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1
two
structural analysis
(statics & mechanics)
Analysis Lecture 2 1 Applied Architectural Structures
lecture
APPLIED ARCHITECTURAL STRUCTURES:
STRUCTURAL ANALYSIS AND SYSTEMS Structural Requirements
- serviceability
- efficiency
- construction
- cost
- other
Analysis Lecture 2 2 Architectural Structures III^ www.pbs.org/wgbh/buildingbig/
Structure Requirements
- strength & equilibrium - safety - stresses not greater than strength - adequate foundation
Structure Requirements
- stability & stiffness - stability of components - minimum deflection and vibration - adequate foundation
2
Analysis 5Lecture 2 Architectural Structures III
- economy and construction - minimum material - standard sized members - simple connections and details - maintenance - fabrication/ erection
Structure Requirements
Analysis 6Lecture 2 Architectural Structures III
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 7Lecture 2 Architectural Structures III
- dead load
- static, fixed, includes material weights, fixed equipment
- live load
- transient and moving loads (including occupants)
- snow load
Structural Loads - STATIC
Analysis 8Lecture 2 Architectural Structures III
- wind loads
- dynamic, wind pressures treated as lateral static loads on walls, pressure or suction
- pressure determined from wind velocity, qh
- dynamic effects include motion from buffeting or “ vortex shedding ”
Structural Loads – STATIC & DYNAMIC
FW Cdqh A
4
Analysis 13Lecture 2 Architect
Structural Math
- quantify environmental loads
- 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 14Lecture 2 A bn
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 15Lecture 2
Units
Volume gallon liter
Temperature
Force
Mass
Length
Units
F C
lb force N, kN
lb mass g, kg
in, ft, mi mm, cm, m
US SI
Analysis 16Lecture 2bn
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
z x
5
Analysis 17Lecture 2 Architectural Structures III
Forces & Reactions
- Newton ’ s 3rd^ law:
- for every force of action there is an equal and opposite reaction along the same line
- external forces act on bodies
- internal forces are
- in bodies
- between bodies (connections)
Force Components
- convenient to resolve into 2 vectors
- at right angles
- in a “ nice ” coordinate system
- is between Fx and F from Fx Fy Fx
F
x
y
Fy Fx
F Fy
Fx
F F cos ^ F x Fy F sin 2 2 F Fx F y
x
y F
F tan
Analysis 19Lecture 2 Architectural Structures III
Load Types
- weight (F = ma)
- concentrated
- distributed
- friction
W tA
x
x/
W
x/
x
2x/
W/ x/
x
x/
W
x/6 x/
W/
0
w x W w w 2^ x ^ W (^2) w w^ 2w
Analysis 20Lecture 2 Architectural Structures III
Load Tracing
- tributary load
- think of water flow
- “ concentrates ” load of area into center
tributary width area
w load
width
7
Supports and Connections
Analysis 26Lecture 2 Architectural Structures III
Centroid
- “ average ” x & y of an area
- for a volume of constant thickness
- where is weight/volume
- center of gravity = centroid of area
A
x A x
A
y A y
W t A^
Moments of Inertia
- 2 nd^ moment area
- math concept
- area x (distance)^2
- need for behavior of
I x Icx Ad y
Internal and Pin Forces
- 3 equations per three-force body
- two-force body forces in line
- 2 reactions per pin + support forces
A
B C
D
G E (^) F
Ax Ay D
8
Analysis 29Lecture 2 Architectural Structures III
Internal Beam V & M (+P)
- maximums needed for design
- Mmax at V = 0
V
L
M
L/
L
+V +M
Analysis 30Lecture 2 Archi
Deflected Shape
- positive bending moment
- tension in bottom, compression in top
- negative bending moment
- tension in top, compression in bottom
- zero bending moment
Analysis 31Lecture 2 Archite
Stress
- stress is a term for the intensity of a force, like a pressure
- internal or applied
- force per unit area
A
P stress
Analysis 32Lecture 2 Architectural Structures III
Stress Types
- normal stress is normal to the cross section
- shear stress parallel to a surface
A
P f (^) t orc
td
P A
P fv
10
Analysis 37Lecture 2 Architectural Structures III
Strain
- materials deform
- axially loaded materials change length
- bending materials deflect
• STRAIN:
- change in length over length L
L strain
Analysis 38Lecture 2 Archit n
Problem Solving
1. STATICS:
equilibrium of external forces, internal forces, stresses
- GEOMETRY: cross section properties, deformations and conditions of geometric fit, strains
- MATERIAL PROPERTIES: stress-strain relationship for each material obtained from testing
Stress to Strain
- important to us in - diagrams:
- straight section
- LINEAR-ELASTIC
- recovers shape (no permanent deformation)
f
f E
AE
PL
Analysis 40Lecture 2 Architectural Structures III
Behavior Types
11
Plastic Behavior
at yield stress
Analysis 42Lecture 2 Architectural Structures III
2 2
max max
f A
P f o
v
- if we need to know where max f and f v happen:
Maximum Stresses
F
A o
P 0 cos 1 f^ max
45 cos sin 0. 5
Analysis 43Lecture 2 Architectural Structures III
Thermal Deformation
- - the rate of strain per degree
- UNITS : ,
- length change:
- thermal strain:
- no stress when movement allowed
T T L
T T
F C
Analysis 44Lecture 2 Architectural Structures III
Beam Deflections
y max( x ) actual allowable
EI
M R
(^1)
EI dx slope M ( x )
EI
curvature M ( x )
EI dx deflection M ( x )
