structural analysis (statics & mechanics) - faculty...
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S2017abnAnalysis 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 2017
S2017abnAnalysis 2
Lecture 2
Applied Architectural Structures
ARCH 631
Structural Requirements
• strength
• serviceability
– deflections
• efficiency
– economy of materials
• construction
• cost
• other
www.pbs.org/wgbh/buildingbig/
S2017abnAnalysis 3
Lecture 2
Applied Architectural Structures
ARCH 631
Structure Requirements
• strength &
equilibrium
– safety
– stresses
not greater
than
strength
– adequate
foundation
S2017abnAnalysis 4
Lecture 2
Applied Architectural Structures
ARCH 631
Structure Requirements
• stability &
stiffness
– stability of
components
– minimum
deflection and
vibration
– adequate
foundation
S2017abnAnalysis 5
Lecture 2
Applied Architectural Structures
ARCH 631
• economy and
construction
– minimum material
– standard sized
members
– simple connections
and details
– maintenance
– fabrication/ erection
Structure Requirements
S2017abnAnalysis 6
Lecture 2
Applied Architectural Structures
ARCH 631
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
S2017abnAnalysis 7
Lecture 2
Applied Architectural Structures
ARCH 631
• dead load
– static, fixed, includes
material weights, fixed
equipment
• live load
– transient and moving
loads (including
occupants)
• snow load
Structural Loads - STATIC
S2017abn
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
Lecture 2
Applied Architectural Structures
ARCH 631
Structural Loads – STATIC & DYNAMIC
S2017abn
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
Lecture 2
Applied Architectural Structures
ARCH 631
Structural Loads - DYNAMIC
S2017abnAnalysis 10
Lecture 2
Applied Architectural Structures
ARCH 631
Dynamic Response
S2017abnAnalysis 11
Lecture 2
Applied Architectural Structures
ARCH 631
Dynamic Response
• period of vibration or
frequency
– wave
– sway/time period
• damping
– reduction in sway
• resonance
– amplification of sway
S2017abnAnalysis 12
Lecture 2
Applied Architectural Structures
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
S2017abnAnalysis 13
Lecture 2
Applied Architectural Structures
ARCH 631
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?
S2017abnAnalysis 14
Lecture 2
Applied Architectural Structures
ARCH 631
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
S2017abnAnalysis 15
Lecture 2
Applied Architectural Structures
ARCH 631
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
S2017abnAnalysis 16
Lecture 2
Applied Architectural Structures
ARCH 631
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
S2017abnAnalysis 17
Lecture 2
Applied Architectural Structures
ARCH 631
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
S2017abn
• 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
Lecture 2
Applied Architectural Structures
ARCH 631
Force Components
Fy
Fx
F
x
y
Fy
Fx
F
Fy
Fx
F
S2017abn
• weight (F = ma)
• concentrated
• distributed
– uniform
– linear
• friction
– F =N
W tA
Analysis 19
Lecture 2
Applied Architectural Structures
ARCH 631
Load Types
x
0
w
𝑤𝑥
2=𝑊
2
S2017abn
load
w tributary widtharea
• tributary load
– think of water flow
– “concentrates” load of
area into center
Analysis 20
Lecture 2
Applied Architectural Structures
ARCH 631
Load Tracing
width
S2017abn
• 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
Lecture 2
Applied Architectural Structures
ARCH 631
Moments
S2017abn
• analytically
• free body diagrams
0x xR F 0y yR F 0M M
http://www.physics.umd.edu
Analysis 22
Lecture 2
Applied Architectural Structures
ARCH 631
Equilibrium
S2017abnAnalysis 23
Lecture 2
Applied Architectural Structures
ARCH 631
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
S2017abnAnalysis 24
Lecture 2
Applied Architectural Structures
ARCH 631
Supports and Connections
S2017abnAnalysis 25
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ARCH 631
Supports and Connections
S2017abn
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
Lecture 2
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ARCH 631
Centroid
S2017abn
2
x cx yI I Ad
• 2nd moment area
–math concept
– area x (distance)2
• need for behavior of
– beams
– columns
Analysis 27
Lecture 2
Applied Architectural Structures
ARCH 631
Moments of Inertia
S2017abnAnalysis 28
Lecture 2
Applied Architectural Structures
ARCH 631
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
S2017abnAnalysis 29
Lecture 2
Applied Architectural Structures
ARCH 631
Internal Beam V & M (+P)
• maximums needed for design
• Mmax at V = 0
V
L
+
M+
-
L/2
L
+V
+M
S2017abnAnalysis 30
Lecture 2
Applied Architectural Structures
ARCH 631
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
S2017abn
Pstress
A
• stress is a term for the intensity of a
force, like a pressure
• internal or applied
• force per unit area
Analysis 31
Lecture 2
Applied Architectural Structures
ARCH 631
Stress
S2017abn
• 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
Lecture 2
Applied Architectural Structures
ARCH 631
Stress Types
S2017abn
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
Lecture 2
Applied Architectural Structures
ARCH 631
Stress Types
S2017abn
• 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
Lecture 2
Applied Architectural Structures
ARCH 631
Bolt Stresses
S2017abn
• tension and compressive stress caused
by bending
• shear stress from bending
b
Mc Mf
I S
v ave
VQf
Ib
Analysis 35
Lecture 2
Applied Architectural Structures
ARCH 631
Bending Stresses
S2017abn
longitudinalV VQ
p I
connected area
connector
VQnF p
I
• plates with
– nails
– rivets
– bolts
• splices
Analysis 36
Lecture 2
Applied Architectural Structures
ARCH 631
Connectors Resisting Shear
x
y
ya
4”
2”
2”
12”
8” p
p p
4.43”
S2017abn
Lstrain
L
Analysis 37
Lecture 2
Applied Architectural Structures
ARCH 631
Strain
• materials deform
• axially loaded materials change
length
• bending materials deflect
• STRAIN:
– change in length over length
S2017abnAnalysis 38
Lecture 2
Applied Architectural Structures
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
S2017abn
• important to us in - diagrams:
– straight section
– LINEAR-ELASTIC
– recovers shape
(no permanent
deformation)
f E
PL
AE
Analysis 39
Lecture 2
Applied Architectural Structures
ARCH 631
Stress to Strain
f
S2017abnAnalysis 40
Lecture 2
Applied Architectural Structures
ARCH 631
Behavior Types
• brittle
• semi-brittle
S2017abnAnalysis 41
Lecture 2
Applied Architectural Structures
ARCH 631
Plastic Behavior
• ductile
at yield stress
S2017abn
• if we need to knowwhere max f and fv
happen:
Analysis 42
Lecture 2
Applied Architectural Structures
ARCH 631
Maximum Stresses
F
1c o s0 oA
Pf max
5.0sincos45
22
maxmax
f
A
Pf
o
v
S2017abn
• - the rate of strain per degree
• UNITS : ,
• length change:
• thermal strain:
– no stress when movement allowed
Analysis 43
Lecture 2
Applied Architectural Structures
ARCH 631
Thermal Deformation
F C
LTT
TT
S2017abn
• curvature, R
Analysis 44
Lecture 2
Applied Architectural Structures
ARCH 631
Beam Deflections
EI
M
R
1
EI
xMcurvature
)(
dxEI
xMslope
)(
dxEI
xMdeflection
)(
a l lo wa b lea ctu a lxy )(max
S2017abn
• short columns
• slenderness ratio = Le/r (L/d)
• radius of gyration =
Analysis 45
Lecture 2
Applied Architectural Structures
ARCH 631
Column Stability
weak axis
aactual
critical FA
Pf
A
Ir
2
2
2
22
rL
E
LA
EAr
A
Pf
ee
criticalcritical
2
e
2
critical
rL
EAP
S2017abn
• when a column gets stubby, Fy will limit the
load
• real world has loads
with eccentricity
• end conditions
Analysis 46
Lecture 2
Applied Architectural Structures
ARCH 631
Column Stresses
LKLe