GATE Civil Formula Sheet 2027: Key Formulas by Subject

Q: Which IS 456 values should I memorise for GATE CE?

The partial safety factors γₛ = 1.15 on steel and γc = 1.5 on concrete, giving 0.87 fy and 0.446 fck; the stress block 0.36 fck b xu acting at 0.42 xu; the limiting ratios xu,max/d of 0.53, 0.48 and 0.46 for Fe250, Fe415 and Fe500; and the limiting concrete strain of 0.0035 in bending.

By MD ANISH AHAMADUpdated 4 Oct 20269 min read

GATE Civil Formula Sheet 2027: Key Formulas by Subject

This GATE Civil formula sheet gathers the formulas and code constants that worked GATE CE questions lean on most, section by section, with the trap that usually goes with each. It is a free selection from the last-minute sheet of the GATE CE 2027 book. Use it to test your recall, then practise each formula on past questions.

In this guide

Key takeaways
How to read this sheet
Engineering Mathematics
Structural Engineering
Geotechnical Engineering
Water Resources Engineering
Environmental Engineering
Transportation Engineering
Geomatics Engineering
Construction Materials and Management
General Aptitude essentials
The four slips made under time pressure
How to revise with this sheet
Subject-wise formula sheets
Quick revision

Key takeaways

Every technical section of the 2027 CE syllabus has a small core of formulas that carries most of its numerical questions.
A formula is only safe with its condition: which logarithm, which unit weight, which length or diameter.
A material partial safety factor always divides strength, never multiplies it.
$0.87 f_y$ belongs to IS 456:2000 for reinforcement, not to IS 800:2007 for steel sections.
You cannot carry a sheet into the hall, so every formula here has to be in your memory (confirm current rules at gate2027.iitm.ac.in).

How to read this sheet

A few symbols repeat across sections, so learn them once. $f_{ck}$ is the characteristic compressive strength of concrete and $f_y$ the characteristic yield strength of steel, both in N/mm². $\gamma_w$ is the unit weight of water. A partial safety factor is the number a code divides a material strength by to get a design strength.

"log" means base 10 and "ln" means natural log. Code values come from IS 456:2000 (concrete), IS 800:2007 (steel) and the IRC guidelines, and only values checked against a worked solution in the book appear here.

The sections run in the order of the syllabus. To see how many marks each one has carried across all sixteen papers from 2019 to 2026, read GATE CE subject-wise weightage, both sets counted.

Engineering Mathematics

Eigenvalues: $\sum \lambda = \text{trace}$ and $\prod \lambda = \det A$ ; if $\lambda$ is an eigenvalue of $A$ , then $p(\lambda)$ is an eigenvalue of $p(A)$
Infinitely many solutions: $\text{rank}(A) = \text{rank}(A \mid b) < n$ , the number of unknowns
$(AB)^T = B^T A^T$ and $(AB)^{-1} = B^{-1} A^{-1}$
Linear first-order ODE $y' + P y = Q$ : integrating factor $e^{\int P\,dx}$
Newton–Raphson: $\displaystyle x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$ ; explicit Euler: $y_{n+1} = y_n + h\,f(x_n, y_n)$
Simpson's 1/3 rule weights $1, 4, 2, 4, \dots, 4, 1$ , with an even number of sub-intervals; trapezoidal weights $1, 2, \dots, 2, 1$
PDE $A u_{xx} + B u_{xy} + C u_{yy} + \dots = 0$ : elliptic if $B^2 - 4AC < 0$ , parabolic if $= 0$ , hyperbolic if $> 0$
Poisson: $\displaystyle P(X = k) = \dfrac{e^{-\lambda} \lambda^k}{k!}$ , and $P(X \ge 1) = 1 - e^{-\lambda}$
Bayes: $\displaystyle P(A \mid D) = \dfrac{P(D \mid A)\,P(A)}{P(D)}$ , where $P(D)$ is summed over every source

Structural Engineering

Formula	Watch out for
$\displaystyle \delta = \dfrac{PL}{AE}$ ; $\displaystyle \sigma = \dfrac{M}{Z}$ with $\displaystyle Z = \dfrac{bd^2}{6}$ for a rectangle	Use the axis about which the section bends
$\displaystyle \dfrac{T}{J} = \dfrac{\tau}{r}$ ; shear flow $\displaystyle q = \dfrac{VQ}{I}$	$Q$ is the first moment of the area beyond the cut
$\displaystyle P_{cr} = \dfrac{\pi^2 E I}{L_e^2}$	The least $I$ governs
No tension: $e \le Z/A$ , so $d/6$ for a rectangle and $D/8$ for a circle	Kern diameter of a solid circle is $D/4$
Plane frame: $D_s = 3m + r - 3j$	Subtract one per internal hinge joining two members
Cantilever tip deflection $\displaystyle \dfrac{PL^3}{3EI}$ ; fixed-end moment for a UDL $\displaystyle \dfrac{wL^2}{12}$	Simply supported and fixed values differ
Member stiffness $4EI/L$ (far end fixed), $3EI/L$ (far end hinged)	Check the far-end condition first
Three-hinged parabolic arch under full UDL: $\displaystyle H = \dfrac{wL^2}{8h}$	$h$ is the rise, not the span
Shape factor $Z_p/Z_e$ : rectangle 1.50, solid circle 1.70, diamond 2.00	I-section is only about 1.12 to 1.15

IS 456:2000 and IS 800:2007 values

Item	Value	Watch out for
Design stress in steel	$f_y/1.15 = 0.87 f_y$	IS 456 only
Design strength of concrete	$0.67 f_{ck}/1.5 = 0.446 f_{ck}$	Stress block force $0.36 f_{ck} b x_u$ at $0.42 x_u$
$x_{u,\max}/d$	Fe250 0.53, Fe415 0.48, Fe500 0.46	Higher grade, smaller ratio
$M_{u,\lim}$	$0.138 f_{ck} b d^2$ for Fe415	0.148 for Fe250, 0.133 for Fe500
Development length	$\displaystyle L_d = \dfrac{\phi\,\sigma_s}{4\,\tau_{bd}}$	$\tau_{bd}$ up 60 per cent for deformed bars
Shear by vertical stirrups	$\displaystyle V_{us} = \dfrac{0.87 f_y A_{sv} d}{s_v}$	$s_v$ is the stirrup spacing
IS 800 $\gamma_{m0}$ , $\gamma_{m1}$	1.10 yielding, 1.25 rupture	Gross section yields, net section ruptures
Welds	$\gamma_{mw}$ 1.25 shop, 1.50 site; throat $t_t = 0.7 \times \text{leg}$	$\displaystyle f_{wd} = \dfrac{f_u}{\sqrt{3}\,\gamma_{mw}}$

Trap: A partial safety factor divides a strength. If your design strength comes out larger than the characteristic strength, you multiplied.

Geotechnical Engineering

Formula	Watch out for
$Se = wG$ ; $\displaystyle \gamma_d = \dfrac{\gamma}{1 + w}$	$w$ as a fraction, not a percentage
Zero air voids: $\displaystyle \gamma_d = \dfrac{G \gamma_w}{1 + wG}$	No compaction reaches this line
$\sigma' = \sigma - u$	Pore pressure measured from the water table
$\displaystyle i_{cr} = \dfrac{G - 1}{1 + e}$ ; flow net $\displaystyle q = kH\,\dfrac{N_f}{N_d}$	$N_f$ counts flow channels, not flow lines
$\displaystyle S_c = \dfrac{C_c H}{1 + e_0} \log_{10} \dfrac{\sigma'_f}{\sigma'_0}$	Stress taken at mid-depth of the layer
$\displaystyle t = \dfrac{T_v d^2}{c_v}$ ; $T_v = 0.197$ at 50 per cent	$d$ is the longest drainage path: half the layer if both faces drain
$\tau_f = c' + \sigma' \tan \phi'$ ; $\displaystyle K_a = \dfrac{1 - \sin \phi}{1 + \sin \phi}$ , $K_p = 1/K_a$	Active thrust $\displaystyle \tfrac{1}{2} K_a \gamma H^2$ acts at $H/3$
Boussinesq on the axis: $\displaystyle \Delta \sigma_z = \dfrac{3Q}{2 \pi z^2}$	Depth squared, not cubed
$\displaystyle q_u = c N_c + \gamma D_f N_q + \tfrac{1}{2} \gamma B N_\gamma$	For $\phi = 0$ : $N_c = 5.7$ , $N_q = 1$ , $N_\gamma = 0$
Pile in clay: $Q_u = \alpha c A_s + 9 c A_p$	Negative skin friction is a downward load

Water Resources Engineering

Formula	Watch out for
$\displaystyle h_f = \dfrac{f L V^2}{2 g D}$	$f$ is the Darcy factor
$\displaystyle y_c = \left(\dfrac{q^2}{g}\right)^{1/3}$	Depends only on $q$ , discharge per unit width
$\displaystyle \dfrac{y_2}{y_1} = \tfrac{1}{2}\left(\sqrt{1 + 8 Fr_1^2} - 1\right)$ ; $\displaystyle \Delta E = \dfrac{(y_2 - y_1)^3}{4 y_1 y_2}$	Use the Froude number upstream of the jump
Manning: $\displaystyle V = \tfrac{1}{n} R^{2/3} S^{1/2}$ , $R = A/P$	Best rectangle: $b = 2y$ , so $R = y/2$
Weir: $\displaystyle Q = \tfrac{2}{3} C_d L \sqrt{2g}\,H^{3/2}$	$H$ is head over the crest
Froude model: $Q_r = L_r^{5/2}$	Not $L_r^{3}$
Risk: $\displaystyle R = 1 - \left(1 - \tfrac{1}{T}\right)^n$	$n$ is the design life in years
Confined well: $\displaystyle Q = \dfrac{2 \pi k b (h_2 - h_1)}{\ln(r_2/r_1)}$ ; unconfined: $\displaystyle Q = \dfrac{\pi k (h_2^2 - h_1^2)}{\ln(r_2/r_1)}$	Natural log; only the ratio of radii matters
$\displaystyle \Delta = \dfrac{8.64 B}{D}$	$\Delta$ in m, $B$ in days, $D$ in ha per cumec

This page is a selection. The book's last-minute sheet carries the full tables for all eight technical sections. The book also works 274 questions in its subject chapters and 650 more in 10 full mock tests. It is part of the GATE CE 2027 book.

Environmental Engineering

Formula	Watch out for
$BOD_t = L_0\left(1 - e^{-kt}\right)$ ; remaining $L_t = L_0 e^{-kt}$	Check whether $k$ is base $e$ or base 10
$\displaystyle \text{SOR} = \dfrac{Q}{A_{\text{surface}}}$	Removal depends on surface area, not depth
Stokes: $\displaystyle v_s = \dfrac{g(\rho_s - \rho) d^2}{18 \mu}$	Diameter squared
$\displaystyle F/M = \dfrac{Q S_0}{V X}$ ; $\displaystyle \text{SVI} = \dfrac{\text{settled mL/L} \times 1000}{\text{MLSS in mg/L}}$	Keep the units as written
Geometric growth: $P_n = P_0 (1 + r)^n$	$r$ per decade if $n$ is in decades
Plume, ground level on the centreline: $\displaystyle C = \dfrac{Q}{\pi u \sigma_y \sigma_z} \exp\left(-\dfrac{H^2}{2 \sigma_z^2}\right)$	$H = h_s + \Delta h$ , the effective height

Hardness converts to CaCO₃ with equivalent weights, and chlorine demand is dose minus residual.

Transportation Engineering

Formula	Watch out for
$\displaystyle \text{SSD} = vt + \dfrac{v^2}{2gf}$	$v$ in m/s
$\displaystyle e + f = \dfrac{V^2}{127 R}$	$V$ in km/h; IRC maximum $e = 0.07$ in plain terrain
$\displaystyle L_s = \dfrac{0.0215 V^3}{C R}$ , $\displaystyle C = \dfrac{80}{75 + V}$	Same units as above
Summit curve, $L > S$ : $\displaystyle L = \dfrac{N S^2}{\left(\sqrt{2h_1} + \sqrt{2h_2}\right)^2}$	Denominator 4.4 for SSD, 9.6 for OSD
Cant: $\displaystyle e = \dfrac{G V^2}{127 R}$	Broad gauge maximum cant 165 mm
$q = k\,v_s$ ; Greenshields $\displaystyle q_{\max} = \dfrac{v_f k_j}{4}$	$v_s$ is space-mean speed, the harmonic mean
Webster: $\displaystyle C_0 = \dfrac{1.5L + 5}{1 - Y}$	$L$ is lost time per cycle
IRC:37: $\displaystyle N = \dfrac{365 A D F\left[(1 + r)^n - 1\right]}{r}$	Result in million standard axles
$\displaystyle \text{PHF} = \dfrac{\text{hourly volume}}{4 \times \text{peak 15-min volume}}$	Fourth-power law: damage $\propto (\text{load}/\text{standard})^4$

Runway length rises 7 per cent per 300 m of elevation above mean sea level.

Geomatics Engineering

Curvature and refraction: $0.0673 D^2$ metres, with $D$ in kilometres
Height of instrument: $\text{HI} = \text{RL} + \text{BS}$ , then $\text{RL} = \text{HI} - \text{sight}$
Latitude $= L \cos \theta$ , departure $= L \sin \theta$ ; Bowditch corrects each leg in proportion to its length
Stadia, horizontal sight: $D = k s + C$
Photo scale $\displaystyle S = \dfrac{f}{H - h}$ ; parallax: height of the camera above the point $\displaystyle = \dfrac{B f}{p}$
Areas scale as the square of the linear scale

Construction Materials and Management

PERT: $\displaystyle t_e = \dfrac{t_o + 4 t_m + t_p}{6}$ ; $\displaystyle Z = \dfrac{T_s - T_e}{\sigma}$ , with $\sigma$ from the variances on the critical path
Total float $= LF - EF = LS - ES$ ; the project duration is the longest path
Cost slope $\displaystyle = \dfrac{\text{crash cost} - \text{normal cost}}{\text{normal duration} - \text{crash duration}}$
Target mean strength (IS 10262): $f_{\text{target}} = f_{ck} + 1.65 s$
Fineness modulus $\displaystyle = \dfrac{\text{sum of cumulative per cent retained}}{100}$

General Aptitude essentials

Average speed over equal distances: $\displaystyle \dfrac{2 v_1 v_2}{v_1 + v_2}$ , the harmonic mean
Two workers together: $\displaystyle T = \dfrac{t_1 t_2}{t_1 + t_2}$ ; add rates, never times
Two-year interest: $\text{CI} - \text{SI} = P r^2$ , with $r$ as a decimal
Arrangements of a word with repeats: $\displaystyle \dfrac{n!}{n_1!\,n_2! \cdots}$
$n$ replacements of a fraction $f$ : solute left $= \text{original} \times (1 - f)^n$

For a quick check, $P = 20{,}000$ at 5 per cent gives $\text{CI} - \text{SI} = 20{,}000 \times 0.05^2 = 50$ . The General Aptitude preparation guide covers the rest of these 15 marks.

Remember: General Aptitude was worth exactly 15 marks in all sixteen counted papers. These five formulas cost an evening and protect marks no technical topic can replace.

The four slips made under time pressure

Gross against net bearing capacity, and the overburden term that separates them.
Which unit weight: bulk, saturated or submerged, and which Terzaghi term takes the submerged value.
Which partial safety factor: 1.10 against 1.25, a shop weld against a site weld.
Which logarithm: base 10 in consolidation, natural in the well equations.

In one line: Most lost numerical marks in Civil come from a correct formula with the wrong condition, not from a forgotten formula.

How to revise with this sheet

Read the sheet once a day in the final two weeks. For each line, say aloud when it applies and what the trap is. Then work one small example from memory, such as $y_c = (3^2/9.81)^{1/3} = 0.97$ m for $q = 3$ m²/s.

Practise every calculation on the on-screen calculator, because the exam allows no other; the virtual calculator guide shows the keys that cost time. Check that no formula you revise belongs to a topic the 2027 syllabus has dropped, such as prestressed concrete; the GATE CE 2027 syllabus changes list what went. To fit this sheet into the final weeks, follow the GATE CE last 2 months strategy.

Subject-wise formula sheets

Each of these goes deeper on one part of the GATE Civil paper, with a worked example and the trap for every formula:

Quick revision

Partial safety factors divide strength: $0.87 f_y$ and $0.446 f_{ck}$ in IS 456, 1.10 and 1.25 in IS 800.
$x_{u,\max}/d$ is 0.53, 0.48 and 0.46 for Fe250, Fe415 and Fe500.
Consolidation uses $\log_{10}$ ; wells use $\ln$ of the radius ratio.
The drainage path $d$ is half the layer when both faces drain.
A flow net counts flow channels; $q = kH N_f/N_d$ .
Critical depth depends only on $q$ : $y_c = (q^2/g)^{1/3}$ .
Fundamental traffic relation uses space-mean speed, the harmonic mean.
Project duration is the longest path; total float $= LF - EF$ .

Frequently asked questions

Can I take a formula sheet into the GATE Civil exam?

No. You cannot carry paper, notes or your own calculator into the hall. The exam gives you an on-screen virtual calculator and rough sheets. A useful habit is to write the few formulas you fear forgetting on the rough sheet in the first five minutes. Confirm the current exam-day rules at gate2027.iitm.ac.in before the exam.

Which IS 456 values should I memorise for GATE CE?

The partial safety factors $\gamma_s = 1.15$ on steel and $\gamma_c = 1.5$ on concrete, giving $0.87 f_y$ and $0.446 f_{ck}$ ; the stress block $0.36 f_{ck} b x_u$ acting at $0.42 x_u$ ; the limiting ratios $x_{u,\max}/d$ of 0.53, 0.48 and 0.46 for Fe250, Fe415 and Fe500; and the limiting concrete strain of 0.0035 in bending.

Which logarithm do GATE Civil formulas use?

It depends on the formula. Consolidation settlement uses base 10: $\displaystyle S_c = \frac{C_c H}{1 + e_0} \log_{10} \frac{\sigma_f}{\sigma_0}$ . The steady well equations use the natural logarithm of the ratio of radii. Mixing the two changes the answer by a factor of about 2.3, so write the base beside each formula when you revise.

What are the partial safety factors in IS 800 for GATE CE?

IS 800:2007 uses 1.10 for yielding of the gross section and 1.25 for rupture of the net section. Shop welds take 1.25 and site welds 1.50; bolts in shop fabrication take 1.25. Every one of them divides the strength. The 0.87 factor belongs to IS 456:2000 for reinforcement and must not be carried into steel design.

How should I revise formulas for GATE Civil?

Read a compact sheet once a day in the last two weeks. For each formula, say in words when it applies and which trap goes with it, then redo one small example from memory. Write the unit next to every number and round only the final answer to the decimals the question asks for.

Sources

Dates, fees and the syllabus are set by the GATE 2027 organising institute and can change. Always confirm at gate2027.iitm.ac.in.