structural analysis (statics &...
Transcript of structural analysis (statics &...
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S2018abnAnalysis 1
Lecture 2
Applied Architectural Structures
ARCH 631
two
structural analysis
(statics & mechanics)
lecture
APPLIED ACHITECTURAL STRUCTURES:
STRUCTURAL ANALYSIS AND SYSTEMS ARCH 631
DR. ANNE NICHOLS
SPRING 2018
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Lecture 2
Applied Architectural Structures
ARCH 631
Structural Requirements
• strength
• serviceability
– deflections
• efficiency
– economy of materials
• construction
• cost
• other
www.pbs.org/wgbh/buildingbig/
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Structure Requirements
• strength &
equilibrium
– safety
– stresses
not greater
than
strength
– adequate
foundation
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ARCH 631
Structure Requirements
• stability &
stiffness
– stability of
components
– minimum
deflection and
vibration
– adequate
foundation
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• economy and
construction
– minimum material
– standard sized
members
– simple connections
and details
– maintenance
– fabrication/ erection
Structure Requirements
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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
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• dead load
– static, fixed, includes
material weights, fixed
equipment
• live load
– transient and moving
loads (including
occupants)
• snow load
Structural Loads - STATIC
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W d hF C q A
• 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”
Analysis 8
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Structural Loads – STATIC & DYNAMIC
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W
ZICWVR
• earthquake loads
– seismic, movement of
ground (3D)
– building mass responds
– static models often used,
V is static shear
• impact loads
– rapid, energy loads
Analysis 9
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Structural Loads - DYNAMIC
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Dynamic Response
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Dynamic Response
• period of vibration or
frequency
– wave
– sway/time period
• damping
– reduction in sway
• resonance
– amplification of sway
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ARCH 631
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
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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?
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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
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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
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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
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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)
http://www.physics.umd.edu
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• convenient to resolve into 2 vectors
• at right angles
• in a “nice” coordinate system
• is between Fx and F from Fx
tany
x
F
F
2 2
x yF F F
sinyF F
cosxF F
Analysis 18
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Force Components
Fy
Fx
F
x
y
Fy
Fx
F
Fy
Fx
F
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• weight (F = ma)
• concentrated
• distributed
– uniform
– linear
• friction
– F =N
W tA
Analysis 19
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Load Types
x
0
w
𝑤𝑥
2=𝑊
2
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load
w tributary widtharea
• tributary load
– think of water flow
– “concentrates” load of
area into center
Analysis 20
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Load Tracing
width
../animations/unesco_load.mov../animations/unesco_load.mov
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• 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
M F d
F
A
C
B
d
Analysis 21
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Moments
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• analytically
• free body diagrams
0x xR F 0y yR F 0M M
http://www.physics.umd.edu
Analysis 22
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Equilibrium
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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
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Supports and Connections
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Supports and Connections
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x Ax
A
y Ay
A
• “average” x & y of an area
• for a volume of constant thickness
– where is weight/volume– center of gravity = centroid of area
W t A
Analysis 26
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Centroid
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2
x cx yI I Ad
• 2nd moment area
–math concept
– area x (distance)2
• need for behavior of
– beams
– columns
Analysis 27
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Moments of Inertia
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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
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Internal Beam V & M (+P)
• maximums needed for design
• Mmax at V = 0
V
L
+
M+
-
L/2
L
+V
+M
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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
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Pstress
A
• stress is a term for the intensity of a
force, like a pressure
• internal or applied
• force per unit area
Analysis 31
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Stress
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• normal stress is normal
to the cross section
• shear stress parallel
to a surface
t or c
Pf
A
v
P Pf
A td
Analysis 32
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Stress Types
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p
P Pf
A td
v
Tf
J
• bearing stress on a
surface by contact in
compression
• torsional stress by shear
from twisting
Analysis 33
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Stress Types
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• single shear
• double shear
• bearing
2
4
vd
P Pf
A
2
4
2 2
2v
d
P PP
fA A
p
projected
P Pf
A td
Analysis 34
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Bolt Stresses
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• tension and compressive stress caused
by bending
• shear stress from bending
b
Mc Mf
I S
v ave
VQf
Ib
Analysis 35
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Bending Stresses
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longitudinalV VQ
p I
connected area
connector
VQnF p
I
• plates with
– nails
– rivets
– bolts
• splices
Analysis 36
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ARCH 631
Connectors Resisting Shear
x
y
ya 4”
2”
2”
12”
8” p
p p
4.43”
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Lstrain
L
Analysis 37
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ARCH 631
Strain
• materials deform
• axially loaded materials change
length
• bending materials deflect
• STRAIN:
– change in length over length
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ARCH 631
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
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• important to us in - diagrams:
– straight section
– LINEAR-ELASTIC
– recovers shape
(no permanent
deformation)
f E
PL
AE
Analysis 39
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ARCH 631
Stress to Strain
f
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Behavior Types
• brittle
• semi-brittle
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Plastic Behavior
• ductile
at yield stress
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• if we need to knowwhere max f and fvhappen:
Analysis 42
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ARCH 631
Maximum Stresses
F
1c o s0 oA
Pf max
5.0sincos45
22
maxmax
f
A
Pf
o
v
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• - the rate of strain per degree
• UNITS : ,
• length change:
• thermal strain:
– no stress when movement allowed
Analysis 43
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Applied Architectural Structures
ARCH 631
Thermal Deformation
F C
LTT
TT
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• curvature, R
Analysis 44
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Applied Architectural Structures
ARCH 631
Beam Deflections
EI
M
R
1
EI
xMcurvature
)(
dxEIxM
slope)(
dxEIxM
deflection)(
a l lo wa b lea ctu a lxy )(max
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• short columns
• slenderness ratio = Le/r (L/d)
• radius of gyration =
Analysis 45
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Column Stability
weak axis
aactual
critical FA
Pf
A
Ir
22
2
22
rL
E
LA
EAr
A
Pf
ee
criticalcritical
2
e
2
critical
rL
EAP
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• when a column gets stubby, Fy will limit the
load
• real world has loads
with eccentricity
• end conditions
Analysis 46
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Column Stresses
LKLe