830 Civil Booster (Civil Ki Goli Publication 9255624029)

Rivets
1
IS 432:1982 Mild steel and medium tensile steel.
Various physical properties of structural steel are given below.
Unit mass of steel,  = 7850 kg/m3
Modulus of elasticity, E = 2 × 105 N/mm2
Modulus of rigidity, G = 0.769 × 105 N/mm2
Poisson ratio,  = 0.3 ( in elastic range ), 0.5 ( in plastic range ).
Coefficient of thermal expansion,  = 12 × 10–6/ºC
Methods of design of steel frame work:
(a) Simple design
 Based on elastic theory & most uneconomical method
 No moment is transferred from the connected member to another
connected member (for design purpose, structure is assumed to be
pin jointed).
 All connection of beams, girder or trusses are virtually flexible.
(b) Semi-rigid design
 It ensure that partial flexibility is available at supports.
 It permits a reduction in maximum bending moments in beams
suitable connected to supports due to partial transfer of moment to
another connected member.
 It is economical than simple design.
(c) Fully rigid design method
 It involves the assumption of the end connections being fully rigid
& capable of transmitting moments & shears.
 The end connection of the members of the frame should have
sufficient rigidity to hold virtually unchanged original angles between
such members & members they connect.
 It is used in convenient cases & given economy in the weight of
steel & saves construction cost.
Wind pressure, P  KV 2
 Design of Steel Structure 831

where, P = Wind pressure (in kg/cm2), V= Velocity of wind (in km/hr)

K = Constant of proportionality.
As per IS 875 (Part III)
Design wind velocity (Vz)
VZ = V b(K 1K 2K 3K 4)
where, Vb= Basic wind speed (m/s), K1 = Probability factor or risk co-
efficient, K2= Terrain, height & structure size factor, K3 = Topography fac-
tor, K4 = Importance factor for cyclonic region.
Design wind pressure,
Pz = 0.6 Vz2
 Connections are made by either Riveting, Bolting or welding.
Riveting
Rivet is made up of a round ductile steel bar piece (mild or high tensile)
called shank, with a head at one end.The size of a rivet is identified by
diameter of shank.

Grip

Length
Manufactured
head
d 
Shank
Initial clearance
Round -Nominal diameter of rivet
countersunk d-Gross diameter of rivet

Rivet is classified as:

1. Hot driven field rivets.
2. Hot driven shop rivets.
3. Cold driven rivets.
Special Point: When the rivets are heated before driving, they are called
hot driven field or hot driven shop rivets, depending upon if they are
placed in the field or in the workshop.
When riveting is done at atmospheric temperature by large pressure to
form the head & complete the driving. Such rivets are called cold driven
rivets.
832 Civil Booster (Civil Ki Goli Publication 9255624029)

4. Hand driven rivets.

5. Power driven rivets
 Cold driven rivets are used for rivet diameter ranging from 12 to 22
mm diameter.
 Cold driven rivets are squeezed of driven to fill the holes & to form
the heads by applicaion of large pressure.
 Most important point is strength of cold driven rivets is more than
hot driven rivets.
 Strength calculation of rivet is done on the basis of gross diameter.
Nominal diameter (d): It is shank diameter in cold condition.Various
diameters available are 8, 10, 12, 16, 18, 20, 22, 25, 30 mm.
Gross diameter (d'): Diameter of the hole, considering that rivet fills
the hole completely.
For d  25 mm, d' = d + 1.5 mm, For d > 25 mm d' = d + 2 mm
As per Unwin's formula d mm  6.01 t mm Where tmm = minimum

thickness of the plates being connected. ( d  1.91 t cm )

Assumptions in riveted Connection
1. Friction between plates is neglected
2. Shear stress is uniform over cross-section of rivet
3. Rivets in group subjected to direct load share the load equally (Infact
stress is equally shared)
4. Bending Strees in rivet is neglected.
5. Distribution of direct stress on the portion of plates between rivet hole
is uniform.
6. Rivets fill the hole completely.

Grip-length

Special Point: When the grip of the rivet is more than 6 times the
diameter of hole, Number of rivets required by normal calculation should be
increased by not less than 1% for each additional 1.5 mm of grip. Grip in
any condition, should not exceed 8 times the diameter of holes.
Basic defination
Pitch: It is the distance between two consecutive rivets in the direction
of force.
 Design of Steel Structure 833

g F

Edge distance
Gauge:- It is the distance between two consecutive rivets perpendicular
to the direction of force.
Special Point:- Diagonal pitch is the distance between centres of two
adjacent rivets in the diagonal direction.
Net area:- It is the area at the root of the thread.
Edge distance:- It is the distance between the edge of a member or
Cover plate & the centre of the nearest rivet hole.
Slip factor:- It is coefficient of friction in friction type joint.
Rivet Value (Rv): It is the minimum strength of rivet in shearing or
bearing
 2
P s = Strength of rivet in shear P  n d' 
s 4 s
n = 1, Single shear case, n= 2, Double shear case
s = Allowable shear stress, d’ = Gross diameter
Pb = Strength of rivet in bearing
P  d't
b br

br = Allowable bearing resistance, d’ = Gross diameter
t = Minimum thickness of (sum of two cover plates, main plate)
Strength of Riveted joint : It is the minimum strength of plate in
shearing, bearing & tearing. The shearing failure can be prevented by
providing sufficient edge distance
Tearing strength: Pt  (B  nd)tat
n = No. of holes
at = Allowable tensile strength of plate.
Efficiency of joint

m in im um R v , S trength of p late 
=
Strength of solid plate
834 Civil Booster (Civil Ki Goli Publication 9255624029)

Number of rivets (n) = Force/Rv

Maximum permissible stress in rivets & bolts (IS 800 : 1984)
Axial tension Shear Bearing
Type of faster
 at  MPa   MPa   MPa 
1. Power driven

( a ) Shop rivets 100 100 300

( b ) Field rivets 90 90 270

2. Hand driven rivets 80 80 250

3. Close tolerance &

120 100 300
turned bolts

4. Bolts in clearance
120 80 250
holes

Permissible Stresses

Types of stresses Notation stress FOS

Axial Tensile stress at 0.6fy 1.67
Max axial Comp Stress ac 0.6fy 1.67
Bending Tensile stress  bt 0.66fy 1.515
Max Bending Comp. stress  bc 0.66fy 1.515
Avg. shear stress  va 0.4fy 2.5
Maximum shear stress  vm 0.45fy 2.22
Bearing Stress P 0.75fy 1.33
Stress in slab base  bs 185 –

Special Point: When wind & earthquake loads are considered, permissible
stresses in steel structures are increased by 33.33% & in rivets & welds
( structure fasteners ) it’s increased by 25%.
Packing:

Packing = t1– t 2
t1
t2

Additional rivet
 Design of Steel Structure 835

If the difference between two plates to be joined ( t1 - t2 > 6 mm), then

additional rivet (over & above that required from normal calculations) shall
be provided on Packing extension.
So, No. of additional rivets = 2.5% of actual no. of rivets obtained from
normal calculation per 2 mm thickness of packing.
Patteren of riveted Joints:

Chain Diamond Staggered

21 21
4321
P P P B
P P P

21 4 21 21

Eccentric Connection:
PAi  Pe ri A i
F Di =  A , FTi   A r 2
i ii

Fri =  FDi 2   FTi 2  2FD FT cos   R v

i i

P
e

column flange

Bracket
plate
Direct load + Twisting
F Di = Direct force in ith rivet & Acts in direction of applied load
FTi = Force in ith rivet due to torsion & Acts perpendicular to the line
joining C.G of rivet group & the rivet under consideration.
r i = distance of ith rivet from C.G

Ai = (d )²
4 i
Most critical rivet has maximum value of r & minimum value of 
Rivets on Section A = Direct loading + Torsion
Rivets on Section B = Direct loading + Bending
836 Civil Booster (Civil Ki Goli Publication 9255624029)

Minimum No. of rivets in Direct load + Twisting case

B A P

P
x x
P
6M x x
n x x
mPR v

B A

M = Pe, Rv = Rivet Value

m = no. of lines of rivets = 2
n = no. of rivets in a line = 3
p = Pitch (Same in both direction)
Special Points :- For rivets subjected to combined shear & Tension

fs ( calculated) f t ( calculated)
 /1.4
s t
f t = Calculated tensile stress, fs = Calculated shear stress
s &  t = Permissible shear & tensile stress.
 IS 808:1989 Dimension for hot rolled steel beam, column,
channel and angle section
 Minimum thickness of any member should not be less than 6 mm
under normal conditions & if it is exposed to weathering, minimum thickness
= 8mm.
 Steel beam theory is used to find the approximate value of the mo-
ment of resistance of a doubly reinforced beam specially when the
area of compression steel is equal to or more than the area of the
tensile steel.
 Design of Steel Structure 837

Limit State Method

of Design 2
Limitations of working stress method (WSM)
1. The main assumption of linear elastic behaviour (stresses can be kept
within permissible limits) is not realistic.
2. In WSM there is no scientific basis for assumption of FOS. but in limit
state method FOS is calculated from reliability analysis.
The partial safety factory takes into account, possible overloads and
possible understrength.
3. WSM fails to discriminate between various types of loads that act
simultaneously , having different degree of uncertainty.

Partial safety factor load   f 

Factored load  Characteristic laod  

f

Limit State of Strength Limit State of

LL Serviceability
Combination DL WL/ WL/ EL
AL DL
Lead- Combined EL Lead- Combined
ing (CL, SL, etc.) ing (CL
etc.

DL+LL +CL 1.5 1.5 1.05 – – 1.0 1.0 1.0 –

DL+LL +CL+ 1.2 1.2 1.05 0.6 – 1.0 0.8 0.8 0.8
WL/EL 1.2 1.2 0.53 1.2 – – – – –

DL+WL/EL 1.5 – – 1.5 – 1.0 – – 1.0

(0.9)*

DL+ER 1.2 1.2 – – – – – – –

(0.9)*

DL+LL +AL 1.0 0.35 0.35 – 1.0 – – – –

838 Civil Booster (Civil Ki Goli Publication 9255624029)

*This value is to be considered when stability against overturning or

stress reversal is critical
DL = Dead Load, SL = Snow Load, CL = Crane Load (Vertical/hori-
zontal), AL = Accidental Load,
WL = Wind Load, EL = Earthquake Load., LL = Imposed Load (Live
Loads), ER = Erection Load,
Partial safety factor for material strength (  m )

Characteristic Strength
Design strength =
m

S.No. Definition Partial Safety factor

1. Resistance, governed 1.10

by yielding m
2. Resistance of member 1.10
to buckling m0
3. Resistance, governed by 1.25
ultimate stress
4. Resistance of connection : Shop Field
Fabrications Fabrications

(i) Bolts-Friction type, mf 1.25 1.25

(ii) Bolts-Bearing type, mb 1.25 1.25
(iii) Rivets, mr 1.25 1.25
(iv) Welds, mw 1.25 1.50

*This value is to be considered when stability against overturning or

stress reversal is critical
Various limit states in designing -
Limit states are the states beyond which the structure no longer satisfies
the performance requirements specified. These are :
(a) Limit state of strength - which associated with failures, under the
action of probable & most unfavourable combinations of loads on the structure.
It includes strength, stability during sway, brittle failure, excessive deformation,
stability against overturning & fracture due to fatigue.
(b) Servicibility limit state - It includes vibration, corrosion & durability,
deflection & deformation and repairable damage due to fatigue cracking
etc.
 Design of Steel Structure 839

Bolts 3
Bolts are used in place of rivets for structures not subjected to vibrations
 A bolt can be loaded in:
(i) Tension, (ii) Shear, (iii) Shear and tension both.

Bolt

A bolt primarily designed to withstand tensile loading. Therefore,

ideally the bolt should only be loaded in tension.Bolts are suitable to carry
axial tension.
Types of bolts

Turned/Finished Black or High strength

Bolt unfinished bolts friction grip bolt
Hexagonal/Square shape
Small (0.15 - 0.5) mm
Made from MS Resistance to load
tolerance
through friction
Used in no slip Used as temporary Designated as
condition fastner normally class 8.8 &
Designated as M20, 10.9 etc.
Used in M24 etc.
Made from medium
dynamic loading carbon steel
 For reversal of stresses, the most suitable bolt generally used in is HSFG
bolt.
840 Civil Booster (Civil Ki Goli Publication 9255624029)

Diameter of bolt holes

For d ( 12 - 14 mm), d' = d + 1 mm, d ( 16 - 24 mm), d' = d + 2 mm,
For d > 24 mm, d' = d + 3 mm
Where d - Nominal diameter, d' - Diameter of hole

Shank area (A sb)

A nb
= 0.78 A sb
Area of root of
thread (A nb)

Types of riveted/bolted joints

Lap joint Butt joint

Double cover Single cover

Single bolted butt joint butt joint
lap joint

Single cover single

Double bolted Double cover bolted
lap joint single bolted

Single cover double

bolted
Double cover
double bolted
S.No. Lap joint Butt joint
1. Lap joint are less strength A butt joint is very strong
2. Number of Rivet required is less Number of rivet required more
3. In Lap joint cover plate are not In Butt joint cover plate are
required required
4. Lap joint is less costly Butt joint is high costly
Single and double cover butt joints both has eccentricity zero but,
lap joint has an eccentricity. As compare to single cover butt joint, double
cover butt joint is more efficient.
 Design of Steel Structure 841

Advantages of welded joint over riveted joint:

(a) As no holes are required for welding, so the loss of member strength is
smaller.
(b) The speed of fabrication is higher with the welding process.
(c) The welded connection look better than the usually bulky riveted joints.
(d) Welded joints are economical as less labour and material are required at
joint.
(e) The welding process requires less working space than the riveting pro-
cess.
(f) No noise is produced in the welding process as in the riveting process.
Specification in bolting
Minimum pitch = 2.5d
Maximum pitch
(a) Tension = min (16t, 200 mm)
(b) Compression = min (12t, 200 mm)
(c) When plates are used in tacking rivets:
(i) The tacking rivets are provided at a pitch in line should not more than 32
times the thickness of outside plate or 300 mm (whichever less)
(ii) Where plates are exposed to weather, the pitch in line should not
more than 16 times the thickness of the outside plate or 200 mm
(whichever less)
(iii) Maximum spacing of taking rivets: Compression zone – 600 mm &
tension zone – 1000 mm.
Minimum edge & end distance
(a) Machine cut = 1.5 × hole diameter
(b) Sheared or hand plane cut edges (Rough) = 1.7 × hole diameter
Maximum edge distance = 12t 

250
Where, 
f y , t = thickness of thinner plate
Special Point: When members are exposed to corrosive environment, then
maximum edge distance  40 mm + 4t (where t  thickness of thinner
plate, mm)
The design strength of bearing bolts under shear is the least of the following
(a) Bearing strength, (b) Shearing strength
842 Civil Booster (Civil Ki Goli Publication 9255624029)

1. Nominal shear strength of bolt (Vnsb)

f ub
Vnsb   n n A nb  n s Asb  ljlgpk
3
Factored shear strength
Vnsb V
 Vsb   , Vsb  nsb
 mb 1.25
Where,
f ub = Ultimate tensile strength of bolt
ns = No. of shear planes without threads intercepting the shear plane
A sb = Nominal shank area
nn = No. of shear planes with threads intercepting the shear plane
A nb = Net tensile stress area
(a)  lj  Reduction factor for long joints, applied when length of
joint > 15 d
0.005l j
lj  1.075 
d
Subjected to limits 0.75   lj  1.0, l j = length of joint

(b)  pk  Reduction factor for packing plates, applied when thickness

of packing plates > 6 mm
 pk  1  0.0125t pk
(c) g - Reduction factor for large grip length, applied when grip length
between 5d and 8d
8d
lg 
3d  lg

2. Nominal Bearing strength of bolt (Vnpb)

 v n pb 
Vnpb  2.5k b dt min f u ,  Vpb  
  mb 
t = aggregate thickness of connected plate experiencing bearing
stress in same direction.
d = nominal diameter of the bolt
f u = ultimate tensile stress of plate in Mpa
 e p f ub 
K b = Smaller of  3d , 3d  0.25, f ,1 , e = end distance
 0 0 u 
 Design of Steel Structure 843
p = pitch, d0 = hole diameter
f ub& fu = ultimate tensile stress of bolt & plate respectively
3. Nominal Tensile Strength of bolt (Tnb)

   Tnb 
Tnb  0.9f ub A nb  f yb A sb  mb  ,  Tdb  
  mo    mb 
A nb = Net tensile area of bolt
A sb = Shank area of bolt
f ub = Ulimate tensile stress of bolt
f yb = Yield stress of bolt
 mb = Partial safety factor for material of bolt = 1.25

 mo = Partial safety factor for material resistance governed by yielding

= 1.10
Strength of plate (lesser of)

Shearing Tearing Block shear failure

Can be prevented Rupture of plate

by sufficient edge

1. Tensile strength of plate:-

fu
Tnd  0.9A n
 m1

A n = Net effective area in mm²

f u = Ultimate strength of material
 m1 = Partial safety factor = 1.25
Net effective area An = (B - ndh )t (For chain bolting)
 m
psi2 
An = B  ndh  
t 1 4g1 
t

(For Staggered bolting)
2. Block Shear failure:- It is the minimum of strength due to
(a) Shear yield + Tension rupture
844 Civil Booster (Civil Ki Goli Publication 9255624029)

A vg P
fy
 A vg
3 0.9f u A tn
 Atn Removed
1.1 1.25
Atn = Net section area in tension
Avg = Gross section area in shearing,
(b) Shear Rupture + Tension yielding

Avn P
f
0.9 u A vn
3 f y A tg
 Atg Removed
1.25 1.1
Avn = Net section area in shearing
A t g = Gross section area in tension
Special Point:- Rupture always occurs on net area but yielding is
considered to occur on gross area.
1.1 and 1.25 are FOS wrt yielding and rupture respectively.
Strength of plate in tearing without any deductions (P)
 fy 0.9f u A tn 
P = min  1.1  A g , 1.25 
 
f yAg
Hence, P As gross-section yielding will always be Critical
1.1
than net section rupture
Efficiency of joint
min imum of  Strength of bolt, Strength of plate 
n=
Strength of plate in tearing without deductions
 Design of Steel Structure 845

Welded
Connections 4
 Welding is the best method for achieving a rigid connection
 Classifications of welded joints based on various factors -
(a) Position of weld - Flat, overhead, vertical & horizontal welds.
(b) Types of weld - Spot, plug , slot, fillet & groove (butt) welds.
(c) Types of joint - Butt, corner, Tee & lap welds.
 In Plug welds small holes are made in one plate and is kept over another
plate to be connected & then entire hole is filled with filler material.

Plug weld

 Fillet welds are provided when two members to be jointed are in different
place (lap joint)
Fillet weld
T
T

 Butt weld is also called groove weld. It is provided when the members to
be jointed are lined up (in one plane) (butt joint)

T T

Single V_ Butt weld

 In Slot weld fillet welding is made along the periphery of hole

Slot
weld
846 Civil Booster (Civil Ki Goli Publication 9255624029)

Square Double Single Single Double

V V U U
Surface is Surface is
not in tension in tension

t t t
s s s

Mitre Concave convex

Types of fillet welds
Method of Representation
Length of weld
finish symbols
Contour and

Other side
Unwelded
length
Shape
Size

Identification line
Reference line Arrow line
10 120

10 mm fillet weld convex

Arrow side finish 50 mm weld length
& 120 mm unwelded length Joint
Weld defects: Incomplete fusion, Slag inclusions, Porosity, Cracks and
under cutting.
Specifications for Butt weld
Effective throat thickness:-
(a) Incomplete penetration:- It is the 7/8 times the minimum
thickness of plates being joined.(But for stress calculation 5/8).
(b) Complete penetration:- It is the thickness of thinner member
joined.
 Design of Steel Structure 847

Design Strength of butt weld (Vdw)

f yw1L w t e
Vdw 
 mw

 f yw 
f yw1 = Smaller of Shear Stress of weld   & the parent
 3 
 fy 
metal 
 3 
 mw = Partial safety factor, Site welding = 1.5
Shop welding = 1.25, fyw = yield stress of weld
te = effective throat thickness in mm
Specifications for fillet weld
 Effective length is the length of the fillet weld for which the specified size
& throat thickness of weld exist. It is considered equal to it’s overall length
minus twice the weld size. The deduction is made to allow for craters to be
formed at the ends of welded length. End returns are made equal to twice
the size of the weld to relieve the latter from high stress concentrations at
their ends.
 Throat of the fillet is the weakest section in a fillet weld:
 The theoretical throat of weld is the shortest distance from the root to
the hypotenuse of the triangle.
Force

Fillet weld Fusion zone

Throat
Leg Weld face toe

Side parallel
to force Side Perpendicular
to force Leg

Note: As per IS 800 – 2007 gives following provision for fillet welds:

Square edge Round edge

1. If a fillet weld is applied to the square edge of a part, the specified

848 Civil Booster (Civil Ki Goli Publication 9255624029)

size of the weld should generally be atleast 1.5 mm less than edge
thickness.
2. If the fillet weld is applied to the round toe of rolled section, the
3th
specified size of weld should not more than of thickness of section
4
at toe.
Minimum size of weld:- depends on thickness of thicker member,
but minimum size of fillet weld is 3 mm.
Thickness of thic ker member Minimum size
0  10 3
10  20 5
20  32 6
8 first run
32  50
10 sec ond run
Size:- It is the minimum weld leg size in the largest right angled triangle
that can be inscribed in the weld. Maximum size of weld is decided by
the thinner member.
Size of weld (mm)

2.4 mm

Special Point: In deep penetration weld where depth of penetration

beyond the root run is minimum 2.4 mm, the size of weld is taken as 2.4
mm + minimum leg size.
Effective throat thickness = K × size of weld

Angle b / w
fusion 60  90 91  100 101  106 107  113 114  120
faces
K 0.7 0.65 0.60 0.55 0.5

Special Point:- Fillet weld is not recommended if the angle of fusion

faces is less than 60° or more than 120°.
Min. Overlap length  max (4 thickness of thinner plate), 40 mm)
 Design of Steel Structure 849

Minimum end return = 2 × size of weld

Min=2s
F

Design strength of fillet weld:

f wn fu
f wd   , throat area = Lw ×KS
 mw 3 mw

fu
so, design strength  L w (KS) 3
mw

where Lw = effective length of the weld

If length of joint in the direction of load transfer is > 150tt, then length is
decrease by a factor 

 lj 
  1.2     1.0
 750t t 

lj = length of joint, tt = KS = throat size of weld

Combination of stresses
(a) Fillet welds subjects to normal and shear stresses
fu
f e  f h2  3f v2  , f v  Direct shear stress
3 m
f h = normal stress (compression or tension)
(b) Combined bearing, bending and shear

f e  f b2  f 2 br  f b f br  3q 2
fb = calculated stress due to bending
f br = calculated stress due to bearing, q = shear stress
850 Civil Booster (Civil Ki Goli Publication 9255624029)

Tension Member 5
1. Net area is effective in tension member
3. Permissible stress is fixed (0.6 fy) and design is straight forward
2. There is no stability problem (as case of compression members) but
slenderness ratio is limited to safeguard against buckling during
transportation & erection.
Maximum slenderness ratio

Maximum
Description

Tension member in which reversal of direct
stress occurs due to live load other than wind 180
or earthquake.
In reversal occurs due to wind or earthquake. 350
A tension member permeanently in tension
400
except in pretensioned members.

Net sectional area (Anet):

P
A net required 
at
at = Permissible axial tensile stress
Anet provided  Anet required
(a) Net area for plate section
 pi2 
A net = b  nd h   t
 4g i 
 Design of Steel Structure 851

(b) Net area of angle section

· Net Effective Area:

t
l2

3  A1
A net  A1  A 2  k1 , k1  A1 (l2 dh  t/2)t
3 A  A
1 2

A 2  (l2  t / 2)  t
 For pair of Angle Placed back to back connected by only one leg of
each angle.

5  A1
 A net  A1  A 2  k 2 ,  K 2 
5  A1  A 2

The area of a web of Tee = Thickness of web × (depth - thickness of

flange)

Tacking rivet

Gusset plate
The non uniform straining of web due to tension is called shear lag.
The shear leg reduces the effectiveness of the tension member ( angle
section) component that is not directly connected to gusset plate. For angle
section, the unconnected leg is known as outstand leg.
Therefore, the contribution of outstanding leg in resisting tension is
less than the connected leg by a factor which is known as k (reduction
factor).
852 Civil Booster (Civil Ki Goli Publication 9255624029)

Permissible stress in design

The direct stress in axial tension on the effective net area should not exceed
at .
where at = 0.6 fy, fy = Minimum yield stress of steel in MPa
Allowable stress at in Axial tension of steel
Form Thickness / diameter mpa
Plates, angles, tees upto & including 20mm 150
& Ibeams, 20mm40mm 140
channels and flats over 40mm 138
Bars (round, Squa  upto & including20mm 150
re & hexagonal over 20 mm 144
Lug Angle: It is a short length of an angle between section used at a
joint to connect the outstanding leg of a member, by reducing the length of
the joint.
• Lug Angle should be provided at the beginning of Joint, so-that It can be
effictively sharing the given load.
• Lug Angle are not very effective to transfer the given load due to eccen-
tricity of C.G. of rivets.
Strength of tension member: It is the minimum of the following
1. Net section rupture
0.9A n f u
(a) For plates Tdn   m1
T dn = Design rupture strength
A n = Net effective area, fu = Ultimate strength of material
 m1 = Partial safety factor in rupture = 1.25
(b) For Angles
0.9A nc f u  A go f y
Tdn  
 ml  m0

 W   bs   f y 
Where,  = 1.4 - 0.076  t   L   f 
  c  u 
 Design of Steel Structure 853

 f   
  u    m0 
  m1   f y 
 0.7
A nc=Net area of connected leg
t = Thickness of angle leg
Ago= Gross area of outstanding
Lc = Length of end connection
W = Outstand leg width
bs = Shear leg width
Special Point: The connected leg undergoes rupture (Net area is used)
but outstanding leg undergoes yielding (gross area is used)

t
t
W
W

bs= w + w1 – t bs= w
w1
Welded Connection
Bolted Connection

For preliminary sizing of a tension member, the tearing strength of net

section may be expressed as

A n fu
Tdn  
 m1
where,  = 0.6 for number of bolts  1 or 2
= 0.7 for 3 bolts
= 0.8 for number of bolts  4
or equivalent weld length
2. Gross - Section yielding

Ag f y
Tdg 
 mo

 mo = 1.1, Ag = Gross-sectional area

854 Civil Booster (Civil Ki Goli Publication 9255624029)

3. Block Shear failure

(a) For Angles:- Appropriate areas in shear & tension should be
considered for failure. But in welded angle tension members, it is
the gross area only that is involved (No concept of net area).
(b) For plates
(1) Shear yielding + Tension rupture.
(2) Tension yielding + Shear rupture.
Special Points -
The gantry girder is designed for the following loads:

Cross head crab

or end girder
Rail

Gantry
girder

Cross travel

(a) Gravity loads : Self weight of the components and reaction from crane
girder, acting vertically downwards.
(b) The lateral thrust, (also called surge loads) due to starting or stopping
of the crab acting horizontally, normal to the gantry girder.
(c) The longitudinal thrust due to starting or stopping of crane, acting in
the longitudinal direction.
Bulb angle
Bulb angles are used in ship building because
(a) When the structure is under extreme stress and starts to buckle, this
shape is highly resistant & increases the longevity of the structure.
(b) They provide better plate stiffening.

Web
h = depth
Flange

b= width
 Design of Steel Structure 855

Compression-
Members 6
 The principal compression member of a crane is called boom.
Effective length:- It’s the distance between point of contra-flexures.

End One end fixed Both end Both end One end fixed
condition one end free Hinged Fixed one end Hinged
Leff 2L L L/2 L
(Theoretical) 2

Leff (As per

2L L 0.65L 0.8L
IS code.)

Effective length of struts

Slenderness ratio (  ): It’s the ratio of effective length to appropriate
radius of gyration.

leff I min
 max  rmin 
rmin , A
Maximum Slenderness Ratio (  max) for compression Members
 Buckling of a member in compression always occurs about minor
principle axis. For minor principle axis, area of moment of inertia is
minimum. So, radius of gyration is minimum. (slenderness ratio is
maximum).
856 Civil Booster (Civil Ki Goli Publication 9255624029)

D escrip tio n  m ax
A stru t co nn ected b y sin g le rivet at eac h en d. 180
In m em b er carryin g lo ad s resu ltin g fro m d ea d
180
lo ad s an d im p o sed lo ad s.
A m em b er sub je cted to co m p re ssiv e fo rce resu ltin g
fro m w in d /ea rth q u ak e fo rc e, p ro v id in g th e d efo rm atio n
250
o f su ch m em be rs d o es n o t a d ve rse ly e ffe ct th e stre ss in
an y p art o f stru ctu re.
C o m p res sio n flan g e o f a b ea m 300
A m e m b e r n o rm ally actin g a s a tie in a ro o f tru ss o r a
b racin g system b u t sub je cted to p o ssib le reve rsal o f 350
stre sses re su ltin g fro m th e a ctio n o f w in d o r e arth q u ick fo rc es.

For prevention of local buckling of flange or web:

K2 E
f cr  2
b
12 1   2   
t
fcr = critical stress, K = depends on support conditions
Special Point: The critical stress at which the plate buckles is inversely
proportional to (b/t)²
To prevent the buckling of flange plate and web plate
b
b > 16 dw
tf > 50
tf tw

tw dw

1. For circular hollow sections

D 88
If  88, then it is a slender section and its Aeff = Ag D/ t2
t 2
 
2. For slender cross-section other than circular hollow section
A eff   Beff t
 t = limiting width to thickness ratio of semi-compact section
Beff = Effective width of slender element, Beff =   L t    b 
250
 = modification factor for yield strength of material  = fy
 Design of Steel Structure 857

Rankine Merchant formula (for permissible compressive stress)

f cc f y
 ac  0.6 1/ n
  f  n   f n 
 cc y


2 E
f cc  Elastic critical stress in compression
2
n = factor assumed as 1.4 (General range b/w 1-3)
Design recommendations for tack riveting
1. Slenderness ratio () of each component between tack riveting should
be such that
  40,   0.6  w hole sec tion 
2. The diameter of tack rivet should not be less than the minimum value
given below
Thickness of member Min.dia
Upto 10mm 16
10 -16 20
 16 mm 22mm
3. Two rows of tack rivets are provided if
(a) Length of leg in angle > 125 mm

l > 125 mm

(b) Web of channel > 150 mm

l > 150 mm

Design recommendations for lacing

(1) The effective slenderness ratio of laced column shall be considered
1.05 times the actual maximum slenderness ratio due to the shear
858 Civil Booster (Civil Ki Goli Publication 9255624029)

deformation effect.

(2) In lacing, length of column is increase by 5% of the effective length

of column.
(3) The lacing shall be design to resist transverse shear Vt, at any point
in the member, which is equal to atleast 2.5% of axial force in the
member. If there are two transverse parallel system, then each
Vt
system should resist of transverse shear..
2
(4) If column is subjected to bending also, then Vt= Bending shear +
2.5% of column force.
(5) As far as possible, the lacing system shall be uniform throughout.
(6) The effective length of single lacing in laced system is equal to the
length between the inner end fasteners.
For double lacing system, it should be considered 0.7 times of this
length.
In welded condition, the effective length shall be taken 0.7 times the
distance between the inner ends of welds connecting the single lacing
bar to the members.
(7) In single lacing system, the direction of lacing on opposite faces should
be shadow of each other. It should not be mutually opposite.
 kL 
(8) The slenderness ratio   for lacing bar shall not exceed 145.
 r 
t
(where r = ) for flats where t = thickness of flat plate.
12
(9) The maximum spacing of lacing bar shall be such that the maximum
slenderness of the main member between consecutive lacing
 Design of Steel Structure 859

connection is not larger than 50 or 0.7 times the most unfavourable

slenderness ratio of the member as a whole.
(10).  should be between 40° - 70°
(11).Minimum width of lacing bar (Generally 3 times of diameter of rivet).

Diameter of rivet Min width

16 50
18 55
20 60
22 65

Design recommendations for batten


b b

d' d

C S'

Intermediate
batten
d1 d'1

1. Effective length of battened column is taken 10% more than the actual
column but for lacing, it is 5% more.
C
2.  50 ,  0.7  Whole Section 
 rmin 
3. Minimum number of battens required is 4 ( 2 end battens & 2
intermediate battens).
4. Thickness t of battens should not be less than 1/50 of the distance
between the inner-most connecting lines of rivets/bolts or welds
perpendicular to main member.
5. Depth:-
3
d1   d >
4
 2 b >2b
860 Civil Booster (Civil Ki Goli Publication 9255624029)

 = distance between C.G of component members.

b = width of member in plane of battens (flange width)
Special Point: Battens are designed only for axial loading as well as
longitudinal shear & moment while but laced columns are designed for
axial force & BM is zero in lacing to ensure zero BM in lacing provide
one rivet at each end as far as possible.

M M
Vt Vt C
2N Vb Vb 2N

Vt Vt
s
2N 2N
Forces acting on batten
1. Designed to carry Bending moment & shear force arising from
transverse shear force Vt which is 2.5% of total axial force on whole
compression member.
Vt C Vt C
2. Vb = ,M=
NS 2N
N = No. of parallel planes dividing the transverse shear force.
Vb = longitudinal Shear force
Vb fy
3. 
At 3 mo
where, At = td, d = Overall depth of batten.
t = thickness of batten
6M f y
4.  bc,cal  
td 2  mo
Column Splice
 A joint provided in the length of the column is called splice.
 Theoretically, a splice plate should be located at the point of
 Design of Steel Structure 861

contraflexure of the column.

 A column splice is used to increase the length of the column.
 Column splices are located just above the floor beam connections,
usually about 2 to 3 feet above the floor.

Web
splice

Column splice Web splice

(Column flange having for shear
complete bearing)
 Column section is spliced when the length of the column is more than
the length of the column section available, a number of pieces are
jointed to funish the full length of the column.
 Laced column are inclined member. Therefore, axial load in laced col-
umn is lesser than as compared to battened column.
Flange splice
 A joint in the flange element provided to increase the length of flange
plate is known as flange splice.It is designed for axial force only.

Flange splice

Web splice

Web splice
 A point in the web plate provided to increase it’s length is known as
web splice. These are designed to resist the shear and moment at the
spliced section.
 The splice plates are provided on each sides of the web.
862 Civil Booster (Civil Ki Goli Publication 9255624029)

Column Bases
and Caps 7
 Sufficient fastening are provided to retain the column firmly on the
base plate and resist all moments and forces (except direct compression in
the column) arising during transit, unloading and erection.
Types of column base

Slab-base Gusseted base Grillage foundation

(a) Gusset plate: It is a thick steel sheet used for joining two or more
than two adjacent structural member when they are intersecting
each other.The thickness for base of club should not be less than 12 mm.
It may be fastened to a permanent member by bolts, welding or
rivets or combination of these three.

Gusset plate Steel column

Angle for connection
Base plate
Concrete Foundation

(b) Anchor plate: It is a steel plate attached to or embedded in a

support & used as an anchor for supporting cables.
(c) Base plate: It is used to connect a column with a RCC foundation
& they are installed below the steel column on RCC foundation.
When we provide a base plate under a column, the load gets dispersed
to a larger area & after that it is transferred to concrete foundation. So
system is safe.
 Design of Steel Structure 863

1. When only axial load is acting

b
a

(a)

t
3w  2 b 2 
a   WSM  t 

2.5w a 2  0.3b 2   LSM 
  ,
 bs  4  f y /  m0
t = slab thickness in mm
P 1.5P
w = B2 assume B (in WSM), w = (in LSM)
  Area
where, B² > P / c and   c  f ck / 4 
a= Greater projection of plate beyond column
b =Lesser projection of plate beyond column
 bs = permissible bending stress in slab bases
= 165 MPa for flanged beams.
= 185 MPa for solid beams
(b) Square slab base under solid round column

d0 B 90W  B 
t  10  
16 bs  B  d 0 

B  1.5  d 0  75  mm
B = Length of the side of the cap or base.
W = Total axial load (KN)
d0 = Diameter of the reduced end.
864 Civil Booster (Civil Ki Goli Publication 9255624029)

Beams (WSM) 8
Beams(WSM)

Laterally restrained beams Laterally unrestrained beams

The compressions flange of the bending compression (bc)

beam is prevented from lateral  bending tension (bt)
buckling.
Primary design criteria for design of beams
.Safety against bending
Maximum permissible compressive or tensile bending stress
 bc   bt  0.66f y
Permissible bending stress (IS 226 - 1975)
Nominal Thickness Yield Stress (fyMpa) bc = bt
of plate

Angle, Tee, I, Channel 250 165

& flat sections upto
and including 20mm

From 20-40 mm 240 158.4

Over 40mm
230 151.8

For designing of rolled Sections for bending moment

Aplate

tf
t

Area due to holes = 2d (t + t)

f
 Design of Steel Structure 865

Zrequired  Zrolled
A p required 
d
my t  Gross area of tension flange 
    bt
Igross  Net area of tension flange 
Net area of tension flange = Gross area for tension – Area due to holes
Safety against Shear
Max. permissible average shear  va = 0.40 fy
Max. permissible shear stress  vm = 0.45 fy
Safety against deflection:-
The maximum allowable vertical deflection under live load for a canti-
Span
lever member supporting brittle cladding in an industrial building is
150
Span Span
Maximum permissile deflection in S.S. steel beam =
,( as per old IS:800)
300 325
Some of the reasons for limiting deflections are:
(a) Excessive deflection may create problems for floor or roof drainage.
(b) Excessive deflection may lead to crack in the plaster of ceilings &
may damage the material attached to or supported by the beam.
(c) There may cause undesirable twisting and distortion of connections
and connected materials.
Web crippling: It occurs due to a concentrated load on the beam, due to
reaction at support, high compressive stresses are produced in the
web near to the upper flange or lower flange.
 The crippling occurs at the root of the radius

Web
crippling
tf

Slope 1
b1 2.5

 According to IS: 800-2007, accepted formulae to find crippling of web.

f yw
f w  (b1  n c )t w 
 mo
866 Civil Booster (Civil Ki Goli Publication 9255624029)

where, br = Stiff bearing length, fyw=Yield stress of web

c = Length obtained by dispersion through the flange to web
junction at a slope of 1 :2.5 the plane at flange.
c = 2.5 tf
Web buckling - It is the sudden sideways deflection of a structural mem-
ber under the application of compressive load used.
b1

45º 45º h
45º

b1 Web buckling

Generally, if the web is safe in crippling, it will be safe in buckling also.

(1) Local flange buckling is due to bending compression.
(2) Web buckling is due to diagonal compression.
(3) Web crippling is due to bearing stress.
Safety against local buckling b

tf

tw dw
b dw
t f  16 t w  50

Safety against web buckling

h2

tw dw dw t
1eff  rmin  w
2 12
h2
 Design of Steel Structure 867

Plate-Girders 9
Generally it is used when bridge span > 20 m.
P
Plange plate

5 mm
Angle section
a 3
Neutral
b 4 axis
a 2 1
c Web plate

R
(a)- larger unsupported length, (b)-Smaller unsupported length
(c)- Spacing between vertical stiffners
(1)-Load bearing stiffner, (2)-Vertical stiffners, (3)-1st horizontal stiffner,
(4)-2nd horizontal stiffner
 The girders having two or more than two webs are called box girder
 Usually a plate girder is called as economical if it corresponds to minimum
weight. L/15 is the depth of plate girder in buildings.
Special Point:
a > 270 tw , b > 180 tw
For stiffned web avg. permissible shear depends upon d/t & c but for
unstiffened web avg. permissible shear is 0.4fy .
Deflection limits for gantry girders
Category Maximum
deflection
Vertical deflection
(a) Manually operated cranes Span/500
(b) Electrically operated cranes upto 500 kN Span/750
(c) Electrically operated creanes over 500 kN Sapn/1000
Relative displacement between 10 mm or
Rails supporting crane span/400
868 Civil Booster (Civil Ki Goli Publication 9255624029)

V
Design of web: Average shear stress in the web  va,cal = d  t
w w
permissible average shear stress,  va .

d1  816 1344 
(i)   , ,85  No stiffener will be required
t w lesser of   va ,cal
 fy 


d2  3200 
(ii) t  lesser of  , 200  . Vertical stiffeners are provided.
 fy 
w  
d2  4000 
(iii) t  lesser of  , 250  . Vertical stiffners alongwith one
 longitudinal stiffener @ 0.2D
w  fy 

d2  6400  Vertical stiffners alongwith two

(iii) t  lesser of  , 400  . longitudinal stiffener @ 0.2D

w  fy  & 0.5D, respectively.
d2 = 2 × clear distance from compression flange angles or plate to the
neutral axis.
Special Point: Vertical stiffener & first horizontal stiffener are provided
at a distance from the compression flange equal to (1/5) of the distance
from the compression flange to the neutral axis
(a) Design of vertical stiffners:-
1. Spacing between vertical stiffeners should be such that average
Vmax
shear is greater than
d.t
2. Vertical stiffeners can be provided at angles on 2 sides of web
plate.
3. The maximum & minimum distance between vertical stiffeners is
1.5d and 0.33d respectively.
4. If it is to be provided on one side only, it is to be staggered.
1.5d 3 t 3
5. Vertical stiffener should have size such that I 
c
For design of connections for vertical stiffener.
t 2
125  p  Rv
h
1. Consider diameter of rivet and get Rivet value Rv
 Design of Steel Structure 869

2. h = outstand of stiffener from the web, h  16t for angles, h  12t

for flats.
3. Then Get pitch p.
(b) Design of horizontal stiffner:-
1. For first horizontal stiffener I  4ct3
2. For second horizontal stiffener I  d 2 t 3w Connection of horizontal
stiffener with web is similar to that of vertical stiffener with web.
(c) Design of load bearing stiffener
 Bearing stiffeners are used to transfer concentrated loads on girder &
heavy reactions at support to the full depth of the web. A bearing stiffener at
the support is called an end bearing stiffener.
 The bearing stiffeners are designed as columns with the length of the
web 20 times the thickness of web on both sides.

Packing plate
Connected leg of
, bearing stiffener
20 t 20 t Outstand lag of
bearing stiffener
20 t 20 t

 Bearing stiffeners should be placed

(a) Straight (it is not crimped) , (b) Tight with the web
(c) In pair of two or four angles, symmetrically placed on both sides
of the web.
I
1. rmin = Shaded Area  A e 

leff
2.   r , where leff = 0.7l
min

3.  ac .A c  P or R as the case may be

(d) Web splices are designed for shear but flange splices are designed for
moment.
870 Civil Booster (Civil Ki Goli Publication 9255624029)

Industrial Building
10
We adopt windows on north side in such a way that diffused light comes
inside the building which creates no shadow and hence dark pockets are
avoided.
2H
Slope of the truss = tan  =
L
Slope  2  Pitch
H
Pitch of the truss =
L
Special Point: For Galvanised Iron sheet pitch is 1/6 but for asbestos
cement sheet, roof covering, flatter pitch of 1/12 is preferred.
Economy of the roof
Cost ofpurlin Cost of roof cov ering
Cost of truss/unit area = Plan area × 2 +
Plan area
span
Spacing of truss (d) - for spans upto 12 m d =
4
span
for spans > 15 m d =
5
Rafter: They supports the purlins. They are mainly compression member
and may be subjected to shear and bending moment if the purlins are not
placed at nodal points.
Struts: The member carrying compressive forces in a roof truss are called
struts.
Sag tie: To reduce deflection and moment due to self weight. Sag rods are
desgined as tension member.
At the crown, sag rod provided is termed as tie rod, it resists the tangen-
tial components from the two sides of the roof truss.
 Design of Steel Structure 871

Tie rod

Sag rod

Purlin

Spceial Point : According to IS specifications, for roofs of slope greater

than 10° for every additional degree rise, the imposed load is reduced by
0.02kN/m2
Purlins: Member of truss which are supported on the principal rafter and
which transverse loads to the truss.
(a) It is a biaxial bending member. It is located at the panel point of the
truss.The span of purlin is centre to centre of truss.
(b) Maximum spacing between purlins < 1.4m
(c) Angle, Channel, I and Z Sections are used for purlins and girders to
support the cladding
(d) If roof slope < 30° and minimum live load of 750 N/m². Then
following reqirements are to be fulfilled (then uniaxial bending design)
L
1. Width of angle leg in the plane parallel to the roof covering 
60
L
2. Width of angle leg in the plane perpendicular to the roof covering 
45
wL2
3. Maximum bending moment in the prulin M = , L = Span of
10
purlin, w = For continuous UDL per unit length of purlin
span
4. Deflection of purlins = (For Brittle cladding)
180
M wL2
5. Zx required =  = mm3
bc 10  165
872 Civil Booster (Civil Ki Goli Publication 9255624029)

Plastic-Analysis 11
Assumptions in Plastic Analysis:
1. Stress-strain curve is elasto-plastic curve (strain hardening is neglected)
2. Relationship between tensile stress and tensile strain, compressive
stress and compressive strain are same.
3. Steel possess ductility so that it could be deformed into plastic state
4. Strain variation is linear from Neutral axis.
5. Joint should be sufficiently strong to transfer the moments
Plastic behaviour of beam

Axis of summery
Y f<fy fy fy fy
Centroidal
Neutral
axis
O axis

Equal
Equal
area
area
Y axis
axis
f<fy fy
fy fy
M<My M=My M<M<M
y p M=M p
(1) (2) (3) (4)

[Development (in stages) of full placticity of a beam section]

fyA
Plastic moment Mp = (y1  y 2 ) = fy Zp,
2
y1 , y 2 = distance of centre of gravity of the area above & below the
neutral axis respectively
y

y1
Z d Z
y2

d2b db 2
Zp(about major zz axis) = , Zp(about major yy axis) =
b
y
4 4
 Design of Steel Structure 873

Plastic Hinge: It is a yielded zone in flexure in a structure in which

infinite rotation can take place at a constant resisting moment Mp of the
section.
1. For analysis plastic hinge is assumed at one section where stress at
every point is fy .
2. In a beam, if plastic hinge is formed at various locations, then they will
form alternatively due to sagging and hogging moment.
3. Moment will be maximum, if radius of curvature is minimum or
curvature (1/R) is maximum.
4. Length of plastic hinge depends on loading and shape of cross-section.
5. Sufficient no. of plastic hinges have to be developed to render the
structure to a collapse or unstable state.
6. No of plastic hinges for complete collapse = Ds + 1
Ds = degree of static indeterminacy.
7. Due to formation of plastic hinge one after another redistribution of
moments takes place.
8. Plastic hinges are expected to form at
(a) At fixed ends
(b) At a section of sudden change in geometry.
(c) At location of point loads
(d) At a point of zero shear in the span loaded with variable loads like
UDL, UVL etc.
9. When two section meet at a point, plastic hinge forms in a section
having smaller Mp .
Mp Plastic moment capacity Zp
Shape factor    = M = Yield moment capacity = Z
y y

It represents the reserve strength of a section beyond the point of first

yield.
Ultimate load  Wu 
Reserve strength    = Load at first yield of structure W
 y
Collapse load  Wc 
Load factor    = Working load W
 o
When all the hinges required for collapse are formed simultaneously,
then
Load factor     Factor of safety  shape factor   
874 Civil Booster (Civil Ki Goli Publication 9255624029)

Shape factors for different shapes

Section Shape factor
1. Rectangular Section 1.5
2. Solid circular Section 1.7
3. Triangular Section 2.34
(vertex upward)
 1  k3 
4. Hollow circular Section 1.7   4 
 1 k 
5. a. Diamand Section Rhombus 2.00
b. Thin Hollow Rhombus 1.50
6. Thin Circular ring Solid 1.27
7. I section
a. About strong Axis 1.12
b. About weak Axis 1.55
8. T Section. 1.90 to 1.95
Length of plastic hinge (Lp):- Length of beam over which the moment
is greater than yield moment. (My)
Support and Elasto-plastic zone Length of ealsto-
loading in beam (fig.) plastic zone (Lp)
condition
(a) Simply W

supported
L/2 L/2
beam with LP Here MP
L/2L/2
concentrated S = shape factor = M
y
load at mid
span MyMPMy

W
(b) Simply
supported L/2 L/2
LP
beam with 
L/2L/2
1
UDL throughout L P  L  1  
 S 
the beam MyMPMy

(c) Cantilever
beam LP
 1
subjected to LP
L
LP  L 1  
 S
point load at MP My

free end
 Design of Steel Structure 875

Conditions to be satisfied for complete collapse

1. Equilibrium condition:-  F = 0,  M = 0, both condition should
be satisfied
2. Yield condition:- At collapse, bending moment at any section must
not more than the fully plastic moment capacity of the section.
3. Mechanism condition:- At collapse, sufficient no. of plastic hinges
must be developed so as to transform a part or whole of structure into
mechanism leading to collapse.
Special Point: If all three conditions are satisfied, we get a unique lowest
value of collapse load.
Method of Analysis:
Plastic moment
condition
(a) Lower bound theorem
(P  Pu )
Equilibirim
condition
(a) Upper bound theorem
(P  Pu )
Mechanism
condition

Method of Analysis

Kinematic method Static method

L P L
2 2
L L
2 2
L P L
2 2 Mp

Mp
  M p/2
Mp Mp
Mp PL
L M + =
P = P × 2  = MP + MP + MP  p
2 4
6MP 6MP
P= P=
L L
876 Civil Booster (Civil Ki Goli Publication 9255624029)

Collapse loads (Wc )

W
4M P
Wc 
L/2 L/2 L
W
8M P
Wc 
L/2 L/2 L

W
2M P L
Wc 
a b ab
W
8M P
Wc 
L L2
W
16M
Wc  2 P
L L

w/m

18 3M P
Wc 
A B L2
L

A 11.66M P
B Wc 
L2

W
A 6M P
L/2 L/2 B Wc 
L