Civil Engineering Formulas — Printable Sheet: RCC, Steel, Geotech

27 key structural, geotechnical and loading formulas across 5 categories — typeset and printable.

RCC (IS 456)

Limiting moment (Fe415)

M_{u, l im} = 0.138 f_{c k} b d^{2}

Singly-reinforced section limit

Limiting moment (Fe500)

M_{u, l im} = 0.133 f_{c k} b d^{2}

Lower x_u/d limit than Fe415

Moment of resistance

M_{u} = 0.87 f_{y} A_{s t} (d - 0.42 x_{u})

Lever-arm method

Neutral axis depth

x_{u} = 0.87 f_{y} A_{s t} / (0.36 f_{c k} b)

From force equilibrium

Development length

L_{d} = ϕ σ_{s} / (4 τ_{b d})

Cl. 26.2.1, σ_s = 0.87 f_y; τ_bd per Cl. 26.2.1.1

Modulus of elasticity

E_{c} = 5000 f_{c k}

Short-term static modulus (MPa)

Nominal shear stress

τ_{v} = V_{u} / (b d)

Compare with τ_c

Min flexural steel

A_{s, min} = 0.85 b d / f_{y}

Cl. 26.5.1.1

Steel (IS 800)

Plastic moment

M_{p} = Z_{p} f_{y} / γ_{m 0}

γ_m0 = 1.10

Tension yield

T_{d g} = A_{g} f_{y} / γ_{m 0}

Gross-section yielding

Slenderness ratio

λ = K L / r

K = effective length factor

Euler buckling load

P_{cr} = π^{2} E I / (K L)^{2}

Elastic critical load

Bolt shear capacity

V_{d s b} = f_{u b} A_{nb} / (3 γ_{mb})

Single shear plane, γ_mb = 1.25

Geotechnical

Terzaghi bearing

q_{u} = c N_{c} + γ D_{f} N_{q} + 0.5 γ B N_{γ}

Strip footing, general shear

Active earth pressure

K_{a} = (1 - s in ϕ) / (1 + s in ϕ)

Rankine, level backfill

Passive earth pressure

K_{p} = (1 + s in ϕ) / (1 - s in ϕ)

Rankine, level backfill

Elastic settlement

S = q B (1 - μ^{2}) I / E

Immediate settlement

Loads

Design wind speed

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

IS 875-3:2015 Cl 6.3 — risk, terrain, topography, cyclonic importance (k_4) factors

Wind pressure

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

Pa, V_z in m/s

Seismic base shear

V_{b} = A_{h} W

IS 1893, W = seismic weight

Seismic coefficient

A_{h} = (Z I S_{a}) / (2 R g)

Design horizontal coefficient

Factored load (LSD)

W_{u} = 1.5 (D L + LL)

Basic gravity combination

Section properties

Rectangle inertia

I = b h^{3} /12

About centroidal axis

Section modulus

Z = I / y

y = extreme-fibre distance

Radius of gyration

r = I / A

Used in slenderness

Circle inertia

I = π D^{4} /64

Solid circular section

Parallel axis

I = I_{o} + A d^{2}

d = axis offset

Code Clause Compass

Read the clause behind each formula — plain-English paraphrases of IS, ACI and Eurocode.

Glossary

Look up the symbols and terms used above — A_st, τ_bd, x_u and more.

Educational quick-reference — always confirm against the governing code clause for your project.

How the Structural Formula Sheet works

This is a one-page reference of the 27 formulas structural engineers look up most often, grouped into five categories: RCC to IS 456 (8 formulas), steel to IS 800 (5), geotechnical (4), loads (5) and section properties (5). Each card shows the formula typeset in standard notation with a short note on where it applies, such as the governing clause or the value of a partial safety factor.

Use the category chips to show one group at a time, or leave it on All to see the whole sheet. The Download PDF button saves every formula, in every category, as a plain-text A4 sheet you can print and keep at the desk or on site.

The sheet does not calculate anything. It gives the expression and the symbols; you supply the values and do the arithmetic, or use the matching calculator on this site. Tabulated inputs that some formulas need, such as τbd, τc or Sa/g, are not reproduced and must be taken from the code.

Limiting moment, Fe415: Mu,lim = 0.138·fck·b·d²
Limiting moment, Fe500: Mu,lim = 0.133·fck·b·d²
Neutral axis depth: xu = 0.87·fy·Ast / (0.36·fck·b)
Development length: Ld = φ·σs / (4·τbd), σs = 0.87·fy
Modulus of elasticity of concrete: Ec = 5000·√fck
Euler buckling load: Pcr = π²·E·I / (K·L)²
Rankine active earth pressure: Ka = (1 − sin φ) / (1 + sin φ)
Wind pressure: pz = 0.6·Vz²

Follows: IS 456:2000 Cl 26.2.1, 26.5.1.1 · IS 800:2007 · IS 875 (Part 3):2015 Cl 6.3 · IS 1893

Worked example: limiting moment of resistance of a singly reinforced beam

Given

Formula from the RCC group: Mu,lim = 0.138·fck·b·d² (Fe415)
Concrete M25, so fck = 25 N/mm²
Beam width b = 300 mm, effective depth d = 450 mm

Steps

d² = 450² = 202,500 mm².
b·d² = 300 × 202,500 = 60,750,000 mm³.
Mu,lim = 0.138 × 25 × 60,750,000 = 209,587,500 N·mm.
Divide by 10⁶ to convert to kN·m: 209,587,500 / 1,000,000 = 209.6 kN·m.
If the factored moment Mu is below 209.6 kN·m, the section can be designed as singly reinforced; above it, increase the depth or provide compression steel.

Result: Mu,lim = 209.6 kN·m for a 300 × 450 mm (effective) M25 / Fe415 beam.

Enter the same values in the calculator above to check each step.

Frequently asked questions

What is the formula for limiting moment of resistance as per IS 456?

For a singly reinforced rectangular section, Mu,lim = 0.138·fck·b·d² for Fe415 and 0.133·fck·b·d² for Fe500. The Fe500 coefficient is lower because its limiting neutral axis depth ratio is smaller. For a 300 mm wide, 450 mm effective depth M25 beam with Fe500, Mu,lim = 0.133 × 25 × 300 × 450² = 202.0 kN·m.

How to calculate modulus of elasticity of concrete as per IS 456?

Use the short-term static modulus Ec = 5000·√fck, with fck in N/mm². M25 concrete gives 5000 × √25 = 25,000 N/mm² and M30 gives 5000 × √30 = 27,386 N/mm².

How to calculate development length of a bar as per IS 456?

Ld = φ·σs / (4·τbd), with σs = 0.87·fy. For a 16 mm Fe415 bar in tension in M20 concrete, τbd = 1.2 N/mm² increased by 60% for deformed bars, so 1.92 N/mm², and Ld = 16 × 0.87 × 415 / (4 × 1.92) = 752 mm, about 47 bar diameters.

What is the active earth pressure coefficient for a soil with φ = 30°?

Rankine's Ka = (1 − sin φ)/(1 + sin φ). With sin 30° = 0.5, Ka = 0.5/1.5 = 0.333, and the passive coefficient Kp = 1.5/0.5 = 3. These apply to a vertical wall with level backfill and no wall friction.

Does the formula sheet calculate values or include the code tables?

No. It lists 27 formulas with notes and does no arithmetic. Table values that some formulas depend on, such as τbd, τc, Sa/g or the wind factors k1 to k4, are not included, so take them from the relevant IS code and confirm each formula against the governing clause.

Formula Sheet