1. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 1 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Engineering 36 Chp09: FluidStatics

2. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 2 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Fluid Statics  The Fish Feels More “Compressed” The Deeper it goes – An example of HydroStatic Pressure  Definition of Pressure Pressure is defined as the amount of force exerted on a unit area of a surface:

3. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 3 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Direction of fluid pressure on boundaries Furnace ductPipe or tube Heat exchanger Dam  Pressure is a Normal Force i.e., it acts perpendicular to surfaces It is also called a Surface Force

4. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 4 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Absolute and Gage Pressure  Absolute Pressure, p: The pressure of a fluid is expressed relative to that of a vacuum (absence of any substance)  Gage Pressure, p g : Pressure expressed as the difference between the pressure of the fluid and that of the surrounding atmosphere (the BaseLine Pressure) which has an Absolute pressure, p 0. p 0 ≡ BaseLine Pressure:

5. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 5 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Pressure Units

6. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 6 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Static Fluid Pressure Distribution  Consider a Vertical Column of NonMoving Fluid with density, ρ, and Geometry at Right  Taking a slice of fluid at vertical position-z note that by Equilibrium: The sum of the z-directed forces acting on the fluid- slice must equal zero

7. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 7 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics The z-Direction Fluid Forces  Consider the Fluid Slice at z under the influence of gravity S is the Top & Bot Slice-Surface Area x y z  Let p z and p z+Δz denote the pressures at the base and top of the slice, where the elevations are z and z+Δz respectively

8. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 8 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics The z-Direction Fluid Forces  A Force Balance in the z-Dir x y z  Solving for Δp/Δz and Taking the limit Δz→0

9. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 9 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Constant Density Case  For Constant ρ  Next Consider the typical Case of a “Free Surface” at Atmospheric Pressure, p 0, and total Fluid Depth, H  If h is the DEPTH, then the z↔h reln:

10. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 10 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Constant Density Case  Then  Sub into Δp Eqn  Now if at h = 0, p = p 0, and Δh = h 2 – 0 = h  In this Case

11. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 11 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Constant Density Case  The Gage Pressure, p g, is that pressure which is Above the p 0 Baseline  Thus the typical Formulation for Liquids at atmospheric pressure The Gage Pressure is used for Liquid-Tanks or Dams

12. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 12 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Specific Weight  Since the acceleration of gravity, g, is almost always regarded as constant, the ρ liq g product is often called the Specific Weight  Then the pressure eqns

13. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 13 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Bouyancy  A body immersed in a fluid experiences a vertical buoyant force equal to the weight of the fluid it displaces  A floating body displaces its own weight in the fluid in which it floats Free liq surf F1F1 F2F2 h1h1 h2h2 ΔhΔh

14. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 14 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Bouyancy  The upper surface of the body is subjected to a smaller force than the lower surface  Thus the net bouyant force acts UPwards Free liq surf F1F1 F2F2 h1h1 h2h2 ΔhΔh  The net force due to pressure in the vertical direction F B = F 2 - F 1 = (p bottom - p top )(ΔxΔy) = Δp(ΔxΔy)

15. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 15 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Bouyancy  The pressure difference  Subbing for Δp in the F B Eqn F B = (ρgΔh)(ΔxΔy)  But ΔhΔxΔy is the Volume, V, of the fluid element  So Finally the Bouyant Force Eqn Free liq surf F1F1 F2F2 h1h1 h2h2 ΔhΔh p bottom – p top = ρg(h 2 -h 1 ) = ρgΔh Δp = ρgΔh

16. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 16 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

17. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 17 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Equivalent Point Load  Consider Fluid Pressure acting on a submerged FLAT Surface with any azimuthal angle θ  In this situation the Net Force acting on the Flat Surface is the pressure at the Flat Surface’s GEOMETRIC Centroid times the Surface area  i.e., F liq = the AVERAGE Pressure times the Area

18. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 18 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Equivalent Point Load  F liq is NOT Positioned at the Geometric Centroid; Instead it is located at the CENTER of PRESSURE  Calculation of the Center of Pressure Requires Moment Analysis as was described last lecture

19. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 19 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Force on a Submerged Surface  Pressure acts as a function of depth  Then the Magnitude of the Resultant Force is Equal to the Area under the curve y 0 dd Resultant, R d ΔyΔy  Face Width of Structure INTO the Screen = b

20. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 20 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Force on a Submerged Surface  The location of the Resultant of the pressure dist. is found the by same technique as used on the horizontal beam  Thus the Location of the Resultant Hydrostatic force Passes Thru the Pressure-Area Centroid y 0 dd Resultant, R d S  The Distance OS is Also Called the Center of Pressure

21. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 21 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Angled Submerged Surface  How does one find the forces on a SUBMERGED surface (red) at an angle?  Construct the FBD  Detach The Triangular Fluid Volume from the bulk fluid to Reveal Forces: Weight of the Volume Pressure at NORMAL Surfaces  Can also ATTACH the Fluid to the Body if desired

22. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 22 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Angled SubMerged Surface  The resulting force distribution without the weight of the water would show Equal to the Trapezoidal Area Under the Pressure Curve Times the Width b

23. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 23 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics NonLinear Submerged Surface  Consider The P-distribution on a non-linear surface.  Liquid Generates Resultant, R on The Surface  Use Detached Fluid Volume as F.B.D.  The Force Exerted by the Surface is Simply −R

24. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 24 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics HydroStatic Free Body Diagram  Examine Support Structure for Non-Rectilinear Surfaces  Detach From Surrounding Liquid, and And from the Structure, a LIQUID VOLUME that Exposes Flat X&Y Surfaces exposed to Liquid Pressure Pressure is –Constant at X-Surfaces (Horizontal) –Triangular or Trapezoidal at Y-Surfaces (Vertical)

25. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 25 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Centroids of Parabolas SectorSpandrel

26. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 26 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics WhtBd Example: P9-122  Determine the resultant horizontal and vertical force components that the water exerts on the side of the dam. Find for R the PoA on the Dam-Face The dam is 25 ft long SW for Water = 62.4 lb/cu-ft

27. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 27 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

28. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 28 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Resultant Location

29. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 29 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics % Bruce Mayer, PE % ENGR36 * 03Dec12 % ENGR36_Dam_Face_P9_122_1212.m % W = 260; % kip P = 487;.5 % kip % Deqn = [P/4 W -(25*P/3+30*W/8)] x = roots(Deqn) y = (x(2))^2/4 xp = linspace(0,10, 500); yp1 = polyval(Deqn, xp); yp = xp.^2/4; xD = x(2) yD = y plot(xD,yD,'s', xp, yp, 'LineWidth', 3), grid, xlabel('x'),... ylabel('y'), title('P9-122 Dam-Face Force Location') Solve By MATLAB Deqn = 1.0e+03 * 0.1218 0.2600 -5.0333 x = -7.5856 5.4500 y = 7.4257 xD = 5.4500 yD = 7.4257

30. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 30 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example  Small Dam  Given Fresh Water dam with Geometry as shown  If the Dam is 24ft wide (into screen) Find the Resultant of the pressure forces on the Dam face BC Assume  Detach the Parabolic Sector of Water

31. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 31 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example: Small Dam  Create FBD for Detached Water-Chunk  The gage Pressure at the Bottom

32. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 32 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example: Small Dam  The Force dF 18 at the base of the Dam with Face Width, b (into paper), consider a vertical Distance dh located at 18ft Deep

33. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 33 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example: Small Dam  Now to Covert dF to w (lb/vertical-Foot) Simply Divide by the vertical distance that generated dF  In this case b = 1ft

34. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 34 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example: Small Dam Digression  For a Submerged Flat surface, with Face Width, b, in a constant density Fluid the Load per Unit-Length profile value, w(h) can found as In Units of lb/ft or lb/in or N/m

35. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 35 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example: Small Dam  The Load Profile is a TriAngle → A = ½BH B = 1123 lb/ft H = 18 ft  Then the Total Water Push, P, is the Area under the Load Profile  Previous Slide

36. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 36 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example: Small Dam  Then the Volume of the Detached Water  And the Weight, W 4, of the Attached Water

37. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 37 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example: Small Dam  And the Location of the of P is the Center of Pressure which is located at the Centroid of the LOAD PROFILE In this case the CP is 6ft above the bottom  Also W4 is applied at the CG of the parabolic sector  From “Parabola” Slide

38. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 38 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example: Small Dam  Now the Dam also pushes on the Detached Water  For Equilibrium the Push by the Dam on the Water-Chunk must be the negative of Resultant of the Load ON the Dam  The Applied Loads are P and W 4  Then the FBD for the Water-Chunk

39. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 39 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example: Small Dam  The Water Chunk FBD  Notice that This is a 3-Force Body Thus the Forces are CONCURRENT and no Moments are Generated  The Force Triangle Must CLOSE Use to Find Mag & Dir for R

40. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 40 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example: Small Dam  The Force Triangle:  The the Load per Unit Width of the Dam 12580 lb/ft-Width Directed 36.5° below the Horizontal and to the Left relative to the drawing

41. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 41 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Resultant Location on Dam  Determine the CoOrdinates, Horizontally & Vertically, for the Point of Application of the 12 580 lb  Take the ΣM C =0 about the upper-left Corner where the water-surface touches the dam

42. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 42 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

43. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 43 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics MATLAB Results W = 7488 P = 10109 k = 0.0309 Deqn = 1.0e+05 * 0.0023 0.1011 -1.6624 x = -56.4769 12.7360 y = 5.0064 xD = 12.7360 yD = 5.0064 x canNOT be Negative in this Physical Circumstance

44. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 44 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics PoA for R on Dam Face

45. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 45 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics MATLAB Code % Bruce Mayer, PE % ENGR36 * 23Jul12 % ENGR36_Dam_Face_1207.m % W = 7488 P = 10109 k = 10/18^2 Deqn = [k*W P -(6*W+12*P)] x = roots(Deqn) y = k*(x(2))^2 xp = linspace(0,18, 500); yp = polyval(Deqn, xp); yp = k*xp.^2; xD = x(2) yD = y plot(xD,yD,'s', xp, yp, 'LineWidth', 3), grid, xlabel('x'),... ylabel('y'), title('Dam Face Force Location‘)

46. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 46 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics WhiteBoard Work  A 55-gallon, 23-inch diameter DRUM is placed on its side to act as a DAM in a 30-in wide freshwater channel. The drum is anchored to the sides of the channel. Determine the resultant of the pressure forces acting on the drum and anchors. Barrel Dam Problem  ATTACH some Water Segments to the Drum

47. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 47 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

48. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 48 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

49. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 49 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

50. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 50 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

51. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 51 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

52. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 52 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

53. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 53 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Bruce Mayer, PE Registered Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Engineering 36 Appendix

54. BMayer@ChabotCollege.edu ENGR-36_Lec-25_FluidStatics_HydroStatics.pptx 54 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics