52436966 lecture-13-plate-girders

N.W.F.P. University of Engineering and
Technology P h
T h l Peshawar

Lecture 13: Plate Girder

By: Prof Dr. Akhtar Naeem Khan
chairciv@nwfpuet.edu.pk

1

Plate Girders
A girder is a flexural member which is required
to carr hea loads on relati el long spans
carry heavy relatively

CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 2

Plate Girder


Plate Girder
Plate girders are typically used as long-span
g yp y g p
floor girders in buildings, as bridge girders, and
as crane girders in industrial structures.
g
Commonly term girder refers to a flexural x-
section made up of a number of elements
elements.
They are generally considerably deeper than the
y g y y p
deepest rolled sections and usually have webs
thinner than rolled sections.


Plate Girder
Modern plate girders are normally fabricated
by welding together two flanges and a web
p
plate.


Plate Girder
Plate girders are at their most impressive in
modern b id construction where main spans of
d bridge t ti h i f
well over 200m are feasible, with corresponding
cross-section d th h
ti depths, haunched over th
h d the
supports, in the range of 5-10m.


Plate Girder

Because plate girders are fabricated
separately, each may be designed
p y, y g
individually to resist the applied
actions using proportions that ensure
low self-weight and high load
resistance.
resistance


Plate Girder
Changes in X-Section

There is also considerable scope for variation
of cross-section in the longitudinal direction.
Ad i
designer may choose t reduce th fl
h to d the flange
thickness (or breadth) in a zone of low
applied moment
moment.
Equally, in a zone of high shear, the designer
might choose to thicken the web plate.


Plate Girder
Changes in Material

Alternatively, higher g
y g grade steel might be
g
employed for zones of high applied moment
and shear, while standard grade would be
used elsewhere. S
d l h So-called "h b id" girders
ll d "hybrid" i d
with different strength material in the flanges
and the web offer another possible means of
more closely matching resistance to
requirements.
requirements


Plate Girder


Plate Girder
Any cross-section of a plate girder is normally
subjected to a combination of shear force and
bending moment.
The primary function of the top and bottom
flange plates of the girder is to resist the axial
compressive and tensile forces arising from
the applied bending moment.
moment
The primary function of the web plate is to
resist the applied shear force.


Plate Girder
Plate girders are normally designed to support
heavy loads over long spans in situations where it
is necessary to produce an efficient design by
providing girders of high strength to weight ratio.
To produce the lowest axial flange force for a
given bending moment, the web depth (d) must be
made as large as possible. To reduce the self
weight, the web thickness (tw) must be reduced to
a minimum
minimum.
As a consequence, in many instances the web
plate is of slender proportions and i th f
l t i f l d ti d is therefore
prone to buckling at relatively low values of
applied shear
shear.

Plate Girder
For efficient design it is usual to choose a
relatively deep girder, thus minimizing the
required area of flanges for a given applied
moment, Msd.
t
This obviously entails a deep web whose
y p
area will be minimized by reducing its
thickness to the minimum required to carry
the applied shear, Vsd.
h li d h
Such a web may be quite slender (
y q (i.e. a high
g
d/tw ratio) and may be prone to local buckling
and shear buckling.


Plate Girder
Web buckling does not determine the
ultimate strength of a plate girder.
lti t t th f l t id
Plate elements do not collapse when they
p y
buckle; they can possess a substantial post-
buckling reserve of resistance.
For an efficient design, any calculation
relating to the ultimate limit state should take
g
the post-buckling action into account.


Design Criteria
Criteria for design of plate girder may be
based on
Elastic bend-buckling strength
Elastic h
El ti shear-buckling strength
b kli t th
Post-bend-buckling
Post bend buckling strength
Post shear buckling(Tension
Post-shear-buckling(Tension field)strength


Design Criteria

The designer has the choice of following four
combinations
1. Elastic bend buckling + Elastic shear buckling
g g
(conventional flexural behavior)
2.
2 Elastic bend buckling + Post shear buckling
3. Post bend buckling + Elastic shear buckling
g g
4. Post bend buckling + Post shear buckling


Elastic Bend Buckling
Strength
The extreme f fiber bending stress at which a
perfectly flat web buckles is given by


Strength
Using a FOS of 1.25 w.r.t service load bending
stress fb gives an eqnuation which is AASHTO
slenderness limit for plat girders webs

Using AASHTO allowable stress fb=0.55Fy
“ h/t=165 f A36 steel “
h/t 165 for t l

Strength
The bend buckling resistance of beam webs can be
increased considerably by reinforcing the slender webs
with Longitudinal stiffeners.
Means webs thinner than those given by the equation can be
used.
used

A typical longitudinally stiffened girder is shown after failure


Web Stiffeners

They usually consists of rectangular
bars to welded to web.
Transverse stiffeners may be in pairs,
one on each side of web, or they may
placed on one side of web.
Longitudinal stiffeners are usually
placed on one side of web.


Web Stiffeners


Web Stiffeners
The main function of the longitudinal stiffeners is
to increase the buckling resistance of the web
with respect o bot s ea a d be d g loads. An
t espect of both shear and bending oads
effective stiffener will remain straight, thereby
sub-dividing the web p
g panel and limiting the
g
buckling to the smaller sub-panels. The resulting
increase in the ultimate resistance of the girder
g
can be significant.


Web Stiffeners
Efficiency of stiffener is a function of its location
in the compression zone
The optimum location for a longitudinal stiffener
has been determined to be at least h/5 from
compression edge.
In this case k=129. The corresponding allowable web
k 129.
slenderness is h/t=330 as compare to 165


Web Stiffeners
Stiffener acts as a beam supported at the ends
where a vertical stiffener holds the web in line
line.
Stiffener acts as a beam column and hence must
be
b proportioned i t
ti d in terms of x-sectional area and
f ti l d
moment of inertia.
AASHTO specifies Is as

Stiffener acts as a beam supported at the ends
where a vertical stiffener holds the web in line.


Web Stiffeners
The stiffeners must also be proportioned to
resist local buckling.
buckling
For plates supported on one longitudinal
p pp g
edge AASHTO require b/t<1625/√fb
Multiple longitudinal stiffeners are used for
large depth webs.
As longitudinal stiffener is also acting as a
column so it must be satisfied for critical
stress (Fcrs>0.6Fcrf)


Post buckling bending
strength

If bending strain increases after Fcr, the upper
g , pp
edge of panels shortens and bottom edge
lengthens.
If web were to remain flat there will be increase in
stress.
Because the web has buckled, the increase in
stress is non linear
non-linear.


strength
As
A variation i post-buckled state i not k
i ti in t b kl d t t is t known,
simplify assumptions are made.
Non-linear compression is replaced with linear
distribution acting on effective depth be.


strength
Point A gives point that enables a girder to reach its full
yield moment(925 /√Fy=154).
If stiffeners at h/5 is provided gives point B
B.

Considering the
post buckling A B
strength, the 0.94
0.82
0 82
point where M/My
reduction in web 0.4
effectiveness 0.18
0 18
begins s taken to
154 315 360
be 980/√Fy=170.
980/√Fy 170.
h/t

strength
Equation connecting the revised point A
with points corresponding t h/t 360 i
ith i t di to h/t=360 is


strength
LRFD

Where


Compression Flange Vertical
buckling


buckling
If plate girder web is too slender the compression
plate-girder slender,
flange may buckle in vertical plane at stress less
than yield stress
stress.
The compression flange is a beam-column
p g
continuous over vertical stiffener as supports
Its stability depends on stiffener spacing and
relative stiffness of the flange and the web. Fcr is


buckling
Slenderness of webs with vertical stiffeners is taken conservatively

AISC ASD/LRFD li it the h/t by the given equation with
limits th b th i ti ith
Aw/Af =0.5


Shear buckling of beam webs
Shear buckling is seldom a determining
factor i d i
f t in design of rolled section b t
f ll d ti but
plate girders have much larger h/t so it
must be considered.
tb id d


Shear buckling of beam webs
Transverse stiffeners are used to
increase th b kli strength b
i the buckling t th by
increasing factor k through a reduction
in aspect ratio a/h.


Transverse Stiffeners
Transverse stiffeners play an important role in
allowing the full ultimate load resistance of a
plate girder to be achieved.
In the first place they increase the buckling
resistance of the web;
Secondly they must continue to remain effective
after the web buckles, to provide anchorage for
p g
the tension field;
finally they must prevent any tendency for the
flanges to move towards one another.


The satisfactory performance of a
transverse stiffener can best be illustrated
by comparing the girders shown, after
shown
testing.

Figure 2
g
Figure 1
Fi

In Figure 1 the stiffeners have remained straight.
g g
In Figure 2 the stiffener has failed and has been
unable to limit the buckling to the adjacent sub-
sub
panels of the girder; instead, the buckle has run
through the stiffener p
g position extending over
g
both panels. Consequently, significant reduction
in the failure load of the girder occurred.
In Figure 1 One can also see the effect of aspect
ratio,i.e greater a/ less k a d s a Fcr.
at o, e g eate a/h ess and small c


The stiffener must be of adequate
rigidity in th di ti perpendicular t
i idit i the direction di l to
the plane of the web to prevent web
buckling. This condition is satisfied
p
provided the stiffener has a second
moment of area Is that satisfies the
following empirical formulae:



AISC/LRFD Moment of Inertia of
stiffener is:
tiff i

where



Transverse stiffeners spacing can be
determined from the following


Tension Field Action
The resulting shear stresses on an
element of a web are equivalent t
l t f b i l t to
principal stresses, one Tensile and one
Compressive, at 45 to the shear stress.



Once a web panel has buckled in shear, it
p ,
loses its resistance to carry additional
compressive stresses.
stresses
On the other hand tensile principal stress
p p
continues to increase in strain in the
diagonal direction.
direction
Such a panel has a considerable p buckling strength,
p post g g ,
since increase in tension is limited only by yield stress.



In this post b ckling range a ne load carr ing
post-buckling range, new load-carrying
mechanism is developed, whereby any additional
shear load is carried by an inclined tensile
membrane stress field. This tension field anchors
against the top and bottom flanges and against the
transverse stiffeners on either side of the web
panel. The load-carrying action of the plate girder
than becomes similar to that of the N-truss
In the post-buckling range, the resistance offered by
the web plates is analogous to that of the diagonal
tie bars in the truss.


Phases of behavior up to collapse of a typical panel in shear

Prior to Buckling Post Buckling Collapse



The load-carrying action of the plate girder
load carrying
than becomes similar to that of the N-truss
In the post-buckling range, the resistance
offered by the web plates is analogous to
that of the diagonal tie bars in the truss.



ft V
V

Vt=Tsinφ
Vt = ft ht cosφ sinφ T=ft ht cosφ

Vt = (1/2)ft ht sin2φ φ

Vt =(1/2) ft ht φ=45
Vty=(1/2) Fy ht………….(1)

Vty =(1/2) Fy ht = Fy

Vy Fvy ht 2Fvy
Vty = √3 Vy = 0.87 Vy
2



The angle φ for which Vt is max



Where



(1)

Taking inelastic and strain hardening range

(2)

(3)


Codal equations are derived from
eqn;(1),(2),(3)



AISC/LRFD

k

a/h


Combined Bending & Shear
of Webs

Interaction diagram is based on Tension-
field f
fi ld of webs
b
If the web is completely yielded in
shear,any accompanying moment must
be
b resisted entirely b fl
i t d ti l by flanges.


Bending & shear Interaction Curve

B B C D
V vyAw)
V/(F A

E 1/√3

0.75 0.83 1.0 1.07 1.12 M/My

1.0

Mu/φMn
0.8

0.6

LRFD I t
Interaction Curve
ti C
0.4

0.2

0.2 0.4 0.6 0.8 1.0 Vu/φVn

Web Proportioning
Notations


Web Proportioning

Depth of girder is influenced by many
factors:
Headroom
Clearance for high water in deck bridges
Traffic passing beneath the bridge


Web Proportioning

Depth: Overall girder depth, h, will
usually be in the range
Lo/12 ≤ h ≤ Lo/8
/8,
occasionally lighter loads may be
accommodated with Lo/20 /20.

Flange:
g

The breadth, b, will usually be in the range
h/5 ≤ b ≤ h/3,


Design Procedure
1. Maximum Moment & Shear for Factored Load
2. Web D i
2 W b Design
1. p girder L/12 ≤ h ≤ L/8
Assume depth of g
2. Depth of Web hw=h-2tf
3. Web slenderness
1. For a/h <5 …………….
2. and for a/h > 5 ……………………
3.
3 hw/tw= 970/√Fy
4. Select optimum tw


Design Procedure

4. Flange Design
1. Find Af
2. Select suitable tf and bf
3.
3 Flange slenderness
1. bf/ 2tf < 65/√Fy …………….Compact


Design Procedure
5.
5 Check trial girder section
1. Web local buckling limit state
1. hw/tw< 640/√Fy…………………..Compact
2. 640/√Fy< hw/tw < 970/√Fy……Non-Compact
3. hw/tw > 970/√Fy…………………..Slender
2. Flange local buckling limit state
1. bf/ 2tf < 65/√Fy …………….Compact
3. Lateral Torsional Buckling
g
1. Calculate Iy
2. A=Af+Aw/6
3. ry= √Iy/A
4. Find Lb/ry
5. λp= 300/√Fy ………….. λ< λp ______Compact

Design Procedure
6. Bending strength
1. Calculate
C l l t Ix
2. Calculate Sxt
3. .

4. .

5. φMn≥ Mu


Procedure for Design
6. Bending strength
1. Calculate Ix
2.
2 Calculate Sxt
3. .

4. .

5.
CE-409: Lecture 13
φMn≥ Mu Prof. Dr Akhtar Naeem Khan

52436966 lecture-13-plate-girders

52436966 lecture-13-plate-girders

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