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This blog post continues from previous blog post which is created to let users understand better towards the graphs provided in the MSMA and derived equation used in MiTS to calculate culvert design.

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Entrance Loss Coefficient

The entrance loss coefficient depends on the inlet/outlet geometry primarily through effect it has on contraction of the flow. This can be referred in MSMA 2nd Edition in Table 18.A1.

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Inlet Control Nomograph for Pipe Culvert and Box Culvert

Headwater depth for inlet control of pipe culvert can be determined by plotting the relationship between diameter and discharge of each of the culvert number with the Inlet Type, which can be referred in Design Chart 18.A2 in MSMA 2nd Edition.

The equation to alternate the graph is as follows;

{ HW }_{ i } = \quad c[{ \frac { Ku.Q }{ A.{ D }^{ 0.5 } } ] }^{ 2 }\quad +\quad Y\quad +\quad Ks.S      (eq. 1.1)

HWi = Headwater depth (m)

D = Diameter of culvert barrel (m)

c = Constant from Table 1.1, Table 1.2

Ku = Unit Conversion, 1.811 SI unit

Q = Discharge (m3/s)

A = Full cross sectional area of culvert barel (m2)

Y = Constant from Table 1.1, Table 1.2

Ks = Slope correction, -0.5

S = Culvert barrel slope (m/m)

Nomograph ScaleInlet TypeKcY
[1]Headwall with Square Edge0.00980.03980.67
[2]Headwall with Socket End0.00180.02920.74
[3]Projecting with Socket End0.00450.03170.69

Table 1.1 Constants for Inlet Control Equation of Pipe Culvert

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Headwater depth for inlet control of box culvert also can be determined by plotting the diameter to the discharge, which can be referred to Design Chart 18.A3 in MSMA 2nd Edition.

Same Equation 1.1 can be used for box calculation, but with different constants, refer to Table 1.2.

Nomograph ScaleWingwall Flare TypeKcY
[1]30° – 75°0.0260.03470.81
[2]90°0.0610.040.8
[3]0.0610.04230.82

Table 1.2 Constants for Inlet Control Equation for Box Culvert

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Relative Discharge, Velocity and Hydraulic Radius in Part-full Pipe Culvert and Box Culvert Flow

Design Chart 18.A5 and Chart 18.A6 in MSMA 2nd Edition are used to determine the discharge, velocity and hydraulic radius within partially full flow of pipe culvert and box culvert, respectively.

These two graphs quite complicated to get the output since several variables need to be determined first before obtaining the end result. The calculation of the partially full flow can be calculated by using Manning equation but with modified n value (H. B. Harlan, Spreadsheet Use for Partially Full Pipe Flow Calculations). The n/nfull as function of y/D is as Table 1.3.

y/Dn/nfull
0 < y/D <= 0.031 + (y/D)/(0.3)
0.03 < y/D <= 0.11.1 + (y/D – 0.03)(12/7)
0.1 < y/D <= 0.21.22 + (y/D – 0.1)(0.6)
0.2 < y/D <= 0.31.29
0.3 < y/D <= 0.51.29 – (y/D – 0.3)(0.2)
0.5 < y/D <= 11.25 – (y/D – 0.5)(0.5)

Table 1.3 n/nfull as function of y/D

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Critical Depth in a Circular Pipe and Box Culvert

For circular pipe, users can plot the critical depth by using Design Chart 18.A7 provided in MSMA 2nd Edition.

Other than plotting, equation also can be used as an alternative to calculate the pipe culvert critical depth.

{ h }_{ c } = \frac { C{ Q }^{ 0.5 } }{ { D }^{ 0.25 } }                                                                                           (eq. 1.2)

hc = Critical depth (m)

C = Constant, 0.562

Q = Discharge (m3/s)

D = Diameter (m)

Critical depth of box culvert also can be determined by using the nomograph Chart Design 18.A8 provided in MSMA 2nd Edition by interrelate the width dimension with discharge.   

Equation 1.3 can be used to calculate box critical depth.

{ h }_{ c\quad } = \quad { \frac { C.Q }{ B } }^{ \frac { 2 }{ 3 } }         (eq. 1.3)

hc = Critical depth (m)

C = Constant, 0.319

Q = Discharge (m3/s)

B = Width of culvert (m)

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Outlet Control Nomograph of Pipe Culvert and Box Culvert

Equation 18.9 in MSMA 2nd Edition is referred to calculate the outlet headwater and the Head is calculated by using Design Chart 18.A9 and Design Chart 18.A10 for pipe culvert and box culvert, respectively. But, users can alternate to equation to calculate those Head for pipe and box culvert. The equations are shown below the graphs.

HWo = ho + H – Ls

H = Head determined by Design Chart 18.A9 and Design Chart 18.A10

ho = Greater of TW and (hc + D)/2

hc = Critical depth (m)

L = Length of culvert

S = Slope (m/m)

Head depth for outlet control of pipe culvert and box culvert can be determined based on MSMA 2nd Edition, accordance with Design Chart 18.A9 and Design Chart 18.A10 respectively.

The following equation can be used to calculate the head depth for both pipe and box culvert, only the constant varies based on the culvert type used.

H\quad = \quad (1\quad +\quad { k }<em>{ e }\quad +\quad \frac { { k }</em>{ u. }{ n }^{ 2 }.L }{ { R }^{ 1.33 } } )(\frac { { V }^{ 2 } }{ 2g } )      (eq. 1.4)

H = Barrel loss (m)

Ke = Constant

Ku = Constant, 19.63 SI Unit

n = Manning rougness coefficient

L = Length of culvert barrel (m)

R = Hydraulic radius of the full culvert barrel (m)

V = Velocity in the barrel (m/s)

g = Gravitational accelaration, 9.81 m/s/s SI unit

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Hydraulic Design of Pipes – Colebrook-White Formula – k = 0.60 mm

Hydraulic design of pipes can be determined by using the complicated Design Chart 18.A2 obtained from MSMA 2nd Edition, whereby users need to plot the 4 axis graph to determine the output.

You can also use the Colebrook-White equation as below;

V\quad = \quad -2(2.g.D.S)log(\frac { k }{ 3.7D } \quad +\quad \frac { 2.51v }{ D(2.g.D.S) } ) (eq. 1.5)

V = Velocity (m/s)

G = Gravitational acceleration, 9.81 m/s/s SI unit

D = Diameter (m)

S = Slope (m/m)

k = Colebrook-White roughness coefficient

𝒗 = Kinematic viscosity of water, 0.00000113

We also provide you the calculator for you to calculate the output.

All graphs and equations explained above are the one that are used in the culvert calculation based on MSMA 2nd Edition. We also carry out several calculation example to benchmark between the equation derived with the graphs provided.