APPLIED ACHITECTURAL STRUCTURES: STRUCTURAL ANALYSIS AND SYSTEMS DR. ANNE NICHOLS SPRING 2017 lecture two structural analysis (statics & mechanics) Analysis 1
Structural Requirements strength serviceability deflections efficiency economy of materials construction cost other Analysis 2 www.pbs.org/wgbh/buildingbig/
Structure Requirements strength & equilibrium safety stresses not greater than strength adequate foundation Analysis 3
Structure Requirements stability & stiffness stability of components minimum deflection and vibration adequate foundation Analysis 4
Structure Requirements economy and construction minimum material standard sized members simple connections and details maintenance fabrication/ erection Analysis 5
Relation to Architecture The geometry and arrangement of the load-bearing members, the use of materials, and the crafting of joints all represent opportunities for buildings to express themselves. The best buildings are not designed by architects who after resolving the formal and spatial issues, simply ask the structural engineer to make sure it doesn t fall down. - Onouy & Kane Analysis 6
Structural Loads - STATIC dead load static, fixed, includes material weights, fixed equipment live load transient and moving loads (including occupants) snow load Analysis 7
Structural Loads STATIC & DYNAMIC wind loads dynamic, wind pressures treated as lateral static loads on walls, pressure or suction pressure determined from wind velocity, q h dynamic effects include motion from buffeting or vortex shedding F C q A W d h Analysis 8
Structural Loads - DYNAMIC earthquake loads seismic, movement of ground (3D) building mass responds static models often used, V is static shear impact loads rapid, energy loads V ZICW R W Analysis 9
Dynamic Response Analysis 10
Dynamic Response period of vibration or frequency wave sway/time period damping reduction in sway resonance amplification of sway Analysis 11
Statics & Mechanics Review how loads affect our structures statics: things don t move forces supports & connections equilibrium mechanics: things can change shape stress & strain deflections buckling Analysis 12
Structural Math quantify environmental loads how big is it? evaluate geometry and angles where is it? what is the scale? what is the size in a particular direction? quantify what happens in the structure how big are the internal forces? how big should the beam be? Analysis 13
Physical Math physics takes observable phenomena and relates the measurement with rules: mathematical relationships need reference frame measure of length, mass, time, direction, velocity, acceleration, work, heat, electricity, light calculations & geometry Analysis 14
Units measures US customary & SI Units US SI Length in, ft, mi mm, cm, m Volume gallon liter Mass lb mass g, kg Force lb force N, kn Temperature F C Analysis 15
Vectors scalars any quantity vectors - quantities with direction like displacements summation results in the straight line path from start to end normal vector is perpendicular to something y Analysis 16 z x
Forces & Reactions Newton s 3 rd law: for every force of action there is an equal and opposite reaction along the same line external forces act on bodies can cause moments internal forces are in bodies between bodies (connections) http://www.physics.umd.edu Analysis 17
Force Components convenient to resolve into 2 vectors at right angles in a nice coordinate system y F is between F x and F from F x F Fcos x F Fsin y F Fx F Fy tan F Analysis 18 2 2 y x F y F F x F F y F y F x F x x
Load Types weight (F = ma) W ta concentrated distributed uniform linear friction F =N 0 wx 2 = W 2 x w Analysis 19
Load Tracing tributary load think of water flow concentrates load of area into center w load area tributary width width Analysis 20
Moments defined by magnitude and direction units: Nm, kft direction: + ccw (right hand rule) F - cw C A value found from F B d and distance M F d d also called lever or moment arm Analysis 21
Equilibrium analytically R R x y F F M M 0 free body diagrams x y 0 0 http://www.physics.umd.edu Analysis 22
Free Body Diagram 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
Supports and Connections Analysis 24
Supports and Connections Analysis 25
Centroid average x & y of an area for a volume of constant thickness W ta where is weight/volume center of gravity = centroid of area x xa A y ya A Analysis 26
Moments of Inertia 2 nd moment area math concept area x (distance) 2 need for behavior of beams columns I I Ad 2 x cx y Analysis 27
Internal and Pin Forces 3 equations per three-force body two-force body forces in line 2 reactions per pin + support forces B A G E C F D A x A y D Analysis 28
Internal Beam V & M (+P) maximums needed for design M max at V = 0 V + L - M + L L/2 +V +M Analysis 29
Deflected Shape positive bending moment tension in bottom, compression in top negative bending moment tension in top, compression in bottom zero bending moment inflection point Analysis 30
Stress stress is a term for the intensity of a force, like a pressure internal or applied force per unit area stress P A Analysis 31
Stress Types normal stress is normal to the cross section f t or c P A shear stress parallel to a surface f v P A P td Analysis 32
Stress Types bearing stress on a surface by contact in compression f v T J f p P A torsional stress by shear from twisting P td Analysis 33
Bolt Stresses f v single shear f v P A double shear bearing P P P P 2 2 2A A 2 d Analysis 34 d f 2 p 4 4 A P projected P td
Bending Stresses tension and compressive stress caused by bending f b Mc I M shear stress from bending f vave VQ Ib S Analysis 35
Connectors Resisting Shear plates with nails rivets bolts splices Analysis 36 nf x connector y 2 4.43 4 y a 2 Vlongitudinal p VQ connected area I 8 12 VQ I p p p p
Strain materials deform axially loaded materials change length bending materials deflect STRAIN: change in length over length strain L L Analysis 37
Problem Solving 1. STATICS: equilibrium of external forces, internal forces, stresses 2. GEOMETRY: cross section properties, deformations and conditions of geometric fit, strains 3. MATERIAL PROPERTIES: stress-strain relationship for each material obtained from testing Analysis 38
Stress to Strain important to us in - diagrams: straight section LINEAR-ELASTIC f E recovers shape (no permanent deformation) PL AE f Analysis 39
Behavior Types brittle semi-brittle Analysis 40
Plastic Behavior ductile at yield stress Analysis 41
Maximum Stresses if we need to know where max f and f v happen: F P 0 c o s 1 f max A o 45 cos sin 0.5 Analysis 42 f vmax 2 P A o f 2 max
Thermal Deformation - the rate of strain per degree UNITS :, F C length change: thermal strain: T T T T L no stress when movement allowed Analysis 43
Beam Deflections curvature, R 1 R M EI slope curvature deflection M ( x) EI M ( x) EI dx M ( x) EI dx y max ( x ) a ctu a l a llo w a b le Analysis 44
Column Stability short columns f critical actual slenderness ratio = L e /r (L/d) P A F a f radius of gyration = critical Analysis 45 P critical A 2 A EAr 2 r I A 2 2 L e L e 2 r E P critical weak axis 2 EA L e r 2
Column Stresses when a column gets stubby, F y will limit the load real world has loads with eccentricity end conditions L e K L Analysis 46