Concept explainers
The cross section of a concrete dam is as shown. For a 1-m-wide dam section determine (a) the resultant of the reaction forces exerted by the ground on the base AB of the dam, (b) the point of application of the resultant of part a, (c) the resultant of the pressure forces exerted by the water on the face BC of the dam.
Fig. P5.81
(a)
The reaction forces exerted by the ground on the base of the concrete dam
Answer to Problem 5.81P
The resultant reaction forces acts on the base of the dam is
Explanation of Solution
Given that the width of the dam section
The free-body diagram consists of dam and the triangular section
The length
Write the equation for weight force acts on the dam.
Here, the weight of the dam is
Replace
Here, the thickness of the dam section is
Write the equation for the weight of the dam represented by the weights of its components.
Here, the weight of the dam by the components of fist section is
Substitute
Write the equation for the weight of the dam represented in the triangular section.
Here, the weight of the dam by the components of second section is
Substitute
Write the equation for the weight of the dam represented in the parabola section by the weights of its components.
Here, the weight of the dam by the components of third section is
Substitute
Write the equation of the force pressure exerted by the ground on the base of the dam.
Here, the reaction force exerted on the dam is
Replace
Write the equilibrium equation for the section of dam acts along x axis (Refer Fig 1).
Here, the reaction force exerted by the ground on the base
Write the equilibrium equation for the section of beam acts along y axis and then calculate the reaction force (Refer Fig 1).
Here, the reaction force exerted by the ground on the base
Conclusion:
Substitute
Substitute
Substitute
Therefore, the resultant reaction forces acts on the base of the dam is
(b)
The point of forces acts on the base
Answer to Problem 5.81P
The point in which the forces acts on the base
Explanation of Solution
The distance from the base of the dam to the point
The distance from the base of the dam to the side
The distance from the base of the dam to the point
Write the equilibrium equation for the section on the base
Here, the different section of the dam is represented as
Conclusion:
Substitute
Solve the above equation for
Therefore, the point in which the forces acts on the base
(c)
The resultant pressure force exerted by the water on the face
Answer to Problem 5.81P
The resultant pressure force exerted by the water on the face
Explanation of Solution
The free body diagram of the water section
Write the equilibrium equation for the s resultant pressure force exerted by the water on the face
Here, the resultant pressure force exerted by the water on the dam is
Solve for the angle of resultant force exerted by the water on the dam by using trigonometric relation (Refer fig 3).
Conclusion:
Substitute
Substitute
Therefore, the resultant pressure force exerted by the water on the face
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Chapter 5 Solutions
Vector Mechanics for Engineers: Statics
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