Post on 20-May-2020
1
F2009abn
two
structural analysis
(statics & mechanics)Analysis 1
Lecture 2
Applied Architectural Structures
ARCH 631
lecture
APPLIED ARCHITECTURAL STRUCTURES:
STRUCTURAL ANALYSIS AND SYSTEMS
ARCH 631
DR. ANNE NICHOLS
SPRING 2018
F2009abn
Structural Requirements
• serviceability
– strength
– deflections
• efficiency
– economy of materials
• construction
• cost
• other
Analysis 2
Lecture 2
Architectural Structures III
ARCH 631
www.pbs.org/wgbh/buildingbig/
Analysis 3
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Structure Requirements
• strength &
equilibrium
– safety
– stresses
not greater
than
strength
– adequate
foundation
Analysis 4
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Structure Requirements
• stability &
stiffness
– stability of
components
– minimum
deflection and
vibration
– adequate
foundation
2
Analysis 5
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
• economy and
construction
– minimum material
– standard sized
members
– simple connections
and details
– maintenance
– fabrication/ erection
Structure Requirements
Analysis 6
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
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 7
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
• dead load
– static, fixed, includes
material weights, fixed
equipment
• live load
– transient and moving
loads (including
occupants)
• snow load
Structural Loads - STATIC
Analysis 8
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
• 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
AqCF hdW
3
Analysis 9
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
• earthquake loads
– seismic, movement of
ground (3D)
– building mass responds
– static models often used,
V is static shear
• impact loads
– rapid, energy loads
Structural Loads - DYNAMIC
WRZICWV
Analysis 10
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Dynamic Response
Analysis 11
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Dynamic Response
• period of vibration or
frequency
– wave
– sway/time period
• damping
– reduction in sway
• resonance
– amplification of sway
Analysis 12
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
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
4
Analysis 13
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
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 14
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
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 15
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Units
• measures
– US customary & SI
litergallonVolume
Temperature
Force
Mass
Length
Units
CF
N, kNlb force
g, kglb mass
mm, cm, min, ft, mi
SIUS
Analysis 16
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
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
xz
5
Analysis 17
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Forces & Reactions
• Newton’s 3rd law:
– for every force of action there is an equal and opposite reaction alongthe same line
• external forces act on bodies
– can cause moments
• internal forces are
– in bodies
– between bodies (connections)
Analysis 18
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Force Components
• convenient to resolve into 2 vectors
• at right angles
• in a “nice” coordinate system
• is between Fx and F from FxFy
Fx
F
x
y
Fy
Fx
F
Fy
Fx
FcosFFx
sinFFy
22
yx FFF
x
y
F
Ftan
Analysis 19
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Load Types
• weight (F = ma)
• concentrated
• distributed
– uniform
– linear
• friction
– F=N
tAW
x
x/2
W
x/2
x
2x/3
W/2
x/3
x
x/2
W
x/6 x/3
W/2
0
Wxw w w 22
Wxw
w
2w
Analysis 20
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Load Tracing
• tributary load
– think of water flow
– “concentrates” load of
area into center
widthtributaryarea
loadw
width
6
Analysis 21
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Moments
• defined by magnitude and direction
• units: Nm, kft
• direction:
+ ccw (right hand rule)
- cw
• value found from Fand distance
• d also called “lever” or “moment” arm
F
A
C
B
dFM
d
Analysis 22
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Equilibrium
• analytically
• free body diagrams
0 xx FR
0 yy FR
0MM
Analysis 23
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
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 24
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Supports and Connections
7
Analysis 25
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Supports and Connections
Analysis 26
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Centroid
• “average” x & y of an area
• for a volume of constant thickness
– where is weight/volume
– center of gravity = centroid of area
A
Axx
A
Ayy
AtW
Analysis 27
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Moments of Inertia
• 2nd moment area
–math concept
– area x (distance)2
• need for behavior of
– beams
– columns
2
ycxx AdII
Analysis 28
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
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 29
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Internal Beam V & M (+P)
• maximums needed for design
• Mmax at V = 0
V
L
+
M+
-
L/2
L
+V
+M
Analysis 30
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
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 31
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Stress
• stress is a term for the intensity of a
force, like a pressure
• internal or applied
• force per unit area
A
Pstress
Analysis 32
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Stress Types
• normal stress is normal
to the cross section
• shear stress parallel
to a surface
A
Pf cort
td
P
A
Pfv
9
Analysis 33
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
• bearing stress on a
surface by contact in
compression
• torsional stress by
shear from twisting
Stress Types
td
P
A
Pf p
J
Tfv
Analysis 34
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
• single shear
• double shear
• bearing
Bolt Stresses
4
2dv
P
A
Pf
4
2
22
2 dv
P
A
P
A
Pf
td
P
A
Pf
projected
p
Analysis 35
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
• tension and compressive stress caused
by bending
• shear stress from bending
Bending Stresses
S
M
I
Mcfb
Ib
VQf avev
Analysis 36
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
• plates with
– nails
– rivets
– bolts
• splices
Connectors Resisting Shear
x
y
ya
4”
2”
2”
12”
8” p
p p
4.43”
I
VQ
p
V allongitudin
pI
VQnF
areaconnected
connector
10
Analysis 37
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Strain
• materials deform
• axially loaded materials change
length
• bending materials deflect
• STRAIN:
– change in length over length
L
Lstrain
Analysis 38
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
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 39
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Stress to Strain
• important to us in - diagrams:
– straight section
– LINEAR-ELASTIC
– recovers shape
(no permanent
deformation)
f
Ef
AE
PL
Analysis 40
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Behavior Types
• brittle
• semi-brittle
11
Analysis 41
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Plastic Behavior
• ductile
at yield stress
Analysis 42
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
22
maxmax
f
A
Pf
o
v
• if we need to knowwhere max f and fv
happen:
Maximum Stresses
F
oA
Pf max1cos0
5.0sincos45
Analysis 43
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Thermal Deformation
• - the rate of strain per degree
• UNITS : ,
• length change:
• thermal strain:
– no stress when movement allowed
LTT
TT
F C
Analysis 44
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Beam Deflections
• curvature, R
allowableactualxy )(max
EI
M
R
1
dxEI
xMslope
)(
EI
xMcurvature
)(
dxEI
xMdeflection
)(
12
Analysis 45
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Column Stability
• short columns
• slenderness ratio = Le/r (L/d)
• radius of gyration =A
Ir
aactual
critical FA
Pf
weak axis
2
2
2
22
rL
E
LA
EAr
A
Pf
ee
criticalcritical
2
e
2
critical
rL
EAP
Analysis 46
Lecture 2
Architectural Structures III
ARCH 631
F2008abn
Column Stresses
• when a column gets stubby, Fy will limit the
load
• real world has loads
with eccentricity
• end conditions LKLe