Design of Steel Structures – Complete Study Notes

Comprehensive chapter-wise notes covering every aspect of Design of Steel Structures as per IS 800 — Working Stress Method principles, structural fasteners (rivets, bolts, welds), tension and compression member design, beams and beam-column connections, built-up sections, plate girders, industrial roof systems, and plastic analysis. All formulae, IS code clauses, SVG diagrams, worked examples and exam-focused tables included.

GATE ESE / IES SSC JE State PSC RRB JE

Ch 1 · Structural Fasteners Ch 2 · Tension Members Ch 3 · Compression Members Ch 4 · Beams Ch 5 · Plate Girders & Industrial Roofs Ch 6 · Plastic Analysis ★ Quick Revision

1Structural Fasteners – Rivets, Bolts & Welds►

1.1 Introduction to IS 800 and WSM

Steel design in India is governed primarily by IS 800:2007 (Limit State Method) and the older IS 800:1984 (Working Stress Method). For competitive exams (GATE, ESE, SSC JE), both versions are relevant; traditional syllabi still emphasise WSM heavily.

ℹ️ Working Stress Method (WSM): Stresses in the structure are kept below permissible values which are a fraction of yield/ultimate strength. FOS is applied to material strength directly. Elastic analysis is used; no redistribution of forces. Key permissible stresses as per IS 800:1984 — σ_at = 150 MPa (tension), σ_ac = depends on slenderness, τ_va = 100 MPa (shear).

1.2 Riveted Connections

Rivets are permanent fasteners driven hot and cooled in place, forming a head on the free end. Though largely replaced by bolts and welds in modern construction, they remain important for exam purposes.

Modes of Failure of a Riveted Joint

Shearing of rivet: Rivet shank shears across the interface
Crushing (bearing) of rivet: Rivet or plate material crushes under bearing pressure
Tearing of plate: Net section of plate fractures in tension
Shear-out (end shear) of plate: Plate shears out behind the end rivet
Splitting of plate: Plate splits at edge of hole (insufficient edge distance)

Strength of a Single Rivet (WSM)

Strength in single shear: P_s = (π/4) × d² × τ_va
Strength in double shear: P_s = 2 × (π/4) × d² × τ_va
Strength in bearing: P_b = d × t × σ_pb
Strength in tearing: P_t = (p − d) × t × σ_at

Where:
d = gross diameter of rivet (shank dia + 1.5 mm for hot-driven rivets)
d = nominal dia + 1.5 mm (per IS 1929); use gross dia for shear, net for tearing
τ_va = permissible shear stress in rivet = 100 MPa (IS 800)
σ_pb = permissible bearing stress = 300 MPa (IS 800)
σ_at = permissible axial tensile stress in plate = 150 MPa
p = pitch of rivets, t = thickness of plate

Rivet value R = min(P_s, P_b) [for single rivet in a lap/butt joint]

Efficiency of a Riveted Joint

η = (Strength of joint per pitch) / (Strength of solid plate per pitch)
η = min(P_s, P_b, P_t) / (p × t × σ_at) × 100%

For a single-riveted lap joint with one rivet per pitch:
η_tearing = (p − d) / p
η_shearing = P_s / (p × t × σ_at)
Balanced joint: η_tearing = η_shearing → optimal pitch

1.3 Rivet Spacing Rules (IS 1929 / IS 800)

Dimension	Minimum	Maximum	IS 800 Clause
Pitch (p)	3d (d = gross rivet dia)	16t or 200 mm (in tension) / 12t or 200 mm (in compression)	Cl. 10.2
Edge distance (e)	1.5d (sheared edge), 1.25d (rolled/sawn)	12t (12 × plate thickness)	Cl. 10.2.4
Back pitch / gauge (g)	3d	Same as pitch max	—

1.4 Types of Riveted Joints

Fig. 1.1 — Lap joint (single shear rivet); load path is eccentric

Fig. 1.2 — Double-cover butt joint (double shear); rivets in double shear → twice the shear strength

1.5 High-Strength Friction Grip (HSFG) Bolts

HSFG bolts (IS 3757) are tightened to a high preload (proof load), creating clamping friction between surfaces. Load transfer is by friction, not shear of the bolt shank — no slippage at working loads. Superior fatigue performance; used in bridges and dynamically loaded structures.

Slip resistance of HSFG bolt:
P_sf = μ × T_f × n_e (per bolt per interface)
Where: μ = slip factor (0.45 for grit-blasted surfaces; 0.2 for as-rolled)
T_f = proof load of bolt (from IS 1367 tables)
n_e = number of effective interfaces (1 for single shear, 2 for double)

For M20 bolt Grade 8.8: T_f = 144 kN; proof stress = 628 MPa

1.6 Black Bolts vs HSFG Bolts

Feature	Black Bolt (IS 1364)	HSFG Bolt (IS 3757)
Load transfer	Shear of bolt shank + bearing on hole	Friction between clamped surfaces
Clearance hole	2–3 mm larger than bolt	1–2 mm larger; tight tolerances
Slip at working load	May slip into bearing (not ideal for fatigue)	No-slip; rigid connection
Grade designations	4.6 (UTS 400 MPa, Fy 240); 8.8; 10.9	8.8 (most common)
Use	Ordinary structures, non-fatigue loads	Bridges, cranes, fatigue loading
Tightening	Snug tight (hand-tight)	Full proof load via torque wrench / turn-of-nut

1.7 Welded Connections

Welds are classified into butt welds (groove welds) and fillet welds. For competitive exams, fillet weld design is the most important.

Fillet Weld Design (WSM)

Effective throat thickness: t_e = 0.707 × s (for 90° fillet welds; s = weld size = leg length)
Note: IS 816 uses effective throat = K × s; K = 0.7 for 90° weld angle

Permissible shear stress in weld (on throat): τ_w = 110 MPa (IS 816, E41 electrode)

Strength of fillet weld per unit length:
q = t_e × τ_w = 0.707 × s × 110 (N/mm per mm run of weld)

Weld size limits:
Minimum weld size: depends on thicker part being joined:
Up to 10 mm plate → 3 mm weld; 10–20 mm → 5 mm; 20–32 mm → 6 mm; > 32 mm → 8 mm
Maximum weld size: t − 1.5 mm for plates ≤ 6 mm; = t for plates > 6 mm (rounded corner)

Minimum length of fillet weld: max(4s, 40 mm)
Effective length = total length − 2s (deduct for starting/stopping craters)

Butt Welds

Full penetration butt weld (CJP): effective throat = thickness of thinner plate joined
Strength = throat × length × σ_at (permissible stress same as parent metal)
No reduction for joint efficiency in full-penetration butt welds → 100% efficient

Fig. 1.3 — Fillet weld: throat t_e = 0.707s; lap joint uses two fillet welds in longitudinal shear

1.8 Eccentric Connections — In-plane Eccentricity

When load P acts at eccentricity e from the CG of bolt/weld group:
Moment: M = P × e

For bolt group, force on most critical bolt:
Direct shear: F_d = P / n (n = number of bolts)
Torsional shear: F_t = M × r_max / (Σr²)
where r_max = distance of farthest bolt from CG, Σr² = sum of squared distances

Resultant on critical bolt: F_R = √(F_d² + F_t² + 2·F_d·F_t·cosθ)
θ = angle between F_d and F_t directions at the critical bolt

1.9 Beam-Column Connections

Connection Type	Moment Transfer	Rotation	Use
Simple (pin) connection	No moment (shear only)	Free rotation	Non-moment frames; beam-to-girder web
Semi-rigid connection	Partial moment	Partial rotation	Composite frames; web cleats + top cleat
Rigid (fixed) connection	Full moment	No relative rotation	Portal frames; moment-resisting frames

📝 GATE Tip: For WSM, rivet value R = min(shear strength, bearing strength). Efficiency η = R / (p × t × σ_at) × 100%. Maximum efficiency for single rivet lap joint is ~100% theoretically but practically 70–80% due to minimum pitch constraints.

2Tension Members►

2.1 Introduction

Tension members are structural elements carrying axial tensile forces — bottom chords of trusses, hangers, cable stays, tie rods, bracing members. Their design is governed by avoiding yielding of gross section and fracture at net section.

2.2 Gross Area and Net Area

Gross area: A_g = full cross-sectional area (ignoring holes)

Net area at a bolt/rivet hole section:
A_n = A_g − n × d_h × t
Where: n = number of holes at the critical section, d_h = hole dia = rivet dia + 1.5 mm (IS 800)
t = thickness of element

For staggered holes — net width (Cochrane formula):
w_n = w_g − Σd_h + Σ(s²/4g)
s = stagger (longitudinal spacing between holes), g = gauge (transverse spacing)
A_n = w_n × t

Apply to all possible failure paths and use the minimum net area.

Fig. 2.1 — Plate with staggered holes: check both Path A (straight) and Path B (diagonal); take minimum net area for design

2.3 Permissible Tensile Stress (WSM — IS 800:1984)

Permissible axial tensile stress: σ_at = 0.6 × f_y
For Fe250 (mild steel, f_y = 250 MPa): σ_at = 150 MPa

Design tensile capacity:
T_dg = A_g × σ_at [based on gross area yielding]
T_dn = A_n × 0.9 × f_u / γ_m1 [LSD approach for fracture at net section]

WSM: T_allowed = A_n × σ_at (net area governs when holes present)

2.4 Shear Lag and Net Section Efficiency

When only one leg of an angle is connected, the outstanding (unconnected) leg does not carry full stress — this is shear lag. The net effective area is reduced by a shear lag factor.

Net effective area for outstanding leg (IS 800:1984):
A_eff = A_1 + A_2 × k
k = 3A_1 / (3A_1 + A_2)

Where:
A_1 = net area of connected leg (after deducting hole area)
A_2 = gross area of unconnected leg

Alternatively per IS 800:2007:
Shear lag factor β = 1 − w_s/(L_c) [for connection length L_c, shear lag width w_s]

2.5 Types of Tension Members

Type	Typical Section	Advantage	Limitation
Single angle	ISA (unequal/equal)	Simple; easy to connect	Eccentric connection; shear lag; unsymmetric
Double angle (back-to-back)	2×ISA with gusset	Symmetric; greater area	Need lacing/batten plates for compound section
T-section	IST (rolled T)	Direct connection to gusset	Limited sections available
Channels	ISMC, ISSC	Larger area; stiff laterally	Single-sided connection eccentricity
Built-up box section	2 channels + plates	Large area; symmetric; good torsional stiffness	More fabrication; expensive
Wire ropes / rods	Circular	Very high tension capacity; flexible	No compression capacity; needs turnbuckles

2.6 Lug Angles

A lug angle is a short angle cleat welded or bolted to the outstanding leg of a main angle member near the connection point to transfer part of the load directly, reducing shear lag effect and the required connection length.

With lug angles, the effective net area = full gross area (shear lag eliminated)
Lug angle force = (A_2 / A_total) × P where A_2 = unconnected leg area
Lug angle connection must carry ≥ 120% of the force in the attached leg (IS 800)

📝 GATE Tip: For a single angle tension member connected through one leg: A_eff = A_1 + k·A_2 where k = 3A_1/(3A_1+A_2). If both legs are connected (via gusset both sides), A_eff = full A_g. Memorise the k formula — it appears every 2–3 years in GATE.

3Compression Members►

3.1 Introduction

Compression members (struts, columns, top chords of trusses, stanchions) must resist yielding of the cross-section and flexural buckling (Euler-type instability). As slenderness ratio increases, buckling becomes the governing criterion.

3.2 Permissible Axial Compressive Stress (IS 800:1984)

Permissible compressive stress σ_ac depends on slenderness ratio λ = l/r

From IS 800 Table 5.1 (for f_y = 250 MPa):
λ = 0: σ_ac = 150.0 MPa (same as σ_at; gross section yields)
λ = 50: σ_ac = 139.0 MPa
λ = 100: σ_ac = 107.5 MPa
λ = 150: σ_ac = 64.4 MPa
λ = 180: σ_ac = 46.5 MPa
λ = 200: σ_ac = 38.4 MPa (maximum permitted λ = 180 for main members)

Maximum slenderness ratio (IS 800:1984):
Main compression members: λ_max = 180
Secondary members (bracing): λ_max = 200

Design: P_allowed = A_g × σ_ac(λ)

3.3 Effective Length of Compression Members

End Condition	Effective Length L_eff	IS 800 Factor
Both ends pin (pin-pin)	L	1.0 L
Both ends fixed	0.5 L	0.5 L
One fixed, one pin	0.7 L (≈ L/√2)	0.7 L
One fixed, one free (flagpole)	2.0 L	2.0 L
Fixed-pin with restraint against sway	0.85 L	0.85 L
Both fixed with sway possible	1.2 L	1.2 L

3.4 Column Sections and Built-Up Columns

Fig. 3.1 — Column sections: I-sections have weak y-y axis; box sections have nearly equal stiffness in both directions

Fig. 3.2 — Laced vs Battened column: lacing transfers shear diagonally; battens are horizontal plates connecting main components

3.5 Lacing and Batten Plate Design

Lacing

Lacing is designed for a transverse shear force V = 2.5% of P (axial column load)
(IS 800 Cl. 7.6.6.1: V = 2.5% of P acts on the whole column)

For N-lacing (single), force in each lacing bar:
F_lace = V / (2 × sinθ)
where θ = angle of lacing with axis perpendicular to member axis

Lacing bar slenderness: l/r_min ≤ 145 (single lacing)
Lacing angle to longitudinal axis: 40° to 70° (optimal ~50°–60°)
Minimum width of flat lacing: l/40 but not < 25 mm
Minimum thickness: l/60 for single; l/40 for double (l = lacing length)

Batten Plates

Battens are designed for:
Longitudinal shear: V_L = V × C / (2 × n_b × S)
Moment in batten: M = V × C / (2 × n_b)
where C = distance between centroids of connections on each main member
n_b = number of bays, S = distance between panel points

Batten dimensions (IS 800):
Depth ≥ 0.75b (b = distance between inner edges of main members)
Thickness ≥ 1/50 × distance between inner edges, min 6 mm
Spacing: not more than 0.7 times the min radius of gyration of main member × 50

3.6 Splices in Compression Members

A splice is a joint connecting two lengths of the same member. Compression splices must transfer the full compressive force plus any accidental bending.

If ends are machined flat (milled ends): 50% of load transferred by direct bearing, 50% by fasteners
If ends not milled: full load transfer through fasteners
Splice plate design: sized to carry full design load in compression
IS 800 requires splice to carry at least 50% of P (even if milled ends assumed)

📝 GATE Tip: Maximum slenderness ratio for main compression members = 180; for secondary/bracing members = 200. Lacing angle optimum = 40°–70°. Transverse shear for lacing design = 2.5% of P. These three values appear repeatedly in GATE and ESE.

4Beams – Design of Steel Beams►

4.1 Classification of Steel Beams

Type	Section	Span Range	Application
Simple rolled beam	ISMB, ISWB, ISHB	Up to 8–10 m	Floors, bridges, crane girders
Compound beam	Rolled section + cover plates	10–15 m	When standard sections insufficient
Built-up girder	Plate girder (web + flanges)	15–30 m	Heavy loads, longer spans
Composite beam	Steel beam + RC slab	8–25 m	Modern buildings; bridges
Castellated beam	ISMB with hexagonal web cut-outs	10–20 m	Light loads; service passage through web

4.2 Permissible Bending Stress (WSM — IS 800:1984)

For compact sections with full lateral support:
σ_bt = σ_bc = 0.66 × f_y = 165 MPa (for Fe250, f_y = 250 MPa)

Required section modulus:
Z_required = M_max / σ_bc

Shear stress check:
Average shear stress: τ_va = V / (d × t_w) ≤ 100 MPa
Maximum shear stress: τ_vm = 1.5 × τ_va ≤ 115 MPa (for thin webs)

Deflection check:
δ_max ≤ L/325 (IS 800) for beams carrying plaster or sensitive finishes
δ_max ≤ L/360 for roofs (without plaster)

4.3 Lateral Torsional Buckling (LTB)

When a beam's compression flange is not adequately restrained laterally, it may buckle sideways (lateral torsional buckling) at a load below the full plastic/elastic capacity. The effective length for LTB governs the permissible bending stress.

Fig. 4.1 — Lateral Torsional Buckling: compression flange deflects laterally while tension flange remains in place; section twists. Prevented by lateral restraint.

Permissible bending compressive stress σ_bc (IS 800:1984) depends on:
1. D/T (depth-to-flange thickness ratio)
2. l/r_y (slenderness for LTB; l = effective length between lateral supports)

From IS 800 Table 6.1B: σ_bc decreases as l/r_y or D/T increases

Full plastic moment available only when l/r_y ≤ permissible limit;
otherwise reduced σ_bc from code tables applies.

Effective length for LTB (IS 800 Table 6.3):
Compression flange fully restrained (both ends): l = 0.7 × L (L = span)
Compression flange partially restrained: l = 0.85 × L
Compression flange unrestrained: l = 1.0 × L

4.4 Web Buckling and Web Crippling

Web Buckling (under concentrated load or reaction):
Dispersal angle through flange: 45° (IS 800)
Bearing length at web: b_1 = load dispersal width
Load-bearing capacity: P_wb = (b_1 + n_1) × t_w × σ_ac(λ)
λ = 2.5 × d / t_w (slenderness of web in bearing)

Web Crippling (local yielding at support/point load):
P_wc = (b + 2.5 × T_f) × t_w × f_yw
b = bearing length, T_f = flange thickness, t_w = web thickness, f_yw = yield stress of web

4.5 Compound and Cover-Plated Beams

When the required Z exceeds available rolled sections, add cover plates to flanges:
Z_compound = Z_rolled + A_plate × ȳ_plate

Horizontal shear per unit length at the plate-flange interface (shear flow):
q = V × Q / I
Where Q = A_plate × ȳ_plate (first moment of cover plate about NA)
Weld/rivet spacing to resist shear flow: spacing = R / q
R = weld strength per unit length or rivet value

4.6 Beam-Column Design (Combined Axial + Bending)

Interaction equation (IS 800:1984 WSM):
P/P_a + M_x/(M_ax) + M_y/(M_ay) ≤ 1.0

Where:
P = applied axial load, P_a = axial capacity = A_g × σ_ac
M_x = bending moment about major axis, M_ax = Z_x × σ_bc
M_y = bending moment about minor axis, M_ay = Z_y × σ_bc

Additional check for yielding at extreme fibre (interaction):
P/P_e + C_m × M_x / [(1 − P/P_e) × M_ax] ≤ 1.0
P_e = π²EI_x / (L_eff)² (Euler load for major axis)

📝 GATE Tip: Permissible bending stress for laterally supported beams = 0.66 f_y = 165 MPa (Fe250). For unsupported beams, σ_bc is reduced — read from IS 800 Table 6.1B based on D/T and l/r_y. The shear flow formula q = VQ/I is used for connecting cover plates and also for composite beams.

5Plate Girders & Industrial Roofs►

5.1 Plate Girder — Introduction

A plate girder is a built-up flexural member fabricated by welding (or riveting) flat plates to form an I-shape — a web plate and two flange plates. Used when spans exceed economical rolled section range (15–100 m) or loads are too heavy for standard sections.

5.2 Components of a Plate Girder

Fig. 5.1 — Plate girder: top + bottom flange plates, web plate, and transverse stiffeners at intervals; stiffeners prevent web buckling between loads

5.3 Design of Plate Girder Web

Web depth d (economic depth — minimises total steel weight):
d_eco = (M / σ_bc) × (12 / t_w)^(1/3) [approximate; varies with flange assumptions]

Web thickness t_w limits (IS 800):
Without horizontal stiffeners: d/t_w ≤ 200
With horizontal stiffeners: d/t_w ≤ 250

Average web shear stress: τ = V / (d × t_w) ≤ τ_va = 100 MPa

For slender webs (d/t_w > 85): tension field action may be considered

5.4 Flange Design

Flange area required (approximate — web carries moment contribution ≈ 1/6):
A_f = M / (σ_bc × d) − t_w × d / 6

OR more precisely:
I_required = M × y_max / σ_bc (full section MI from extreme fibre stress)
I_web = t_w × d³ / 12
I_flange_needed = I_required − I_web
A_f = I_flange_needed / (d/2)² [approx, flange as thin plate at d/2 from NA]

Flange width b_f limits (outstand criterion to prevent local buckling):
b/t ≤ 256/√f_y (plate outstand from web face; IS 800 WSM)
For f_y = 250 MPa: b/t ≤ 256/√250 ≈ 16.2

5.5 Stiffeners in Plate Girders

Stiffener Type	Location	Purpose	Design For
Bearing stiffener (load-bearing)	Over supports and under concentrated loads	Prevent web crippling; transfer reaction	Axial compression from reaction/point load
Intermediate transverse	At regular intervals along web	Prevent shear buckling of web	Shear buckling (d/t_w, panel aspect ratio)
Longitudinal stiffener	Horizontal, in compression zone of web	Prevent bending buckling of web	d/t_w > 200 (prevents need for thicker web)
Flange stiffener	Attached to flange (plates)	Reduce flange outstand	Local flange buckling control

Transverse stiffener spacing (IS 800 — to prevent shear buckling):
c/d ≤ 1.5 (aspect ratio of each panel)
Without diagonal tension: shear stress τ ≤ τ_cr (critical shear buckling stress)
τ_cr = k_s × π²E / (12(1−ν²)) × (t_w/d)²
k_s = shear buckling coefficient ≈ 5.35 + 4.0/(c/d)² for intermediate panels

5.6 Flange-to-Web Connection

Horizontal shear flow at flange-web interface:
q_fw = V × A_f × ȳ_f / I

Weld size required:
s_w = q_fw / (2 × 0.707 × τ_w) [two fillet welds, one each side of web]
= q_fw / (2 × 0.707 × 110) [E41 electrode, τ_w = 110 MPa]

5.7 Industrial Roof Structures

Industrial buildings use steel roof trusses (spanning 10–30 m) or lattice girders for large clear spans. The roof system must carry dead loads (sheeting, purlins), live loads, wind, and snow/maintenance loads.

Types of Roof Trusses

Fig. 5.2 — Roof truss types: Pratt (diagonals in tension); Howe (diagonals in compression); Fink/Fan; North Light (sawtooth); Lean-to

Roof Truss Loads

Load Type	Source	Typical Value	Application
Dead load (roofing)	Sheeting, purlins, truss self-weight	0.4–1.0 kN/m² (horizontal)	Permanent; full span
Live load (roof)	Maintenance, minor snow	0.75 kN/m² (IS 875 Part 2)	Reduced for slopes > 10°
Wind load	External pressure / suction	Per IS 875 Part 3; Cp values	May cause uplift on windward slope
Snow load	Accumulation (applicable zones)	0.25–2.5 kN/m² (IS 875 Part 4)	Hilly/cold regions; Jammu, NE India
Crane girder loads	Overhead traveling cranes	Wheel loads per crane capacity	Industrial buildings with cranes

Purlins

Purlins are secondary beams spanning between roof trusses, supporting the roof covering (sheeting). They are typically angle sections, channel sections, or Z-sections.

Purlin load: w = (dead + live + wind) × spacing of purlins × cos θ
(component perpendicular to roof surface governs bending)

Biaxial bending check:
M_u / Z_xx + M_v / Z_yy ≤ σ_bc
M_u = wL²/8 × cosα (in plane of roof)
M_v = wL²/8 × sinα (perpendicular to roof)
α = roof slope angle; sag rods reduce M_v by providing intermediate support in minor axis

5.8 Crane Girder Design

Crane girder must be designed for:
1. Vertical loads: wheel loads with impact factor (IS 875 Part 2)
P_design = P_wheel × (1 + impact factor)
Impact factor = 10–25% of lifted load (depending on class of crane)

2. Horizontal lateral loads (surge): 10% of wheel load (side thrust from crane travel)
3. Fatigue considerations for frequently operated cranes

Deflection limit for crane girder: δ ≤ L/750 (running clearance requirement)

📝 GATE Tip: Economic depth of plate girder d_eco ≈ 1.1 × (M/σ_bc)^0.5 × (1/t_w)^(1/3). Web d/t_w ≤ 200 (without horizontal stiffener). Flange b/t ≤ 16 (for Fe250). North light truss gives diffuse daylight — preferred for textile industries. These facts are classic MCQ fodder.

6Plastic Analysis (Ultimate Load Design)►

6.1 Introduction — Plastic vs Elastic Design

Elastic design assumes the structure fails when the most stressed fibre yields. Plastic (ultimate load) design recognises that after first yield, load redistribution continues until a plastic collapse mechanism forms. This allows more economical design because unused capacity of redundant structures is utilised.

⭐ Key advantage of plastic design: Ductile steel structures can sustain loads well beyond first yield. A fixed-end beam can carry 1.5× the load that caused first yield, before collapse — this extra 50% is the benefit of plastic design over elastic design for a fixed-fixed beam.

6.2 Plastic Bending — Key Concepts

Elastic limit: σ_y at extreme fibre; NA at geometric centroid
At plastic limit (full section yielded):
Plastic moment: M_p = f_y × Z_p
Elastic moment: M_y = f_y × Z_e (Z_e = elastic section modulus = I/y)

Plastic section modulus: Z_p = first moment of area about equal-area axis
Z_p = A/2 × (ȳ_top + ȳ_bot) [where ȳ = centroid of top/bottom half from PNA]

Shape factor (form factor): f = M_p / M_y = Z_p / Z_e
Rectangle: f = 1.5
Solid circle: f = 1.698 ≈ 1.7
Diamond: f = 2.0
I-section: f ≈ 1.12–1.15 (close to 1.0 for deep, slender I)
Hollow circle: f ≈ 1.27 (thin-walled)

Fig. 6.1 — Progression from elastic to fully plastic stress distribution; at full plastic: rectangular stress blocks ±σ_y; M_p = f_y × Z_p

Fig. 6.2 — Plastic collapse of fixed-fixed beam under central point load: 3 plastic hinges form (at both ends + midspan); P_u = 8M_p/L

6.3 Plastic Hinge

A plastic hinge forms at a cross-section where the full plastic moment M_p has been reached. Unlike a real hinge (no moment), a plastic hinge can rotate freely while maintaining M_p. After formation, it acts as a pin in further load analysis — i.e., it transmits constant M_p while allowing rotation to continue.

Number of plastic hinges needed for collapse (mechanism):
n_h = DSI + 1 (for a one-degree mechanism)
where DSI = degree of static indeterminacy

Simply supported beam (DSI=0): 1 hinge for collapse
Propped cantilever (DSI=1): 2 hinges for collapse
Fixed-fixed beam (DSI=3 total; 3 bending redundants → need 3 hinges for sway or 4 for combined)
Actually: n_h = DSI + 1 per independent mechanism
Fixed-fixed beam under one load: needs DSI_bending + 1 = 2+1 = 3 hinges ✓

6.4 Collapse Load — Standard Cases

Beam Type	Loading	Collapse Load (P_u or w_u)	Elastic Load (P_y or w_y)	Load Factor f
SS beam	Central P	4M_p/L	4M_y/L	f = M_p/M_y = shape factor
SS beam	UDL w	8M_p/L²	8M_y/L²	f = shape factor
Propped cantilever	Central P	6M_p/L	5.03M_y/L (first yield at fixed end)	~1.19×M_p/M_y
Propped cantilever	UDL w	11.66M_p/L²	8M_y/L² (fixed end)	~1.46×M_p/M_y
Fixed-fixed beam	Central P	8M_p/L	6M_y/L	= (4/3)×(M_p/M_y)
Fixed-fixed beam	UDL w	16M_p/L²	12M_y/L² (ends)	= (4/3)×(M_p/M_y)

6.5 Methods of Plastic Analysis

Static (Lower Bound) Method

Rules (Lower Bound Theorem):
1. Equilibrium: the moment diagram must be in equilibrium with the applied loads
2. Yield: |M| ≤ M_p at all sections
3. The collapse load so determined is a lower bound (safe side)

Procedure: Assume plastic hinge locations → equilibrium gives P_u → check |M| ≤ M_p everywhere

Kinematic (Upper Bound / Mechanism) Method

Rules (Upper Bound Theorem):
1. Mechanism: assume a collapse mechanism (sufficient hinges to turn structure into mechanism)
2. Energy balance: external work done = internal energy dissipated at plastic hinges
3. The collapse load so found is ≥ true collapse load (unsafe side unless exact mechanism found)

Energy method: Σ(P_i × δ_i) = Σ(M_p × θ_j)
P_i = applied loads, δ_i = displacements, M_p = plastic moment, θ_j = hinge rotation

6.6 Worked Example — Propped Cantilever with UDL

Fixed-pinned beam, UDL w over full span L:

Plastic hinge forms first at fixed end (max hogging BM).
After first hinge: system becomes SS beam (propped cantilever rotates at fixed end).
Second hinge at location of max +ve BM.

Energy method for mechanism (2 hinges):
Let hinge 1 at fixed end A (rotation θ₁), hinge 2 at x = x₀ from A (rotation θ₁ + θ₂)
At B (roller), δ_B = 0 → compatibility: x₀θ₁ = (L−x₀)θ₂ → θ₂ = x₀θ₁/(L−x₀)

External work = ½ × w × L × δ_max = ½ × w × L × x₀θ₁/2 ... [integration needed]
Actually: External work = w × x₀θ₁/2 × x₀ [triangular deflection shape]

Internal work = M_p × θ₁ + M_p × (θ₁ + θ₂) = M_p × (2θ₁ + θ₂)

Minimising w_u w.r.t. x₀: dw_u/dx₀ = 0 → x₀ = 0.414L
∴ w_u = 11.66 M_p / L² (exact result)

6.7 Load Factor in Plastic Design

Load factor = Collapse load / Working load = P_u / P_working
Minimum load factor (IS 800 plastic design): λ_min = 1.85 (dead + live)

Design approach:
P_u = λ_min × P_working
Determine M_p required for each member from collapse analysis
Select section with Z_p ≥ M_p / f_y

Requirements for plastic design (IS 800 Cl. 9):
1. Material: f_y ≤ 450 MPa; elongation ≥ 15%; f_u/f_y ≥ 1.2
2. Section: compact section (b/t ≤ 8.9√(250/f_y) for flanges)
3. No LTB: adequate lateral support at plastic hinge locations
4. Deflection: check under service loads (elastic analysis)

6.8 Shape Factors and Z_p Values

Section	Z_e	Z_p	Shape Factor f
Rectangle (b × d)	bd²/6	bd²/4	1.5
Solid circle (D)	πD³/32	D³/6	16/(3π) ≈ 1.698
Thin circular tube (D, t)	πD²t/4	D²t	4/π ≈ 1.27
Diamond (diag. 2a × 2b)	ab²/3	ab²/2	2.0
I-section (approx)	I/(d/2)	b_f·t_f·(d−t_f)+t_w·d_w²/4	1.12–1.15
Triangle (b, h)	bh²/24	bh²/12	2.0 (about apex); varies

⚠️ Plastic design is only applicable to structures made of ductile steel with adequate rotation capacity. It must NOT be used for structures subject to repeated loading (fatigue), structures where deformations are critical, or members where LTB or local buckling would limit rotation at hinges.

★Quick Revision – All Formulae, Tables & Mnemonics►

IS 800:1984 Permissible Stresses (Fe250, f_y = 250 MPa)

Tension: σ_at = 0.6 f_y = 150 MPa
Bending (compact): σ_bc = σ_bt = 0.66 f_y = 165 MPa
Shear in rivet: τ_va = 100 MPa; Max shear: τ_vm = 115 MPa
Bearing on rivet: σ_pb = 300 MPa
Shear in weld: τ_w = 110 MPa (E41 electrode)
Axial compression: σ_ac = from Table 5.1 (depends on λ = L_eff/r)

Rivet and Bolt Design

Rivet value R = min(P_s, P_b)
P_s (single shear) = (π/4)d²τ_va | P_s (double shear) = 2 × single shear
P_b (bearing) = d × t × σ_pb
P_t (tearing) = (p−d) × t × σ_at
Gross rivet dia = nominal + 1.5 mm (hot-driven)
Pitch: min 3d; max 16t or 200 mm (tension), 12t or 200 mm (compression)
Edge distance: min 1.5d (sheared); 1.25d (rolled)

Fillet Weld

Throat = 0.707s | Strength per unit length q = 0.707s × 110 (N/mm)
Min length = 4s or 40 mm | Effective length = total − 2s (start/stop deduction)
Max weld = t (plate thickness); Min weld: 3mm (≤10mm plate) → 8mm (>32mm plate)

Tension Members

Net area A_n = A_g − n·d_h·t (no stagger)
Staggered holes: w_n = w − Σd_h + Σ(s²/4g) A_n = w_n × t
Shear lag (angle, 1 leg connected): A_eff = A_1 + k·A_2
k = 3A_1/(3A_1+A_2) [A_1 = connected leg net, A_2 = unconnected leg gross]
T_allow = A_eff × σ_at = A_eff × 150 MPa

Compression Members

λ = L_eff / r (slenderness ratio)
Max λ: main members = 180; secondary/bracing = 200
σ_ac from IS 800 Table 5.1 (decreases with λ)
Effective lengths: PP=L, FF=L/2, FP=0.7L, FFree=2L
Lacing shear = 2.5% P; Lacing angle = 40°–70°
Batten depth ≥ 0.75b; thickness ≥ (inner edge dist)/50; min 6 mm

Beams

Z_req = M_max / σ_bc (= M_max / 165 MPa for laterally supported beam, Fe250)
Avg shear: τ_va = V/(d×t_w) ≤ 100 MPa
Deflection: δ ≤ L/325 (IS 800 for plaster); ≤ L/360 (roofs)
LTB: σ_bc reduced from IS 800 Table 6.1B based on D/T and l/r_y
Shear flow (cover plate/composite): q = VQ/I

Plate Girder

Web: d/t_w ≤ 200 (no horizontal stiffener); ≤ 250 (with H-stiffener)
Flange outstand: b/t ≤ 16 (Fe250); b/t ≤ 256/√f_y
Stiffener spacing: c/d ≤ 1.5 (transverse stiffeners)
Flange-to-web weld: q_fw = V·A_f·ȳ_f / I (shear flow; two welds)

Plastic Analysis

M_p = f_y × Z_p | M_y = f_y × Z_e
Shape factor f = Z_p/Z_e: Rectangle=1.5; Circle=1.698; I-section≈1.12; Diamond=2.0
Hinges for collapse = DSI + 1 (per mechanism)
Energy: Σ(P×δ) = Σ(M_p×θ) [external work = internal dissipation]
Collapse loads: SS-central P → 4M_p/L; Fixed-fixed central P → 8M_p/L; PP-cantilever → 6M_p/L
Load factor min = 1.85 (IS 800 plastic design)

Key Mnemonics

Rivet failure modes: "SCTSE"
Shearing of rivet | Crushing (bearing) | Tearing of plate | Shear-out | Edge split

Rivet pitch limits: "3d to 16t or 200 (tension) / 12t or 200 (compression)"

Slenderness ratio limits: "180 – 200"
Main members ≤ 180; Bracing/secondary ≤ 200
"Main members May (180); Bracing can Be a bit more (200)"

Effective lengths: "2 – 1 – 0.7 – 0.5"
Fixed-Free: 2L | Pin-Pin: 1L | Fixed-Pin: 0.7L | Fixed-Fixed: 0.5L
"Worst to best: 2, 1, 0.7, 0.5"

Shape factors (descending order):
Diamond 2.0 → Triangle ≈ 2.0 → Circle 1.70 → Hollow Circle 1.27 → Rectangle 1.50 → I-section 1.12
"Diamonds are Round; Rectangles are Half-way; I-sections barely Improve"
More precisely: Diamond=2.0, Circle≈1.7, Rectangle=1.5, I≈1.12

Fillet weld throat: "0.707 × size"
"707 — like Boeing 707, always at 45°"

Plate girder web limit: "200/250"
Without H-stiffener: d/t ≤ 200 | With H-stiffener: d/t ≤ 250
"Add a Horizontal stiffener to get 250 — just 50 more"

Plastic collapse: "Hinges = DSI + 1"
"Need one more hinge than redundants to make a mechanism (one more move to checkmate)"

Exam-Angle Comparison Table

Topic	GATE Focus	ESE Focus	SSC JE Focus
Fasteners	Rivet value, efficiency, fillet weld strength per unit length	HSFG bolt design; eccentric bolt/weld group; combined shear + tension	Pitch/edge distance rules; failure modes
Tension Members	Net area; shear lag k formula; A_eff	Lug angle; staggered holes detailed calc; block shear	Gross vs net area concept
Compression Members	σ_ac from table; lacing design; buckling load	Batten plate design; stability of laced column; load eccentricity	Max λ = 180; effective length; types
Beams	Z_req, σ_bc, LTB concept; shear check	Beam-column interaction; compound beam design; crane girder	Z_req formula; deflection limits
Plate Girder	Web d/t limits; flange b/t; stiffener spacing	Full plate girder design from scratch; tension field action	Component names; web limit
Industrial Roofs	Roof truss types; purlin design in biaxial bending	Complete purlin, rafter, truss design; wind load Cp values	Truss type identification; load types
Plastic Analysis	Shape factor; collapse load for standard cases; number of hinges	Virtual work method; multi-span frame collapse; load factor	Shape factor values; plastic hinge concept