APPLIED ARCHITECTURAL STRUCTURES: STRUCTURAL ANALYSIS AND SYSTEMS Structural Requirements ARCH 631 • serviceability DR. ANNE NICHOLS FALL 2013 – strength – deflections lecture two • efficiency – economy of materials • construction structural analysis • cost (statics & mechanics) • other www.pbs.org/wgbh/buildingbig/ Analysis 1 Applied Architectural Structures F2009abn Analysis 2 Architectural Structures III F2009abn Lecture 2 ARCH 631 Lecture 2 ARCH 631

Structure Requirements Structure Requirements • strength & • stability & equilibrium stiffness – safety – stability of – stresses components not greater – minimum than deflection and strength vibration – adequate – adequate foundation foundation

Analysis 3 Architectural Structures III F2008abn Analysis 4 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631

1 Structure Requirements Relation to Architecture • economy and “The geometry and arrangement of the construction load-bearing members, the use of – minimum material materials, and the crafting of joints all represent opportunities for buildings to – standard sized express themselves. The best members buildings are not designed by – simple connections architects who after resolving the and details formal and spatial issues, simply ask – maintenance the structural engineer to make sure it – fabrication/ erection doesn’t fall down.” - Onouy & Kane

Analysis 5 Architectural Structures III F2008abn Analysis 6 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631

Structural Loads - STATIC Structural Loads – STATIC & DYNAMIC • dead load • wind loads – static, fixed, includes – dynamic, wind pressures material weights, fixed treated as lateral static equipment loads on walls, pressure or suction • live load – pressure determined – transient and moving from wind velocity, q loads (including h – dynamic effects include occupants) motion from buffeting or FW  Cd qh A • snow load “vortex shedding”

Analysis 8 Architectural Structures III F2008abn Analysis 7 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631

2 Structural Loads - DYNAMIC Dynamic Response • earthquake loads – seismic, movement of ground (3D) – building mass responds – static models often used, V is static shear • impact loads ZICW – rapid, energy loads V  RW

Analysis 9 Architectural Structures III F2008abn Lecture 2 ARCH 631 Analysis 10 Architectural Structures III F2008abn Lecture 2 ARCH 631

Dynamic Response Statics & Mechanics Review • period of vibration or • how loads affect our structures frequency – statics: things don’t move – wave • forces – sway/time period • supports & connections • equilibrium • damping – mechanics: things can change shape – reduction in sway • stress & strain • resonance • deflections – amplification of sway • buckling

Analysis 11 Architectural Structures III F2008abn Analysis 12 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631

3 Structural Math Physical Math • quantify environmental loads • physics takes observable phenomena – how big is it? and relates the measurement with rules: • evaluate geometry and angles mathematical relationships – where is it? • need – what is the scale? – reference frame – what is the size in a particular direction? – measure of length, mass, time, direction, • quantify what happens in the structure velocity, acceleration, work, heat, – how big are the internal forces? electricity, light – how big should the beam be? – calculations & geometry

Analysis 13 Architectural Structures III F2008abn Analysis 14 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631

Units Vectors • measures • scalars – any quantity – US customary & SI • vectors - quantities with direction – like displacements Units US SI – summation results in Length in, ft, mi mm, cm, m the “straight line path” Volume gallon liter from start to end Mass lb mass g, kg – normal vector is perpendicular to y Force lb force N, kN something Temperature F C z x Analysis 15 Architectural Structures III F2008abn Analysis 16 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631

4 Forces & Reactions Force Components • Newton’s 3rd law: • convenient to resolve into 2 vectors – for every force of action • at right angles there is an equal and • in a “nice” coordinate system F opposite reaction along y F the same line •  is between F and F from F y x x  x F • external forces act on bodies F Fx Fx  F cos Fy – can cause moments F F  F sin y • internal forces are y Fx 2 2 Fx – in bodies F  Fx  Fy F – between bodies (connections) tan  y Analysis 18 F Architectural Structures III F2008abn Analysis 17 Architectural Structures III F2008abn Lecture 2 x ARCH 631 Lecture 2 ARCH 631

Load Types Load Tracing • weight (F = ma) • tributary load W  tA – think of water flow • concentrated – “concentrates” load of area into center • distributed – uniform  load  w   tributary width – linear  area  width

w  x W 2w • friction w  x  W  w 2 2 w w – F=N 0 x x x W W Analysis 19 Architectural Structures III F2008abnW/2 W/2Analysis 20 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631

x/2 x/2 2x/3 x/3 x/2 x/6 x/3 5 Moments Equilibrium • defined by magnitude and direction • analytically • units: Nm, kft Rx   Fx  0 • direction: Ry   Fy  0 + ccw (right hand rule) F M  M  0 - cw C  • value found from F A B • free body diagrams and  distance d M  F d • d also called “lever” or “moment” arm

Analysis 21 Architectural Structures III F2008abn Analysis 22 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631

Free Body Diagram Supports and Connections • FBD (sketch) • tool to see all forces on a body or a point including – external forces – weights – force reactions – external moments – moment reactions – internal forces

Analysis 23 Architectural Structures III F2008abn Analysis 24 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631

6 Supports and Connections Centroid • “average” x & y of an area • for a volume of constant thickness – W  tA where  is weight/volume – center of gravity = centroid of area

xA x  A yA y  A Analysis 26 Architectural Structures III F2008abn Analysis 25 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631

Moments of Inertia Internal and Pin Forces • 2nd moment area – 3 equations per three-force body – math concept – two-force body forces in line – area x (distance)2 – 2 reactions per pin + support forces • need for behavior of

– beams B – columns G C E 2 F I x  Icx  Ad y A D Ax Ay D

Analysis 27 Architectural Structures III F2008abn Lecture 2 ARCH 631 Analysis 28 Architectural Structures III F2008abn Lecture 2 ARCH 631

7 Internal Beam V & M (+P) Deflected Shape • maximums needed for design

• Mmax at V = 0

V + L • positive bending moment - – tension in bottom, compression in top • negative bending moment +V M – tension in top, compression in bottom + L +M L/2 • zero bending moment – inflection point

Analysis 29 Architectural Structures III F2008abn Lecture 2 ARCH 631 Analysis 30 Architectural Structures III F2008abn Lecture 2 ARCH 631

Stress Stress Types • stress is a term for the intensity of a • normal stress is normal force, like a pressure to the cross section • internal or applied P f  • force per unit area t or c A • shear stress parallel P to a surface stress  A P P f   v A td

Analysis 31 Architectural Structures III F2008abn Analysis 32 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631

8 Stress Types Bolt Stresses • bearing stress on a • single shear surface by contact in P P f   compression P P v d 2 f p   A  4 A td • double shear • torsional stress by P P P shear from twisting f   2  2 v d 2 T 2A A  4 fv  P P J • bearing f p   Aprojected td Analysis 33 Architectural Structures III F2008abn Analysis 34 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631

Bending Stresses Connectors Resisting Shear

• tension and compressive stress caused • plates with p y p by bending – nails p 8” 2” Mc M – rivets 4.43” x 4” f   ya b – bolts 2” I S 12” • shear stress from bending • splices Vlongitudinal VQ VQ  fvave  p I Ib VQ nF  connected area  p connector I

Analysis 35 Architectural Structures III F2008abn Analysis 36 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631

9 Strain Problem Solving • materials deform 1. STATICS: • axially loaded materials change equilibrium of external forces, length internal forces, stresses • bending materials deflect 2. GEOMETRY: cross section properties, deformations and conditions of geometric fit, strains 3. MATERIAL PROPERTIES: • STRAIN: L stress-strain relationship for each material – change in length over length strain  L obtained from testing Analysis 37 Architectural Structures III F2008abn Lecture 2 ARCH 631 Analysis 38 Architectural Structures III F2008abn Lecture 2 ARCH 631

Stress to Strain Behavior Types • important to us in f - diagrams: • brittle – straight section – LINEAR-ELASTIC f  E  – recovers shape (no permanent • semi-brittle deformation) PL   AE

Analysis 39 Architectural Structures III F2008abn Lecture 2 ARCH 631 Analysis 40 Architectural Structures III F2008abn Lecture 2 ARCH 631

10 Plastic Behavior Maximum Stresses • ductile • if we need to know where max f and f v F happen: at yield stress P   0  cos 1 fmax  Ao   45  cos  sin  0.5

P fmax fvmax   2Ao 2 Analysis 42 Architectural Structures III F2008abn Analysis 41 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631

Thermal Deformation Beam Deflections •  - the rate of strain per degree • curvature, R 1 M M (x)  curvature  • UNITS : , R EI EI F C M (x)   slope  dx • length change:  EI    T L T   M (x)   deflection   dx • thermal strain: EI T  T 

– no stress when movement allowed ymax (x)   actual   allowable

Analysis 43 Architectural Structures III F2008abn Analysis 44 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631

11 Column Stability Column Stresses

• short columns • when a column gets stubby, Fy will limit the P f  actual  F load critical A a • real world has loads with eccentricity • slenderness ratio = Le/r (L/d) • end conditions Le  K  L I • radius of gyration = r  A weak axis 2 2 2 2 Pcritical  EAr  E  EA fcritical   2  2 Pcritical  2 A AL   L   L  e  e   e   r   r  Analysis 45 Architectural Structures III F2008abn Analysis 46 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631

12