Steel Structures — Design of Structures

Design of Steel Structures applies the analysis results from Theory of Structures to size real structural members as per IS 800 — from the Working Stress Method fundamentals, through structural fasteners (rivets, bolts, welds), tension and compression member design, beam design and lateral torsional buckling, plate girders and industrial roof systems, up to plastic (ultimate load) analysis. Every formula, IS code clause, diagram, solved example and exam-pattern table is included.

Learning Objectives

After studying this chapter you will be able to:

• Design riveted, bolted (black bolt & HSFG) and welded connections, and check the strength of eccentric bolt/weld groups
• Compute gross and net area of tension members, apply the shear-lag factor for single-angle connections, and design lug angles
• Determine the permissible compressive stress from slenderness ratio and design laced/battened built-up compression members
• Design steel beams for bending, shear and deflection, and identify when lateral torsional buckling governs the permissible bending stress
• Proportion the web, flanges and stiffeners of a plate girder, and design purlins and crane girders for industrial roof structures
• Apply plastic analysis — shape factor, plastic hinge formation, static and kinematic methods — to find the collapse load of indeterminate beams

Exam Relevance

Concept Map

Where This Fits

Prerequisite: Theory of Structures (determinacy, analysis methods and BM/SF diagrams computed there are the direct inputs to member sizing here). Related: RCC/PCC Design covers the equivalent design process for reinforced/plain concrete members.

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.

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: \(P_s = (\dfrac{\pi}{4}) \times d^{2} \times \tau_{va}\). Strength in double shear: \(P_s = 2 \times (\dfrac{\pi}{4}) \times d^{2} \times \tau_{va}\). Strength in bearing: \(P_b = d \times t \times \sigma_{pb}\). Strength in tearing: \(P_t = (p - d) \times t \times \sigma_{at}\).

Where: d = gross diameter of rivet (nominal dia + 1.5 mm for hot-driven rivets, per IS 1929; use gross dia for shear, net for tearing); \(\tau_{va}\) = permissible shear stress in rivet = 100 MPa (IS 800); \(\sigma_{pb}\) = permissible bearing stress = 300 MPa (IS 800); \(\sigma_{at}\) = 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 a single rivet in a lap/butt joint.

Efficiency of a Riveted Joint

η = (Strength of joint per pitch) / (Strength of solid plate per pitch). For a single-riveted lap joint with one rivet per pitch: \(\eta_{tearing} = (p - d) / p\) and \(\eta_{shearing} = P_s / (p \times t \times \sigma_{at})\). A balanced joint has \(\eta_{tearing} = \eta_{shearing}\), giving the optimal pitch.

1.3 Rivet Spacing Rules (IS 1929 / IS 800)

1.4 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 (per bolt per interface): \(P_{sf} = \mu \times T_f \times n_e\), where \(\mu\) = 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 an M20 bolt Grade 8.8: \(T_f = 144\) kN; proof stress = 628 MPa.

1.5 Black Bolts vs HSFG Bolts

1.6 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: \(t_e = 0.707 \times s\) (for 90° fillet welds; s = weld size = leg length). IS 816 uses effective throat \(= K \times s\), K = 0.7 for a 90° weld angle. Permissible shear stress in weld (on throat): \(\tau_w = 110\) MPa (IS 816, E41 electrode). Strength of fillet weld per unit length: \(q = t_e \times \tau_w = 0.707 \times s \times 110\) (N/mm per mm run of weld).

Weld size limits — minimum weld size depends on the 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 × \(\sigma_{at}\) (permissible stress same as parent metal). There is no reduction for joint efficiency in full-penetration butt welds → 100% efficient.

1.7 Eccentric Connections — In-plane Eccentricity

When load P acts at eccentricity e from the CG of a bolt/weld group, the moment \(M = P \times e\). For a bolt group, the force on the most critical bolt has two components: Direct shear \(F_d = P / n\) (n = number of bolts) and Torsional shear \(F_t = M \times r_{max} / (\Sigma r^{2})\), where \(r_{max}\) = distance of the farthest bolt from the CG and \(\Sigma r^2\) = sum of squared distances. Resultant on the critical bolt: \(F_R = \sqrt{F_d^{2} + F_t^{2} + 2\cdot F_d\cdot F_t\cdot \cos\theta}\), where \(\theta\) = angle between \(F_d\) and \(F_t\) directions at the critical bolt.

1.8 Beam-Column Connections

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 the gross section and fracture at the 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: \(A_n = A_g - n \times d_h \times t\), 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): \(w_n = w_g - \Sigma d_h + \Sigma(\dfrac{s^{2}}{4g})\), where \(s\) = stagger (longitudinal spacing between holes), \(g\) = gauge (transverse spacing); \(A_n = w_n \times t\). Apply this to all possible failure paths and use the minimum net area.

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

Permissible axial tensile stress: \(\sigma_{at} = 0.6 \times f_y\). For Fe250 (mild steel, \(f_y\) = 250 MPa): \(\sigma_{at} = 150\) MPa. Design tensile capacity: \(T_{dg} = A_g \times \sigma_{at}\) (based on gross area yielding); \(T_{dn} = A_n \times 0.9 \times f_u / \gamma_{m1}\) (LSD approach for fracture at net section). WSM: \(T_{allowed} = A_n \times \sigma_{at}\) (net area governs when holes are 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): \(A_{eff} = A_1 + A_2 \times k\), where \(k = 3A_1 / (3A_1 + A_2)\); \(A_1\) = net area of connected leg (after deducting hole area), \(A_2\) = gross area of unconnected leg. Alternatively per IS 800:2007, shear lag factor \(\beta = 1 - \dfrac{w_s}{L_c}\) (for connection length \(L_c\), shear lag width \(w_s\)).

2.5 Types of Tension Members

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 the shear lag effect and the required connection length.

With lug angles, the effective net area = full gross area (shear lag eliminated). Lug angle force = \((A_2 / A_{total}) \times P\), where \(A_2\) = unconnected leg area. The lug angle connection must carry ≥ 120% of the force in the attached leg (IS 800).

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 \(\sigma_{ac}\) depends on slenderness ratio \(\lambda = \dfrac{l}{r}\). From IS 800 Table 5.1 (for \(f_y = 250\) MPa):

At \(\lambda = 0\), \(\sigma_{ac} = \sigma_{at} = 150\) MPa since the gross section simply yields. Maximum permitted \(\lambda\) for main members is 180 (\(\lambda=200\) is only reached for secondary/bracing members). Maximum slenderness ratio (IS 800:1984): main compression members \(\lambda_{max} = 180\); secondary members (bracing) \(\lambda_{max} = 200\). Design: \(P_{allowed} = A_g \times \sigma_{ac}(\lambda)\).

3.3 Effective Length of Compression Members

3.4 Lacing and Batten Plate Design

Lacing

Lacing is designed for a transverse shear force \(V\) = 2.5% of \(P\) (axial column load), per IS 800 Cl. 7.6.6.1 — this shear is assumed to act on the whole column. For N-lacing (single), force in each lacing bar: \(F_{lace} = \dfrac{V}{2\sin\theta}\), where \(\theta\) = angle of the lacing bar with the axis perpendicular to the member axis.

Lacing bar slenderness: \(\dfrac{l}{r_{min}} \le 145\) (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 \(V_L = V \times C / (2 \times n_b \times S)\) and moment in batten \(M = V \times C / (2 \times n_b)\), 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 the main member × 50.

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

3.6 Column Bases (Base Plates)

A column base transfers the column load to the concrete pedestal/footing over an area large enough to keep the bearing pressure within the permissible value for concrete. Two principal types:

Slab base design (IS 800 Cl. 7.4): the plate thickness is governed by the cantilever bending of the plate projection under the uniform bearing pressure \(w\):

$$ t_s = \sqrt{\frac{2.5\,w\,(a^2 - 0.3b^2)\,\gamma_{m0}}{f_y}} $$

where \(a, b\) = larger and smaller projections of the plate beyond the column, \(w\) = uniform pressure under the plate. Anchor bolts (holding-down bolts) connect the base to the foundation and resist uplift and shear; a minimum of four bolts is provided for stability. For bases carrying moment, the plate is designed for the resulting non-uniform (trapezoidal or triangular, with anchor-bolt tension) pressure distribution.

4.1 Classification of Steel Beams

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

For compact sections with full lateral support: \(\sigma_{bt} = \sigma_{bc} = 0.66 \times f_y = 165\) MPa (for Fe250, \(f_y\) = 250 MPa). Required section modulus: \(Z_{required} = M_{max} / \sigma_{bc}\).

Shear stress check: average shear stress \(\tau_{va} = V / (d \times t_w) \leq 100\) MPa; maximum shear stress \(\tau_{vm} = 1.5 \times \tau_{va} \leq 115\) MPa (for thin webs). Deflection check: \(\delta_{max} \le \dfrac{L}{325}\) (IS 800) for beams carrying plaster or sensitive finishes; \(\delta_{max} \le \dfrac{L}{360}\) 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.

Permissible bending compressive stress \(\sigma_{bc}\) (IS 800:1984) depends on: (1) \(\dfrac{D}{T}\) (depth-to-flange thickness ratio); (2) \(\dfrac{l}{r_y}\) (slenderness for LTB; \(l\) = effective length between lateral supports). From IS 800 Table 6.1B: \(\sigma_{bc}\) decreases as \(\dfrac{l}{r_y}\) or \(\dfrac{D}{T}\) increases. Full plastic moment is available only when \(\dfrac{l}{r_y} \le\) the permissible limit; otherwise a reduced \(\sigma_{bc}\) from the code tables applies.

Effective length for LTB (IS 800 Table 6.3): compression flange fully restrained (both ends) → \(l = 0.7 \times L\) (L = span); compression flange partially restrained → \(l = 0.85 \times L\); compression flange unrestrained → \(l = 1.0 \times L\).

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 \(P_{wb} = (b_1 + n_1) \times t_w \times \sigma_{ac}(\lambda)\), where \(\lambda = 2.5 \times d / t_w\) (slenderness of web in bearing).

Web Crippling (local yielding at support/point load): \(P_{wc} = (b + 2.5 \times T_f) \times t_w \times f_{yw}\), where \(b\) = 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 the flanges: \(Z_{compound} = Z_{rolled} + A_{plate} \times \bar y_{plate}\). Horizontal shear per unit length at the plate-flange interface (shear flow): \(q = V \times Q / I\), where \(Q = A_{plate} \times \bar y_{plate}\) (first moment of the cover plate about the NA). Weld/rivet spacing to resist shear flow: \(spacing = R / q\), where R = weld strength per unit length or rivet value.

4.6 Beam-Column Design (Combined Axial + Bending)

Interaction equation (IS 800:1984 WSM): \(\dfrac{P}{P_a} + \dfrac{M_x}{M_{ax}} + \dfrac{M_y}{M_{ay}} \le 1.0\), where \(P\) = applied axial load, \(P_a\) = axial capacity \(= A_g \times \sigma_{ac}\); \(M_x\) = bending moment about the major axis, \(M_{ax} = Z_x \times \sigma_{bc}\); \(M_y\) = bending moment about the minor axis, \(M_{ay} = Z_y \times \sigma_{bc}\).

Additional check for yielding at the extreme fibre (interaction): \(\dfrac{P}{P_e} + C_m \times M_x / [(1 - \dfrac{P}{P_e}) \times M_{ax}] \le 1.0\), where \(P_e = \pi^{2}EI_x / (L_{eff})^{2}\) (Euler load for the major axis).

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 the economical rolled-section range (15–100 m) or loads are too heavy for standard sections.

5.2 Design of Plate Girder Web

Web depth d (economic depth — minimises total steel weight): \(d_{eco} = \dfrac{M}{\sigma_{bc}} \times \left(\dfrac{12}{t_w}\right)^{1/3}\) [approximate; varies with flange assumptions].

Web thickness \(t_w\) limits (IS 800): without horizontal stiffeners \(\dfrac{d}{t_w} \le 200\); with horizontal stiffeners \(\dfrac{d}{t_w} \le 250\). Average web shear stress: \(\tau = \dfrac{V}{d \times t_w} \le \tau_{va} = 100\) MPa. For slender webs (\(\dfrac{d}{t_w} > 85\)): tension field action may be considered.

5.3 Flange Design

Flange area required (approximate — web carries moment contribution ≈ 1/6): \(A_f = M / (\sigma_{bc} \times d) - t_w \times d / 6\). OR more precisely: \(I_{required} = M \times y_{max} / \sigma_{bc}\) (full section MI from extreme fibre stress); \(I_{web} = t_w \times d^{3} / 12\); \(I_{flange,needed} = I_{required} - I_{web}\); \(A_f = I_{flange,needed} / (\dfrac{d}{2})^2\) [approx, flange as thin plate at \(\dfrac{d}{2}\) from NA].

Flange width \(b_f\) limits (outstand criterion to prevent local buckling): \(\dfrac{b}{t} \le \dfrac{256}{\sqrt{f_y}}\) (plate outstand from web face; IS 800 WSM). For \(f_y = 250\) MPa: \(\dfrac{b}{t} \le \dfrac{256}{\sqrt{250}} \approx 16.2\).

5.4 Stiffeners in Plate Girders

Transverse stiffener spacing (IS 800 — to prevent shear buckling): \(\dfrac{c}{d} \le 1.5\) (aspect ratio of each panel). Without diagonal tension, shear stress \(\tau \le \tau_{cr}\) (critical shear buckling stress): \(\tau_{cr} = k_s \times \pi^{2}E / (12(1-\nu^{2})) \times (\dfrac{t_w}{d})^{2}\), where \(k_s\) = shear buckling coefficient \(\approx 5.35 + \dfrac{4.0}{(c/d)^2}\) for intermediate panels.

5.5 Flange-to-Web Connection

Horizontal shear flow at the flange-web interface: \(q_{fw} = V \times A_f \times \bar{y}_f / I\). Weld size required: \(s_w = q_{fw} / (2 \times 0.707 \times \tau_w) = q_{fw} / (2 \times 0.707 \times 110)\) [two fillet welds, one each side of web; E41 electrode, \(\tau_w = 110\) MPa].

5.6 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.

Roof Truss Loads

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: \(w = (\text{dead} + \text{live} + \text{wind}) \times \text{spacing of purlins} \times \cos\theta\) (the component perpendicular to the roof surface governs bending). Biaxial bending check: \(\dfrac{M_u}{Z_{xx}} + \dfrac{M_v}{Z_{yy}} \le \sigma_{bc}\), where \(M_u = \dfrac{wL^2}{8} \times \cos\alpha\) (in plane of roof) and \(M_v = \dfrac{wL^2}{8} \times \sin\alpha\) (perpendicular to roof); \(\alpha\) = roof slope angle. Sag rods reduce \(M_v\) by providing intermediate support in the minor axis.

5.7 Crane Girder Design

Crane girder must be designed for: (1) Vertical loads — wheel loads with impact factor (IS 875 Part 2): \(P_{design} = P_{wheel} \times\) (1 + impact factor); impact factor = 10–25% of the 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: \(\delta \le \dfrac{L}{750}\) (running clearance requirement).

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.

6.2 Plastic Bending — Key Concepts

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

Plastic section modulus \(Z_p\) = first moment of area about the equal-area axis: \(Z_p = \dfrac{A}{2} \times (\bar y_{top} + \bar y_{bot})\), where \(\bar y\) = centroid of the top/bottom half from the plastic neutral axis. 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).

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 under one load: needs \(DSI_{bending} + 1 = 2+1 = 3\) hinges ✓ (3 bending redundants → need 3 hinges for a single-load mechanism).

6.4 Collapse Load — Standard Cases

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 — \(|M| \le M_p\) 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 \(|M| \le M_p\) everywhere.

Kinematic (Upper Bound / Mechanism) Method

Rules (Upper Bound Theorem): (1) Mechanism — assume a collapse mechanism (sufficient hinges to turn the structure into a mechanism); (2) Energy balance — external work done = internal energy dissipated at plastic hinges; (3) The collapse load so found is ≥ the true collapse load (unsafe side unless the exact mechanism is found). Energy method: \(\Sigma(P_i \times \delta_i) = \Sigma(M_p \times \theta_j)\), where \(P_i\) = applied loads, \(\delta_i\) = displacements, \(\theta_j\) = hinge rotation.

6.6 Worked Example — Propped Cantilever with UDL

Fixed-pinned beam, UDL w over full span L: a plastic hinge forms first at the fixed end (max hogging BM). After the first hinge, the system becomes an SS beam (the propped cantilever rotates at the fixed end); a second hinge forms at the location of max +ve BM.

Energy method for the mechanism (2 hinges): let hinge 1 be at fixed end A (rotation \(\theta_1\)), hinge 2 at \(x = x_0\) from A (rotation \(\theta_1 + \theta_2\)). At B (roller), \(\delta_B = 0\), so compatibility gives \(x_0\theta_1 = (L-x_0)\theta_2 \to \theta_2 = x_0\dfrac{\theta_1}{L-x_0}\). Internal work = \(M_p \times \theta_1 + M_p \times (\theta_1 + \theta_2) = M_p \times (2\theta_1 + \theta_2)\). Minimising \(w_u\) with respect to \(x_0\): \(\dfrac{dw_u}{dx_{0}} = 0 \to x_{0} = 0.414L\), giving the exact result \(w_u = 11.66\,M_p / L^{2}\).

6.7 Load Factor in Plastic Design

Load factor = Collapse load / Working load = \(P_u / P_{working}\). Minimum load factor (IS 800 plastic design): \(\lambda_{min} = 1.85\) (dead + live). Design approach: \(P_u = \lambda_{min} \times P_{working}\); determine \(M_p\) required for each member from collapse analysis; select a section with \(Z_p \ge \dfrac{M_p}{f_y}\).

Requirements for plastic design (IS 800 Cl. 9): (1) Material — \(f_y \le 450\) MPa, elongation ≥ 15%, \(\dfrac{f_u}{f_y} \ge 1.2\); (2) Section — compact section (\(\dfrac{b}{t} \leq 8.9\sqrt{\dfrac{250}{f_y}}\) 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

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