Design of Steel Structures (CE-801) Unit 1 Notes

UNIT 1: Design of Steel Structures

1. Introduction and Design Principles

Properties of Structural Steel:

Yield Stress ($$\displaystyle f_y $$): Stress at which plastic deformation begins.
Ultimate Stress ($$\displaystyle f_u $$): Maximum stress before failure.
Ductility: Ability to undergo large plastic deformation (measured by elongation, reduction in area).
Toughness: Ability to absorb energy (impact resistance).

Comparison with Cast Iron & Wrought Iron:

Property	Steel	Cast Iron	Wrought Iron
Tensile Strength	High	Low	Moderate
Ductility	High	Brittle	High
Weldability	Excellent	Poor	Poor
Uniformity	Homogeneous	Heterogeneous	Non-uniform
Corrosion Resistance	Moderate	Good	Good

[!TIP] Exam Focus: Steel's high strength-to-weight ratio, ductility, and weldability make it superior for modern structures.

Design Philosophies:

Working Stress Design (WSD)/Allowable Stress Design (ASD): Stresses under working loads < allowable stresses (factor of safety on yield).

$$f_{actual} \leq f_{allowable} = \frac{f_y}{\gamma_m}$$

Limit State Design (LSD)/Load and Resistance Factor Design (LRFD): Factored loads < design strength (partial safety factors on load & material).

$$\gamma_f F \leq \frac{F_R}{\gamma_m}$$

IS 800:2007 adopts LSD.

Relevant IS Codes:

IS 800:2007: General construction in steel.
IS 808: Rolled steel beams.
IS 814: Rivets.
IS 875: Imposed loads (Parts 1-5).
IS 1893: Earthquake loads.
IS 883:1994: Timber classification (Group A, B, C based on modulus of elasticity & bending stress).

Loads & Combinations (IS 875):

Dead Load (DL)
Imposed/Live Load (LL)
Wind Load (WL)
Seismic Load (EL)
Snow Load (SL)
Temperature Effects

Load Combinations (LSD - IS 800):

$1.5(DL + LL)$
$1.2(DL + LL + WL/EL)$
$1.2(DL + WL/EL) + 1.6LL$
$1.5(DL + WL/EL)$

Advantages of Wide Flange Beams (ISMB) over Narrow Flange (ISLB):

Higher moment of inertia about strong axis.
Better lateral stability.
Efficient for bending.
Easier connection to flanges.

2. Connections

2.1 Bolted Connections

Bolt Grades (ISO):

4.6: $$\displaystyle f_u = 400 $$ MPa, $$\displaystyle f_y = 0.6 \times f_u = 240 $$ MPa.
4.8: $$\displaystyle f_u = 400 $$ MPa, $$\displaystyle f_y = 320 $$ MPa.
8.8: $$\displaystyle f_u = 800 $$ MPa, $$\displaystyle f_y = 640 $$ MPa.
Black Bolts: Non-finished, used in bearing type.
High-Strength Bolts (HSFG): Precision made, used in friction grip.

Types of Joints:

Lap Joint: Single/double, plates overlap.
Butt Joint: Single cover, double cover (cover plate on one/both sides).
Tee Joint: Plates perpendicular.
Seat Connection: Beam seated on angle/plate.
Framed Connection: Cleat angles connect beam web to column flange.
End Plate Connection: Plate welded to beam end, bolted to column.

Failure Modes of Bolted Joints:

Failure Mode	Description
Shear Failure of Bolt	Bolt fails in shear across shank.
Bearing Failure	Crushing of plate around bolt hole.
Tensile Failure of Plate	Net section rupture.
Splitting Failure	Edge shear-out of plate material.
Block Shear Failure	Combination of shear along bolt line & tension on net section.

Design of Bearing-Type Connections (IS 800):

Bolt Shear Strength (Single Shear):

$$V_{sb} = \frac{f_u}{\gamma_{mb}} \cdot A_{sb} \quad \text{or} \quad V_{db} = \frac{f_u}{\gamma_{mb}} \cdot n \cdot A_{nb}$$

where $$\displaystyle A_{sb} = \frac{\pi}{4}d^2 $$, $$\displaystyle A_{nb} = \frac{\pi}{4}(d - 0.5)^2 $$ (net area at threads).

Bearing Strength:

$$B_{db} = \frac{2.5 k_b d t f_u}{\gamma_{mb}}$$

$$\displaystyle k_b = \min\left(1, \frac{e}{3d_0}, \frac{p}{3d_0} - 0.25\right) $$, $$\displaystyle d_0 = d + 2 $$ mm.

Net Section Tension:

$$T_{dn} = \frac{0.9 f_u A_{nc}}{\gamma_{m0}}$$

Pitch & Edge Distance: $p \geq 2.5d$, $e \geq 1.7d$ (min), $$\displaystyle p_{min} = 2.7d $$ for multiple bolts.

High-Strength Bolts (HSFG):

Friction Grip: Pre-tensioned, load transferred by friction. Slip-resistant.
Bearing Type: Same as black bolts after slip.
Slip Factor: Depends on surface treatment (grade 8.8 bolts, Class A surface).

Efficiency of Riveted/Bolted Joint:

$$\eta = \frac{\text{Strength of joint}}{\text{Strength of un-pierced plate}} \times 100\%$$

Lug Angles:

Purpose: Provide additional bolts when connecting single angle tension member to gusset.
Design: Lug angle size ≥ main angle, bolts in two rows.
Sketch:
DiagramCANVAS: Lug angle attached to main angle and gusset plate, bolts in two rows along lug
Applications: Heavy tension members, limited gusset space.

2.2 Welded Connections

Types:

Fillet Weld: Triangular, most common. Size = leg length ($s$). Throat thickness $$\displaystyle t = 0.7s $$.
Groove Welds: Butt, corner, edge, V/U/J-groove. For full strength.

Shop vs Site Welding:

Shop: Controlled environment, better quality.
Site: Weather dependent, inspection difficult.

Design of Fillet Weld (IS 800):

Effective Throat: $$\displaystyle t_e = 0.7 \times \text{leg length} $$ (minimum).
Design Strength per unit length:

$$f_{wd} = \frac{f_u}{\sqrt{3} \gamma_{mw}}$$

Total Strength: $$\displaystyle R_t = f_{wd} \times l_{eff} $$
Effective Length: $$\displaystyle l_{eff} = \text{length} - 2 \times \text{end return} $$ (end return ≤ 2 × size).
Size Limitations: $s \leq \text{thinner plate thickness}$, $$\displaystyle s_{min} = 3 $$ mm, $$\displaystyle s_{max} = \text{min}(t, t' - 2 $$ mm).

Groove Weld Design: Full penetration weld strength = base metal strength.

Welding Symbols (AWS/IS):

DiagramSEARCH: welding symbols arrow side other side

2.3 Riveted Connections (Historical)

Types: snap head, countersunk.
Efficiency: $$\displaystyle \eta = \left( \frac{n-0.6}{n} \right) \times 100\% $$ for single riveted lap joint.
Largely replaced by bolts.

2.4 Pin Connections

Types: Pin joint, hinge, clevis.
Applications: Trusses, bridges, machinery (allow rotation).
Design: Check bearing, shear, bending on pin.

2.5 Design Steps for Connections

Bolted:

Determine factored load.
Select bolt grade & diameter.
Check shear, bearing, net section.
Provide pitch, edge distance.
Check grip length, number of bolts.

Welded:

Determine factored load.
Select weld type & size.
Calculate effective throat & length.
Check weld strength.
Provide end returns, weld size ≤ plate thickness.

3. Tension Members

3.1 Single Angle Sections

Net Area: $$\displaystyle A_{nc} = A_g - n \cdot d \cdot t $$ (for holes in one leg).
Shear Lag Effect: Effective net area $$\displaystyle A_{ne} = \beta \cdot A_{nc} $$, $$\displaystyle \beta = 1 - \frac{\bar{x}}{l_v} \geq 0.7 $$ for single angle.
Block Shear Failure:

$$T_{db} = \frac{1}{\gamma_{m0}} \left( \frac{A_{tn} f_u}{\sqrt{3}} + \frac{A_{tv} f_y}{\gamma_{m0}} \right) \quad \text{or} \quad \frac{A_{tn} f_u}{\gamma_{m0}} + 0.9 \frac{A_{tv} f_y}{\gamma_{m0}}$$

Connection: Welded to gusset (fillet weld on two sides) or bolted (lug angles often needed).

3.2 Double Angle Sections

Back-to-back: Long legs connected (common), short legs connected (rare).
Tack Bolts: At close intervals (≤ 600 mm) to hold angles together.
Net Area: $$\displaystyle A_{nc} = 2(A_g - n \cdot d \cdot t) $$ (if holes in both angles).
Connection: Gusset plate on one side (angles on same side) or both sides.

3.3 Channel & I-Section Tension Members

Net area: subtract hole area from web/flange as applicable.
Block shear possible if holes in flange and web.

3.4 Design Strength Calculation

Yielding: $$\displaystyle T_{dg} = \frac{f_y A_g}{\gamma_{m0}} $$
Rupture: $$\displaystyle T_{dn} = \frac{0.9 f_u A_{nc}}{\gamma_{m0}} $$
Block Shear: $$\displaystyle T_{db} $$ as above.
Slenderness Limit: $$\displaystyle \lambda = \frac{l_{eff}}{r_{min}} \leq 400 $$ (for truss members).

3.5 Lug Angles in Tension Members

Purpose: Increase bolt row capacity, reduce eccentricity.
Design: Lug angle ≥ main angle, bolts in two rows, check strength of lug angle.
Sketch:
DiagramCANVAS: Tension member with lug angle bolted to gusset, two rows of bolts

4. Compression Members

4.1 Single Sections

Design Strength: $$\displaystyle P_{d} = \frac{A_g f_y}{\gamma_{m0}} \cdot \frac{1}{\phi + \sqrt{\phi^2 - \lambda^2}} $$ (inelastic) or Euler for long columns.
Slenderness Ratio: $$\displaystyle \lambda = \frac{l_e}{r} $$, $$\displaystyle l_e = \text{effective length} $$.
Buckling Curves (IS 800):
- a: I-sections (major axis)
- b: I-sections (minor axis), channels
- c: Angles, tees
- d: Hollow sections
Imperfection Factor $\alpha$ from curve.
Euler Load: $$\displaystyle P_{cr} = \frac{\pi^2 E I}{(l_e)^2} $$

4.2 Double Angle Sections

Back-to-back: Long legs connected, tack bolts @ ≤ 600 mm.
Radius of Gyration: $$\displaystyle r = \sqrt{\frac{I}{2A_g}} $$ (about axis perpendicular to connecting legs).
Effective Length: Depends on end conditions & lacing.

4.3 Laced Columns

Single Lacing: Lacing bars in one plane, angle sections usually.
Double Lacing: Two planes, better stability.
Inclination of Lacing: 60°–70° (optimal).
Design of Lacing Bars:
- Compressive strength: $$\displaystyle P_{ld} = \frac{A_{lc} f_y}{\gamma_{m0}} \cdot \frac{1}{\phi + \sqrt{\phi^2 - \lambda_{lc}^2}} $$
- Tensile strength: $$\displaystyle T_{ld} = \frac{0.9 f_u A_{ln}}{\gamma_{m0}} $$
Connections: Bolted or welded to chords.
Design Steps:
1. Choose section for axial load.
2. Check slenderness of lacing bars.
3. Design lacing bar cross-section.
4. Design connections (bolts/welds).
5. Check shear in chords at lacing points.

4.4 Battened Columns

Batten Plates: Flat plates connecting chords.
Design: Batten subjected to shear & bending. Thickness ≥ 1/50 of distance between centroids.
Comparison: Laced columns more economical for long spans; battened for shorter.

4.5 Column Splices

Bearing Splices: Flanges in bearing, web bolted/welded.
Splice Plates: Welded/bolted to flanges & web.
Design: Transfer axial load & moment (if any).

4.6 Eccentric Loading on Columns

Combined Axial Load & Bending: Check interaction.

$$\frac{P_u}{P_{dz}} + \frac{M_u}{M_{dz}} \leq 1.0 \quad \text{(for yielding)}$$

Design: Increase section to resist additional moment.

4.7 Failure Modes of Columns

Elastic Buckling (Euler): Long, slender.
Inelastic Buckling: Intermediate slenderness.
Material Failure (yielding/rupture): Stocky sections.

Plastic/Compact Sections: Better moment redistribution, higher ductility.

5. Beams

5.1 Laterally Restrained vs. Unrestrained Beams

Laterally Restrained: Compression flange supported against lateral-torsional buckling (by slab, bracing, etc.). Design based on moment capacity only.
Laterally Unrestrained: Must check lateral-torsional buckling. Moment capacity reduced.
Conditions for Restraint:
- Concrete slab on top flange (composite action).
- Lateral bracing at close intervals.
- Deep beams with high $$\displaystyle d/t_w $$.

5.2 Design of Laterally Supported Beams

Select Section based on $$\displaystyle M_u $$ & $$\displaystyle V_u $$.
Check Moment Capacity: $$\displaystyle M_{dz} = \beta \cdot Z_p \cdot f_y / \gamma_{m0} $$ (plastic) or $$\displaystyle Z_e \cdot f_y / \gamma_{m0} $$ (elastic).
Check Shear Capacity: $$\displaystyle V_{dz} = \frac{A_w f_y}{\sqrt{3} \gamma_{m0}} $$
Deflection Check (if required): $$\displaystyle \delta_{max} \leq L/250 $$ (IS 800).
Check Web Buckling: If $$\displaystyle d/t_w > 67 \varepsilon $$ for simply supported.

5.3 Plate Girders

Components: Web, flanges, stiffeners (transverse, longitudinal).
Design:
- Web: Check shear buckling ($$\displaystyle V_{dw} = \frac{d t_w f_y}{\sqrt{3} \gamma_{m0}} $$), provide stiffeners if needed.
- Flanges: Welded to web, check moment capacity.
- Stiffeners:
  - Transverse: At supports, concentrated loads, every 0.5d–0.75d.
  - Longitudinal: If web depth > 1.5 times depth of web without stiffeners.
Design Steps:
1. Determine loads & moments.
2. Preliminary depth & flange width.
3. Check web shear & buckling.
4. Design flanges.
5. Design stiffeners.
6. Check deflection.

5.4 Beam Connections

Framed Connection: Cleat angles, transfer shear.
Seat Connection: Seated angle, supports beam bottom.
End Plate Connection: Plate welded to beam end, bolted to column.
Design: For shear & moment (if any).

6. Purlins & Gantry Girders

6.1 Purlins

Types: Z, C, I-sections (hot-rolled or built-up).
Design Loads (on sloping roof):
- Dead Load (sheeting, purlin weight).
- Live Load (maintenance).
- Wind Load (suction/pressure, perpendicular to slope).
Spacing: Truss spacing (main) & purlin spacing (secondary).
Slope Effect: UDL on slope → normal to purlin = $w \cdot \cos\theta$.
Design Procedure:
1. Calculate characteristic loads.
2. Apply load combinations.
3. Determine max $$\displaystyle M_u $$, $$\displaystyle V_u $$.
4. Select I-section (Z/C also possible).
5. Check moment, shear, deflection ($L/200$ for purlins).
6. Check web buckling if needed.

6.2 Gantry Girders

Profile: I-section with wider flange or built-up section.
Crane Loads:
- Vertical: Wheel load × impact factor (1.25–1.5).
- Horizontal: Braking/traction force.
Deflection Limits (IS 800):
- Vertical: $L/500$ (for crane).
- Horizontal: $L/500$.
Design Considerations: Fatigue, lateral stability, wheel load positioning.

7. Bases & Foundations

7.1 Slab Base (Ungrouted)

Base Plate: Thick plate welded to column base.
Design:
- Bearing Pressure: $$\displaystyle p = \frac{P}{A_{bp}} \leq f_{bc} $$ (allowable bearing on concrete).
- Base Plate Thickness: $$\displaystyle t_{bp} = \sqrt{\frac{3P}{f_y \cdot \text{perimeter of column}}} $$ (for cantilever model).
Weld: Column to base plate (full penetration).

7.2 Gusseted Base

Components: Base plate, gusset plates, angles.
Design: For axial load with moments (eccentric loading).
Concrete Pedestal: Size based on bearing pressure.

7.3 Grillage Foundation

Used for: Heavy loads, poor soil.
Design with I-sections: Grillages (layers of I-beams) at top & bottom.
Steps:
1. Calculate load & soil bearing capacity.
2. Determine base plate size.
3. Design grillage beams (I-sections) for bending.
4. Check bearing on soil & concrete.
Bearing Capacity of Soil: $$\displaystyle q_{allow} = \frac{q_{ult}}{FOS} $$.

7.4 Types of Base Plates

Slab base, gusseted base, grillage base, pocket base.

8. Roof Trusses

8.1 Types of Roof Trusses

King Post: Central vertical, for short spans.
Queen Post: Two verticals, medium spans.
Fink: For pitched roofs, common in residential.
Howe/Pratt: Diagonal members in tension/compression.
Scissor: For vaulted ceilings.
Selection: Span, roofing material, aesthetics.

8.2 Components (10 with Sketch)

Top Chord: Compressive, inclined.
Bottom Chord: Tensile, horizontal.
Web Members: Diagonals/verticals.
King Post: Central vertical in king post truss.
Queen Post: Pair of verticals.
Strut: Compression web.
Tie: Tension web.
Purlin: Secondary member on top chord.
Rafter: Main sloping beam (sometimes part of truss).
Bracing: Diagonal in plane (wind bracing).
Gusset Plate: Connects members at joints.
Sag Rod: Vertical tension rod at bottom chord.

8.3 Joints in Trusses

Welded Joints: Shop, for simplicity.
Bolted Joints: Field, with gusset plates.
Gusset Plate Design: Check shear, bearing, block shear.

8.4 Design of Roof Trusses

Load Calculation: Dead (self-weight, roofing), live, wind.
Geometry: Span, pitch, panel length.
Analysis: Method of joints/sections.
Member Design: As tension/compression members.
Connection Design: Bolted/welded to gusset.

9. Miscellaneous Theoretical Topics

9.1 Block Shear Failure

Mechanism: Rupture along a path combining shear (along bolt line) and tension (perpendicular).
Equation (IS 800):

Occurs in: Tension members with holes, gusset connections.
Diagram:
DiagramCANVAS: Block shear path: tension area A_tn, shear area A_tv along bolt line

9.2 Shear Lag

Concept: Non-uniform stress distribution in tension members with connections.
Effect: Reduced effective net area $$\displaystyle A_{ne} = \beta A_{nc} $$.
Factors: Connection length, width of flange, number of bolts.
$\beta$ for I-section: $$\displaystyle \beta = 1 - \frac{\bar{x}}{l_v} \geq 0.7 $$ (for flanges).
Diagram:
DiagramCANVAS: Shear lag in I-section: stress concentration near connection, uniform away

9.3 Advantages of Steel over Cast Iron & Wrought Iron

Higher strength, ductility, weldability.
Uniform quality, predictable behavior.
Recyclable, faster construction.

9.4 Advantages of Bolted Connections

Reusable, no skilled labor like welding.
No fire hazard, no distortion.
Easy inspection, suitable for field.

9.5 Efficiency of Joints

$$\displaystyle \eta = \frac{\text{Strength of joint}}{\text{Strength of un-pierced plate}} \times 100\% $$.
Riveted: $$\displaystyle \eta = \left( \frac{n-0.6}{n} \right) \times 100\% $$ (lap).
Bolted: Similar, but higher due to less hole damage.

9.6 Pin Connections

Types: Clevis, turnbuckle, hinge pin.
Applications: Truss joints, bridge articulation, machinery linkages.
Design Checks: Bearing, shear, bending on pin.

10. Design Steps & Procedures (Summary)

Tension Member:

Determine $$\displaystyle T_u $$.
Select section (angle, I, etc.).
Check yielding, rupture, block shear.
Design connection (bolted/welded).
Check slenderness ($\lambda \leq 400$).

Compression Member (Single):

Determine $$\displaystyle P_u $$, $$\displaystyle l_e $$.
Select section, calculate $$\displaystyle r_{min} $$.
$$\displaystyle \lambda = l_e / r_{min} $$, get buckling curve.
Compute $$\displaystyle P_{dz} $$ (yielding/buckling).
Check $$\displaystyle P_u \leq P_{dz} $$.

Laced Column:

Design main chords for axial load.
Check chord slenderness.
Design lacing bars (compression/tension).
Design lacing connections.
Check shear in chords at lacing points.

Laterally Supported Beam:

Determine $$\displaystyle M_u $$, $$\displaystyle V_u $$.
Select section (ISMB/ISLB).
Check $$\displaystyle M_{dz} \geq M_u $$, $$\displaystyle V_{dz} \geq V_u $$.
Check deflection if required.
Check web buckling if $$\displaystyle d/t_w > 67\varepsilon $$.

Plate Girder:

Preliminary depth & flange.
Check web shear & buckling → stiffeners.
Design flanges for moment.
Design stiffeners (transverse/longitudinal).
Check deflection, fatigue.

Bolted Connection (Bearing Type):

Determine $$\displaystyle F_u $$.
Select bolt grade & diameter.
Check bolt shear: $$\displaystyle V_{sb} \geq F_u / \text{no. of bolts} $$.
Check bearing: $$\displaystyle B_{db} \geq F_u / (\text{no. of bolts} \times \text{thickness}) $$?
Check net section of plate.
Provide $p \geq 2.5d$, $e \geq 1.7d$.

Welded Fillet Connection:

Determine $$\displaystyle F_u $$.
Choose weld size $s$ (≤ thinner plate).
Calculate $$\displaystyle l_{eff} $$ (minus end returns).
Check $$\displaystyle R_t = f_{wd} \times l_{eff} \geq F_u $$.
Provide end returns (≤ 2s).

Slab Base:

Determine $$\displaystyle P_u $$.
Assume base plate size, check $$\displaystyle p = P_u / A_{bp} \leq f_{bc} $$.
Calculate $$\displaystyle t_{bp} = \sqrt{\frac{3P}{f_y \cdot \text{perimeter}}} $$.
Design weld to column.

Grillage Foundation:

$$\displaystyle P_u $$, soil $$\displaystyle q_{allow} $$ → base area $$\displaystyle A = P_u / q_{allow} $$.
Base plate size based on column dimensions.
Design grillage beams (I-sections) for bending.
Check bearing on soil & concrete.

Purlin:

Calculate loads (dead, live, wind) on slope.
UDL on purlin = load × spacing × $\cos\theta$.
Max $$\displaystyle M_u = wL^2/8 $$, $$\displaystyle V_u = wL/2 $$.
Select I-section, check $$\displaystyle M_{dz} $$, $$\displaystyle V_{dz} $$, deflection.
Check web buckling if needed.

Roof Truss:

Loads & geometry.
Analysis (method of joints/sections).
Design chords & webs (tension/compression).
Design gusset & connections.
Design purlins & bracing.

[!TIP] Common Pitfalls:

Forgetting shear lag in tension members (especially single angles).

Incorrect net area calculation (hole diameter = bolt dia + 2 mm).

Neglecting end returns in weld length.

Wrong effective length for laced column chords.

Not checking deflection for purlins/gantry girders.

Confusing factored vs. working loads in WSD/LSD.