Distributed Force Acting on a Boundary

Transcription

1 Outline: Centres of Mass and Centroids Centre of Mass Integration Centroids Composite Bodies Distributed Force Acting on a Boundary Line (Beams External Effects) Areas 1

2 Up to now all forces have been concentrated forces (force applied at single point) But forces are actually applied over some contact area If contact area is significant compared to the other dimensions we must account for it Sum the effects of the distributed force over the contact region using mathematical integration 2

3 Integration is the inverse of differentiation The integral of a function is a measure of the area between it and the x-axis The symbol of integration is, an elongated S (the S stands for "sum"). The integral is written as: and is read "the integral of f-of-x with respect to x." b a f ( x) dx 3

4 Example: If we know that (dy/dx)=3x 2 and we need to know the function this derivative came from, then we "undo" the differentiation process. y = x 3 is ONE antiderivative of 3x 2 (since the derivative of any constant is zero) In general: y = x 3 + C, is the indefinite integral, C is called the constant of integration Indefinite integrals deal with algebraic quantities (define an entire function) 4

5 Integrals with a definite bound are definite integrals The definite integral is written as: and is read "the integral from a to b of f-of-x with respect to x." a and b are the lower limit and upper limit, respectively, of integration, defining the domain of integration; f is the integrand, to be evaluated as x varies over the interval [a,b]; and dx is the variable of integration. b a f ( x) dx 5

6 Back to the example: to evaluate the integral of 3x 2 between x=1 and x=5: 5 1 3x 2 dx Some integration rules and properties: x (5) 3 (1) 3 dx adx 124 x C a dx dx dv dx dv x n dx n1 x n 1 C 6

7 How does it balance? Image from Accessed March

8 The center of mass is a unique point whose location is a function of the distribution of mass throughout the body It is the point through which the total weight vector acts B A B A G G 8

9 If an object is acted on by gravity then the Weight, W (the gravitational force acting on the object) is W = Mg (where g is the gravitational field intensity) The Center of Mass is where the weight vector acts. Find the location of an object s center of mass such that a concentrated mass placed at that location creates a moment equivalent to the net moment created by all of the particles masses. 9

10 Consider a 2D body: Divide the plate into n-small elements (particles) each with weight DW and co-ordinates x,y 10

11 co-ordinates for the center of mass: W x dw xdw W y ydw W z zdw 11 W

12 Weight is a function of mass and the gravitation field intensity (W=mg) When g is constant (like on earth) then the locations of the center of mass and the centre of gravity are the same Mass is a function of object size (volume, area or length) and object density (ρ) When ρ is constant (the object is homogeneous) then the locations of the center of mass and the geometrical centroid are the same. 12

13 Centroid For a Wire: W=pgAL W = pga L X Y Z c c c xdw W ydw W zdw W ρ (density), g, and A (the crosssectional area) are constant over the length (L) ga X Y Z c c c xdl ydl L L zdl 13 L

14 Centroid For an Area: W=pgtA W = pgt A X Y Z c c c xdw W ydw W zdw W ρ (density), g, and t (the thickness) are constant over the surface area (A) gt X Y Z c c c xda yda A A zda 14 A

15 Centroid For a Volume: W=pgV W = pg V X Y Z c c c xdw W ydw W zdw W ρ (density), and g are constant over the volume (V) g X Y Z c c c xdv ydv V V zdv 15 V

16 Steps to solve: Choose a differential element for integration Express da, dl or dv in terms of 1 variable Express x, y or z (location of centroid of element) in terms of 1 variable Substitute into equations (from previous slides) and integrate Use axes of symmetry whenever possible 16

17 Find the centroid of the area under y=x 2. 17

18 Expression for differential element (tiny rectangle) Integrate to get expression for A Chapter 5 ENGR

19 ANSWER: The centroid of the area is 3 4, 3 10 m Chapter 5 ENGR

20 Find the centroid of the cone, with radius a and height h. 20

21 An irregularly shaped body may be divided into smaller shapes with known G s (see Appendix A3 for centroids of common shapes) Summing moments as before: ( W X W x W x W x 1 W2 W3 )

22 Steps to solve a composite body: Choose a reference frame Divide the composite area/volume into parts whose centroids you know (Appendix A3) or can easily determine Determine the co-ordinates of the centroids and the volume/area/length for each part; watch for cases of symmetry that can simplify calculations 22

23 Find the centroid of the L-shaped bar of negligible thickness. 1 m 1 m 23

24 The centroid is at (0.75, 0.25) m 24

25 Find the centroid of the irregularly-shaped plate below. All dimensions in cm. y r= x 25

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27 PART Ai Xi Yi Xi * Ai Yi * Ai Sum Chapter 5 ENGR

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29 Locate the center of mass of the composite body made of a conical frustrum and a hemisphere, shown below. 29

30 PART Vi Zi Zi * Vi Sum 30

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32 Three types of Force Distribution: Line Distribution - vertical load supported along a cable Intensity is in N/m Area Distribution - hydraulic water pressure against dam face Intensity is in N/m 2 or Pascals (Pa) Called pressure for action of fluid forces Called stress for internal distribution in solids Volume Distribution - gravitational attraction Intensity is in N/m 3 For gravity the intensity is the specific weight (density times gravitational field strength (ρg )) 32

33 Beams are bars of material that support lateral loads (perpendicular to the axis of the beam) Beams are probably the most important structural member Beams can support both concentrated and distributed loads Analyze load carrying capacity of beams for: Equilibrium and external reactions Internal resistance (strength characteristics) next chapter 33

34 Distributed loads: intensity ω of a distributed load is expressed as force per unit length of the beam (can be constant or variable) The resultant of the distributed load is equal to the area formed by the intensity ω and the length The resultant of the distributed load passes through the centroid of the area 34

35 For a more general load distribution integrate over the span (of load application) Total Force: R = w dx X c Moment Produced by total force: X c R = x w dx Gives location (Xc) of load centroid 35

36 To Determine External Reactions for Beams: Reduce each distributed load to an equivalent concentrated load (find total force applied and location) Use given concentrated loads and equivalent concentrated loads with 2-D equations of equilibrium 36

37 Determine the equivalent concentrated load(s) for the simply supported beam subject to the distributed load shown. Also, determine the external reactions if the beam is in equilibrium. 37

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41 Determine the equivalent concentrated load for the cantilever beam subject to the distributed load shown. Also, determine the external reactions if the beam is in equilibrium. 41

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