Lecture-05
Example of Calculating Maximum Vertical Stresses, Principal Stress, and Shear Stress Intensity in Dam Structures
Figure shows the section
of gravity dam built of concrete.
Calculate
·
The maximum vertical stresses at the heel
and toe of the dam
·
The major principal stress at toe of the
dam
·
The intensity of shear stress on a
horizontal plane near the toe.
Assume weight of concrete 23.5 kN/m3 and unit length of dam. Allowable stress in concrete may be taken 2500 kN/m2. Neglect earthquake effect.
Solution:
Various forces and moment acting on dam:
Vertical forces and moment about toe
Uplift pressure and moment about toe
Horizontal pressure and moment about toe
Net 𝛴V = W1 + W2 + W3 - U1 - U2 - U3 - U4 = 43050
Net 𝛴H
= P1 - P2 = 31215
Net 𝛴M
= MW1 + MW2 + MW3 - MU1 - MU2 - MU3 - MU4 - MP1 + MP2
= 777639 kN-m
[(+) sign for
anti-clockwise and (-) sign for clockwise moments]
Distance of Resultant from toe
X = 𝛴M/𝛴V = 777639/43050 =
18.06 m
Eccentricity = e = B/2 – x = 56/2 – 18.06 = 9.94 m
Vertical stress
Pv = 𝛴V/B
* [ 1 ± 6e/B]
= [43050/56] *[1±
(6 x 9.94/56)]
= 768.8 [1 ± 1.065]
Maximum Vertical stress at toe
= Pmax = 768.8 [1+1.065] = 1587.6 kN/m2
Minimum Vertical stress at heel =
Pmin = 768.8 [1-1.065] = - 49.97 kN/m2
σat
Toe = Pv. Sec2α – (P’ – Pe’)
tan2α
[As no earthquake effect Pe’ = 0, P’ = 58.9 kN/m2
tanα = 2/3, sec2α = 1 + tan2α = 1+(2/3)2 = 13/9]
σat Toe = 1587.6 x 13/9 - 58.9 x 4/9 =2267 kN/m2 < 2500 kN/m2 (ok)
Intensity of shear stress
𝝉 toe = (Pv – P’) tanα = [1587.6 – 58.9] * 2/3 = 1019.1 kN/m2