This document describes how to derive a required time (T) unit hydrograph from a given time (D) unit hydrograph when T is not a multiple of D using the S-curve method. It explains that an S-curve hydrograph is generated by continuous, uniform effective rainfall and rises continuously in the shape of an S until equilibrium is reached. The ordinates of the S-curve can be calculated using the equation S(t) = U(t) + S(t-D), where S(t) is the ordinate of the S-curve at time t, U(t) is the ordinate of the given unit hydrograph at time t, and S(t-D) is the

1. S-CurveHydrograph
Prepared by
Prof. S. G. Taji
Dept. of Civil Engineering
S.R.E.S’s Sanjivani College of Engineering,
Kopargaon

2. Type-III (B): Derivation of req. T-hr UH from Given
D-hr UH (where T is not multiple integral of D)
 In this type, S-curve approach should be used for calculation.
S-curve or S-Hydrograph
 A ‘S’ hydrograph is nothing but a hydrograph generated by a
continuous effective rainfall occurring at an uniform rate for an
indefinite period.
 It is called ‘S’ hydrograph because the shape of the hydrograph
comes out like alphabet ‘S’ though slightly deformed. Figure
shows a typical ‘S’ hydrograph.
 It can be derived by summation of the ordinates of an infinite
series of unit hydrographs of same .unit duration spaced at the
same unit duration apart and hence the name summation
hydrograph.
 It is a curve which rises continuously in the form or shape of the
letter S, till a constant discharge value i.e. equilibrium is reached.

4.  The discharge of the S-curve at the time of equilibrium,
A = Area of catchment in hectares
I = Constant rate of effective rainfall in cm/hour
If ‘A’ is taken in Sq. Km., then above eq. becomes
 The ordinates of S-curve can be plotted by using eq.
S(t) = U (t) + S (t-D)
Where, S(t)= req. ordinate of S-curve hydrograph at time
duration t.
U(t) = Ordinate of Given UH at time duration t.
t = required duration (hr)
D= Given duration of UH.
Therefore, S(t-D) = s-curve ordinate of preceding interval

5.  Q. Derive S-curve ordinates from given UH

6.  Ordinates of S-curve
 S(t) = U (t) + S (t-D)
 For S (0): At t = 0, U (t) = 0, S(t-D) = 0, Thus, S (0) = 0
(D=4hr i.e. given duration of UH)
 For S (4): At t = 4, U (t) = 10 m3/s, S(t-D) = S(4-4) = S (0) = 0,
 Thus, S (4) = 10 + 0 = 10 m3/s
 For S (8): At t = 8, U (t) = 30, S(t-D) = S(8-4)= S (4) = 10,
 Thus, S (4) = 30 + 10 = 40 m3/s
 For S (12): At t = 12, U (t) = 25, S(t-D) = S(12-4)= S (8) = 40,
 Thus, S (4) = 25 + 40 = 65 m3/s
 ……… continue till the end.
 These calculation can also be perform in table as shown below.

8.  Q. Determine 12 hr UH from given 4 hr UH by using S-curve
method.
 Soln:
 In this case, D=given duration= 4 hr and T=req. duration = 12hr
 Step-1: Draw two columns next to given UH (i.e. col. 3 and col. 4). These
two columns are filled consequently as follows:
 Initially, mark dash in first rows up to preceding interval of D hr of 3rd
col (i.e. in this case D=4hr, thus mark dash up to t=0 only)
 Step-2.1: Now, make addition col.2 and col. 3 and write it in col. 4.
 Step-2.2:Then, write that result at next row in col. 3 (as shown by
arrow).
 Step-2.3: Again make addition col.2 and col. 3 and write it in col. 4.
 Step-2.4: Then again, write that result at next row in col. 3 (as shown by
arrow).

9.  For ex. At time 8 hr, drag last ordinate from col.4 in col.
3 (i.e. 20 as shown by arrow), now do addition of 80 +
20 = 100 and write in col. 4.
 Continue this operation till the end of col. 3 and col. 4.
 Step-3: In col. 4, we get S-curve ordinates which are
nothing but the running summation of flow.
 Now, lag these s-curve ordinates by T-hr as shown in
col. 5 (in this case, T=12hr)
 Note: Always be lag your S-curve ordinate by that
duration for which you are going to derive UH (for ex. If
you going to plot 8 hr UH from given 24 hr UH, then in
step-3 lag your S-curve ordinates by 8 hr and vice
versa)

10.  Step-4: Now deduct that lagged column from S-
curve ordinates (i.e. col. 4 – col. 5 = col. 6)
 This is necessary because we have a summed valued
which should be converted into single flow value.
This will give us DRH of T/D cm ER (i.e. in this case,
T/D = 12/4 = 3cm ER)
 Thus, col. 6 is DRH of T/D = 12/4 = 3 cm ER.
 Step-5: This is the last step, in which divide last
column by ratio of T/D. (i.e. in this case, T/D = 12/4
= 3 cm) for getting req. 12 hr Uh.

12. Thank You……….