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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

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