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Structural Engineering
Formula Reference

A comprehensive collection of essential equations for structural analysis and design. Covering beams, columns, slabs, foundations, and steel connections based on IS codes.

πŸŒ‰ Beams & Flexure
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Simple Bending Equation
Οƒ = (M Γ— y) / I
Relationship between bending stress, moment, and section geometry.

Οƒ = Bending stress

M = Bending moment

y = Distance from neutral axis

I = Moment of Inertia

Transverse Shear Stress
Ο„ = (V Γ— Q) / (I Γ— b)
Shear stress distribution at a point in the cross-section.

V = Shear force

Q = First moment of area above point

I = Moment of Inertia

b = Width of section

Max Deflection (SS Beam, UDL)
Ξ΄ = (5 Γ— w Γ— L⁴) / (384 Γ— E Γ— I)
Maximum deflection at mid-span for simply supported beam with Uniformly Distributed Load.

w = Load per unit length

L = Span length

E = Modulus of Elasticity

I = Moment of Inertia

Deflection (Cantilever, Point Load)
Ξ΄ = (P Γ— LΒ³) / (3 Γ— E Γ— I)
Deflection at the free end of a cantilever with a point load P.

P = Point load

L = Length

E, I = Material/Section props

Angle of rotation ΞΈ = PLΒ²/(2EI)
Moment of Inertia (Rectangular)
I = (b Γ— dΒ³) / 12
Moment of inertia for a solid rectangular section about its centroidal axis.

b = Width

d = Depth

πŸ›οΈ Columns & Compression
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Euler's Critical Load
Pcr = (π² Γ— E Γ— I) / (K Γ— L)Β²
Theoretical buckling load for long slender columns.

E = Modulus of Elasticity

I = Min Moment of Inertia

K = Effective length factor

L = Actual length

For pinned-pinned K=1.0; Fixed-fixed K=0.5
Slenderness Ratio
Ξ» = Le / r
Measure of column slenderness. Le = Effective length, r = Radius of gyration.

r = √(I/A)

A = Cross-sectional area

Limit for steel columns: Ξ» ≀ 180 (compression), ≀ 250 (wind/seismic reversal)
Rankine's Empirical Formula
PR = (fc Γ— A) / (1 + Ξ± Γ— (L/r)Β²)
Combines crushing and buckling failure modes. Useful for intermediate columns.

fc = Crushing stress

Ξ± = Rankine constant (material dependent)

Secant Formula (Eccentric Load)
Οƒmax = (P/A) [1 + (eΓ—c/rΒ²) sec( (L/2r)√(P/EA) )]
Stress in columns with eccentric loading. Accounts for additional moment due to deflection.

e = Eccentricity

c = Distance to extreme fiber

🟫 Slabs & Two-Way Action
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Yield Line Method (Square Slab)
w Γ— LΒ² = 8 Γ— mp
Ultimate load capacity for a square slab with fixed or simply supported edges assuming yield lines form diagonals.

w = Ultimate load per area

mp = Plastic moment capacity per unit width

Short Span Moment (IS 456)
Mx = Ξ±x Γ— w Γ— LxΒ²
Coefficient method for two-way slabs. Ξ± values depend on edge conditions and Ly/Lx ratio.

Ξ±x = Short span coefficient (Table 26 IS 456)

Lx = Short span length

Long Span Moment
My = Ξ±y Γ— w Γ— LxΒ²
Moment along the longer direction using coefficient Ξ±y.

Ξ±y = Long span coefficient

Punching Shear Stress
Ο„v = Vu / (b0 Γ— d)
Shear stress around a column supporting a flat slab.

Vu = Factored shear force

b0 = Perimeter of critical section (typically d/2 from column face)

d = Effective depth

βš™οΈ Steel Connections (IS 800)
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Shear Strength of Bolt
Vdsb = (fub Γ— Anb) / (√3 Γ— Ξ³mb)
Design shear strength of a high-strength bolt in bearing type connection.

fub = Ultimate tensile stress of bolt

Anb = Net tensile area

Ξ³mb = Partial safety factor (1.25)

Bearing Strength of Bolt
Vdpb = 2.5 Γ— kb Γ— d Γ— t Γ— fu / Ξ³mb
Capacity limited by bearing of plate against bolt hole.

kb = Lesser of e/(3dβ‚€), p/(3dβ‚€)-0.25, fu/fub, 1.0

d = Nominal diameter

t = Thickness of connected plate

Weld Strength (Fillet)
Rw = lw Γ— tt Γ— (fu / (√3 Γ— Ξ³mw))
Strength of a fillet weld per unit length.

lw = Effective length of weld

tt = Throat thickness (= 0.7 Γ— weld size)

Ξ³mw = Safety factor (1.25 shop, 1.5 field)

Block Shear Strength
Tdb = min[ (AvgΓ—fy)/(√3Γ—Ξ³m0) + (0.9Γ—AtnΓ—fu)/Ξ³mb, ... ]
Failure mode involving tension rupture and shear yielding along bolt lines.

Avg = Gross shear area

Atn = Net tension area

πŸ—οΈ Foundations
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Terzaghi's Safe Bearing Capacity
qult = cNc + Ξ³DNq + 0.5Ξ³BNΞ³
Ultimate bearing capacity for shallow strip footing.

c = Cohesion

Nc,q,Ξ³ = Bearing capacity factors

D = Depth

B = Width

Immediate Settlement
Si = q Γ— B Γ— (1-Ξ½Β²) / E Γ— If
Elastic settlement calculation for flexible footings.

q = Contact pressure

If = Influence factor

E = Young's modulus

Passive Earth Pressure
Pp = Kp Γ— Οƒv + 2c√Kp
Pressure exerted by soil resisting movement of a retaining structure.

Kp = (1+sinφ)/(1-sinφ)

Οƒv = Vertical effective stress