Refer to Fig. 3-17. What is the minimum width b for the base of a dam 100 ft high if upward pressure beneath the dam is assumed to vary uniformly from full hydrostatic head at the heel to zero at the toe, and also assuming an ice thrust F1 of 12,480 Ib per linear foot of dam at the top? For this st

Question

Refer to Fig. 3-17. What is the minimum width b for the base of a dam 100 ft high if upward pressure beneath the dam is assumed to vary uniformly from full hydrostatic head at the heel to zero at the toe, and also assuming an ice thrust F₁ of 12,480 Ib per linear foot of dam at the top? For this study make the resultant of the reacting forces cut the base at the downstream edge of the middle third of the base (at O) and take the weight of the masonry as 2.50 γ.

Accepted Answer

Shown in the diagram are the H and V components of the reaction of the foundation, acting through O. Consider a length of 1 ft of dam, and evaluate all the forces in terms of [latex]\gamma [/latex] and b, as follows:

[latex]F_{H} [/latex] = [latex]\gamma [/latex] (50) (100 x 1) = 5000[latex]\gamma [/latex] Ib

[latex]F_{V} [/latex] = area of the loading diagram

= [latex]\frac{1}{2} [/latex](100[latex]\gamma [/latex])(b x 1) = 50[latex]\gamma [/latex]b lb

[latex]W_{1} [/latex] = 2.50[latex]\gamma [/latex](20 x 100 x 1) = 5000[latex]\gamma [/latex] Ib

[latex]W_{2} [/latex] = 2.50[latex]\gamma [/latex] [[latex]\frac{1}{2} [/latex] x 100(b - 20) ] x 1

= 125[latex]\gamma [/latex](b - 20) Ib = (125[latex]\gamma [/latex]6 - 2500[latex]\gamma [/latex]) Ib

[latex]F_{1} [/latex] = 12,480 Ib, as given for the ice thrust

To find the value of b for equilibrium, take moments of these forces about axis O. Considering clockwise moments positive,

[latex]5000\gamma \left(\frac{100}{3} \right) +50\gamma b \left(\frac{b}{3} \right) -5000\gamma \left(\frac{2}{3}b-10 \right)-(125\gamma b-2500\gamma )\left[\frac{2}{3}(b-20)-\frac{b}{3} \right] +12,480(100)=0 [/latex]

Simplifying and solving, 3[latex]b^2 [/latex] + 100[latex]b [/latex] - 24,400 = 0 and [latex]b [/latex] = 75 ft wide.

Question 3.19: Refer to Fig. 3-17. What is the minimum width b for the base...

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