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Civil Engineering Formulas and Handbook

 

Click on Section name to go to directly to that section

1) Surveying

2) Transportation Engineering

3) Mechanics of Materials

4) Hydraulics

5) Stresses in Structures

6) Steel Design

7) Concrete Design

8) Geotechnical Engineering

9) Sewerage and Sewage disposal

10) Water Supply and Purification

11) Building Materials

Section 1) Surveying

Traverse Surveying

 Correction For Sag $$ {C_s} = {{{w^2}{l^3}}\over {24.P^2}} $$ Note:-sag correction is subtractive

Where 

w = weight of tape per meter

l= distance between supports

P = Applied Tension

Correction For Tension $$ {C_p} = {{(P-{P_0})l}\over AE} $$

P₀ is tension for which the tape is standardized

A = Cross sectional area

E = Modulus of elasticity of tape material

Correction due to Temperature $${C_T} = L{\alpha}({T_m}-{T_0})$$

T_m is temperature at time of measurement

T₀ is temperature at time of standardization of tape

⍺ is coefficient of thermal expansion

L is measured length

Latitude : Projection of a line on N - S direction is called Latitude => L = l.cosӨ

Departure : Projection of a line on E - W direction is called Departure => D = l.sinӨ

Closing Error:

    if ΣL ≠ 0 and ΣD ≠ 0 then there is closing error

closing error (e) $$e = \sqrt{{(\sum L)}^2 + {(\sum D)}^2 }$$

Direction of Closing error

$$ {\delta} = {tan}^-1 \left({\sum D\over \sum L}\right) $$

The sign of ΣL and ΣD will thus define the quadrant in which closing error lies

Rules of Balancing the Survey

1) Compass Rule :It states that the correction to be applied to the Latitude and Departure of any course is to the total error in Latitude and Departures as the length of the course is to the length of the traverse.

Bowditch method : Error in linear measurement ∝ √L where L is length of line

                                Error in Angular measurement ∝ √(1/L) 

Note:- This method is mostly used to balance a traverse where linear and angular measurements have been taken with equal precision.

correction to a particular line 

$${C_L} = \left({l\over {\sum l}}\right){\times}{\sum L} $$

$${C_D} = \left({l\over {\sum l}}\right){\times}{\sum D} $$

Where, 

C_L = correction in Latitude of a line

ΣL = Total error in Latitude

C_D = correction in Departure of a line

ΣD = Total error in Departure

Σl = Sum of length of all lines

(Compass rule is most commonly applied)

2) Transit Rule :It states that the correction to be applied to the Latitude and Departure of any course is to the total error in Latitude and Departures as the length of the course is to the Arithmetical sum of all the Latitude and Departures in the Traverse.

Note:- This method is suitable where angular measurement are more precise than linear measurements.

Correction in Latitude of a line $${C_L} = \left({L\over {L_T}}\right){\times}{\sum L} $$

Correction in Departure of a line $${C_D} = \left({D\over {D_T}}\right){\times}{\sum D} $$

Where, 

L_T = Sum of all Latitudes without considering sign

D_T = Sum of all Departures without considering sign

ΣL = Total error in Latitude

ΣD = Total error in Departure

L = Latitude of a line

D = Departure of a line

3) Axis Method :

Note:- This method is suitable where Angles are measured very accurately. so correction are done in length of line only, bearing of lines are not changed.

Correction to any length = That length x 1/2 closing error / Length of Axis

Levelling

Height of Instrument = H.I. = R.L. + B.S.

R.L. = H.I. - F.S.

Arithmetic check

ΣB.S. - ΣF.S. = ΣRise - ΣFall = Last R.L. - First R.L.

Reciprocal levelling

Used to check whether instrument is correct or not

Checking --> 

Setup instrument at point A

    Reading on staff at A = a

    Reading on staff at B = b

Setup instrument at point B

    Reading on staff at A = c

    Reading on staff at B = d

if instrument is correct then (a- b)=(c- d)

True difference between RL A and B (H) 

$$H={{(b-a)+(d-c)}\over 2} $$

Section 2) Transportation Engineering

Stopping site distance

 Stopping site distance (SSD in 'metres') of a vehicle is the sum of :

1) The distance travelled by the vehicle during the total reaction time known as lag distance and 

$${\text{Lag distance}}=0.278Vt$$

2) The distance travelled by the vehicle after the application of the brakes, to a dead stop position which is known as braking distance.

$${\text{Braking distance}}={{V^2}\over {254(f\pm S \%)}}$$

$$\text{SSD = Lag distance + Braking distance}$$

$$SSD = 0.278Vt + {{V^2}\over {254(f\pm S \%)}}$$

Where S% is gradient '+' sign for ascending gradient & '-' sign for descending gradient

Overtaking Site Distance (OSD)

The minimum distance open to the vision of a driver of a vehicle intending to overtake slow vehicle ahead with safety against the traffic of opposite direction is known as minimum overtaking sight distance (OSD) or safe passing site distance available.

OSD = d₁+ d₂+ d₃  ...... Where OSD is overtaking site distance in 'm'

$${d_1}={0.278 V_b t}$$

$${d_2}=b+2S$$

OR

$${d_2}={0.278 V_b t}+{{a T^2}\over 2}$$

Where, S = minimum spacing between 2 vehicles

$$S = 0.2{V_b}+6$$ 

here V_b is in km/hr.

$$T = {\sqrt{4s\over a}}$$

Where, a = acceleration in 'm/s²'

$$T = {\sqrt{14.4s\over a}}$$

Where, a = acceleration in 'km/hr/sec'

$${d_3}={0.278 V_c T}$$

Where, d₃  = distance travelled by on coming vehicle from C₁ to C₂  during overtaking operation

V_c = V= Speed of overtaking vehicle or design speed (km/hr)

if V_b is not given then

V_b = (V-16) ..................in km/hr

V_b = (V-4.5) ..................in m/s

For 1 way traffic 

OSD = d₁+ d₂

For 2 way traffic 

OSD = d₁+ d₂+ d₃

Minimum length of overtaking zone = 3 x OSD

Desirable overtaking zone = 5 x OSD

Terrain Catogory	Over taking Sight Distance	Safe stopping sight for minimum design speed	Safe stopping sight for Rulling design speed
Plain Terrain	475	120	170
Rolling Terrain	310	90	120
Hilly Terrain	-	50	60
Steep	-	40	50

Super Elevation (e)

The Super Elevation 'e' is expressed as the ratio of the height of outer edge with respect to horizontal width.

$$e={NL\over ML} = tan\theta = {{V^2}\over 225R}$$

$$e+f = {{V^2}\over 127R}$$

Where, 

V= speed in km/hr

R = Radius in 'm'

f = design value of lateral friction = 0.15

e = rate of super elevation

Maximum Super Elevation

emax	Terrain Type
0.07	Plain and Rolling Terrain
0.10	Hilly Area
0.04	Urban roads with frequent intersections

Ruling minimum radius of the curve

$${R_{Ruling}} = {{V^2}\over {127(e+f)}}$$

Radii for horizontal curves in meters

Terrain Catogory	Absolute	Rulling
Plain Terrain	250	370
Rolling Terrain	155	250
Hilly Terrain	50	80
Steep	30	50

Extra Widening (E_w)

Extra widening of pavement on horizontal curves is divided into 2 parts

1) Mechanical Widening (W_m):- The widening required to account for the off-tracking due to the rigidity of wheel base is called mechanical widening.

$$W_m = {{n{l^2}}\over 2R}$$

2) Psychological Widening (W_p):- Extra width of pavement is also provided for psychological reason such as to provide for greater maneuverability of steering at higher speeds, to allow for extra space requirements for the overhangs of the vehicles and to provide greater clearance for crossing and overtaking vehicles on the curves. Psychological widening is important in pavements with more than 1 lane.

$$W_p = {V\over {9.5\sqrt{R}}}$$

Combined Extra widening required on curves

$$E_w = W_m + W_p $$

$$E_w = {{n{l^2}}\over 2R} +{V\over {9.5\sqrt{R}}}$$

Transition Curve

The Indian road congress (IRC) reccomends the use of spiral as transition curve in the horizontal alignment of highways due to following reasons:

(i) The Spiral Curve satisfies the requirements of ideal transition.

(ii) The geometric properties of spiral is such that the calculations and setting out the curve in the field is simple and easy.

Length of transition curve (L)

(i) According to rate of change of centrifugal acceleration

$$L = {{0.0215 V^3}\over CR}$$

Where, V = Speed of vehicle in (km/hr)

                C = Allovable rate of change of centrifugal acceleration in (m/s²)

                R = Radius of curve in (m)

                L = Length of transition curve in (m)

(ii) According to rate of change of super elevation

$$L = {\text {150x → For Plain & Rolling Terrain}}$$

$$L = {\text {100x → For Built up area}}$$

$$L = {\text {60x → For Hilly area}}$$

Whre, x = Raise of outer line of road

$$x = (w + E_w )e \text{....... if the pavement rotated about inner side}$$

$$x = (w + E_w ){e \over 2} \text{....... if the pavement rotated about centre line}$$

(iii) According to empirical formula

$$L = {{2.7 V^2}\over R}\text{...........For Plain and Rolling terrain}$$

Note:- This is as per IRC and its minimum value of Transition curve.

$$L = {{V^2}\over R}\text{...........For Hilly Area}$$

To find out offset length at desired chainage 'x' the equation of cubical parabola

$$Y = {{x^3}\over {6RL}}$$

$${Shift}= {{{L_s}^2}\over {24R}}$$

$${L_s}={{E\times N}\over 2}$$

$${E\over 2} = {\text {Total Raise of outer edge of Pavement}} ={{eB}\over 2}$$

$$N ⟶{\text {Rate of Distribution of super elevation}}$$

Set Back Distance (m)

The clearance distance or set back distance required from the centre lines of a horizontal curve to an obstruction on the inner side of the curve to provide adequate sight distance depends upon the following factors :

(i) Required Sight distance (SSD)

(ii) Radius of horizontal curve, (R)

(iii) Length of the curve (L_c)

1. For Single lane road

(a) When L_c > SSD

 $$m = R - R.{Cos{{\alpha} \over 2}}$$

&

$${\alpha \over 2}={{180.s}\over {2\Pi R}}$$

Where, L_c = Length of Curve & s = SSD

(b) L_c < SSD

$$m=R\left({1-cos{\alpha \over 2}}\right) + {{S - L_c}\over 2} sin{\alpha \over 2}$$

&

$${\alpha \over 2}={{180.L_c}\over {2\Pi R}}$$

2. For two lane road

(a) When L_c > SSD

 $$m = R - (R-d){Cos{{\alpha} \over 2}}$$

&

$${\alpha \over 2}={{180.s}\over {2\Pi (R-d)}}$$

Where, L_c = Length of Curve & s = SSD

(b) L_c < SSD

$$m=R-(R-d)\left({1-cos{\alpha \over 2}}\right) + {{S - L_c}\over 2} sin{\alpha \over 2}$$

&

$${\alpha \over 2}={{180.L_c}\over {2\Pi (R-d)}}$$

Grade Compensation

$$\text{Grade Compensation}={{30 + R}\over R}\%$$

And 

$$\text{Maximum value of Grade Compensation}={{75}\over R}\%$$

Where, R = radius of curve in meter

Vertical Curve

Due to changes in grade in vertical alignment of highway, it is necessary to introduce vertical curves at the intersections of different grades to smoothen out the vertical profile and thus ease off changes in gradients for fast moving vehicles.

The vertical curves used in highways may be classified into two categories :

(i) Summit curves or crest curves with convexity upwards

(ii) Valley or sag curves with concavity upwards.

Summit curves (Crest curve with convexity upwards):

Summit curves with convexity upwards are formed in any one of the case illustrated in fig. The deviation angles between the two interacting gradients is equal to the algebraic difference between them. Of all the cases, the deviation angles will be maximum when an ascending gradient meets with a descending gradient. i.e. N = n₁ - (- n₂) = n₁ + n₂

(i) Length of Summit curve for SSD

(a) When L > SSD

$$L = {{NS^2}\over {(\sqrt{2H} + \sqrt{2h})^2}}$$

$$L = {{NS^2}\over 4.4}$$

Where, L = Length of summit curve in meter

            S = SSD (m)

            N = Deviation Angle

                = Algebraic deference of grade

            H = Height of eye level of driver above road way surface ≈ 1.2 m

            H = Height of object above Pavement surface ≈ 0.15 m

(a) When L < SSD

$$L = 2S-{{(\sqrt{2H} + \sqrt{2h})^2}\over N}$$

$$L = 2S - {4.4\over N}$$

(ii) Length of Summit curve for Safe Overtaking Distance (OSD) or Intermediate Sight Distance (ISD)

(a) When L > OSD

$$L = {{N{S_o}^2}\over {(\sqrt{2H} + \sqrt{2h})^2}}$$

$$L = {{N{S_o}^2}\over 9.6}$$

Where, S_o = Overtaking or Intermediate sight distance

(b) When L < OSD

$$L = 2S-{{(\sqrt{2H} + \sqrt{2h})^2}\over N}$$

$$L = 2{S_o} - {9.6\over N}$$

Valley Curves (Sag curve with Concavity Upward):

(i) Length of Valley curve as per comfort condition (Transition curves are provided back to back)

$$L=2\left[NV^3 \over C\right]^{1\over 2}$$

if C = 0.6 m/s² then 

$$L=0.38(NV^3)^{1\over 2}$$

(ii) Length of Valley curve for head light sight distance (Parabolic Curve is provided)

Summit curves (Crest curve with convexity upwards):

Summit curves with convexity upwards are formed in any one of the case illustrated in fig. The deviation angles between the two interacting gradients is equal to the algebraic difference between them. Of all the cases, the deviation angles will be maximum when an ascending gradient meets with a descending gradient. i.e. N = n₁ - (- n₂) = n₁ + n₂

(i) Length of Summit curve for SSD

(a) When L > SSD

$$L = {{NS^2}\over {(2h_1 + 2S.tan\alpha)}}$$

$$L = {{NS^2}\over {1.5+0.035S}}$$

Where, L = Total Length of Valley curve in meter

            S = SSD (m)

            N = Deviation Angle

                = Algebraic deference of grade

            α = Beam Angle ≈ 1^o

            h₁ = Avg. Height of headlight above Pavement surface ≈ 0.75 m

(a) When L < SSD

$$L = 2S-{{(2h_1 + 2S.tan\alpha)}\over N}$$

$$L = 2S - {{1.5+0.035S}\over N}$$

Section 3) Mechanics of Materials

 

Section 4) Hydraulics

 The depth at partial flow can be evaluated from equation -

$$ d = ({D\over 2}-{D\over 2}.cos{\alpha\over 2}) $$

Area of cross section while running partially full 

$$ a ={\pi D^2\over 4}({\alpha\over 360^\circ}-{sin\alpha\over 2\pi})$$

Wetted perimeter, while running partially full 

$$P = \pi D{\alpha\over 360^\circ}$$

Hydraulic mean depth, while running partially full

$$ r = {a\over P} $$

Velocity of flow is given by Manning's formula

$$ v = {1\over N}.R^{2\over 3}.s^{1\over 2}$$

Discharge when pipe is running partially full,

q =av

Proportionate discharge  = $$ {q\over Q} = {{a.v}\over {A.v'}}$$

=$$ {q\over Q} =({\alpha\over 360^\circ}-{sin\alpha\over 2\pi}).{(1-{{{360^\circ}sin\alpha}\over {2 \pi \alpha}})}^{2\over 3}$$

for detailed derivation go to link

The broad-crested weir is a type of weir used in open channel flow measurements, which has a rectangular notch on top. The discharge over a broad-crested weir can be calculated using the following equation:

$$Q = CLH^{3/2}$$

where:

• Q is the discharge over the weir [m^3/s],
• C is the discharge coefficient, which depends on the shape and size of the weir,
• L is the width of the weir [m],
• H is the height of the water above the weir crest [m].

To calculate the value of C, you can use one of the following equations:

• For a rectangular weir:

$$C = 1.84 - 0.5\left(\frac{H}{L}\right)^{1.5} + 0.25\left(\frac{H}{L}\right)^3$$

For Trapezoidal weir:-

$$C = \frac{2}{3}\left(1 + \frac{2h}{b}\right)\left(1 - \frac{h}{H}\right)^{3/2}$$

where b is the bottom width of the trapezoidal notch, and h is the height of the notch.

The sharp-crested weir is another type of weir commonly used in open channel flow measurements. Unlike the broad-crested weir, it has a sharp edge or crest at the top of the notch. The discharge over a sharp-crested weir can be calculated using the following equation:

Rectangular weir equation:

$$Q = CLH^{3/2}$$ $$C = 1.33\sqrt{\frac{2g}{H}}\left(\frac{L}{H}\right)^{1/2}$$

where g is the acceleration due to gravity (approximately 9.81 m/s^2)

V-notch weir equation:

$$Q = CLH^{3/2}$$ $$C = \frac{1}{2}\sqrt{\frac{2g}{H}}\tan\left(\frac{\theta}{2}\right)$$

Where, Ө is the V-notch angle in degrees.

Note that you will need to provide values for L, H, and either assume a value for C (for a rectangular weir) or measure the V-notch angle Ө (for a V-notch weir) in order to calculate the discharge Q.

The Chezy formula is an empirical equation used to calculate the average velocity of flow in an open channel. The formula is given as:

$$V = C\sqrt{RS}$$

where:

• V is the average velocity of flow [m/s],
• C is the Chezy coefficient, which is a dimensionless constant that depends on the roughness of the channel,
• R is the hydraulic radius, which is the cross-sectional area of flow divided by the wetted perimeter [m],
• S is the slope of the energy line, which is the change in energy per unit length of the channel [m/m].

Note that you will need to provide values for C, R, and S in order to calculate the average velocity of flow V. The value of C can be determined using experimental data or reference tables, and R can be calculated from the dimensions of the cross section of the channel.

The loss of head due to friction is an important concept in fluid mechanics and is often used in the design of pipelines and other fluid transport systems. The loss of head due to friction is given by the Darcy-Weisbach equation:

$$h_f = \frac{fLV^2}{2gd}$$

where:

• h_f is the loss of head due to friction [m],
• f is the Darcy-Weisbach friction factor, which depends on the Reynolds number and the roughness of the pipe,
• L is the length of the pipe [m],
• V is the average velocity of flow in the pipe [m/s],
• g is the acceleration due to gravity (approximately 9.81 m/s^2), and
• d is the diameter of the pipe [m].

Note that you will need to provide values for f, L, V, g, and d in order to calculate the loss of head due to friction h_f. The value of f can be determined using experimental data or reference tables, and V, L, and d can be measured or determined from the design of the pipe system.

Manning's formula is an empirical equation used to calculate the average velocity of flow in an open channel. The formula is given as:

$$V = \frac{1}{n}R^{2/3}S^{1/2}$$

where:

• V is the average velocity of flow [m/s],
• n is the Manning's roughness coefficient, which is a dimensionless constant that depends on the roughness of the channel,
• R is the hydraulic radius, which is the cross-sectional area of flow divided by the wetted perimeter [m],
• S is the slope of the energy line, which is the change in energy per unit length of the channel [m/m].

Section 5) Stresses in Structures

 Equation of Pure torsion

$${T\over {I_p}}={\tau \over r}={{G \theta} \over L}$$

Where, 

τ --> Shear stress for example shear stress on wall of a tube subjected to torsion

Ө --> Angle of twist

$$G = {E\over {2(1+\mu)}}$$

1) Circular tube subjected to torsion

$$\text{Polar Moment of Inertia }{I_p} = \text{Area of The section }\times{r^2}$$

$$(\pi D t)\times \left({D \over 2}\right)^2  = {{\pi {D^3} t}\over 4}$$

2) Thin Square tube subjected to torsion

Maximum shear stress τ is given by

$$\tau = {T\over {2A.t}}$$

Section 6) Steel Design

 

Section 7) Concrete Design

 The moment of resistance of a reinforced concrete beam as per the Indian Standard (IS) code is given by:

$$M_R = \frac{Z_{xx} \times f'_{ck} \times A_c}{\gamma_{mb} \times 10^6} + \frac{Z'_{xx} \times f_{yk} \times A_{st}}{\gamma_{ms} \times 10^6} \times (d - \frac{a}{2})$$

where:

• M_R is the moment of resistance [N·m],
• Z_{xx} is the section modulus of concrete, which is a geometric property of the beam [m^3],
• f'_{ck} is the characteristic compressive strength of concrete [Pa],
• A_c is the area of concrete in the cross-section of the beam [m^2],
• ɤ_{mb} is the partial safety factor for the material and bending moment as per the IS code,
• Z'_{xx} is the effective section modulus of steel, which is a geometric property of the beam [m^3],
• f_{yk} is the characteristic yield strength of steel reinforcement [Pa],
• A_{st} is the area of steel reinforcement in the cross-section of the beam [m^2],
• d is the effective depth of the beam [m],
• a is the distance from the edge of the compression zone to the centroid of the tension reinforcement [m],
ɤ_{ms} is the partial safety factor for the material and the steel reinforcement.

The moment of resistance of a reinforced concrete beam as per the Indian Standard (IS) code using the limit state method is given by:

$$M_R = \frac{0.87 \times f_y \times Z'_{xx} \times (d - \frac{a}{2})}{\gamma_{m} \times 10^6} + \frac{0.36 \times f'_c \times A_C \times (d - \frac{k}{2})}{\gamma_{m} \times 10^6}$$

The moment of resistance of a rectangular reinforced concrete beam as per the Indian Standard (IS) code using the limit state method is given by:

$$M_R = \frac{0.87 \times f_y \times Z'_{xx} \times (d - \frac{a}{2})}{\gamma_{m} \times 10^6} + \frac{0.36 \times f'_c \times b \times d^2 \times (0.5 - \frac{k}{6d})}{\gamma_{m} \times 10^6}$$

Section 8) Geotechnical Engineering

 Cohesive soils:-

Soil consisting of the finer products of rock weathering. Cohesion is derived from large number of water films associated with the fine grained particles in the soil.

Total force of Attraction between two particle (f) =

$$f={{2 {\pi} rT}\over {1+tan \left({\theta}\over 2 \right) }}$$ 

where   r => Radius of Particle

            T => Surface Tension

Liquidity index:-

The difference between natural moisture content and plastic limit expressed as percentage ratio of the plasticity index. 

$$LI={{m-PL}\over {LL-PL}}\times 100$$ 

m => moisture content

Plasticity Index:-

The numerical difference between liquid limit and plastic limit of a soil.

$$\text{Plasticity Index}={\text{Liquid Limit }-\text{ Plastic Limit}} $$

Section 9) Sewerage and Sewage disposal

 

Section 10) Water Supply and Purification

 

Section 11) Building Materials

SR. No.	Items	Unit	Materials
			Particulars	Quantity	Unit
1	Brick Work	100 cu.m	Bricks	50000	Nos
			Mortar	30	cu.m
2	Rubble stone Masonry	100 cu.m	Stone	125	cu.m
			Mortar	42	cu.m
3	25 mm Cement Concrete 1:2:4	100 sq.m	Stone grit	2.4	cu.m
			sand	1.2	cu.m
			cement	24	Bags
4	12 mm thick plastering	100 sq.m	Mortar	2	cu.m
5	Pointing in brickwork	100 sq.m	Mortar	0.6	cu.m
6	White washing one coat	100 sq.m	Lime	10	Kg
7	Distempring
	i) First coat	100 sq.m	Dry distemper	6.5	Kg
	ii) Second coat	100 sq.m	Dry distemper	5	Kg
8	Oil Painting one coat	100 sq.m	Ready-made paint	10	Litres
9	Oil Painting one coat	100 sq.m	Paint stiff	10	Kg
10	20 mm thick Damp-proof course of 1:2:4 cement Mortar	100 sq.m	Cement	27	Bags
			Sand	1.8	Cu.m
			Imperno	27	Kg
11	Painting with bitumen
	i) First coat	100 sq.m	Bitumen	150	Kg
	ii) Second coat	100 sq.m	Bitumen	100	Kg
12	C.G.I. Sheet Roofing	100 sq.m	C.G.I. sheets	128	Sq.m
13	A.C. Sheet Roofing	100 sq.m	A.C. sheets	115	Sq.m
14	Panneled door shutter 40 mm thick	100 sq.m	Timber	4.5	Cu.m
15	Battened door shutter 40 mm thick	100 sq.m	Timber	5	Cu.m
16	Partialy Panneled and Glazed shutter 40 mm thick	100 sq.m	Timber	3	Cu.m
17	Fully glazed shutter 40 mm thick	100 sq.m	Timber	2	Cu.m