Upload
trinhxuyen
View
219
Download
3
Embed Size (px)
Citation preview
S2016abnMechanics of Materials 1
Lecture 4
Elements of Architectural Structures
ARCH 614
ELEMENTS OF ARCHITECTURAL STRUCTURES:
FORM, BEHAVIOR, AND DESIGN
ARCH 614
DR. ANNE NICHOLS
SPRING 2016
four
mechanics
of materials
lecture
www.carttalk.com
S2016abnMechanics of Materials 2
Lecture 4
Elements of Architectural Structures
ARCH 614
Mechanics of Materials
• MECHANICS • MATERIALS
S2016abnMechanics of Materials 3
Lecture 4
Elements of Architectural Structures
ARCH 614
Mechanics of Materials
• external loads and their
effect on deformable bodies
• use it to answer question if structure
meets requirements of
– stability and equilibrium
– strength and stiffness
• other principle building requirements• economy, functionality and aesthetics
S2016abnMechanics of Materials 4
Lecture 4
Elements of Architectural Structures
ARCH 614
Knowledge Required
• material properties
• member cross sections
• ability of a material to resist breaking
• structural elements that resist excessive
– deflection
– deformation
S2016abnMechanics of Materials 5
Lecture 4
Elements of Architectural Structures
ARCH 614
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
S2016abnMechanics of Materials 6
Lecture 4
Elements of Architectural Structures
ARCH 614
A
Pfstress
Stress
• stress is a term for the intensity of a
force, like a pressure
• internal or applied
• force per unit area
S2016abnMechanics of Materials 7
Lecture 4
Elements of Architectural Structures
ARCH 614
• materials have a critical stress value
where they could break or yield
– ultimate stress
– yield stress
– compressive stress
– fatigue strength
– (creep & temperature)
Design
acceptance
vs. failure
S2016abnMechanics of Materials 8
Lecture 4
Elements of Architectural Structures
ARCH 614
allowableactual Ff • we’d like
• stress distribution may
vary: average
• uniform distribution
exists IF the member is
loaded axially
(concentric)
Design (cont)
S2016abnMechanics of Materials 9
Lecture 4
Elements of Architectural Structures
ARCH 614
Scale Effect
• model scale
– material weights by volume,
small section areas
• structural scale
– much more material weight,
bigger section areas
• scale for strength is not
proportional:L
L
L
2
3
S2016abnMechanics of Materials 10
Lecture 4
Elements of Architectural Structures
ARCH 614
Normal Stress (direct)
• normal stress is normal
to the cross section
– stressed area is
perpendicular to the
load
A
Pf cort
S2016abnMechanics of Materials 11
Lecture 4
Elements of Architectural Structures
ARCH 614
• stress parallel to a surface
Shear Stress
td
P
A
Pfv
ave
S2016abnMechanics of Materials 12
Lecture 4
Elements of Architectural Structures
ARCH 614
• stress on a surface by
contact in compression
Bearing Stress
td
P
A
Pf p
S2016abnMechanics of Materials 13
Lecture 4
Elements of Architectural Structures
ARCH 614
• normal stress caused by bending
Bending Stress
S
M
I
Mcfb
S2016abnMechanics of Materials 14
Lecture 4
Elements of Architectural Structures
ARCH 614
• shear stress caused by twisting
Torsional Stress
J
Tfv
S2016abnMechanics of Materials 15
Lecture 4
Elements of Architectural Structures
ARCH 614
• what structural elements see shear?
– beams
– bolts
– splices
– slabs
– footings
– walls
• wind
• seismic loads
Structures and Shear
connections
V
S2016abn
• connected members in tension cause
shear stress
• connected members in
compression cause
bearing stress
Bolts
Mechanics of Materials 16
Lecture 4
Elements of Architectural Structures
ARCH 614
S2016abn
• seen when 2 members are connected
Single Shear
4
2dv
P
A
Pf
Mechanics of Materials 17
Lecture 4
Elements of Architectural Structures
ARCH 614
S2016abn
Double Shear
F=
• seen when 3 members are connected
• two areas
4d
v 2
2P
A
2P
A2
Pf
Mechanics of Materials 18
Lecture 4
Elements of Architectural Structures
ARCH 614
S2016abn
• compression & contact
• projected area
Bolt Bearing Stress
td
P
A
Pf
projected
p
F=
Mechanics of Materials 19
Lecture 4
Elements of Architectural Structures
ARCH 614
S2016abnMechanics of Materials 20
Lecture 4
Elements of Architectural Structures
ARCH 614
Strain
• materials deform
• axially loaded materials change
length
• bending materials deflect
• STRAIN:
– change in length
over length + UNITLESS L
Lstrain
(S)
S2016abnMechanics of Materials 21
Lecture 4
Elements of Architectural Structures
ARCH 614
Shearing Strain
• deformations
with shear
• parallelogram
• change in angles
• stress:
• strain:
– unitless (radians)
s
L
tanL
s
S2016abnMechanics of Materials 22
Lecture 4
Elements of Architectural Structures
ARCH 614
Shearing Strain
• deformations
with torsion
• twist
• change in angle of line
• stress:
• strain:
– unitless (radians)
L
S2016abnMechanics of Materials 23
Lecture 4
Elements of Architectural Structures
ARCH 614
Load and Deformation
• for stress, need P & A
• for strain, need & L
– how?
– TEST with load and
measure
– plot P/A vs.
S2016abnMechanics of Materials 24
Lecture 4
Elements of Architectural Structures
ARCH 614
Material Behavior
• every material has its own response
– 10,000 psi
– L = 10 in
– Douglas Fir vs.
steel?
S2016abnMechanics of Materials 25
Lecture 4
Elements of Architectural Structures
ARCH 614
Behavior Types
• ductile - “necking”
• true stress
• engineering stress
– (simplified)
A
Pf
oA
Pf
S2016abnMechanics of Materials 26
Lecture 4
Elements of Architectural Structures
ARCH 614
Behavior Types
• brittle
• semi-brittle
S2016abnMechanics of Materials 27
Lecture 4
Elements of Architectural Structures
ARCH 614
Stress to Strain
• important to us in - diagrams:
– straight section
– LINEAR-ELASTIC
– recovers shape
(no permanent
deformation)
f
S2016abnMechanics of Materials 28
Lecture 4
Elements of Architectural Structures
ARCH 614
Hooke’s Law
• straight line has constant slope
• Hooke’s Law
• E
– Modulus of elasticity
– Young’s modulus
– units just like stress
f
E
1
Ef
S2016abnMechanics of Materials 29
Lecture 4
Elements of Architectural Structures
ARCH 614
Stiffness
• ability to resist strain
• steels
– same E
– different
yield points
– different
ultimate strength
uf
S2016abnMechanics of Materials 30
Lecture 4
Elements of Architectural Structures
ARCH 614
Isotropy & Anisotropy
• ISOTROPIC
– materials with E same at
any direction of loading
– ex. steel
• ANISOTROPIC
– materials with different E
at any direction of loading
– ex. wood is orthotropic
S2016abnMechanics of Materials 31
Lecture 4
Elements of Architectural Structures
ARCH 614
Elastic, Plastic, Fatigue
• elastic springs back
• plastic has permanent
deformation
• fatigue caused by
reversed loading
cycles
S2016abnMechanics of Materials 32
Lecture 4
Elements of Architectural Structures
ARCH 614
Plastic Behavior
• ductile
at yield stress
S2016abnMechanics of Materials 33
Lecture 4
Elements of Architectural Structures
ARCH 614
Lateral Strain
• or “what happens to the cross section
with axial stress”
• strain in lateral direction
– negative
– equal for isometric materials
E
f xx
0 zy ff
zy
S2016abnMechanics of Materials 34
Lecture 4
Elements of Architectural Structures
ARCH 614
Poisson’s Ratio
• constant relationship between
longitudinal strain and lateral strain
• sign!
x
z
x
y
strainaxial
strainlateral
E
f xzy
5.00
S2016abnMechanics of Materials 35
Lecture 4
Elements of Architectural Structures
ARCH 614
Calculating Strain
• from Hooke’s law
• substitute
• get
Ef
LE
A
P
AE
PL
(L)
S2016abnMechanics of Materials 36
Lecture 4
Elements of Architectural Structures
ARCH 614
Orthotropic Materials
• non-isometric
• directional values of
E and
• ex:
– plywood
– laminates
– polymer
composites
S2016abnMechanics of Materials 37
Lecture 4
Elements of Architectural Structures
ARCH 614
• why we use fave
• increase in stress atchanges in geometry
– sharp notches
– holes
– corners
–
Stress Concentrations
S2016abnMechanics of Materials 38
Lecture 4
Elements of Architectural Structures
ARCH 614
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
S2016abnMechanics of Materials 39
Lecture 4
Elements of Architectural Structures
ARCH 614
Maximum Stresses
S2016abnMechanics of Materials 40
Lecture 4
Elements of Architectural Structures
ARCH 614
Deformation Relationships
• physical movement
– axially (same or zero)
– rotations from axial changes
• relates to P
steel 20 kN
aluminum
AE
PL
S2016abnMechanics of Materials 41
Lecture 4
Elements of Architectural Structures
ARCH 614
Deformations from Temperature
• atomic chemistry reacts
to changes in energy
• solid materials• can contract with decrease in temperature
• can expand with increase in temperature
• linear change can
be measured per
degree
S2016abnMechanics of Materials 42
Lecture 4
Elements of Architectural Structures
ARCH 614
Thermal Deformation
• - the rate of strain per degree
• UNITS : ,
• length change:
• thermal strain:
– no stress when movement allowed
LTT
TT
F C
S2016abnMechanics of Materials 43
Lecture 4
Elements of Architectural Structures
ARCH 614
Coefficients of Thermal Expansion
Material Coefficients () [in./in./F]
Wood 3.0 x 10-6
Glass 4.4 x 10-6
Concrete 5.5 x 10-6
Cast Iron 5.9 x 10-6
Steel 6.5 x 10-6
Wrought Iron 6.7 x 10-6
Copper 9.3 x 10-6
Bronze 10.1 x 10-6
Brass 10.4 x 10-6
Aluminum 12.8 x 10-6
S2016abnMechanics of Materials 44
Lecture 4
Elements of Architectural Structures
ARCH 614
Stresses and Thermal Strains
• if thermal movement is restrained
stresses are induced
1. bar pushes on supports
2. support pushes back
3. reaction causes internal
stressE
LA
Pf
S2016abnMechanics of Materials 45
Lecture 4
Elements of Architectural Structures
ARCH 614
Superposition Method
– can remove a support to
make it look determinant
– replace the support with a
reaction
– enforce the geometry
constraint
S2016abnMechanics of Materials 46
Lecture 4
Elements of Architectural Structures
ARCH 614
Superposition Method
– total length change restrained to
zero0 TP
0 LTAE
PL
ETA
Pf
LTT AE
PLp
constraint:
sub:
S2016abnMechanics of Materials 47
Lecture 4
Elements of Architectural Structures
ARCH 614
Dynamics
• kinematics
– time, velocity, acceleration
– linear motion
– angular rotation
• kinetics
– forces causing motion
W = m·g
– work
– conservation of energy
2
21)0()( attvts
S2016abnMechanics of Materials 48
Lecture 4
Elements of Architectural Structures
ARCH 614
Dynamic Response
S2016abnMechanics of Materials 49
Lecture 4
Elements of Architectural Structures
ARCH 614
Dynamic Response
• period of vibration or
frequency
– wave
– sway/time period
• damping
– reduction in sway
• resonance
– amplification of sway
S2016abnMechanics of Materials 50
Lecture 4
Elements of Architectural Structures
ARCH 614
Frequency and Period
• natural period of vibration
– avoid resonance
– hard to predict seismic period
– affected by soil
– short period• high stiffness
– long period• low stiffness
“To ring the bell, the sexton must pull
on the downswing of the bell in time
with the natural frequency of the bell.”
S2016abnMechanics of Materials 51
Lecture 4
Elements of Architectural Structures
ARCH 614
Design of Members
• beyond allowable stress...
• materials aren’t uniform 100% of the
time
– ultimate strength or capacity to failure may
be different and some strengths hard to
test for
• RISK & UNCERTAINTY
A
Pf u
u
S2016abnMechanics of Materials 52
Lecture 4
Elements of Architectural Structures
ARCH 614
Factor of Safety
• accommodate uncertainty with a safety
factor:
• with linear relation between load and
stress:
SF
loadultimateloadallowable
.
stressallowable
stressultimate
loadallowable
loadultimateSF .
S2016abnMechanics of Materials 53
Lecture 4
Elements of Architectural Structures
ARCH 614
Load and Resistance Factor Design
• loads on structures are
– not constant
– can be more influential on failure
– happen more or less often
– UNCERTAINTY
- resistance factor
- load factor for (D)ead & (L)ive load
nLLDDu RRRR