1.1 The Equations of Motion

Transcription

1 1.1 The Equations of Motion In Book I, balance of forces and moments acting on an component was enforced in order to ensure that the component was in equilibrium. Here, allowance is made for stresses which var continuousl throughout a material, and force equilibrium of an portion of material is enforced. One-Dimensional Equation Consider a one-dimensional differential element of length and cross sectional area A, Fig Let the average bod force per unit volume acting on the element be b and the average acceleration and densit of the element be a and. Stresses act on the element. () A b,a ( ) Figure 1.1.1: a differential element under the action of surface and bod forces The net surface force acting is ( ) A ( ) A. If the element is small, then the bod force and velocit can be assumed to var linearl over the element and the average will act at the centre of the element. Then the bod force acting on the element is Ab and the inertial force is Aa. Appling Newton s second law leads to ( ) A ( ) A ba aa ( ) ( ) b a (1.1.1) so that, b the definition of the derivative, in the limit as 0, d b a 1-d Equation of Motion (1.1.2) d which is the one-dimensional equation of motion. Note that this equation was derived on the basis of a phsical law and must therefore be satisfied for all materials, whatever the be composed of. The derivative d / d is the stress gradient phsicall, it is a measure of how rapidl the stresses are changing. Eample Consider a bar of length l which hangs from a ceiling, as shown in Fig

2 z l Figure 1.1.2: a hanging bar The gravitational force is F mg downward and the bod force per unit volume is thus b g. There are no accelerating material particles. Taking the z ais positive down, an integration of the equation of motion gives d g 0 gz c (1.1.3) dz where c is an arbitrar constant. The lower end of the bar is free and so the stress there is zero, and so Two-Dimensional Equations g l z (1.1.4) Consider now a two dimensional infinitesimal element of width and height and unit depth (into the page). and Looking at the normal stress components acting in the direction, and allowing for variations in stress over the element surfaces, the stresses are as shown in Fig (, (, (, (, Figure 1.1.3: varing stresses acting on a differential element Using a (two dimensional) Talor series and dropping higher order terms then leads to the linearl varing stresses illustrated in Fig (where, and the partial derivatives are evaluated at,, which is a reasonable approimation when the element is small. 4

3 Figure 1.1.4: linearl varing stresses acting on a differential element The effect (resultant force) of this linear variation of stress on the plane can be replicated b a constant stress acting over the whole plane, the size of which is the average stress. For the left and right sides, one has, respectivel, 1 2, 1 2 (1.1.5) One can take awa the stress ( 1/ 2) / from both sides without affecting the net force acting on the element so one finall has the representation shown in Fig (, Figure 1.1.5: net stresses acting on a differential element Carring out the same procedure for the shear stresses contributing to a force in the direction leads to the stresses shown in Fig (, b, a (, Figure 1.1.6: normal and shear stresses acting on a differential element Take a, b to be the average acceleration and bod force, and to be the average densit. Newton s law then ields 5

4 b a (1.1.6) which, dividing through b and taking the limit, gives b a (1.1.7) A similar analsis for force components in the direction ields another equation and one then has the two-dimensional equations of motion: b b a a 2-D Equations of Motion (1.1.8) Three-Dimensional Equations Similarl, one can consider a three-dimensional element, and one finds that z z z z z z z zz b b b z a a a z 3-D Equations of Motion (1.1.9) These three equations epress force-balance in, respectivel, the,, z directions. 6

5 Figure 1.1.7: from Cauch s Eercices de Mathematiques (1829) The Equations of Equlibrium If the material is not moving (or is moving at constant velocit and is in static equilibrium, then the equations of motion reduce to the equations of equilibrium, z z z b z z b z zz bz z D Equations of Equilibrium (1.1.10) These equations epress the force balance between surface forces and bod forces in a material. The equations of equilibrium ma also be used as a good approimation in the analsis of materials which have relativel small accelerations Problems 1. What does the one-dimensional equation of motion sa about the stresses in a bar in the absence of an bod force or acceleration? 2. Does equilibrium eist for the following two dimensional stress distribution in the absence of bod forces? 3 48 / zz z z z z 7

6 3. The elementar beam theor predicts that the stresses in a circular beam due to bending are 4 M / I, V ( R ) / 3I ( I R / 4) and all the other stress components are zero. Do these equations satisf the equations of equilibrium? 4. With respect to aes 0 z the stress state is given in terms of the coordinates b the matri 2 z ij z z z 2 z z zz 0 z z Determine the bod force acting on the material if it is at rest What is the acceleration of a material particle of densit 0.3kgm, subjected to the stress z 2 4 ij 2 2 2z 2 4 2z 2z 2z z and gravit (the z ais is directed verticall upwards from the ground). 6. A fluid at rest is subjected to a hdrostatic pressure p and the force of gravit onl. (a) Write out the equations of motion for this case. (b) A ver basic formula of hdrostatics, to be found in an elementar book on fluid mechanics, is that giving the pressure variation in a static fluid, p gh where is the densit of the fluid, g is the acceleration due to gravit, and h is the vertical distance between the two points in the fluid (the relative depth). Show that this formula is but a special case of the equations of motion. 8