Roadway Bridge

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Chapter 14

• Roadway Bridge

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contents

• Roadway Bridge Floor
•   Side walks and Railings
• Bridge Bracings
• Design of lateral support at top chord of through
  pony bridge
• Cross Sections for wind Bracing
• End X-frame in deck bridges
• Transmission of the braking forces the bearing

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• Truss Bridges
• A.     Types of bridge trusses
• B- Determination of forces in
  truss members
• c. Proportioning of truss
  members
• D- Box section for bridge
  trusses Top chords
• Lacing bars, batten plates
• Bottom chords
• Diagonals
• Verticals
• Design of compression member
• Design of Tension Members

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• Design of Bolted Joint
• Design of Battens and Diaphragms
• Design of End Portals

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Roadway Bridge Floor

• The floor of a Roadway Bridges consists of
• 1.      A wearing surface or Roadway Covering.
• 2.      Sub floor transmitting the loads to the
  stringers and X-girders.
• The sub floor is similar to the solid floor of a
  ballasted Railway Bridges. It may be timber,
  steel floor or R.C. floor.

Back
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•         Timber floor (Type 1)
• For bridges, generally two layers of flanks are
  provided. For calculating these flanks we assume
  that the maximum wheel load is distributed over
  two flanks.

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        Reinforced concrete floors (R.C.
floor) It may be supported by the main girders
only, the X.G. only or by stringers and X.
girders. The span of the slab may be 2.5 to 3.5
m, and thickness of slab to be 20 cm nearly. The
R.C. slab reinforcement, generally 12 bars are
used at least per one meter.
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        Wearing surface         The wearing
surface for roadway covering consists of timber
blocks, hard bricks, asphalt bricks, stone
blocks, asphalt or concrete. The choice of
material depends on the traffic, the span of
bridge, the cost and climate
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Side walks and Railings
The side walks are placed either inside or
outside of the main girder. If they are arranged
outside, they must be supported on cantilever
brackets situated in the plane of the X.G. so
that the ve bending moment of the bracket is
transmitted to the X.G. The floor of these side
walks should be a precast R.C. slab (6cm) thick
resting on the side walk stringers. The wearing
surface is a 2 cm layer of asphalt. In through
bridges the curb should be at least 50 cm inside
the main girder
Back
13
Hand railings and brackets withstand the effect
of a transverse horizontal force of 150 kg/ m in
cases of Railway bridges, Roadway bridges, and
foot bridges, supposed acting at top level of
hand rail. This horizontal is transmitted from
the hand rail to the main posts and from their
connections to the cantilever brackets.
Side walks parts 1.      Slab
Take strip 1 m and statical system as continuous
beam supported on side walk stringer (one way
slab), take t 8 cm and get As. The applied
loads are considered 500 kg/m2 or one
concentrated load 5 t.
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1.      Side walk stringer Simple beam span
distance between two brackets (take channel X.
sec.)
2.      Hand rail Simple beam span distances
between two brackets (take angle or channel X.
sec.).
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4     Post Cantilever beam (take 2-angle or
2-channel X. sec.).
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5    Connections Double shear bolts

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6      Bracket Calculate M N Q at center of
bracket. In case of beam loaded alone we must
calculate Fl.t.b and the check that the actual
stress Fc is less than the
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allowable stress Fp.b. For ST 37 If ? ? 100 If ?
? 100
? l/ i , where I for compression flange
only.         for bracket l/ b ? 2 l/ b        
assume unequal angle 80?120         Bolts
subjects to shear
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Bolts subjects to tension
? 0.8 Ft
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Bridge Bracings

• The bridge is provided with horizontal and
  vertical bracings-
• 1. The stringers are connected together by
  stringer bracing given before.
• 2. The chords of the main girders are jointed
  together by an upper and lower horizontal bracing
  called wind bracings.

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• These transmit to the bearings of the bridge
• a.       The lateral forces due to wind.
• b.      The lateral shock 6t.
• c.       Centrifugal force.
• 3 - Special horizontal bracings for the braking
  forces.
• 4-Two vertical and transverse bracing called
  X-frames or portals (in case of through)
  transmitting reaction of the upper lateral
  bracing to the bearings of bridge.
• 5- Some intermediate vertical transverse bracings
  called intermediate X-frames or intermediate
  portals for the rigidity of the structure.
• It isnt necessary to find all these bracings in
  every bridge, there existence depend upon the
  type of the bridge, the span and the floor.

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  I-The Deck Bridge
The upper wind bracing transmits the wind
pressure WT on the train WF on the floor ½ WG
on the wind ward side of the main girder.
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The lower wind bracing transmits the wind
pressure ½ WG on the wind ward side of the main
girder. W 100 kg/ m2 in case of loaded
bridge W 200 kg/ m2 in case of unloaded
bridge The wind pressure WT on the train
produces in addition to horizontal loading of the
upper wind bracing. a vertical loading,
to the main girder.
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In case of a truss bridge, only the exposed area
of the members is considered. This area is
equivalent to 40 of the hole area of the
surface of the truss. In all bridges with an
upper and a lower wind bracing, their shall be
provided
at each end a X-frame to transmit to the
bearings,the horizontal reactions of the upper
wind bracing. The horizontal reactions of the
lower wind bracing are transmitted directly to
the bearings
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        The end X-frames in deck bridges shall
be of rigid type. In all railways and in roadway
deck bridges there shall be intermediate
transverse bracing at least at every third panel
point to increase the stiffness of the bridge.
These intermediate X-frames will release the end
X-frame from a part of the horizontal reaction of
the upper wind bracing. Yet it is recommended not
to consider that release unless the bridge as the
space structure.
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ii-The Through Bridge
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In through bridge two horizontal wind bracings
should be arranged if possible. In the plate
girder through bridges we cant arrange an upper
wind bracing in the bony truss Roadway Bridge we
have only a lower wind bracing which transmits
all the wind loads to bearings. The force WF on
the floor will be considered to act on a solid
surface as the plate girder. The through Railway
Bridge shown above is provided with two
horizontal wind bracings.
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The upper wind bracing transmits the wind
pressure ½ WG on the wind ward side of the main
girder. The lower wind bracing transmits the wind
pressure ½ (WG), WT on the train WF on the
floor on the wind ward side of the main girder.
At the connection of each X-G to the main girder,
stiffness bracket shall be arranged.
Back
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Design of lateral support at top chord of through
pony bridge

• C force in flange Af?Ft
• The U-frame formed by the two vertical stiffeners
  and the horizontal stiff X-girder is acted upon
  by a horizontal transverse force C/100 at the
  centroid of compression flange as well as the
  wind pressure between two consequence X-girders.
  The maximum stressed section is mn. The
  compression stress at point n ? Fltb. If the
  stress isnt safe, we either increase the
  thickness of the bracket plate or add a
  stiffening angle.

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The connection between the X-girder and bracket
is designed on the shearing force A that between
X-girder and the bracket and horizontal gusset of
wind bracing on force B. if the X-G is built up
section the bracket connection is designed as a
web splice.
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Back
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Cross Sections for wind Bracing

• The diagonal of wind bracing
• The diagonal of wind bracing shall have stiff
  section to prevent vibration and to help in
  reducing the deflection of main girder due to
  eccentric loading (space frame treatment). The
  section should have a depth not less than L/40.
  The recommended sections are given in Fig.(5.).

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The choice of the section depends more or less
upon the span of the diagonal, the two channel
section is convenient for too long spans executed
in Banha and Samanood bridges. The two channels
are connected together by latticing or batten
plates. ? in compression ? in tension
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In case of one diagonal member only Case of the
warren system which designed on a force S
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Case of the N-system which designed on a force S
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Case of the K-system , Rhombic And Multiple
which designed on a force S
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If the bracing is made up of crossed diagonal and
struts, the calculation is made under the
assumption that the tension diagonal are only
acting. The struts here receive compression
force. If the multiple systems of wind bracing a
further reduction of 20 in the allowable
stresses given before, shall be made to account
for approximation of solution that both systems
(tension and compression member) equally share
the lateral loads. in case of one angle in
compression the allowable compression stress
shall be reduced by 40 of Fc.
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Hence, the approximations in allowable stresses
are
Back
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End X-frame in deck bridges The compression
diagonal is assumed in acting and we design the
tension diagonal, also we assume that the X-frame
is resting at a movable support at one end and
the hinge support at other end.
Back
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Transmission of the braking forces the bearing
In Railway bridges especial bracing should be
arranged to transmit the longitudinal forces from
the stringers to the panel points of the main
girder, hence they are transmitted through the
main girder to the hinged bearings. Some times a
bracing is arranged at every panel point. But
generally two bracings at the quarter points of
bridge are sufficient. The braking bracing system
shown in sketch is statically indeterminate but
the loads are symmetric about perpendicular axes
to X-X. Therefore the diagonal Bn cn are
zeros since they correspond to themselves. Also,
the loads are antisymmetric about axes Y-Y and
thus members mb mc nb nc are zeros, If
special bracing of the longitudinal forces is
omitted, these forces are transmitted
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from stingers to main girder by bending of the
X-girder.
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Truss Bridges b ? L/ 20, b ? h/ 3 Where, b
bridge width distance between center lines of
two main girders L span of bridge The depth of
trusses shall be chosen in such away that the
elastic deflection due to L.L (without dynamic
effect) shouldnt exceed L/800 for Railway
bridges and L/600 for Roadway bridges.
Back
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h? (simple ) (continue)
Road. h? (simple )
(continue) Rail.
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A.     Types of bridge trusses
Either both chords are straight and parallel or
only one of them. In a through bridge the upper
chord may polygonal, in a deck bridge the lower
chord may be polygonal. Curved chord should not
be used in bridge trusses on a account of the
additional bending stresses. The loads are
transmitted to the panel point of the truss by a
system of stringers and cross girder. No load
except the own weight of the truss members should
act between the panel point.
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1.Trusses with horizontal chords
      They suitable for span up to 60 m. the
joints are simpler than in trusses with polygonal
chords. The depth is h ? L/ 8 for Railway Bridge,
or h ? L/ 10 for Roadway Bridge. For continuous
and cantilever trusses the depth may be taken h ?
L/ 10 for Railway bridges, h ? L/ 12 for Roadway
bridges. Some times a greater depth is used to
allow an upper wind bracing. The arrangement of
web members may be N-system or warren system. The
warren system trusses require generally less
material than the N-shaped trusses, since the
vertical members have smaller forces, the number
of joints and changes of cross section in warren
system are also less. Shop work for warren
trusses will be cheaper.
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2-Trusses with polygonal chords
They are used for spans up to 60 m. the
economical depth at middle is h L/7. The web
system is either N-shaped or warren. The
economical inclination of the diagonal to
horizontal ? 40 - 60?. A polygonal chord
trusses lighter than a truss with horizontal
chords since the forces in the diagonals are
smaller. On the other hand the shop work is more
complicated which means a higher cost.
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3-Trusses with subdivided panels (e), with
Rhombic diagonal (f) and K-system These kinds are
economical for span over 80 m. The panel length
is reduced in all this system and thus the cost
of the floor is less, but the increased number of
joint increase the cost of shop work. A truss
with Rhombic diagonal has a good appearance a
truss with subdivided panels has big secondary
stresses. K-system trusses have the smaller
secondary stress.
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4- Trusses with multiple web system These
were used in past where the tension diagonals
consist of flat bars. Now they are again used for
main girders, but type h with crossed diagonals
is frequently used for wind bracing. For
approximate calculation, the common assumption is
that the truss may be divided up into two or more
component trusses with the same chords but with
different web system. The loads also are divided
and placed upon this component trusses. Then the
stress in a web member is determined as its
stress calculated in the truss of which it is a
part. The chords are a part of all component
trusses, hence the stresses in a chord member is
obtained by adding it partial stress from each
component truss.
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5- Trusses with 3 chords The arch truss with
a tie (k) and the truss reinforced by a hinged
arch (Pow string truss) are supported on a hinged
bearing at one end and a movable bearing at the
other. They are therefore externally static
determinant but internally they are static
indeterminate. These trusses have good appearance
but they more expansive than trusses with two
chords.
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Back
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b- Determination of forces in truss members
We determine the forces in the truss members on
the assumption that the member are connected by
hinges, so that loads applied at the panel point
produce only axial forces in the truss member.
The secondary stress which are the bending stress
induced by the rigidity in connection, are
generally neglected. In our specification it is
required to calculate the secondary stress in the
following cases 1- For trusses with
subdivided panels. 2-For member whose width in
the plane of the truss is more than 1/10 of its
length. 3-For loads acting between the panel
point.
Back
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c. Proportioning of truss members For the
chord member we can use either sections with one
web plate (T section) or sections with two web
plates (Box section). T-sections are used only
for small bridge. Box sections have grates moment
of inertia about axis y-y and are better used for
the connection of gusset plate. The sections of
all chords and web member should be symmetrical
about axis y-y in the plane of the truss.
Diagonals and verticals are usually symmetrical
about axis x-x also. The required area of the
chords change at every panel point of the truss
and in choosing the different cross section we
must try to get simpler connection and splices at
the panel point.
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D- Box section for bridge trusses Top
chords The minimum section consists of a
horizontal plate and 2 channels or a horizontal
plate, 2 vertical plates and four angles. Depth
of top chord h (1/12 1/15) of the panels
length ? Width a (0.75 1.25) h
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To avoid local buckling, the minimum thickness of
web and cover plates should be as follows The
unsupported width of a plate measured between
adjacent lines of rivets or welds connection the
plate to other parts of the section should not
exceed
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t thickness of a single plate or of 2 or more
plate provided that this plates are adequately
tacked together.
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Only excess over this width should not be
included in the effective sectional area in
computing direct compressive stresses. The center
of gravity axis x-x for the different section
should not change too much. In drawing the truss
we use an average value y (y1 y2 y3
....)/ n.
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It is good practice to use a cover plate over the
whole length of the top chord even if the end
members have excessive cross section.
Back
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Lacing bars, batten plates The two plates of the
compression members shall be connected together
by diaphragms and the open side of the box
section shall be provided with batten plate close
to the gusset plate and with intermediate batten
plates or lacing bars to avoid lateral buckling
of their component parts. The slenderness ratio
of each component part between consequent
connections of lacing bars or batten plates shall
be not more than 50 (and 2/3 l/i of the whole
section).
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Bottom chords The depth of the bottom chord is
equal to that of the top chords h (1/12 1/15)
of the panels length, or slightly more (2 4
cm). No horizontal plate is provided at the
bottom of the section to avoid water packets. In
continuous and cantilever bridges where some
bottom chord members are in compression,
horizontal plate may be used and it must be
provide with drainage holes (4 5 cm) ?. The two
component parts of tension member shall be
connected together by diaphragms and batten
plates similar to these of the compression
members, but their thickness may be taken 25
lighter (t2 ? lmember/ 15).
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Diagonals For appearance the width of the
diagonal should not be more than that of the top
chord and should decrease from the end to the
middle of the bridge. The compression diagonal at
end of the warren truss has a section similar to
that of the top chord.
Back
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Verticals In trusses with a N-shape web system,
the vertical have similar sections as diagonals.
In warren trusses the vertical may consist of a
web plate 4 flange plates or an I-beam
(B.F.I.B). For diagonal or vertical tension
member t2 ? lmember/ 15
( D V ) t2 ? (lmember)/
30 Railway bridges ( D V C ) t2 ?
(lmember)/ 35 Roadway bridges ( D V C ) t2 ?
(lmember)/ 10 (C ) D ? Diagonal V?
Vertical C ? Chord
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Back
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Design of compression member The slenderness
ratio l/ i of compression member of main girder
shall not exceed 90 for Railway bridges and 110
for Roadway bridges. E.  Effective buckling
lengths Table 4.5
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Table (4.5)
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Effective buckling lengths
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Unbraced compression chords a- For simply
supported truss, with laterally unsupported
compression chords and with no cross-frames but
with each end of the truss adequately restrained
(Figure 4.1), the effective bucking length (kL),
shall be taken equal to 0.75 of the truss span,
(clause 4.3.2.2). b- For a bridge truss
where the compression chord is laterally
restrained by U-frames composed of the cross
girders and verticals of the trusses, the
effective buckling length of the compression
chord (Lb) is
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E the Youngs modulus 2100 t/cm2 Iy the
moment of inertia of the chord member about the
Y-Y axis. a the distance between U-frames (cm)
S
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d1 the distance from the centroid of the
compression chord nearest face of the cross
girder of the U-frame dw Hx.G. d2 the
distance from the centroid of the compression
chord to the centroidal axis of the cross girder
of the U-frame dw Hx.G./2 I1 the second
moment of area of the vertical member forming the
arm of the U-frame about the axis of bending.
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I2 the second moment of area of X-G about the
axis of bending IX B the distance between
centers of consecutive Main Girders connected by
the U-frame
Back
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Design of Tension Members
Tension members shall always be of rigid
construction and their slenderness ratio l/ i
shall not exceed 160 for Railway bridges and 180
for Roadway bridges. The effective net section
area shall be taken for all tension members. This
area shall be the least that can be determined
from any plane or planes cutting each component
plate or sections ? to its axis,
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diagonally or following zig-zag line through
adjacent rivet holes. In each case all holes of
line to meet with shall be deducted from the
gross sectional area where any portion of the
sectional area is measured for a diagonal plane
adding (S2/ 4g) for each gauge space. The minimum
sectional area should not be less than that
obtained by assuming all the holes to be in one
perpendicular plane.
Back
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Design of Bolted Joints Connection of web member
to gusset plate and splices of chord member
shall have a strength equal to the maximum
strength of the connected members. The bolts
shall be arranged symmetrical about the center
line of the member. The connection to either
direct or part of it is indirectly connected
by- 1- Splice plates or lug angles. 2- If
breaking is along section S-S bolts (1 single 2
double 3 double) shear, must carry the load.
3- The strength of the splice plate should be
enough to carry a force corresponding to bolts (4
5) single shear.
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Connections for members The connection shall be
designed for a capacity based on the maximum
of- 1- The average between the actual force
and the maximum strength of the member of not
less than 0.75 the maximum strength of the
member. 2 The bolts between the chord and the
gusset plate must correspond to the algebric sum
of the horizontal components of the strength of
the diagonals S1, S2   L S1?Cos ?
S2?Cos ?
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3. The number of bolts should correspond to
the effective strength of the two diagonal, i.e.,
the number of ? (number of bolts in member 1
number of bolts in member 2)?Cos ?.
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Example
L750
In a lower chord panel, if the maximum force in
chords and diagonals are as given, design
suitable cross section, connections and splices.
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Member S1 94 t 2 No 24 Bolts M22, ?
24 Anet 2?(42.3 2.4?0.95 - 2?2.4?1.3) 67.56
cm2
Fact 94/ 67.56 1.39 t/ cm2 lt 1.6 t/ cm2
Maximum force 67.56?1.60 108.10 t Rleast
Rs. sh qb 0.25 Fub For bolts of grade (
8.8), Rs. sh qb ?As ?n 0.25?8.0?3.03 ?1
6.06 t (n No. of shear planes)
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Bolts connecting diagonal to gusset (Maximum
force/ Rleast) ?1.15 (108.1/
6.06)?1.15 20.52 bolts Use 24 bolt M22
Member S2 - 70 t 2 No 22 Bolts M22, ? 24
Agross 2?(37.40)
74.80 cm2 ?y ly/ iy (0.85?750/ 11) ?1.20
69 (1.20 due to lacing bars) ?x lx/ ix
(0.70?750/ 8.5) ?1.20 61
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?max 69 ? Fp.b 1.60 0.000085 (l/ i)2
1.119 t/ cm2 Fact 70/ 74.80 0.94 t/ cm2 lt
1.119 t/ cm2 Maximum force 74.80?1.119
83.70 t Rleast Rs. sh qb 0.25 Fub For
bolts of grade ( 8.8), Rs. sh qb ?As ?n
0.25?8.0?3.03 ?1 6.06 t (n No. of shear
planes) Bolts connecting diagonal to gusset
(Maximum force/ Rleast) ?1.15 (83.7/
6.06)?1.15 15.88 bolts Use 16 bolt M22
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Member S3 126 t 2 No 30 Bolts M22, ?
24 Anet2?(58.80 2?2.4?1.00 - 2?2.4?1.6)
92.64 cm2 Fact 126/ 92.64 1.36 t/ cm2 lt 1.6
t/ cm2    
90
Member S4 196 t 2 No 30 2 PL 240?12
Bolts M22, ? 24 Anet for 2 2?(58.80
2?2.4?1.00 - 2?2.4?1.6) 92.64 cm2 Anet for 2PL
2?(24 2?2.4) ?1.2 46.08 cm2
Anet for 22PL 92.64 46.08 138.72 cm2 Fact
196/ 138.72 1.413 t/ cm2 lt 1.6 t/ cm2
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Force to be transmitted from gusset plate to
bottom chord Fmax Fmax (S1 S2)?Cos ?
(67.56?1.6 74.80?1.119) ?1/?2 135.62 t Bolts
connecting diagonal to gusset (135.62/
6.06)?1.15 25.74 bolts Use 28 bolt M22 The
bolts in the framing angle are not counted as
they used to transmit the reaction of the cross
girder.
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Splice of chord Splices of tension or compression
chord shall be designed on the maximum strength
of the member. For straight chord the splice
shall be outside the gusset plate. For broken
chord the splice will be within the gusset
plate. Member S3 2 No 30 Bolts M22, ?
24 Net area of flange (10 23) ?1.60 12.30
cm2 Net area of splice plate (10 23) ?1.60
12.30 cm2 Number of single shear field bolts
(12.30?1.6/ 6.06) ?1.15 3.25 bolts
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Net area of web (30 2?1.60) ?1.00 22.20
cm2 Use 2 splice plates 30?0.80 24?0.80 Net
area of 2 splice plates (30 2?2.3) ?0.80
(24 - 2?2.30) ?0.8 35.80 cm2 Rleast RD. sh
or Rb Rb Fb ? d ? min? t Fb ? ? Fub ?
0.60 for end distance (S?1.5d), (Table 6.2) Fb
0.60 ? 8.00 4.80 t/ cm2
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Rb 4.80 ? 2.20 ? 1.00 10.56 t qb 0.25 Fub
For bolts of grade ( 8.8), Rs. sh qb ?As ?n
0.25?8.0?3.03 ?2 12.12 t
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Rb 10.56 t Number of double shear field bolts
(22.20?1.6/ 10.56) ?1.15 3.87 bolts Use 4
bolt M22 The splices in the chords are placed in
the side of the- 1. Smaller cross section
except in cases where the erection is done by the
cantilever method. 2. Gusset plate in a
polygonal chord in order to avoid the bending of
plates, angles and channels the splice plate is
placed at the brake of the chord on the gusset
plates. 3.Gusset shall be proportion to
withstand the effective forces in the web member
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The thickness of the gusset is determined
from the critical section abcd. This section
should be at least 15 20 stronger than the
diagonal it self generally all gusset are made of
the same thickness. The thickness of gusset plate
shall be at least 12 mm in Railway bridges and 10
mm in Roadway bridges.
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4- At the chord panel point the cover plate
should be connected to a gusset plate by special
connection angle to make the center of gravity of
the rivet group between gusset plate and top
chord nearer to the center of gravity of the top
chord.
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Back
102
Design of Battens and Diaphragms The two parts of
the box section must be connected together in
such away that they act ass one unit. For
compression member stronger details are necessary
than for ten member. Diaphragms Diaphragms are
transverse plates or channel connected to the two
webs of the box section by angles. They are
necessary to assume the rectangular shop of the
box section. For the chords wee have at least one
diaphragm between the two panel points. In the
diagonals, we arrange at least one diaphragm near
each end.
103
Batten plate One the open sides of the box
section we have batten plates as close to the
gusset plate as possible one intermediate batten
plate or lacing bars to avoid lateral buckling of
the unsupported flange for the calculation of the
lattice bars of a compression member we assume a
transverse force 2 of the longitudinal force
in the member. If there is a continuous plate at
the upper side of the box section, latticing will
be in the lower side only and transverse force
will be according to cross section of the lower
side only. In tension member a lattice system and
batten plates 25 lighter is used.
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Battening of compression member The number of
batten is such that we get at least 3 bays batten
shall be of plates, channels, I-section bolted or
welded to resist the following forces- The
member as a whole can be considered as a
vierandeen girder or we can assume hinges at mid
distances and change it to statically determine
system. Shear in batten plate Q?d/ a
106
Bending moment in batten Q?d/ 2
107
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Design of End Portals End transverse bracing are
called portals. The portals are placed either in
the vertical plan of the end post (1), in the
plan of the first vertical (2), or in the
inclined plan of the end diagonal (3). In case
(2) the first panel of the lower wind bracing is
affected by the reaction transmitted by the end
portals. The arrangement of end portal in case
(2) is stiffer than in case (3). The portal must
not interfere the clearance line.
108
The shop of the portal depends on the depth and
on the clearance line. The portals are generally
static indeterminate closed frames in which the
post over subject to bending stresses.
109
Approximate calculation of portals The point of
inflection of the post are situated according to
the relative stiffness of the cross girders,
post, and the upper strut at height between 1/3
1/2 of the force height h. We can replace the
point of inflection by two hinges at C and C
each of them transmits ½ W1. Then the portal can
be calculated as static determine frame. If the
portal is in the plan of inclined end diagonal,
the points of inflection C, C should net be more
than h/ 3 from A and A.
110
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