N.W.F.P. University of Engineering and
Technology Peshawar
Lecture 10: Beams
By: Prof Dr. Akhtar Naeem Khan
chairciv@nwfpuet.edu.pk
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Beam
 A beam is generally considered to be
any member subjected principally to
transverse gravity or vertical loading.
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Beam
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Beam
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Types of Beams
 Girders usually the most important beams.
 Stringers Longitudinal bridge beams
spanning between floor beams.
 Floor Beams In buildings, a major beam
usually supporting joists; a transverse beam
in bridge floors.
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Types of Beams
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Types of Beams
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Types of Beams
 Joists A beam supporting floor construction but
not major beams.
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Types of Beams
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Types of Beams
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Types of Beams
 Purlins Roof beam spanning between trusses.
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Types of Beams
 Girts Horizontal wall beams serving
principally to resist bending due to wind
on the side of an industrial building.
 Lintels Member supporting a wall over a
window or door opening.
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Sections used for Beams
 Among the steel shapes that are used as
beam include:
 W shapes, which normally prove to be the
most economical beam sections and they
have largely replaced channels and S
sections for beam usage.
 Channels are sometimes used for beams
subjected to light loads, such as purlins
and at places where clearances available
require narrow flanges
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Sections used for Beams
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Design Approaches
Elastic Design
 For many years the elastic theory has
been the bases for the design and
analysis of steel structures. This theory is
based on the yield stress of a steel
structural element.
 However, nowadays, it has been
replaced by a more rational & realistic
theory the ultimate stress design that is
based on plastic capacity of a steel
structure.
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Design Approaches
Elastic Design
 In the elastic theory the maximum load
that a structure could support is assumed
to equal the load that cause a stress
somewhere in the structure equal the
yield stress of the Fy of the material.
 The members were designed so that
computed bending loads for service loads
did not exceed the yield stress divided a
factor of safety (e.g. 1.5 to 2)
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Design Approaches
Elastic Design Versus Ultimate Design
 According to ASD, one FOS is used that
accounts for the entire uncertainty in loads &
strength.
 According to LRFD(probability-based) different
partial safety factors for different load and
strength types are used.
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Design Approach
Elastic Design Versus Ultimate Design
 Engineering structures have been designed for
many years by the allowable stress design(ASD),
or elastic design with satisfactory results.
 However, engineers have long been aware that
ductile members(e.g. steel) do not fail until a
great deal of yielding occurs after yield stress is
first reached.
 This means that such members have great
margin of safety against collapse than the elastic
theory would seem to suggest.
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Bending Behavior of Beams

Assumptions & Conditions
1. Strains are proportional to the distance from the
neutral axis.
2. The stress-strain relationship is idealized to
consist of two straight lines.
3. Deformations are sufficiently small so that
ø = tanø
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Bending Behavior of Beams
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Bending Behavior of Beams
Rectangular Beam: Elastic Bending
d / 2
M    b dy y
d / 2
e

dF =  b dy

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Bending Behavior of Beams
Bending Stresses
 If the beam is subjected to some bending
moment the stress at any point may be
computed by usual flexural formula
• It is important to remember that this
expression is only applicable when the
maximum computed stress in the beam is
below elastic limit.
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Bending Behavior of Beams
Bending Stresses
 The value of I/c is a constant for a
particular section and is known as section
modulus.
 The flexural formula may then be written
as
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Bending Behavior of Beams
Bending Stresses
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Bending Behavior of Beams
Internal Couple Method
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Bending Behavior of Beams
Internal Couple Method
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Bending Behavior of Beams
Plastic Moment
• Stress varies linearly from neutral axis to extreme fibers.
• When moment increases there will also be linear increase in
moment and stress until yield.
• When moment increases beyond yield moment the outer fiber
will have the same stress but will yield.
• The process will continue with more and more parts of the beam
x-section stressed to yield point until finally a fully plastic
distribution is approached.
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Bending Behavior of Beams
Plastic Moment
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Bending Behavior of Beams
Plastic Moment
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Bending Behavior of Beams
Plastic Moment
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Bending Behavior of Beams
Plastic Moment
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Bending Behavior of Beams
Plastic Moment
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Bending Behavior of Beams
Plastic Moment
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Bending Behavior of Beams
Plastic Moment
Progression of Yield Zone
Leading to Fully Plastic Hinge
and Collapse
• Stresses reach Yield Magnitude at
extreme fibers
• Yield Zones spreads towards
Neutral axis
• Yield Zones join, are now spread
through entire x-section
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Bending Behavior of Beams
Plastic Hinges
 The effect of plastic hinge is assumed to
be concentrated at one section for
analysis purpose.
 However, it should be noted that this effect
may extend for some distance along the
beam.
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Bending Behavior of Beams
Plastic Moment
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Bending Behavior of Beams
Plastic Modulus
 The resisting moment at full plasticity
can be determined in a similar manner.
 The result is the so-called plastic
moment Mp.
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Bending Behavior of Beams
Plastic Modulus
b
d
d/2
Fy
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Bending Behavior of Beams
Plastic Modulus
 The plastic moment is equal to the yield
stress Fy times the Plastic modulus Z.
 From the foregoing expression for a
rectangular section, the plastic modulus Z
can be seen equal to bd2/4.
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Bending Behavior of Beams
Shape Factor
 The shape factor which is equal to
 So, for rectangular section the shape factor is
equal to 1.5
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Bending Behavior of Beams
Shape Factor
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Bending Behavior of Beams
Shape Factor
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Bending Behavior of Beams
Neutral Axis for Plastic Condition
 The neutral axis for plastic condition is
different than its counterpart for elastic
condition.
 Unless the section is symmetrical, the
neutral axis for the plastic condition will
not be in the same location as for the
elastic condition.
 The total internal compression must equal
the total internal tension.
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Bending Behavior of Beams
Neutral Axis for Plastic Condition
 As all the fibers are considered to have
the same stress Fy in the plastic
condition, the area above and below the
plastic neutral axis must be equal.
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Bending Behavior of Beams
Plastic Modulus
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Bending Behavior of Beams
Plastic Modulus: Unsymmetrical Shape
 The areas above and below the
neutral axis must be equal for
Plastic analysis
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Bending Behavior of Beams
Plastic Modulus: Assignment
 Determine the yield moment My, the Plastic Mp and the
plastic modulus Z for the simply supported beam having
the x-section as given.
 Also calculate the shape factor.
 Calculate nominal load Pn acting transversely through
the mid span of the beam. Assume the Fy=36 ksi
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Bending Behavior of Beams
Advantages of Plastic Design
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Bending Behavior of Beams
Advantages of Plastic Design
There is 50% increase in strength above the computed elastic
limit (stage !) due to plasticization of the x-section
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Bending Behavior of Beams
Advantages of Plastic Design: Wide Flange Section
My = Fy S
Mp = Fy Z
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Bending Behavior of Beams
Advantages of Plastic Design
Shape factor is one source of reserve strength beyond
elastic limit.
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Bending Behavior of Beams
Advantages of Plastic Design: Shape Factors
Mp/My
f/fy
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Bending Behavior of Beams
Advantages of Plastic Design
 Another source of reserve strength in
indeterminate structure loaded beyond
the elastic limit is called re-distribution
of moments.that is the process of
moment transfer due to successive
formation of plastic hinges which
continues until ultimate load is reached.
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Bending Behavior of Beams
Advantages of Plastic Design
1
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2
3
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Bending Behavior of Beams
Advantages of Plastic Design
Load
Wu
2
3
1
Deflection
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Thanks
Design of Steel Beams
The development of a plastic stress distribution
over the cross-section can be hindered by two
different length effects:
(1) Lateral Torsional buckling of the unsupported
length of the beam/member before the crosssection develops the plastic moment Mp.
(2) Local buckling of the individual plates (flanges
and webs) of the cross-section before they
develop the compressive yield stress Fy.
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Lateral Torsional Buckling
 A simply supported beam can be subjected to
gravity transverse loading.
 Due to this loading the beam will deflect
downward and its upper part will be placed in
compression and hence will act as compression
member.
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Lateral Torsional Buckling
 Beams are generally proportioned such
that moment of inertia about the major
principal axis is considerably larger than
that of minor axis.
 This is done to make Economical Beams.
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Lateral Torsional Buckling
 As result they are weak in resistance to
Torsion and Bending about the Minor axis.
 If its Y-axis is not braced perpendicularly,
it will buckle laterally at much smaller load
than would otherwise have been required
to produce a vertical failure.
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Lateral Torsional Buckling
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Lateral Torsional Buckling
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Lateral Torsional Buckling
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Lateral Torsional Buckling
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Lateral Torsional Buckling

A laterally unsupported compression flange
will behave like a column and tend to buckle
out of plane between points of lateral support.
However because the compression flange is
part of a beam x-section with a tension zone
that keeps the opposite flange in line, the xsection twists when it moves laterally. This
behavior is referred to as lateral torsion
buckling. Simply it is a sidewise buckling of
beam accompanied by twist.
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Lateral Torsional Buckling
 Embedment of top flange in concrete slab
provides lateral support to beam, except
when the beam is cantilever.
 Lateral bracing will be adequate (both for
strength & stiffness) if each lateral brace is
designed for 2% of compressive force in
the flange of beam it braces.( this thumb
rule is based on lab test results).
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Lateral Torsional Buckling
 Consider a doubly symmetric prismatic beam
 Both ends simply supported w.r.t x & y axis but
 Held against rotation about z-axis.
 It is subjected to a uniform bending moment Mx
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Lateral Torsional Buckling
Mx
•Moment at which Lateral Torsional buckling begins is given by:
Mn = Mcr =
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
Lb
2
  E 
  I y  C w
E  I y  G  J  
 Lb 
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Lateral Torsional Buckling
 The critical moments for beams with end
moments and beams with transverse
loads acting through shear center can be
given by
Where Cb is a coefficient which depends on variation in
moments along the span and K is an effective length coefficient
depending on restraint at supports.
Values of Cb and K are given in table 5-1
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Lateral Torsional Buckling
 Values of Cb developed by curve fitting to data from
numerical analysis of LTB of simple beams acted only by
end-moments is given by:
Cb=1.75 + 1.05(M1/M2) + 0.3(M1/M2)2  2.3
• Another equation obtained by working on numerical test data
of beam-column behaviour is
Cb= 1/ [0.6 – 0.4(M1/M2) ]  2.3
• Where M1 is smaller of two end moments.
• M1/M2 is +ve for reverse curvature. M1
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M2
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Lateral Torsional Buckling
• Accurate equation for Cb, if moment diagram within the un
braced length is not a straight line
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Inelastic LTB
 If stress is proportional to strain, Mx,cr for
elastic LTB is valid as given.
 But for critical stress, Fcr exceeding Fy,
Mx,cr is given by
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Inelastic LTB
 The equation can be solved in a
simplified manner by using an
equivalent radius of gyration which is
obtained by equating the critical
bending stress to the tangent modulus
critical stress for columns
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Local Buckling of Beam
Elements
 Concept of

Compact,

Non-Compact, And

Slender Elements and Sections.
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Local Buckling of Beam
Elements
 For establishing width-thickness ratio
limits for elements of compression
members, the LRFD classification
divides members into three distinct
classification as follows.
 Compact
 Non-compact
 Slender
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Local Buckling
 Compact Elements

M
If the slenderness ratio (b/t) of the plate
element is less than lp, then the
element is compact. It will locally buckle
much after reaching Fy
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p
M
r
lp lr
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Local Buckling
 Non-compact Elements

If the slenderness ratio (b/t) of the plate
element is less than lr but greater than
lp, then it is non-compact. It will locally
buckle immediately after reaching Fy
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M
p
M
r
lp lr
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Local Buckling
 Slender Elements

M
If the slenderness ratio (b/t) of the plate
element is greater than lr then it is
slender. It will locally buckle in the
elastic range before reaching Fy
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p
M
r
lp lr
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Local Buckling
 Compact Sections

A section that can develop fully plastic moment Mp
before local buckling of any of its compression
element occurs.
 Non-compact Sections

A section that can develop a moment equal to or
greater than My, but less than Mp, before loca
buckling of any of its element occurs.
 Slender sections

If any one plate element is slender, the section is
slender.
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Local Buckling
Important Note
 Thus, slender sections cannot develop Mp due to
elastic local buckling. Non-compact sections can
develop My but not Mp before local buckling
occurs. Only compact sections can develop the
plastic moment Mp.
•All rolled wide-flange shapes are compact with the
following exceptions, which are non-compact.
•W40x174, W14x99, W14x90, W12x65, W10x12,
W8x10, W6x15 (made from A992)
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Local Buckling contd;
 If the beam x-section is to develop the yield moment
My, the compression flange must be able to reach
yield stress and the web/webs, must be able to
develop corresponding bending stresses.
Local Buckling of the flange and/ or web can prevent
these limits from being attained.
More restrictive limits must be observed if a beam xsection is to attain the fully plastic moment Mp.
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Local Buckling
 For uniformly compressed laterally simply
supported on one unloaded edge and free on
the other, the critical stress is
2
 Plated used in structural members are long
enough to warrant neglecting the second term,
so
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Local Buckling
 Following limits of late slenderness (b/t) which
preclude premature local buckling of
compression flange of beams are available.
Projecting Element
Flange of Box
Since these limits are not well defined, they differ somewhat
from one specifications to another refer table 5-3
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Local Buckling
Limiting values of beam flange and web slenderness
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