Ch 7. Centroid/Distributed Loads/Inertia Multimedia Engineering Statics Centroid: Line Area Vol Centroid: Composite Distributed Loads Area Moment of Inertia
 Chapter 1. Basics 2. Vectors 3. Forces 4. Moments 5. Rigid Bodies 6. Structures 7. Centroids/Inertia 8. Internal Loads 9. Friction 10. Work & Energy Appendix Basic Math Units Sections Search eBooks Dynamics Statics Mechanics Fluids Thermodynamics Math Author(s): Kurt Gramoll ©Kurt Gramoll

 STATICS - CASE STUDY SOLUTION Pressure Distribution Solving for a Single Slice Water Pressure Resultant Force The pressure at a point on the submerged arch is perpendicular to the surface and depends on the water depth at that point. If an axis system is oriented at the center of the arch, the pressure at any height y can be determined as      p = ρg(b - y) where ρ is the water density, g is the acceleration of gravity and b is the total depth of the water to the floor. If the angle θ is as shown, then      y = a sinθ and      p = ρg(b - a sinθ) The pressure at any angle θ can be broken into x and y components as      px = -p cosθ           = ρg(a sinθ - b) cosθ      py = -p sinθ           = ρg(a sinθ - b) sinθ Resultant Force Integrate Over Surface Area S The resultant force is found by integrating the pressure over the arch surface, S, which means integrating θ from 0 to π and multiplying by the length of the arch.       For the x component: Integrate or use integral tables to find      Rx = 270 ρg [a/2 sin2θ - b sinθ]0π Evaluating the integral for Rx gives      Rx = 0 lb The y component is Integrating and evaluating for Ry      Ry = 270ρg [a/2 (-cosθ sinθ + θ) b cosθ]0π      Ry = 270ρg (aπ/2 - 2b) lb Finally, substitute for the known density, water depth, and arch radius to give      Ry = -436,200 lb = -436 kips Therefore, in vector form, the resultant force is      R = 0i - 436j + 0k kips

Practice Homework and Test problems now available in the 'Eng Statics' mobile app
Includes over 500 problems with complete detailed solutions.
Available now at the Google Play Store and Apple App Store.