BASIC
IS 800 : 2007 Used for steel design
Steel ρ = 7850 kg / m³
E = 2 x 10⁵ N/mm² = 2000 Kg/mm²
G = 0.769 x 10⁵ N/mm²
K = 1.6 x 10⁵ N/mm²
1 MPa = N/mm² = 10 kg/cm²
μ = 0.3 (elastic range) & μ = 0.5 (plastic range)
α = 12 x 10⁻⁶ /°C → Nearly same as concrete
Mild steel < 0.25% Carbon
Carbon ↑es → ductility↓es & fu↑es
Gauge length = 5.65√Ao
Excess sulphur produces Red shortness in steel
Thickness of steel member = 6mm → Not exposed to weather
Adding manganese → ↑es strength & hardness & used in Rails
Chromium & Nickel → ↑es Resistance to corrosion & Temp ex. invar tape
Quality of structure steel → by yield stress
Permissible bending stress = 1850kg/cm² → Steel slab plate
High strength quenched and tempered steels → fy = 550 - 700 Mpa
Gross area → Bending & Compressive stress
Net Area → Tensile stress
Torsion → Box type sectⁿ
Fe250 (mild steel) → fu = 420 N/mm², fy = 250 N/mm²
Brittle material→ FOS = Fu/working stress
Ductile material → FOS = Fy/working stress
Yield moment → Just produce yield stress in outermost fibre of the sectⁿ
Composite construction → interface slipping is prevented by using Shear connector
Minimum thickness of plates in a steel stack = 6mm
Classification Of Rolled Steel Section
Designation → Depth of section and weight per meter
ISLB 500@735.8N/m → I-sectⁿ 500 mm deep self wt 735.8N/m length
I-sectⁿ → Most efficient & economical sectⁿ used as steel beam.
ISMB is most appropriate hot rolled indian std sectⁿ → used in steel girder
Buckling class A → Hot rolled section of any type/shape about any axis
Buckling class B → Cold formed section of any type/shape about any axis
Rolled steel tubes → Referred by Outer diameter, Buckling class A
Tensile and compressive residual stress in rolled section may reach → 0.3fy and 0.5fy
i. Beam
in fives series i.e ISJB, ISLB, ISMB, ISWB, ISCS.
ISJB: indian standard junior beams
ISLB: indian standard Lightweight beams
ISMB: indian standard medium weight beams
ISWB: indian standard wide flange beams
ii. Column or Heavy weight Beams
ISHB (indian standard Heavy weight beams) → Having largest MOI for given depth
ISCS: indian standard columns sectⁿ
iii. Channels
ISJC: indian standard junior Channels
ISLC: indian standard Lightweight Channels
ISMC: indian standard medium weight Channels
ISMCP : indian standard medium weight parallel flange Channels
iv. Rolled steel angle sectⁿ
Divided into three parts: ISA → Equal angles, Unequal angle, ISBA → Bulb angle
Bulb angles are used in ship building
Gauge steel sheet thickness
14 gauge steel = 1.6 mm
16 gauge steel = 1 mm
20 gauge steel = 0.80 mm
Steel Beam Theory
Used for Doubly Reinforced sections
to find MOR of doubly reinforced section especially when Area Compression steel ≥ Tensile steel
IS 800 : 2007 (LSM)
Specifications by is 800:2007(LSM)
Working shear stress on Gross area of a Rivet = 1020 kg/cm²
Design Compression member by Perry- Robertson formula for axial load.
Secant formula → allowable stress in axial compression
Beams shall be designed and checked for Stiffness, Bending strength and Buckling.
Mechanical properties:
Yield stress(fy), Tensile or ultimate stress(fu), max % elongation
Physical properties
Unit mass, modulus of elasticity, Poisson ratio, modulus of rigidity, coefficient of thermal expansion.
i. Limit State of strength or collapse
Loss of equilibrium, Loss of stability (overturning), Rupture of structure, Fracture due to fatigue, Brittle failure, Torsion, Buckling, Sliding.
ii. Limit State of Serviceability
Deformation & deflection, Vibration, Corrosion & Durability, Cracks due to fatigue or repairable damage, Fire
Partial factor
For Materials = 1.05
Resistance governed by yielding or buckling → mo=1.10
Resistance governed by ultimate stress → m1=1.25
Permissible Stresses
Avg. Shear = 0.40fy
Max shear = 0.45fy
Axial or Direct tensile(σ at) & Compression(σ ac) = 0.60fy
Bending tensile or compressive, Crippling = 0.66fy
Bearing stress = 0.75fy
Combined bearing and bending = 0.90fy
WL & EQ increases stress by 33.3% in steel str & 25% in rivets & weld
Bearing stress → load is transferred through one surface to another surface in contact
IS 875
IS 875 → Standard load is described
Live loads for Residential building (IS 875 Part-II)
UDL → Dwelling house = 2 kN/m², House, hotel and hospital = 3 kN/m²
Design wind velocity (Vz) (IS 875 Part-III)
Vz = Vb K1 K2 K3
Vb = Basic wind speed (m/s)
K1 = 1 → Probability or risk coeff
K2 = 0.8 → Terrain, ht & str size factor
K3 = 1.0 → Topography factor
K4 → importance factor in cyclonic zone
Wind pressure
P = KV²=0.6Vz2
P-Kg/cm², V-km/hr, K- coeff
RIVETS & BOLT
Shear connection → Number of rivets on the web of a beam and the number of rivets on the flange of a column be calculated from the strength of rivet
BOLT
Bolts are most suitable to carry Axial tension
Used in place of rivers for str not subjected to vibrations
High strength bolt → used when Subjected to reversal of stress
M20 bolt → Shank Dia = 20mm
Grade 4.6 → fu = 4 x 100 = 400 Mpa & fy = 0.6 x 400 = 240 Mpa
Types of rivet & Bolted joint
Lap joint → 2 member overlapped and connected, Single bolted or double bolted lap joint, Always in single shear
Butt joint → 2 members are placed end to end are joined by cover plates,
Single cover butt → Cover plate on one side
Double cover butt joint → Load is not eccentric, free from bending stress, Cover plate on both side, Double shear
Patterns
Diamond pattern → Max efficiency, used in Structural units
Basic definitions
Pitch (p) → in directⁿ of force
Gauge (g) → Perpendicular/transverse directⁿ of force
Staggered pitch → Distance b/w one rivet line to another rivet line.
g > p → Zigzag failure
g < p → Failure ⟂ to sectⁿ
g = p → Dia of hole ↑es
Proof load → initial tension in HSFG bolts
Rivet line or scrieve line or back line An imaginary line along which rivets are placed
Gross dia (d') or Dia of hole (WSM)
d' = d + 1.5mm (d ≤ 25mm)
d' = d + 2 mm (d > 25mm)
d' = Gross dia or Dia after driven or Dia of hole or Rivet hole
LSM
d' = d + 1 mm (d ≤ 14 mm)
d' = d + 2 mm (d =16- 24mm)
d' = d + 3 mm (d ≤>24 mm)
Unwin's formula
dmm = 6.01 tmin in mm → t > 8
t → Thinner plate in mm, d → Rivet dia in mm
T < 8 → sdt=d2b/4
Specification
min pitch (Spacing)
P ≥ 2.5 d (d ← nominal dia)
η = (p - d)/p=(2.5d-d)/2.5d=60%
max pitch
Tension, Plate exposed to weather = min of (16t, 200mm)
Compression, Non- staggered = min (12t, 200mm)
Staggered = min (18t, 300mm) → increased by 50 %
Tacking rivets = min of (32t, 300mm) → For both compression and tension
Tacking rivet in Tension member ≤ 1000mm
Tacking rivet in Compression member ≤ 600mm
min edge & end distance
To avoid tearing, shearing, splitting and bearing failure of plate
Machine cut = 1.5 x hole dia
Sheared or hand cut edges(Rough) = 1.7 x hole dia
Reduction factor for shear capacity
Long joint Bolt (Lj > 15d) → =1.075-(0.5%)Lj/d 0.751
Long joint Weld (Lj > 150tt)=1.2-0.2Lj/150tt
Grip length → (lg > 5d)=8d/(3d+lg)
Packing plates (tpk > 6mm) → =1-0.0125tpk
Nominal Bearing strength of Bolt
Vnpb = 2.5 Kb d t fu/1.25
Kb = min of (e/3do , p/3do - 0.25 , fub/fu ,1.0)
e = end distance, do = hole dia, p = pitch, fub = ultimate tensile stress of bolt, fu = strength of plate
Rupture strength of plate = 90% of ultimate
f proof = 0.70 x ultimate strength of bolt
Prying forces
Additional tensile force due to Flexibility of connected parts → HSFG bolts
PF = Mp/n → Mp = plastic moment, n = no of bolts
Tensile capacity of Bolt
Min of → {0.9 An.fu/1.25, Ag.fy/1.1}
Bolt subjected to both Shear and Tension
(Vu/Vd)2+(Ta/Tb)21
RIVETS
Size by shank dia
Rivet in tension with countersunk heads → Tensile value should be Reduced by 33.33 %
Area of cover plates of a built beam, an allowance for rivet holes to be added is = 0.13 (13%)
Working shear stress on gross area of a river as per IS = 100N/mm² = 1025Kg/cm²
Anet = (B - nd')t
Classification
Hot driven field rivets :
Hot driven shop rivets :
Cold driven rivets: Dia = 12 - 22mm.
Hand driven rivets
Power driven rivets
Strength → Cold driven > hot driven rivets.
Assumption In Rivet Connection
Rivets are stressed equally
Frictⁿ b/w plates neglected
Shear force is uniform over c/s of rivets
Distⁿ of direct stress on portⁿ of plates b/w rivet hole is uniform
Group - load/stress equally shared
Bending stress uniform b/w plate and rivet
BM or Bending of rivet are neglected
Rivets fills hole completely
Failure in Rivet joint
06 Types
Shearing, Bearing, Tension of Rivets
Splitting, Bearing, Shear of Plates
Rupture of net section → tension
Axis of load lies in the plane of rivet group → Rivet are subjected to only shear
Eccentricity of connection → introduces secondary stresses
Strength of R joint
Rivet value Rv = min of (Ps & Pb)
Shearing strength of rivet → Ps = nπd'²σs/4 → n = 1(single shear), n = 2(double shear), n = 4(Double riveted double covered butt joint).
Bearing strength of rivet → (Pb) = d't σ br
Tearing strength of plate → Pt = (B - nd')t σ at
Efficiency → η = min of (Ps ,Pb & Pt)/P = (p - d')/p = strength of rivet joint/strength of solid plate
Strength of Solid plate → P = pt σ at = Bt σ at
No of rivets = Force/Rv
Connection of gusset plate → No of rivet ≥ 2
Type | σ at | σs = τ | σ br|
Power shop = 100 | 100 | 300 |
Power field = 90 | 90 | 270| ≈ 90% |
Hand driven = 80 | 80 | 250 | ≈ 80% |
For WL/EQ load → increase above values by 25 %
Working τ on gross area of rivet acc to IS = 1020 kg/cm².
Working τ on gross area of Power driven rivets = 945kg/cm²
Pd = 4 Ps ← Pd = Per pitch, Ps = Strength of 1 rivet in shear
no of rivets n = √(6M/mpRv).
m = 2 ,
M = P x e
Bending stress = M/Z
Z = I/y
Tacking Rivets
Used when min distance between two adjustment rivets > 12t or 200mm
Not considered to calculate stress
Provided throughout the length of a compression member composed of two components back to back
WELDED CONNECTION
Weld Defects → incomplete fusion, Slag inclusions, Porosity, Cracks & Under cutting
Filler → Metal added at the joint while welding
Flux → The fusible material used in welding to dissolve and facilitate the removal of oxides and other undesirable substances
Strength of the welded joint may be more than strength of the elements
Partial FOS
Shop weld = 1.25
Field shop = 1.5
Types of Welded joint
Position of weld → Flat, Vertical, Horizontal & Overhead weld
Type of Weld → Fillet, Spot, Plug, Groove or Butt & Slot weld
Type of joints → Butt, Corner, Tee & Lap weld
Square, double vee, single vee, single U, Double U
Spot → Two plates are placed One butting against the other
Plug : 🔴 →
Slot : ⭕ → Overlapping length of weld < Required weld length
Welded joints vs bolted/riveted
The loss of member strength → Welded < Riveted
Fatigue failure → Weld fail earlier than riveted
FILLET WELD
Triangular c/s & join at 90° wood
Two members in diff places (Lap joint) → Joins 2 surface at approximate right angle to each other
Strength = 80 - 95 % of the main member
Fillet welds are easy to make, require less material preparation & are easier to fit than the butt welds
Fillet weld Always fails → in Shear along a plane through the throat of the weld
Transverse fillet weld is designed for Tensile strength
Standard fillet weld → isosceles triangle with θ = 45, S : t = √2 : 1 = 1.414 : 1
Long joint → Weld > 150t , Rivet > 50d, t → throat thickness
Maximum stress → at the throat
Types of fillet weld
Mitre, Concave, convex fillet weld
Side fillet weld
Load axis is parallel to the weld axis
Weld subjected to shear
Weld shear strength is limited → Half the weld metal tensile strength
Ductility is high
End fillet weld
Load axis is perpendicular to the weld axis
Weld Strength develop = Value of weld metal
Note → End fillet weld stronger than side weld → But for calculation and analysis both are taken same
Specifications for fillet weld
Size → Eff throat thickness or Smaller side of triangle of fillet
Throat is weakest sectⁿ
Width or dia > (3t or 25 mm)
i. Max size of weld
Based on thickness of Thinner plate/member
Square plate = t - 1.5mm → t = thickness of thinner plate
Rounded edge < 3t/4 → t = nominal thickness of round edge
Eff Area = Eff length x Eff throat thickness
ii. Min size of weld → 3568
Min size → To avoid risk of cracking in absence of prehealing
Based on thickness of Thicker plate/member
0 -10 mm = 3mm
(10 - 20) = 5mm
(20 - 32) = 6mm
(32 - 50) = 8mm (1st run) & 10mm (2nd run)
Thicker plate > 50mm → Special precaution like preheating of plate will be taken
iii. Eff throat thickness (t)
Shortest distance from the root of the fillet weld to the face of the weld
t = K x Size of weld = KS → A function of angle b/w fusion faces
Size of weld (s) ≈ Thickness of thinner member when two members meet
t/S < 0.707
Angle of fusion(θ)
θ↑es → K↓es
60° ≤ θ ≤ 120°
fillet weld is not recommended → if (θ) < 60° & > 120°.
Size of right angled fillet weld = 0.414 x throat thickness
Size of fillet weld with unequal legs = smaller leg length
iv. Effective length
Leff = L - 2s L=Leff+2S
Leff ≥ (4S or 40mm)
Lap length → Leff ≥ (4t or 40mm) → t = thinner
Min end return = 2 x Size of weld =2S
vi. Clear spacing b/w eff L of intermittent Fillet
Compression ≤ 12t or 200 mm
Tension ≤ 16t or 200mm
Length of intermittent fillet weld = max of (4t or 40mm)
t = thickness of thinner member
Design shear strength of fillet weld
Based on throat area
Shop → Lefftfu/(31.25)
Field → Lefftfu/(31.50)
t=ks, fu → Smaller of ultimate strength of weld and parent metal
Design shear stress is same for → Plug, Slot and fillet weld
Reduction factor for long joint in side fillet weld
(Lj > 150tt)=1.2-0.2Lj/150tt1
Lj → Length of joint or length of side fillet weld in direction of force
Combination of stress in fillet weld
feq=fa2+3q2 Design shear strength of fillet weld
fa → Normal stress, q → Shear stress
No need to check for combination of stress → if i) Side fillet weld joining cover plates and flange plates ii) Fillet weld where Normal stress + shear stress < Design shear strength of fillet weld
Butt weld (Groove weld)
When member to be joined are in one plane
Specified → By penetration thickness
Specifications for butt weld
Butt weld shall be treated as parent metal with a thickness = throat thickness
Size of buttweld = throat dimension (eff throat thickness)
Stress < Permitted in parent metal
i. Reinforcement
To increase efficiency
0.75mm < R < 3mm
Reinforcement increase throat thickness by 10 %
In calculation neglect reinforcement
ii. Eff throat thickness
incomplete penetration = 5/8 of t → Single V
Complete penetration = t → Double U, V
t = thickness of thinner member
iii. Length of butt weld
Eff length = length of full size weld
Min Length = 4 Size of weld
Intermittent butt welding
Min eff length & Spacing = 4S and <16tmin
Intermittent weld shall not be used in position subjected to dynamic, repetitive and alternating stresses
iv. Axial and Shear Strength of butt weld
Govern by yielding
Shop → Leffttfy/(31.25)
Field → Leffttfy/(31.50)
fy → Smaller of yield strength of weld and parent metal
Combination of stress
For combined Axial tension & Bending → fc/fat + fbty/fabty + fbt/fabt ≤ 1
equivalent stress → fe ≤ 0.9 fy
fe = √(fb² + fp² + fb²fp² + 3τb2) → Bearing, bending and shearing
Check for a combination of stress in butt weld need not to be carried out if
Butt welds are axial loaded
In Single and double bevel weld
Sum of normal + Shear stress < Design normal stress
Shear stress < 50 % of design shear stress
Method of inspection of welded joint
i. Magnetic particle method: iron filling is spread over the weld & it is then subjected to an electric current.
ii. Dye penetration method: Dye is applied over the weld surface
iii. Ultrasonic method:
iv. Radiography: X- ray or γ ray are used to locate defects, used in butt welds only
TENSION MEMBERS
Net area is effective in tension member
Working stress is of order of 150 N/mm
Projection of plate or flange beyond its connected to a web < 25t
Max slenderness ratio
To check the lateral vibration of the member, for local buckling
Various form of tension member
Rods & Bar → Used when Length of tension member is too long, small tension members welded/threaded/bolted to gusset plate
Wire ropes → Used for moderate span of truss bridges, Hosting purpose
Ties → Steel members designed to carry axial tensile load, in a truss tie is a horizontal beam connecting 2 rafters
Cables → Suspension bridge, Negligible flexural stiffness
Plates & Flats → Transmission tower, foot bridge, lacing flat, batten plates, end tie plate
Tension Member (Bracing) is pretensioned to avoid sag, need not to satisfy max slenderness ratio
Net Sectⁿ Area
Required → An = f/ σat
σat is Permissible Axial Tensile stress
An Provided ≥ Required
Factor considered → Ductility factor, Geometry factor, Shear lag factor
i. Plate Sectⁿ
An = (B - nd' + Σ p²/4g )t
d' → Hole or gross dia
ii. Angles Sectⁿ
An = A1 + k x A2.
k = (3 x A1)/( 3 x A1 + A2) ← Single angle
k = (5 x A1)/( 5 x A1 + A2) ← Pair of angle back to back.
A1 → Connecting leg, A2 → Outstanding leg
Lug Angle
short length of an angle b/w sectⁿ used at a joint to connect the outstanding leg of a member
Reduce the length of connection → Save gusset plate
Reduce shear lag effect → increases efficiency, stress-strain uniform no shear lag
Min number of rivets used for attaching the lug angle to the gusset or other supporting member = 2
Design for 40 % excess force carried by the outstanding leg of main angle sectⁿ & for 20%..................channel sectⁿ.
Used → With single angle, with channel member & not used with double angle member
Shear lag effect:
Non uniform stress distribution → Stress in one part legs behind the other part of section
Connected leg will have highest stress at failure than outstanding leg
Less in large length of connection
Unequal angle with long leg connected is preferred
Reason → I sectⁿ with bottom flange connected to gusset plate, Angle with one leg connected to gusset plate , Two angles connected back to back on both sides of the gusset plate
Strength Of TM
min of below 1 ,2 & 3
1. Net - Section rupture
Tdn = (α An fu) / γ m1
For Plate → Tdn = 0.9 An.fu/1.25 = (0.9 An.fu) / γ m1
Partial safety factor → γ m1 = 1.25
α = 0.6(bolt ≤ 2), = 0.7 (3 bolts), = 0.8 ( ≥ 4 bolt)
2. Gross - Section yielding
Tdg = Ag.fy/1.1 = Ag.fy / γ mo
Partial safety factor → γ mo = 1.1
Design strength in yield = fy / γ mo = fy / 1.1
3. Block shear failure
Ag.fy/1.1 + 0.9 An.fu/1.25
For plate: → Shear yielding + Tension rupture or Tension yielding + Shear rupture
Block shear at an end connection of plate → Shear along two planes, tension along one plane
Modes of TM. failure
Net - section rupture
Gross - section Yielding
Block shear failure
Tension splice section
Splices → Designed for max factored tensile load & 0.3 x design strength of TM
Splices cover → Designed to develop net Tensile strength of main member
COMPRESSION MEMBER
Most economical sectⁿ for steel column → Tubular sectⁿ
Most preferred section pov of strength → Box
Example → Strut, Raftar, Boom(Part of crane)
Strut → Compression in a direction parallel to its longitudinal axis
Channel → 1 Web & 2 Flange
Best double angle sectⁿ in case of CM → Unequal angles with long legs back to back
MOI → Most important property of the section in a compression member of any steel structure.
Web crippling → Generally occurs at the point where concentrated load acts & it is a phenomenon of Local Buckling.
Outstanding length of a compression member consisting of a channel is measured as Nominal width of the sectⁿ.
Torsional Buckling → Torsional rigidity < Bending rigidity
Flexural Buckling → Due to Bending alone.
Design Strength for Buckling = Ultimate Strength / 1.1
λ > 180 → Steel column fails in buckling
Unsupported length of transverse tie < 48 x Dia of tie in two direction
Lap length in column = Development length
The channels aur angles in the compression chords of the Steel truss girder bridges are turned outward → To increase radius of gyration
Assumption made while designing a compression member(or column)
ideal column is absolutely straight having No crookedness
Modulus of elasticity is assumed to be constant in a built-up section
Secondary stresses (which may be of the order of even 25% - 40% of primary stresses) are neglected.
Euler's theory
Pcr = π²EI/leff² = π²EA/λ² (80 ≤ λ) → Only for long column
I = Ar²
Crippling, Buckling & Critical load(Pcr) → All are the same.
Buckling load for column depends on → Both length and least lateral dimension
Critical Buckling stress fcr = Pcr/A
λ = 0 when the column is spreading throughout its length ( leff = 0)
Secant formula: allowable stress in axial compression.
λmax = leff / r min, (leff = 0.85L,0.65L,1L,2L).
Radii of gyration: r = √(I/A)
λ↑ → r↓ → Sectⁿ will buckle about r min
Buckling load ∝ 1/λ ∝ r min.
For Hanger bar (Ceiling fan rod) → λ = 160
Effective length
SSB ends restrained against torsion & ends of compression flange partially restrained against lateral bending = 0.85L
if both flanges fully restrained = 0.70L
Imperfection Factor
Class a = 0.21, b = 0.34, c = 0.49, d = 0.76
Depends on → Shape & c/s of column ,directⁿ in which buckling can occur & fabrication process (Hot rolled, Welded)
Buckling Class a sectⁿ carries max axial Compressive stresses
Perry Robertson formula
Design CS of an axially loaded compression member is based on the Perry Robertson formula
Pd = Ae Fcd
LACING
lacing is subjected to Compression & tension both
Best for eccentric loading
Designed to resist Shear force
Lacing members → Rolled section, tube of equal strength, ISF, ISA, ISLB
Design → As slender compression member(truss member)
λ ≤ 145
eff λ = 1.05 x λ column → 5% increase
C/rmin < (50 & 0.7whole) → if fail increase θ and provide double lacing
40° < θ < 70° → Angle of inclination
Leff → Single lacing = L, Double or welded lacing = 0.71L
Thickness → Single lacing ≥ L/40, Double lacing t ≥ L/60
Design → resist Transverse shear = 2.5% the axial force or load in column
Tie plates at end of lacing system → To prevent distortion of built up c/s due to unbalance horizontal force
Force in each lacing → F=V/Nsin
N → Single = 2, Double lacing = 4
No of rivet required → 2Fcos/Rv
Width ≥ 3 x Nominal dia of rivet or bar
BATTEN
Only axial load, not for eccentric loading, Design as frame
Designed and Subjected → longitudinal SF & BM arising from transverse shear
Flat plate are used for batten
Eff Length and Slenderness ratio = 10% more than laced column
Thickness = 1/15th of the distance b/w the innermost connecting lines of rivets, bolts or welds
Min no. of batten = 4 → member divided into 3 ways
Min no. of intermediate batten = 2
Transverse SF → V = 2.5 % of axial load
Eff depth → d = D - 2 x edge distance
Minimum thickness → t > S1/50, S1 = transverse dist b/w centroid of inner end bolt/rivet group
F = VC/NS, M = VC/2N
Laced column is stronger then battened column → for same P, Leff, end conditions
Buckling of strut component
Tack bolted (T-24)
l1/rmin< 40 (0.6 whole)
Strength depends on Slenderness ratio of strut
Slab base
t=2.5w(a2-0.3b2)/(fy/1.1)
Built up Column
Two channel section: Clear distance is designed by MOI about major = minor axis.
Splices & short column
Joint to ↑es length of column
Splices are designed → As Short columns
Splices shall be provided at point of contra flexure
Lap splices not recommended → Rebar dia > 36 mm
Splice plate → Design as a intermediate column
Note
Eff sectⁿ in compression: thin hollow circular cylinder
Bending : I-sectⁿ
Torsional rigidity < Bending rigidity →Torsional buckling.
BEAM & PLATE GIRDER
Local flange buckling → due bending compression
Web crippling → More bearing stress at root of fillet
Web buckling → Diagonal compression due to shear
Normally if web is safe in crippling it will be safe in buckling also
Vertical deflection limits
Excessive deflection may lead to crack in plaster and may damage the material attached or supported by beam
i) Cantilever
Elastic cladding < Span/120
Brittle cladding < span/150
ii) SSB
elastic < span /240
brittle < span/300
in general < L/325
iii) Other
Timber beam supporting brittle covering ≤ Span/360.
For purlins,girts beams in industrial Building
Brittle cladding Vertical deflection ≤ L/180
Elastic cladding Vertical Deflection ≤ L/150
Outstand of Flange plates
Compression flange < 16t
Tension flange < 20t
PLATE GIRDER
For larger and heavy gravity load
Riveted or bolted plate girder → Provide angles (unequal leg with longer leg horizontal)
Welded plate girder → Provide falter plate section, 5-15% less wt then riveted/bolted plate girder
Components of Plate Girder
Web → SF → Vertical member
Flange → BM → Horizontal member
Web splices → Connects webs, SF & BM, Design for Shear and moment, provided on each side of plate
Flange splices → Connects flange, BM & Axial force, provided at Quarter Span sectⁿ, Design for axial force only
Eff flange area in compression = Af + Aw/6
Eff flange area in tension = Af + 0.75(Aw/6)
Stiffeners
To prevent buckling of web plate
Transverse/Vertical/Stability stiffener → ↑es buckling resistance of web due to shear, intermediate vertical stiffeners are joggled
Horizontal/Longitudinal stiffener → Web buckling due to Bending Compression.
Bearing/Load/End bearing stiffeners → Provided at supports & prevents Buckling of web
Diagonal stiffeners → Safe web against shear+bearing
Torsional stiffeners → Transmit tensile forces applied to web through a flange
web stiffeners are provided within D/2 of plastic hinge location where Concentrated load exceeds 10% of Shear capacity of the member
Min distance b/w VS = 0.33d
Max distance b/w VS = 1.5d
HS are provided at a distance = 0.2d from compression end
Min unsupported length of stiffeners = 180tw
Max unsupported length of stiffeners = 270tw
Web Design
d/t ≤ 85 → No need of stiffeners or unstiffened web plate.
d/t > 85 → Web plate with VS or TS
85 < d/t < 200 → Provide intermediate VS or TS only
200 < d/t < 250 → 1VS with 1LS or 1HS
250 < d/t < 400 → 1VS with 2LS or 2HS (2nd HS at N.A.)
d/t > 400 → Redesign
d = Clear depth, t = eb thickness
d/tw > 65ε → check web for shear buckling.
d/tw < 65ε → design unstiffened girder i.e. no girder required.
HS web t < D/20
Depth of girder
Economical depth → d = 1.1 M / σ tw
Deep girder > 750mm
Shallow plate girder ≤ 750mm
Gusset plate
Connect two or more structural members
t ≥ 12mm
For Less load → Slab base & For Heavy load → Gusset plate
Single angle discontinuous strut is connected to a gusset plate with one rivet → Leff = L, Permissible stress = 80 %
COLUMN BASE & FOOTINGS
Column load → Base plate → Biaxial loading
Weaker soil → Grillage footing, Column footing, Raft footing
Perforated Cover plates
for built up sectⁿ → four angle box sectⁿ.
Area of cover plates of a built up beam 13% Area allowance is given to Rivet Holes.
Column Base
Base plate Area A = P/.45fck = Load/Bearing capacity
Generally subjected to Bending & Compression
Thickness of base plate is determined from Flexural strength of plate
Pressure under footing → q = P/A ± ML/2I
e = M/P
For compression stress → max pressure = min pressure
Permissible bending stress in steel slab plate = 1890 kg/cm²
Avg shear stress for rolled beam sectⁿ = 1020 Kg/cm²
Permissible tensile stress in bolts = 120 N/mm²
Max bearing strength < 0.60 fck
Allowable working stress corresponding to λ
Double angle placed back to back & connected to one side of a guesser plate = 0.8 σ ac ( to 80%) → Discontinuous
Single angle Discontinuous strut = 0.8 σ ac ( to 80%)
For other conditions remains σ ac (100%)
Grillage foundation
Grillage foundation beams → Check for SF, BM, Web buckling & Web crippling
It is a Spread foundation
Heavy load on low bearing soil
I-sectⁿ → Heavily loaded isolated column, 2 sets of perpendicularly placed steel bars
in grillage footing max SF occurs at edge of base plate & max BM occurs at Centre of base plate
ROOF TRUSS
RT are subjected to DL, LL, SL, WL & Transmit these loads to the walls
Spacing = ⅓ - ⅕ of the Span → Generally = 10ft - 15ft
Require very light members, to reduce the DL & to make structure stable
Economical for Span > 6m
No. of bolt or rivets ≥ 2
Min angle used → ISA 50 x 50 x 6
Gusset plate used → t ≥ 6mm
Bracing λ ≤ 120
Permissible bending stress in steel slab plate = 185 MPa = 1850 kg/cm²
Rivet = 5% total weight of Roof Truss
Load on truss
Dead load or weight DL = (L/3 + 5) x 10 → L = Span of truss
θ ≤ 10° → LL = 0.75 kN/m² (Access not provided), LL = 1.5 kN/m² (Access provided)
θ > 10° → LL = 0.75-(-10°)0.02
LL ≥ 0.4kN/m²
a) Slope of the truss
S = tanθ = H/L = 2 x Pitch
Pitch↑es → Load capacity↑es
Pitch = (¼ - ⅙) to its slope
Economical Spacing = (⅓ - ⅕) of span
b) Economy of truss
Overall cost to be minimum
C = 2P + R
Cost of truss/unit area = 2 x (Cost of purlin/plan area) + (Cost of roof covering/plan area)
Cost of truss = 2 x Cost of purlin + Cost of roof covering
i. Width of angle leg
Plane parallel to The roof covering ≥ L/60
Plane perpendicular to the roof covering ≥ L/45
ii. Max BM
as a continuous beam = wL²/10
as a SSB = wL²/8
iii. Deflection
Purlin & Girts ≤ Span/180
Girt → Unsymmetrical bending
Components of roof truss
i. Rafter
Support covering material
ii. Purlins:
A horizontal beam
Design as continuous beam (flexural member)
Subjected to biaxial bending & runs perpendicular to Truss
max BM = wL²/10
δ = Span/180
Angle sectⁿ as purlin if slope of roof < 30°.
Purlins are supported by Principal Rafter.
iii. Principal rafter: it is the top chord subjected to Compressive force only, it supports purlins.
iv. Bracings: resists lateral load due to WL,EQ parallel to the ridge
Types of Truss
King post Truss → Span = 5 - 8m
Queen post Truss → Span = 8 - 12m
Pratt Truss → 6 - 10m
Howe Truss → 6 - 30m
GANTRY GIRDER & CRANES
Gantry Girder
To carry Cranes, Subjected to unsymmetrical bending due to lateral truss
Lateral Load → Due to moving and stopping of crab
longitudinal load → Movement of truss of rails, Starting and stopping of crane
Transverse/Vertical load → DL, gravity load
No wind load
it is designed → by I-sectⁿ, channel sectⁿ & box Girder
It can be designed as a laterally supported or laterally unsupported beam used in industrial building
Vertical deflection under DL+LL
manually operated = Span/500
charging car = L/600 for other moving load
electrically operated < 50 tons = Span/750
electrically operated > 50 tons = Span/1000
Lateral deflection
Absolute = Span/400
Relative displacement b/w rails supporting crane = 10mm
Vertical & lateral ∆ shall be calculated without considering the impact factor or Dynamic effect
PLASTIC DESIGN
Plastic theory → Rigid frame str generally
Plastic neutral axis → Equal area axis
Plastic moment → Mp = fy.Zp
for Rectangular → Zp = bd²/4
Plastic modulus of sectⁿ → Zp = A(y̅1 + y̅2)/2
Elastic modulus of sectⁿ → Ze = I/y
No. of independent mechanism=Hinge - Redundancies
At fully plastic section infinite rotation can occur → at constant plastic moment
Equilibrium condition should be satisfied both in elastic and plastic analysis
Plastic Hinge
Yielded zone in flexure, infinite rotation, Constant Mp
No. of plastic hinge required → N = Ds + 1 → Static indeterminacy + 1
No. of independent mechanism → I = Possible N - Ds
Curvature or rate of change of slope at plastic limit = infinity
Load factor
Load factor = Ultimate collapse load/working load
Load factor = factor of safety x Shape factor → Q = F x S
Plastic design method → LF = 1.7 - 2
Dl & LL steel str design → LF = 1
LF & FOS depends on → Geometry of c/s area, Mode of failure, Support condition, Nature of loading
Shape Factor
α = Mp/My = Zp/Ze
Depends on c/s area
Triangle = 2.34 → Vertex upward
Diamond, Rhombus = 2
T section= 1.9
Circle = 1.7
Square = Rectangular = 1.5
Ring or Tubular = 1.27 = 4/π
I section = 1.1 - 1.55
Std. Rolled Beam I-sectⁿ = 1.1 - 1.2
Length of Plastic Hinge (L of elastic-plastic zone)
Zone of yielding (Mp to My)
Depends on → Loading (udl, pont), Geometry (Shape factor), length of beam
Point load → Lp = L(1 - 1/α) → Ssb(mid) or Cantilever(free end)
UDL → Lp = L[√(1 - 1/α)] → SSB or Cantilever
Lp = L → Cantilever (Moment at free end)
Collapse Load (Wc)
4Mp/L → SSB point load at mid
8Mp/L² → SSB with UDL
8Mp/L → Fix beam P at mid
16Mp/L² → Fix beam P at mid & UDL all span
Propped cantilever Point load at mid → 6Mp/L
Propped cantilever with UDL → 11.656Mp/L²
Theories of plastic analysis
i. Upper bound/Kinematic thᵐ
Based on mechanism cndⁿ
Load obtained ≥ Collapse load Pu
ii. Lower bound/Static thᵐ
Based on yield condtⁿ
Load obtained ≤ Collapse load Pu
iii. Uniqueness thᵐ
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