting torsion can be catastrophic (collapse, Kobe example).
Layout decisions made early (by architects/owners) lock in structural performance and cost—requires collaborative discussion.

Keep the resultant gravity line inside the middle third for overturning safety.
Lateral drift is controlled by E I and grows rapidly with building height.
Moment of inertia hinges on orientation; every section has strong & weak axes.
Torsion arises from eccentricity between load path and stiffness centroid; quantify with weighted-average formulas.
Symmetry, distributed walls, and joints are the designer’s primary torsion cures before resorting to “ginormous piles”.