5 Distributed Forces 5.1 Introduction

Transcription

1 5 Distributed Forces 5.1 Introduction - Concentrated forces are models. These forces do not exist in the exact sense. - Every external force applied to a body is distributed over a finite contact area. Example: Force exerted by the pavement on an automobile tire. Two cases to be considered: - Force is applied to the tire over its entire area of contact distributed force - Force acts on the car as a whole concentrated force Internal loads are always distributed forces stresses

2 Types of distributed forces 1. Line distributed force - distributed along a line - the intensity of this force is expressed as force per unit length of line (N/m), lb/ft 2. Area distributed force - These forces are distributed over an area. - The intensity of these forces is expressed as force per unit area - This intensity is called pressure for the action of fluid forces and stress for the internal distribution of forces in solids.

3 - The basic unit for pressure or stress in SI is the newton per square meter (N/m 2 ), which is called pascal (Pa). - In the U.S. customary system of units, the unit for pressure or stress is pound per square inch (lb/in. 2 ). 3. Volume distributed force - These forces are distributed over the volume of a body and are called body forces. - Examples of body forces are gravitational attraction and the weight. - The intensity of gravitational force is the specific weight γ = ρg, where ρ is the density of the body and g is the acceleration due to gravity. - The units for the intensity of body forces are N/m 3 in SI units and lb/in 3 in the U.S. customary system.

4 Section A: Center of Mass and Centroids 5.2 Center of Mass Center of gravity It exists no unique center of gravity in the exact sense. Determining the center of gravity Assume: A uniform and parallel force field due to the gravitational attraction of the earth. - To determine mathematically the location of the center of gravity of any body, we apply the principle of moments to the gravitational forces. - Principle of moments: The moment of the resultant gravitational force W about any axis equals the sum of the moments about the same axis of the gravitational force dw acting on all particles treated

5 as infinitesimal elements of the body (The sum of moments equal the moment of the sum). - Principle of moments about the y-axis: xdw = xw - For all three coordinates of the center of gravity G we get: - With W = mg and dw = gdm, we get:

6 or in vector form: - With dm = ρ dv, we obtain Center of Mass versus Center of Gravity - The equations in which g not appears define the center of mass - The center of mass coincides with the center of gravity as long as the gravity field is treated as uniform and parallel. - The center of mass is unique. - It is meaningless to speak of the center of gravity of a body which is removed from the gravitational field of the earth, since no gravitational forces would act on it. - The calculation of the position of the center of mass may be simplified by: - The intelligent choice of the position of reference axis.

7 - The type of the coordinates (rectangular, polar) - Consideration of symmetry. - Whenever there exists a line or plane of symmetry in a homogeneous body, a coordinate axis or plane should be chosen to coincide with this line or plane. The center of mass will always lie on such a line or plane. 5.3 Centroids of Lines, Areas, and Volumes - Assume ρ = const. ρ will cancel from the previous equations. - The remaining expressions of the equations a purely geometrical property of the body the centroid. - The term centroid is used when the calculation concerns a geometrical shape only. - If the density is uniform throughout the body, the positions of the centroid and center of mass are identical.

8 Calculation of centroids: 1. Lines: - Consider a wire or rod of length L, with constant cross-sectional area A and constant density ρ. - The element has a mass dm = ρadl. - The coordinates of the centroid are given by: - In general, the centroid C will not lie on the line.

9 2. Areas: - When a body of density has a small but constant thickness t, we can model it as a surface area A. - The mass of an element becomes dm = ρ tda. - If ρ and t are constant over the entire area, then the coordinates of the centroid may be given as: - The centroid C for the curved surface will in general not lie on the surface.

10 3. Volumes - For a general body of volume V and density ρ, the element has a mass dm = ρdv. - If ρ is constant over the entire volume then the coordinates of the centroid may be given as: Choice of Element for Integration The principal difficulty with a theory often lies not in its concepts but in the procedure for applying it. The following five guidelines will be useful for the choice of the differential element and setting up the integrals.

11 1. Order of Element Whenever possible, a first-order differential element should be selected. Choose da = ldy not da = dxdy Choose dv = πr 2 dy not dv = dxdydz

12 2. Continuity Whenever possible, we choose an element which can be integrated in one continuous operation to cover the figure. Horizontal strip da = ydx requires only one integral. Vertical strip da = xdy requires two separate integrals because of discontinuity at x = x Discarding Higher-Order Terms Higher-order terms may always be dropped compared with lower-order terms. Select da=ydx not da= ydx + 0.5dxdy In the limit, of course, there is no error.

13 4. Choice of Coordinates We choose the coordinate system which best matches the boundaries of the figure. Choose rectangular coordinate system for this figure Choose rectangular coordinate system for this figure

14 5. Centroidal Coordinate of Element When a first- or second-order differential element is chosen, it is essential to use coordinate of the centroid of the element for the moment arm in expressing the moment of the differential element. It is essential to recognize that the subscript c serves as a reminder that the moment arms appearing in the numerators of the integral expressions for moments are always the coordinates of the centroids of the particular element chosen.

15 5.4 Composite Bodies and Figures; Approximations When a body or figure can be conveniently divided into several parts whose mass centers are easily determined, we use the principle of moments and treat each part as a finite element of the whole. For the x-coordinate of the center of mass of the body shown in the figure we get: In general, the coordinate of the mass center are given as:

16 An approximation method Centroidal coordinates:

17 5.6 Beams External Effects - Beams are structural members which offer resistance to bending. - Most beams are long prismatic bars. - The loads are usually applied normal to the axes of the beams. - To analyze the load-carrying capacities of beams we must: - Determine the external loading and reactions acting on a beam as a whole. - Calculate the distribution along the beam of the internal force and moment.

18 Types of Beams Statically determinate beams: Equilibrium Equations Statics only Statically indeterminate beams Equilibrium Equations + Elastic Equations Statics + Mechanics of Materials In the following only statically determinate beams will be considered.

19 Distributed Loads Constant distributed load Linear distributed load Trapezoidal load Broken into a rectangular and a triangular load. General load distribution: