d EG1108 Lecture Slides for Week 4 - Node Analysis to Maximum Power
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Chapter 1: Steel Materials & Fundamentals
of Steel Design
Why Design and Build Structures Using Steel?
Advantages of Steel
High strength-to-weight ratio High ductility and energy absorption (good for seismic applications) Slender members capable of very long spans Equal strength and modulus in tension and compression Excellent shear strength Versatile for construction of complex and unique structures No need for labor intensive formwork or shoring Can serve structural & architectural functions
Disadvantages of Steel
Slender sections prone to buckling and vibration problems Some details are susceptible to fatigue failure Material and fabrication costs can be high Susceptible to corrosion Temperature variations can cause distortion of slender members Final structure is sensitive to construction tolerances
Can beavoided with proper design
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Basics of Steel Fabrication
Iron is melted and mixed with other alloying elements. The melted iron
is cast into large slabs, blooms or billets and cooled gradually until it
hardens. Primary steel making uses pig iron, a partly processed form
of iron ore, as the main precursor. In contrast, secondary steel making
uses scrap metal as the main precursor and is generally achieved using
an electric arc furnace.
Hot Rolling
Steel is heated to a red hot condition and passed through a series of
rollers to form gradually to the desired shape. This distorts the crystal
structure of the steel. Gradual cooling allows recrystalization of the
steel grains.
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Hot-Rolling Process and Hot-Rolled Sections
Wide flange (beams) W
H Piles C Channels Angles L Plate
Bar
Hollow Structural Sections HSS
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Residual Stresses Due to rolling, differential cooling, and welding
Yielding will occur when yrcAP
rc = residual compressive stress Residual stresses have major implications on inelastic buckling of
compression members as we will see later.
Cold Forming
Thin sheets or plates of steel can be mechanically formed to the desired
shape using a press or a brake without heating. This process, known as
cold working, typically results in increased strength and hardness, but
reduced ductility.
Comp CompTen.
Has been measure as high as 20ksi
uneven cooling
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C channels, Z purlins, Sigma sections, sheet piles, steel decking
Steel Material Characteristics
Steel is a metallic alloy composed primarily of Iron (Fe) and Carbon (C).
While at the macro-scale steel is a homogeneous material, at the micro-
scale the granular structure of steel is clearly evident. The individual
grains are ordered crystals of iron, carbon and other alloying elements.
The evolution of this crystal structure during processing gives steel its
unique characteristics. The properties of the steel can be widely varied
by altering the fundamental crystal structure. This can be done during
fabrication, by heat treatment, by including various alloys in the steel
and by varying the carbon content.
(www.prosmetal.com) (www.structuresmag.org)
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Carbon Content
Structural steel is produced by melting iron (Fe) and combining it with
various alloying elements. Iron is a ductile, soft, and weak metallic
element. Besides iron, carbon (C) is the most common element in
typical mild structural steels. Carbon is a hard, strong, and brittle non-
metallic element. Combining these two elements, in different
proportions, yields steel with different properties. The relationship
between carbon content, temperature, and crystal structure is defined by
the phase diagram.
Individual grains bcc crystal structure
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Crystal structure and phase diagram for iron-carbon alloys (Campbell, 2008)
(http://threeplanes.net/toolsteel.html)
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(http://www.gowelding.com/met/carbon.htm)
As it cools from the liquid state, pure iron (C < 0.008%) forms a body
centered cubic (bcc) structure known as ferrite ( iron) at a temperature of approximately 1540oC. With continued cooling, the crystal
undergoes a shift to a face centered cubic (fcc) structure called austenite
( iron) at a temperature of about 1400oC. Continued cooling results in a second shift back to a bcc ferrite structure ( iron).
However, pure iron is too soft to be useful for typical structural
applications. As such, it is commonly combined with carbon to provide
strength and hardness. The carbon content of most structural steels is
typically within the range of 0.1% - 0.5%. Upon cooling, steels form
crystals of ferrite and cementite or iron carbide (Fe3C), a hard brittle
compound. Increasing the carbon content increases the hardness and
strength of steel while reducing its ductility, toughness, and weldability.
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The effect of carbon content on several steel properties is illustrated
below.
Effect of carbon content on steel properties (Davis et al., 1982)
Crystal Structure & Grain Size
Perfectly ordered crystal structures are typically quite brittle. The
characteristic ductility of steel results from the presence of
discontinuities, or dislocations, in the crystal structure. Yielding occurs
as these dislocations move along slip plains through the crystal structure.
Plastic deformation due to movement of dislocations (Campbell, 2008)
~ 50 ksi
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As steel cools, crystals begin to form around nucleation sites. As these
crystals grow, they begin to intersect forming individual grains with
different orientations. As such, the mechanical properties of steel are
also influenced by the size of the grains that form upon cooling. Fine-
grained steels generally have higher yield strengths, ductility, and
fracture strength than coarse grained steels. Therefore, it is often
desirable to fabricate steel in such a way as to produce a fine-grained
microstructure.
The relationship between grain size and yield strength for different
metals is given by the Hall-Petch relationship (illustrated below).
Reducing grain size is very effective in increasing yield strength for iron
(Fe) while it is less effective for other metals. Decreasing grain size also
increases toughness and decreases the ductile-brittle transition
Coarse-grained steel (short, direct slip planes)
Fine-grained steel (longer, winding slip planes)
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temperature (DBTT), discussed below. All of these are generally seen
as positive features of fine grained steels.
Heat Treatment
Grain size and microstructure can be controlled by subjecting steel to
different types of heat treatment and carefully controlling heating and
cooling rates to achieve the desired mechanical properties:
Annealing Heating to 1500oF, hold temperature and gradually cool.
o Relieves internal stresses which form during mechanical working
o Increases ductility and toughness of steel o Reduces steel strength and hardness
d = 0.25 mm d = 0.01 mm
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Hardening Heating to 1500oF followed by rapid cooling (quenching) in suitable fluid such as water or oil.
o Rearranges atomic structure of steel o Increases steel hardness and strength o Reduces ductility and toughness
Tempering Heating to between 400oF and 1000oF followed by gradual or rapid cooling. Typically done after hardening to restore
ductility and toughness.
Alloying Elements
Steels with different mechanical properties (stainless steel, tool steel etc.) can be formed by alloying steel with various other elements. Some common alloying elements and their function are (Davis et al., 1982): Aluminum (Al) helps expel gasses from molten steel (Al killed
steels)
Chromium (Cr) produces stainless and heat resisting steel, increases hardness and strength
Copper (Cu) enhances corrosion resistance Manganese (Mn) removes impurities, improves rollability,
slightly increases hardness and strength
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Nickel (Ni) produces finer grain structure, makes quenching more effective, increases strength with little loss of ductility
Silicon (Si) deoxidizer, increases strength without reducing ductility, increases hardness slightly
Vanadium (V) increases elastic and tensile strengths, produces fine grained clean metal
Other alloying agents have been adopted to enhance the workability of
steel and to give steel various other properties
Stress-Strain Response
Steel is a ductile material. Its stress-strain response is idealized by an
elastic-perfectly plastic relationship. Many of the principles that we
implement in design are based on the inherent characteristics of the steel
and the simplifications that we make in representing this behavior.
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Stress-Strain Relationship of ASTM A572 Steel
0
10
20
30
40
50
60
70
80
0 0.1 0.2 0.3 0.4
Stre
ss (M
Pa)
Strain (in/in)
0
10
20
30
40
50
60
70
80
0 0.01 0.02 0.03 0.04 0.05
Stre
ss (M
Pa)
Strain (in/in)
T
T
Strain hardening
region
Plastic region
Elastic region
Strain softening
region
rupture
u = ultimate strain
Fu = ultimate strength
Strain hardening
region
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Stress-Strain Relationship of ASTM A572 Steel
This stress-strain behavior is characteristic of low-carbon, or mild
structural steels near room temperature. At extreme temperatures, the
mechanical properties of steel are quite different.
0
10
20
30
40
50
60
70
80
0 0.002 0.004 0.006 0.008 0.01
Stre
ss (M
Pa)
Strain (in/in)
E = Elastic modulus,
1 Fy = Yield
Strength
y = Fy/E Yield Strain
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(Bruneau et al., 2011)
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Fracture Toughness
Toughness is the capacity of steel to dissipate energy during
deformation. In steel it is commonly measured using the Charpy V-
notch test (CVN).
Standard CVN specimen
Toughness = W(h2 h1)
Strain rate effect High strain rate
Slow bending test
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Steel toughness is
dramatically affected
by temperature. At
higher temperatures
steel exhibits ductile
behavior with
significant energy
absorption. However,
below the ductile-
brittle transition
temperature (DBTT)
steel becomes brittle
with low energy
absorption capacity.
This makes steel
particularly susceptible
to fatigue damage at
low temperatures.
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Strain Aging
Strain aging is a phenomenon that develops due to cold working of steel
materials. If steel is loaded, unloaded and immediately reloaded, it
typically follows a similar loading path as shown by path 1 below. The
reloaded steel does not exhibit an inelastic plateau if it was previously
loaded into the strain-hardening range. However, if the steel is loaded
and unloaded and the left unstressed for a time, particularly at elevated
temperatures, a phenomenon called strain aging occurs. In this case,
the inelastic plateau of the steel is re-established and the material
becomes stronger and more brittle (path 2 below).
Strain aging is generally caused by the diffusion of carbon atoms (that
arent locked in iron carbide crystals), and nitrogen atoms through the
crystal structure of the steel in the spaces between atoms (interstitials).
Stre
ss
Strain
Failure (Rupture)
Permanent Set
Load
Unload
Reload
path (1)
Failure (Rupture)
path (2)
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These atoms move through interstitials and collect near dislocations.
The presence of these interstitial atoms near dislocations pins the
dislocations making it harder for them to move. This increases the yield
and ultimate strength of the material and reduces its toughness. The
process is accelerated at elevated temperatures because the increased
energy facilitates movement of carbon atoms through the crystal lattice
structure.
This process can be particularly problematic in cold worked steel
structures that are required to resist repeated cyclic loads or are required
to have significant ductility (bridges and transportation infrastructure,
cold worked and galvanized structures).
Buckling of Compression Members
Buckling occurs when relatively slender elements are subjected to
compression loading. At low load levels, the compression element
exhibits only one stable configuration. As the load increases, once the
applied load reaches a critical value, the compression element can
remain in equilibrium in one of two configurations: the original, un-
deflected configuration or a buckled, deflected configuration. This is
known as bifurcation or buckling instability.
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Global Buckling
For slender, elastic compression members, the phenomenon of buckling
was first studied by Euler. His formulation lead to the well known
Euler buckling load
2
2
cr )kL(EIP
where E and I are the elastic modulus of the material and the moment of
inertia of the section about the axis of buckling, respectively and kL is
the effective length (distance between the inflection points of the
P < Pcr P = Pcr
Lateral displacement,
App
lied
load
, P
P = Pcr
L
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buckled element). The effective length factor k depends on the
boundary conditions of the member.
The buckling capacity of a member can be effected by three primary
factors:
1. Residual stresses primary reason 2. Initial out-of-straightness
3. Load eccentricity
Question: How would these 3 factors affect the P- relationship of a compression member? Illustrate on the P- graph on the previous page.
For non-slender members, the existence of high levels of residual
stresses can lead to premature yielding of portions of the cross section at
load levels lower than the Euler buckling load. In this case the member
may exhibit inelastic buckling prior to yielding but at a load lower than
the elastic buckling load.
Once believed to be primary causes
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Local Buckling
Similarly to global buckling, local elements (web or flange) of a cross-
section can buckle under compressive stresses. This is based on
consideration of plate bending and leads to an expression for the critical
buckling stress of:
y22
2
cr F)t/b)(1(12EkF
where k in this case is a parameter (different from the effective length
factor described previously) that depends on the boundary conditions of
the plate element. This expression is used to establish limiting values of
flange and web slenderness, bf/2tf and h/tw respectively, which define the
boundaries between compact, non-compact, and slender elements and
cross-sections.
Fy F
r/
2t
2
r/EF
22
e r/EF
Elastic buckling Inelastic buckling
Limiting slenderness ratio
Limiting buckling
stress
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Local buckling limits for elements in axial compression
For compression members, the primary consideration for local buckling
relates to how much of the cross section is rendered ineffective due to
local buckling. For members with non-slender elements ( < r), the entire cross-section is effective. For members with slender elements ( > r), a reduction factor is applied to account for the lost efficiency of the section due to local buckling.
Response of Flexural Members
Flexure of steel beams is assumed to conform to the basic assumptions
of beam theory (plane sections remain plane, normals remain normal,
symmetric sections). Additionally, to simplify the analysis and
design, an elastic-perfectly plastic stress-strain relationship is assumed.
Fy Elastic local buckling
slenderness ratio, bf/2tf for FLB h/tw for WLB
Critical stress
Transition curve
~2 Fy
r, Limiting value from AISC Spec Table B4.1a
Efficiency of section reduced due to local buckling (AISCS E7)
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Based on these assumptions we can track the evolution of bending
stresses in a steel beam as follows:
Note: to achieve the fully plastic moment capacity large strains, (7y to 9y), must be developed in the compression flange.
y y y y
y y y >>y
N.A.
N.A.
yMM yMM py MMM pMM
Initial yielding
Fully plastic
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Regardless of whether or not they achieve their full plastic capacity, Mp,
steel beams ultimately fail due to buckling in one of three modes:
1) Flange local buckling (FLB)
2) Web local buckling (WLB)
3) Beam lateral-torsional buckling (LTB)
Four different types of behavior are illustrated below:
1) The beam achieves its plastic moment capacity, Mp, and
exhibits significant inelastic deflection before ultimately failing
due to buckling (Compact section)
2) The beam achieves its plastic moment capacity, Mp, but fails
due to lateral-torsional buckling prior to achieving significant
inelastic deflection (Compact section with inelastic LTB of
member)
3) The beam fails due to inelastic buckling (which is affected by
the presence of residual stresses) prior to achieving its full
~10~18% yM
deflection
SFM yy
LC
p y
y
M F Z 1.10 to 1.18M
pM
M M
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plastic moment capacity, Mp. (Non-Compact section, or
Compact section with inelastic LTB of member)
4) The beam buckles elastically prior to the onset of any inelastic
behavior. (Slender section, or Non-Compact or Compact
Section with elastic LTB of member)
Therefore, when designing and analyzing doubly symmetric steel beams,
we must consider the behavior at two levels: 1) local buckling at the
cross-section level and 2) lateral-torsional buckling at the member level.
Local buckling limits for elements in flexural compression
Similarly to compression members, local buckling of steel elements in
flexural compression can be controlled by limiting the slenderness ratio
of the flange and the web. The flange local buckling (FLB) behavior of
steel beams is illustrated below where:
Inelastic due to residual stresses, Mr
inelastic
elastic
deflectionLC
Moment
1 2
4
Complete yielding, Mp
3Initial yielding, My
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Mr = 0.7FySx
Fy = yield strength
Sx = Ix/y (elastic section modulus )
Ix = moment of inertia about x-axis
y = distance from neutral axis to most extreme fibers in the section
The moment Mr is the reduced moment at which inelastic behavior
initiates due to the effect of residual stresses.
All hot-rolled I shaped sections have compact webs for the range of
yield strengths used in building construction. Welded, built-up sections
with non-compact or slender webs are classified as plate girders and are
Bending Moment
Mp
p r Limiting values from
AISC Spec Table B4.1b
slenderness ratio, bf/2tf for FLB
Mr My Effect of residual stresses
Slender Compact
Non-compact
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designed as such. For these types of members, the slenderness of the
web can reduce the flange local buckling strength of the section.
Why?
Lateral-Torsional Buckling (LTB)
Lateral-torsional buckling occurs in members that do not have adequate
lateral support to prevent global instability of the compression region of
the beam. As such, the LTB capacity of a beam is governed by its
unbraced length, Lb, the distance between lateral supports.
Bending Moment
Mp (or less)
Lp Lr
Limiting values from AISC Spec Table B4.1b
Unbraced Length, Lb
Mr My Effect of residual stresses
Elastic LTB Section capacity governs
Inelastic LTB
Constant moment Non-constant moment
(Cb factor)
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Ductility
Ductility is defined as the ability of a material deform under tensile
stress. In steel, ductility comes from yielding and plastic flow and is
associated with increased energy dissipation or toughness. Ductility can
be defined at the material level, the section level and the structure level.
While there are many definitions, they generally relate the behavior at
ultimate to the behavior at yielding. As such, at the various levels,
ductility can be defined as:
Material ductility: strain ductility = u/y Section ductility: curvature ductility = u/y Joint ductility: rotation ductility = u/y Member ductility: deflection ductility = u/y
Where subscripts u and y represent values at ultimate and yield
respectively.
Steel is an inherently ductile material with a strain ductility ratio on the
order of 150. However, at the section level, ductility can be reduced by
local buckling. Further, at the member level, ductility can be reduced
due to the localization of inelastic behavior as illustrated below.
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0
10
20
30
40
50
60
70
80
0 0.1 0.2 0.3 0.4
Stre
ss (M
Pa)
Strain (in/in)
Material Level
u/y 100-160
Curvature,
Moment, M
y u
u/y 7 9
Section Level
My Mp
Member Level
Deflection,
Load, P
y u
u/y 3 5
Py
Pp
P
Bending Moment, M
Mp My My
Beam Span, L
Curvature,
Plastic hinge region (~0.3 0.4 L)
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Calculation of the Plastic Moment Capacity
CA = area in compression
TA = area in tension
Cy = distance from p.n.a to centroid of comp. force res.
Ty = distance from p.n.a to centroid of tension force res.
0XF TC TyCy AA CA = TA = 2/A
0.. anM
pM = TyTCyC yAyA = yTTCC yAyA )(
pM = yZ where: Z = plastic section modulus = first moment of area w.r.t. the plastic neutral axis(p.n.a.)
= TTCC yAyA = i i PNA A y
cA
cy
ty
y
y
C
T
p.n.a
Mp
tA
(defines plastic neutral axis, for homogeneous cross-sections)
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Design Philosophies Two philosophies of design have been adopted by AISC:
1) Allowable Stress Design (ASD) 2) Load and Resistance Factor Design (LRFD)
While both design methodologies are accepted and widely used, there is a general trend by code and specification writing bodies, and the engineering community to move towards LRFD. Allowable Stress Design The basic principle behind ASD can be summarized as the stress in a member due to the effect of applied loads should not exceed a specified allowable value. Required strength to < specified value support applied loads (allowable strength)
n
aRR
where: Ra = Required Strength Rn = Nominal Strength = Factor of Safety (depends on nature of load applied
& failure mode) Typically, = 1.67 for yielding or buckling = 2.00 for rupture The Factor of Safety defined in this way does not give us any idea about the probability of failure of a structure designed according to this philosophy.
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Load and Resistance Factor Design The LRFD design philosophy is formulated so that members designed according to this formulation will have a known, acceptable probability of failure. This design philosophy accounts for the inherent statistical variability of applied loads, material properties and member dimensions.
Required strength < design strength.
Ru Rn where: Ru = Required strength (LRFD) Rn = Nominal strength (Tn, Pn, Vn, Mn) = Resistance factor Rn = Design strength Required Strength, Ru Ru = i Qi where: i = Load Factor Qi = Effect of applied load Applied Loads: Dead Load (D) Earthquake (E) Fluid Pressure (F) Live Load (L) Snow (S) Flood (Fa) Roof Live Ld. (Lr) Rain (R) Lat. Earth Press. (H) Wind (W) Weight of Ice (Di) Self straining (T)
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Based on ASCE 7, AISC defines seven load combinations:
1. 1.4D 2. 1.2D + 1.6L +0.5(Lr or S or R) 3. 1.2D + 1.6(Lr or S or R) + (0.5L or 0.5W) 4. 1.2D + 1.0W + 0.5L + 0.5(Lr or S or R) 5. 1.2D + 1.0E + L + 0.2S 6. 0.9D + 1.0W Cases when dead load counteracts 7. 0.9D + 1.0E effect of applied loads
Applied loads are inherently variable and uncertain. Some load effects, such as live loads and wind loads, are exhibit more variability than others, such as dead load. Generally, each load effect can be represented by a statistical distribution.
Similarly, the material properties and geometry of members that define their resistance are also statistically variable.
Generally we can define a limit state function of the form: g(x) = Resistance Load = Rn i Qi
Failure is defined by g(x) = 0 Since, the resistance, Rn, and the load effects, Qi, each have their own statistical distributions, the load and resistance factors, i and respectively, are adjusted to achieve an acceptable probability of failure defined by the reliability index, . This process is called code calibration. While the target probability of failure is the subject of debate, many US design codes accept 0.02% as a tolerable probability of failure. This corresponds to a reliability index, = 3.5.
ResistanceLoad Failure region
Prob
abili
ty o
f O
ccur
renc
e