Advanced Civil Engineering Handbook › Chapter 5

Chapter 5 of 14 · 40 formulas · 86 one-liners

Steel Structures (IS 800:2007) — formulas & one-liners

Limit state design of steel tension and compression members, beams, plate girders, bolted and welded connections and column bases as per IS 800:2007, with IS 875 load basics and working stress values still asked in exams.

5.1 Limit State Basics, Material & Partial Safety Factors

Yield stress ratio (used in all b/t limits)

ε = \frac{250}{f _{y}}

f_y = yield stress of the steel (MPa)

ε = 1.0 for E250 steel

For structural steel E = 2 × 10⁵ MPa, G = 0.769 × 10⁵ MPa, Poisson's ratio 0.3 (elastic range), α = 12 × 10⁻⁶ per °C and unit mass 7850 kg/m³ (IS 800 Cl 2.2.4.1).
E250 steel (IS 2062) has f_y = 250 MPa for thickness below 20 mm (240 MPa for 20–40 mm, 230 MPa above 40 mm) and f_u = 410 MPa.
Partial safety factor for material γ_m0 = 1.10 applies to resistance governed by yielding and buckling (IS 800 Table 5).
Partial safety factor γ_m1 = 1.25 applies to resistance governed by ultimate stress, i.e. rupture of the net section.
For bearing-type bolts γ_mb = 1.25 for both shop and field connections.
For slip resistance of friction-type (HSFG) bolts γ_mf = 1.10 when slip resistance is designed at service load and 1.25 when designed at ultimate load (IS 800 Cl 10.4.3).
For welds γ_mw = 1.25 for shop welds and 1.50 for field welds.
Load factor for dead load + imposed load is 1.5 on each (IS 800 Table 4).
Load factor for dead + imposed + wind (or earthquake) load is 1.2 on each load.
Load factor for dead + wind (or earthquake) load is 1.5 on each, but dead load is factored by 0.9 where it relieves the effect (uplift, overturning).

5.2 Section Classification (IS 800 Table 2)

Factor β_b in bending strength

β_{b} = 1.0 (plastic, compact); β_{b} = \frac{Z _{e}}{Z _{p}} (semi-compact)

Z_e = elastic section modulus; Z_p = plastic section modulus

Class 1 (plastic) sections can develop plastic hinges with enough rotation capacity for a plastic collapse mechanism.
Class 2 (compact) sections can develop the full plastic moment but have inadequate rotation capacity for a plastic mechanism.
Class 3 (semi-compact) sections reach yield stress at the extreme fibre but local buckling prevents development of the plastic moment.
Class 4 (slender) sections buckle locally before the extreme fibre reaches yield stress and are designed on an effective section.
Outstand of compression flange of a rolled section: b/t_f ≤ 9.4ε (plastic), 10.5ε (compact), 15.7ε (semi-compact), with b = half the flange width.
Outstand of compression flange of a welded section: b/t_f ≤ 8.4ε (plastic), 9.4ε (compact), 13.6ε (semi-compact).
Web of an I-section with neutral axis at mid-depth: d/t_w ≤ 84ε (plastic), 105ε (compact), 126ε (semi-compact).
An angle in axial compression is semi-compact if b/t ≤ 15.7ε, d/t ≤ 15.7ε and (b + d)/t ≤ 25ε.

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5.3 Tension Members (IS 800 Section 6)

Design strength due to yielding of gross section

T_{d g} = \frac{A _{g} f _{y}}{γ _{m 0}}

A_g = gross area; f_y = yield stress; γ_m0 = 1.10

Design strength due to rupture of net section (plates)

T_{d n} = \frac{0.9 A _{n} f _{u}}{γ _{m 1}}

A_n = net effective area; f_u = ultimate stress; γ_m1 = 1.25

Net area of a plate with staggered holes

A_{n} = [b - n d_{h} + \sum \frac{p _{s i}^{2}}{4 g _{i}}] t

b = plate width; n = holes in the zig-zag section; d_h = hole diameter; p_s = stagger (pitch); g = gauge; t = thickness

Rupture strength of an angle connected through one leg

T_{d n} = \frac{0.9 A _{n c} f _{u}}{γ _{m 1}} + \frac{β A _{g o} f _{y}}{γ _{m 0}}

A_nc = net area of connected leg; A_go = gross area of outstanding leg

Shear lag factor β for angles

β = 1.4 - 0.076 \frac{w}{t} \frac{f _{y}}{f _{u}} \frac{b _{s}}{L _{c}}

w = outstand leg width; t = leg thickness; b_s = shear lag width; L_c = length of end connection

0.7 ≤ β ≤ f_u γ_m0 / (f_y γ_m1)

Block shear strength, case 1 (gross shear yield + net tension rupture)

T_{d b 1} = \frac{A _{v g} f _{y}}{3 γ _{m 0}} + \frac{0.9 A _{t n} f _{u}}{γ _{m 1}}

A_vg = gross area in shear; A_tn = net area in tension

Block shear strength, case 2 (net shear rupture + gross tension yield)

T_{d b 2} = \frac{0.9 A _{v n} f _{u}}{3 γ _{m 1}} + \frac{A _{t g} f _{y}}{γ _{m 0}}

A_vn = net area in shear; A_tg = gross area in tension

Block shear strength = smaller of T_db1 and T_db2

The design tensile strength of a member is the least of gross-section yielding, net-section rupture and block shear strengths.
As a simplification, rupture strength of an angle may be taken as T_dn = α A_n f_u / γ_m1 with α = 0.6 for one or two bolts, 0.7 for three bolts and 0.8 for four or more bolts (or an equivalent weld).
Block shear is a tearing failure along a shear plane parallel to the force combined with a tension plane perpendicular to it at the end connection.
Maximum slenderness ratio of a tension member in which stress reversal occurs due to loads other than wind or earthquake is 180 (IS 800 Table 3).
Maximum slenderness ratio of a member normally acting as a tie but subject to reversal due to wind or earthquake is 350.
Maximum slenderness ratio of a member always in tension (other than pre-tensioned members) is 400.

5.4 Compression Members, Built-up Columns & Column Bases

Design compressive strength

P_{d} = A_{e} f_{c d}

A_e = effective area (gross area for non-slender sections); f_cd = design compressive stress

Design compressive stress

f_{c d} = \frac{f _{y} / γ _{m 0}}{ϕ + ϕ ^{2} - λ ^{2}} \leq \frac{f _{y}}{γ _{m 0}}

f_y = yield stress; γ_m0 = 1.10; φ, λ as below

IS 800 Cl 7.1.2.1

Factor φ for column buckling

ϕ = 0.5 [1 + α (λ - 0.2) + λ^{2}]

α = imperfection factor of the buckling class; λ = non-dimensional slenderness

Non-dimensional effective slenderness ratio

λ = \frac{f _{y}}{f _{cc}}

f_cc = Euler buckling stress

Euler buckling stress

f_{cc} = \frac{π ^{2} E}{( K L / r ) ^{2}}

E = 2 × 10⁵ MPa; KL = effective length; r = radius of gyration about the buckling axis

Thickness of a rectangular slab base

t_{s} = \frac{2.5 w ( a ^{2} - 0.3 b ^{2} ) γ _{m 0}}{f _{y}} \geq t_{f}

w = uniform pressure under the base plate; a, b = larger and smaller projections of the plate beyond the column; t_f = column flange thickness

IS 800 Cl 7.4.3.1

Imperfection factor α is 0.21 for buckling class a, 0.34 for class b, 0.49 for class c and 0.76 for class d (IS 800 Table 7).
Rolled I-sections with h/b_f > 1.2 and t_f ≤ 40 mm fall in buckling class a about the z-z axis and class b about the y-y axis.
Rolled I-sections with h/b_f ≤ 1.2 and t_f ≤ 100 mm fall in buckling class b about z-z and class c about y-y.
Channels, angles, T-sections, solid sections and built-up (laced or battened) members are buckling class c about any axis.
Effective length (IS 800 Table 11) is 0.65L with both ends fixed, 0.8L with one end fixed and the other hinged, and 1.0L with both ends hinged.
Effective length is 1.2L when one end is fixed and the other is restrained in rotation but free to sway, and 2.0L for a cantilever (one end fixed, other free).
Maximum slenderness ratio is 180 for a member carrying compression from dead and imposed loads (IS 800 Table 3).
Maximum slenderness ratio is 250 for a member in compression only under load combinations including wind or earthquake.
Maximum slenderness ratio of the compression flange of a beam against lateral-torsional buckling is 300.
Lacing bars should be inclined at 40° to 70° to the member axis.
Lacing and battens are designed for a transverse shear equal to 2.5% of the axial force in the member.
The slenderness ratio of a lacing bar should not exceed 145.
The effective slenderness ratio of a laced column is taken as 1.05 times its actual value, and of a battened column as 1.10 times.
Battens should divide the member into at least three bays, and the slenderness of each component between battens should not exceed 50 or 0.7 times that of the whole member.
Bearing strength of concrete under a column base is taken as 0.6 f_ck, so the required base plate area is P_u/(0.6 f_ck).

5.5 Beams: Bending, Shear, Lateral-Torsional Buckling & Web

Design bending strength of a laterally supported beam (low shear)

M_{d} = \frac{β _{b} Z _{p} f _{y}}{γ _{m 0}}

β_b = 1.0 (plastic/compact) or Z_e/Z_p (semi-compact); Z_p = plastic modulus

Valid when V ≤ 0.6 V_d

Design shear strength

V_{d} = \frac{A _{v} f _{y w}}{3 γ _{m 0}}

A_v = shear area (h t_w for rolled I and channel sections under major-axis bending); f_yw = yield stress of web

Reduced moment capacity under high shear (V > 0.6 V_d)

M_{d v} = M_{d} - β (M_{d} - M_{f d}), β = (\frac{2 V}{V _{d}} - 1)^{2}

M_fd = plastic design moment of the flanges alone; V = factored shear

Plastic and compact sections

Design bending strength of a laterally unsupported beam

M_{d} = β_{b} Z_{p} f_{b d}, f_{b d} = \frac{χ _{L T} f _{y}}{γ _{m 0}}

f_bd = design bending compressive stress; χ_LT = LTB reduction factor

Reduction factor for lateral-torsional buckling

χ_{L T} = \frac{1}{ϕ _{L T} + ϕ _{L T}^{2} - λ _{L T}^{2}} \leq 1.0

φ_LT, λ_LT as below

Factor φ_LT

ϕ_{L T} = 0.5 [1 + α_{L T} (λ_{L T} - 0.2) + λ_{L T}^{2}]

α_LT = 0.21 for rolled sections, 0.49 for welded sections

Non-dimensional slenderness for LTB

λ_{L T} = \frac{β _{b} Z _{p} f _{y}}{M _{cr}}

M_cr = elastic critical moment for lateral-torsional buckling

Web bearing (crippling) strength

F_{w} = \frac{( b _{1} + n _{2} ) t _{w} f _{y w}}{γ _{m 0}}

b₁ = stiff bearing length; n₂ = dispersion length through flange to web junction at a slope of 1:2.5; t_w = web thickness

To avoid irreversible deformation under service loads, M_d of a simply supported beam is limited to 1.2 Z_e f_y/γ_m0 (1.5 Z_e f_y/γ_m0 for a cantilever).
Lateral-torsional buckling need not be checked when the non-dimensional slenderness λ_LT ≤ 0.4.
For a simply supported beam with both flanges free to rotate in plan at the supports, the effective length for LTB is 1.0L under normal loading and 1.2L under destabilising loading (IS 800 Table 15).
Shear buckling of an unstiffened web must be checked when d/t_w exceeds 67ε.
For web buckling under a concentrated load the web is treated as a strut of effective length 0.7d, giving a slenderness ratio of about 2.5d/t_w, with load dispersed at 45° from the bearing.
Vertical deflection limit for floor and roof members of ordinary buildings is span/300 for elements not susceptible to cracking and span/360 for elements carrying brittle finishes (IS 800 Table 6).
Vertical deflection limit for purlins and girts of industrial buildings is span/150 with elastic cladding and span/180 with brittle cladding.

5.6 Bolted Connections (IS 800 Section 10)

Design shear strength of a bearing-type bolt

V_{d s b} = \frac{f _{u b} ( n _{n} A _{nb} + n _{s} A _{s b} )}{3 γ _{mb}}

f_ub = ultimate stress of bolt; n_n, n_s = shear planes through threads and shank; A_nb = net (tensile stress) area; A_sb = shank area

Multiply by β_lj, β_lg, β_pk where applicable; γ_mb = 1.25

Design bearing strength of a bolt

V_{d p b} = \frac{2.5 k _{b} d t f _{u}}{γ _{mb}}

k_b = least of e/(3d₀), p/(3d₀) - 0.25, f_ub/f_u and 1.0; d = bolt diameter; t = plate thickness in bearing; f_u = ultimate stress of plate; d₀ = hole diameter

Design tensile strength of a bolt

T_{d b} = \frac{0.9 f _{u b} A _{n}}{γ _{mb}} \leq \frac{f _{y b} A _{s b}}{γ _{m 0}}

A_n = net tensile stress area; f_yb = yield stress of bolt; A_sb = shank area

Design slip resistance of an HSFG bolt

V_{d s f} = \frac{μ _{f} n _{e} K _{h} F _{0}}{γ _{m f}}

μ_f = slip factor; n_e = number of effective interfaces; K_h = hole factor; F₀ = minimum bolt tension (proof load) = 0.7 f_ub A_nb; γ_mf = 1.10 (slip at service load), 1.25 (slip at ultimate load)

Reduction factor for long joints

β_{l j} = 1.075 - \frac{l _{j}}{200 d}

l_j = distance between first and last bolts in the direction of load; d = bolt diameter

Applies when l_j > 15d; 0.75 ≤ β_lj ≤ 1.0

Bolt under combined shear and tension

(\frac{V _{s b}}{V _{d b}})^{2} + (\frac{T _{b}}{T _{d b}})^{2} \leq 1.0

V_sb, T_b = factored shear and tension; V_db, T_db = design shear and tension strengths

The net tensile stress area of an ISO-threaded bolt may be taken as about 0.78 times its gross shank area.
Standard clearance holes are 1 mm larger than the bolt for 12 and 14 mm bolts, 2 mm larger for 16 to 24 mm bolts and 3 mm larger for bolts above 24 mm (IS 800 Table 19).
Minimum pitch of bolts is 2.5 times the nominal bolt diameter.
Maximum pitch is 32t or 300 mm, whichever is less, where t is the thickness of the thinner connected plate.
Maximum pitch in the line of force is 16t or 200 mm for tension members and 12t or 200 mm for compression members, whichever is less.
Minimum edge and end distance is 1.7 times the hole diameter for sheared or hand-flame-cut edges and 1.5 times for rolled, machine-flame-cut, sawn or planed edges.
Maximum edge distance is 12tε, where t is the thickness of the thinner outer plate.
Property class 4.6 bolts have f_ub = 400 MPa and f_yb = 240 MPa; class 8.8 bolts have f_ub = 800 MPa and f_yb = 640 MPa.
Hole factor K_h is 1.0 for standard clearance holes, 0.85 for oversized and short slotted holes, and 0.7 for long slotted holes loaded parallel to the slot.
Slip factor μ_f for faying surfaces that are not treated is 0.20 (IS 800 Table 20).
For large grip lengths (l_g > 5d) bolt shear strength is reduced by β_lg = 8d/(3d + l_g), and the grip length should not exceed 8d.

5.7 Welded Connections

Effective throat thickness of a fillet weld

t_{t} = K s

s = size of weld (leg length); K = 0.7 for fusion faces at 60° to 90°

Design strength of a fillet weld

P_{d w} = \frac{L _{w} t _{t} f _{u}}{3 γ _{m w}}

L_w = effective length; t_t = throat thickness; f_u = smaller of weld and parent metal ultimate stress; γ_mw = 1.25 (shop), 1.50 (field)

Equivalent stress in a fillet weld under combined normal and shear stress

f_{e} = f_{a}^{2} + 3 q^{2} \leq \frac{f _{u}}{3 γ _{m w}}

f_a = normal stress; q = shear stress on the throat

Throat factor K for fillet welds is 0.70 for fusion-face angles of 60°–90°, 0.65 for 91°–100°, 0.60 for 101°–106°, 0.55 for 107°–113° and 0.50 for 114°–120° (IS 800 Table 22).
Minimum fillet weld size is 3 mm when the thicker part is up to 10 mm, 5 mm for 10–20 mm, 6 mm for 20–32 mm and 8 mm (first run) for 32–50 mm (IS 800 Table 21).
Effective length of a fillet weld is its overall length minus twice the weld size, and should not be less than four times the weld size.
Maximum fillet weld size along a square edge is the plate thickness minus 1.5 mm, and along the rounded toe of a rolled section 3/4 of the toe thickness.
The effective throat of a fillet weld should not be less than 3 mm and generally not more than 0.7t, where t is the thinner plate thickness.
Fillet welds ending at the ends or sides of parts should be returned around the corner for at least twice the weld size.
In a lap joint the overlap should be at least four times the thickness of the thinner part or 40 mm, whichever is more.
The effective throat of a complete penetration butt weld equals the thickness of the thinner part joined.
The effective throat of an incomplete penetration butt weld is taken as 5/8 of the thickness of the thinner part.

5.8 Plate Girders & Gantry Girders

Shear buckling coefficient of a web panel

k_{v} = 5.35 + \frac{4}{( c / d ) ^{2}} (c / d \geq 1); k_{v} = 4 + \frac{5.35}{( c / d ) ^{2}} (c / d < 1)

c = spacing of transverse stiffeners; d = depth of web

k_v = 5.35 when only end (support) stiffeners are provided

Elastic critical shear stress of a web

τ_{cr, e} = \frac{k _{v} π ^{2} E}{12 ( 1 - μ ^{2} ) ( d / t _{w} ) ^{2}}

E = 2 × 10⁵ MPa; μ = 0.3; d = web depth; t_w = web thickness

Non-dimensional web slenderness for shear buckling

λ_{w} = \frac{f _{y w}}{3 τ _{cr, e}}

f_yw = yield stress of web

Shear buckling stress for a slender web (λ_w ≥ 1.2)

τ_{b} = \frac{f _{y w}}{3 λ _{w}^{2}}

Simple post-critical method; V_cr = A_v τ_b

In the simple post-critical method τ_b = f_yw/√3 when λ_w ≤ 0.8, and τ_b = [1 - 0.8(λ_w - 0.8)] f_yw/√3 when 0.8 < λ_w < 1.2.
The tension field method uses the post-buckling strength of a web panel and requires intermediate transverse stiffeners with adequately anchored end panels.
For serviceability, a plate girder web without intermediate stiffeners (connected to both flanges) should have d/t_w ≤ 200ε.
To avoid compression flange buckling into the web, d/t_w ≤ 345ε_f² when transverse stiffener spacing c ≥ 1.5d.
For electric overhead travelling (EOT) cranes, the vertical impact allowance is 25% of the maximum static wheel load (IS 875 Part 2).
For EOT cranes the lateral (surge) force is 10% of the weight of crab plus lifted load, and the longitudinal force is 5% of the static wheel loads.
For hand-operated cranes the vertical impact allowance is 10%, the lateral force 5% of crab plus lifted load and the longitudinal force 5% of static wheel loads.
Vertical deflection limit of a crane (gantry) girder is span/500 for manually operated cranes, span/750 for electric cranes up to 50 t and span/1000 for electric cranes over 50 t (IS 800 Table 6).

5.9 Loads for Steel Structures (IS 875) & Working Stress Values

Design wind speed at height z (IS 875 Part 3:2015)

V_{z} = V_{b} k_{1} k_{2} k_{3} k_{4}

V_b = basic wind speed; k₁ = risk coefficient; k₂ = terrain roughness and height factor; k₃ = topography factor; k₄ = importance (cyclonic region) factor

Wind pressure at height z

p_{z} = 0.6 V_{z}^{2}

p_z in N/m²; V_z in m/s

Design wind pressure

p_{d} = K_{d} K_{a} K_{c} p_{z} \geq 0.7 p_{z}

K_d = wind directionality factor; K_a = area averaging factor; K_c = combination factor

Wind load on a building surface or element

F = (C_{p e} - C_{p i}) A p_{d}

C_pe, C_pi = external and internal pressure coefficients; A = surface area

Wind directionality factor K_d = 0.9 for buildings, but it is taken as 1.0 in cyclone-affected regions.
Internal pressure coefficient C_pi is ±0.2 for buildings with openings below 5% of wall area, ±0.5 for 5–20% openings and ±0.7 for openings above 20%.
Imposed load on a flat roof (slope up to 10°) is 1.5 kN/m² where access is provided and 0.75 kN/m² where access is not provided (IS 875 Part 2).
For sloping roofs above 10° the imposed load of 0.75 kN/m² is reduced by 0.02 kN/m² for every degree above 10°, subject to a minimum of 0.4 kN/m².
Roof trusses are economically spaced when the cost of trusses equals twice the cost of purlins plus the cost of roof covering (t = 2p + r).
Permissible axial tensile stress in working stress design (IS 800:1984) is 0.6f_y.
Permissible bending stress in tension or compression for laterally supported beams (IS 800:1984) is 0.66f_y.
Permissible average shear stress is 0.4f_y and maximum shear stress 0.45f_y (IS 800:1984).
In the working stress method of IS 800:2007 (Section 11), the permissible axial compressive stress is 0.6f_cd.
Power-driven rivets have a permissible shear stress of 100 MPa and a permissible bearing stress of 300 MPa (IS 800:1984).
Permissible shear stress on the throat of a shop fillet weld is 108 MPa (some texts round it to 110 MPa; IS 816), and field welds are limited to 80% of the shop value.
In working stress design, permissible stresses in structural steel may be increased by 33⅓% when wind or earthquake load is included.

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Values follow the current Indian Standards and IRC / Indian Railways documents named in each item. Always check the latest amendment of a code before using a value in design.