STU 122 – Global Structural Behaviour (Theme 1 | Lecture 2)

Recap of Previous Concepts

• Types of structural loads already covered in the course:

  • Dead load (DL)

  • Live load (LL)

  • Wind load (WL)

  • Seismic load (EL)

  • Others (snow, temperature, foundation settlement, etc.)

• Structural design always combines loads so that the structure and its components avoid:

  • Loss of stability (global or local)

  • Collapse (ultimate limit state)

  • Excessive deflection or vibration (serviceability limit state)

  • Excessive cracking (serviceability & durability)

Essence of Structural Behaviour

• Core purpose of every structural system: create a continuous path for forces to travel safely to the ground.

• Famous architectural illustration: Centre Pompidou (Paris) shows clearly expressed structural and service pathways.

• Practical takeaway: every beam, column, brace or wall serves as one link in that uninterrupted load path.

Loads on Building Structures

• Two broad categories of actions acting on any building:

  • Internal / gravity related: floor loads, partitions, equipment, occupants, self-weight.

  • External / environmental: wind, earthquake, ground-induced movements, differential settlements.

• Designer’s task: trace each individual load from its point of application through members → foundations → soil.

Global Stability

• Stability deals with the building as a whole resisting sliding and overturning under combined vertical & horizontal loads.

• Key horizontal actions considered: wind pressure, suction, uplift.

• Failure modes examined:

  • Sliding (net horizontal reaction inadequate)

  • Overturning (moment equilibrium violated)

• Fundamental equilibrium relationships (cantilever analogy):

  • Overturning moment: $M = H a$

  • Restoring moment (from vertical reactions): $M_R = V d$

  • For overall equilibrium: $H a = V d$

• Insight: increasing the lever arm of vertical reactions or reducing the lever arm of horizontal force lowers foundation reactions.

Comparative Study – Aspect Ratio Effects

• Diagrams show three wall panels with same height $h$ but different base widths $d<em>a, d</em>b, d_c$.

• Thicker base (larger $d$) gives larger restoring lever arm → smaller required vertical reactions $V$ to resist same $H$.

• Rule of thumb: increasing base width is usually more efficient for stability than simply adding weight.

Rule of the Middle Third

• For any prismatic block under combined vertical load $W$ and overturning moment $M$, the resultant compressive stress distribution stays entirely compressive (no tension at base) if and only if the resultant vertical load falls within the middle third of the base.

• Practical meaning:

  • Resultant eccentricity $e$ must satisfy $e \le \frac{d}{6}$ where $d$ = base width.

  • Outside this zone, part of the base sees tension; if uplift occurs → potential rocking/overturning.

Stiffness and Deflection

• Microscopic view: atomic “spring” model – tensile forces stretch bonds, compressive forces shorten them.

• Macroscopic law: deflection is inversely proportional to stiffness.

• Two governing ingredients of stiffness:

  1. Material (modulus of elasticity $E$)

  2. Geometry (member area $A$ for axial; moment of inertia $I$ for bending; torsional constant $J$ for torsion)

• Typical modulus values:

  • Steel: $E = 200\,\text{GPa}$

  • Concrete: $E \approx 28 \text{–} 37\,\text{GPa}$ (depends on strength & age)

  • Timber: $E \approx 7.8 \text{–} 16\,\text{GPa}$ (species-dependent)

Axial Stiffness

• Defined by: $k_{\text{axial}} = \dfrac{A E}{L}$

• Axial deflection of a prismatic bar: $\Delta = \dfrac{P L}{A E}$

Bending Stiffness

• For beams, bending rigidity: $k_{\text{bending}} = \dfrac{E I}{L}$ ; slope/deflection vary with boundary conditions.

• Handy mnemonic: larger $I$ (strong axis) → dramatically lower mid-span deflection.

• Weak vs strong axis illustrated; designers orient I-sections, rectangular beams, etc. to exploit strong axis inertia.

Standard Beam-Formula Extracts (Simply Supported Examples)

• Concentrated load at centre: $\delta_{\max} = \dfrac{P L^{3}}{48 E I}$

• Concentrated load at arbitrary point $a$ from left support:
  $\delta_{\max} = \dfrac{P a b ( 2 L^{2} - 2 L a - b^{2})}{6 L E I}$ where $b = L - a$; maximum often occurs under load if a > b.

• Uniformly distributed load $w$ over entire span:
  $\delta_{\max} = \dfrac{5 w L^{4}}{384 E I}$

• End couple $M$ at right support (simply supported beam): central deflection $\delta = \dfrac{M L^{2}}{16 E I}$.

Deflection Exercise Snippet

• Provided formula for rectangular section inertia: $I = \dfrac{b d^{3}}{12}$

• Reminder: depth has cubic influence → doubling depth cuts deflection by factor of 8 (all else equal).

Torsional Effects

• Torsion in buildings arises when line of action of resultant wind (or seismic) load does not pass through the centre of stiffness (CS) of lateral-load-resisting system.

• Consequences: differential lateral displacements, additional shear in members, discomfort, façade damage.

Design Principles to Minimise Torsion

• Place shear walls, cores, braced frames symmetrically in plan with respect to both axes.

• If asymmetry unavoidable, increase torsional stiffness of distant elements or add supplementary braces.

• Always model 3-D behaviour; check drift and torsional amplification factors per code.

Centre of Stiffness Calculation (Illustrative Grid)

• For walls $W<em>1, W</em>2, W<em>3, W</em>4$ with stiffnesses $K<em>i$ and coordinates $(x</em>i, y<em>i)$: $x</em>{CS} = \dfrac{\sum K<em>i x</em>i}{\sum K<em>i}, \quad y</em>{CS} = \dfrac{\sum K<em>i y</em>i}{\sum K_i}$

• When wind resultant $R$ acts at $(x<em>R, y</em>R)$, torsional moment about CS: $M<em>T = R \times \sqrt{(x</em>R - x<em>{CS})^{2} + (y</em>R - y_{CS})^{2}}$

Practical / Ethical / Philosophical Implications

• Safety: misjudging global stability or torsional behaviour can lead to catastrophic failure; engineers have ethical duty to protect life.

• Economy vs robustness: adding symmetric bracing may appear costly but prevents future retrofits or failures.

• Architectural expression (e.g.e Centre Pompidou) shows structure but still obeys same physics; creativity must align with sound engineering.

Connections to Prior & Future Lectures

• Builds on previous lecture’s load categorisation.

• Next lecture will develop detailed load pathse tracing vertical & horizontal actions from application point to soil.

End-of-Lecture Summary

• Structures = force-paths to ground.

• Lateral systems (shear walls, braced frames, moment frames) resist horizontal loads; must ensure overall global stability.

• Deflection depends on load magnitude, $E$, and geometric properties $A, I$.

• Ability to compute deflection for standard cases critical for serviceability design.

• Torsion appears when resultant lateral force and centre of stiffness misalign; design symmetrical layouts or compensate analytically.