elasticity

FYSP102 / K2. MODULUS OF ELASTICITY Purposes of the work: /square4 To become familiar with properties of solid materia ls in more details than during the lectures /square4 To rehearse performing repeated calculations with a PC Due to limited time available in basic courses, pro perties of materials are touched only very shortly. Therefore in this work, you’ll get acquai nted with the modulus of elasticity for bending- a phenomenon which is more important in th e education of an engineer than that for a physicist. In the data analysis calculations using same formul as are repeated many times, therefore a computer is a useful tool. This work gives you a go od opportunity to rehearse how to use Excel for repeated calculation of quantities. General Elasticity properties of materials are discussed in the text book Young & Freedman (11 th ed.), sc. 11.4 and 11.5. The home work related to this laboratory work is in the end theory section. The home work should be worked out before measurements. 1. Theory 1.1. Hooke’s law Solid bodies tend to conserve their shape and volum e. Under induced stress they behave differently compared to gases and liquids. A simple example is a solid rod (length L, cross section A) fixed from one end and an external, axial force F acting on the other end. The FYSP102_K2 – 2 – force F acts equally on every cross section A independent of its position. The pulling tension in each position is F/A . The tension F/A , depending of its direction either stretches or co mpresses the rod. The amount of relative extension ΔL/L obeys in linear approximation Hooke’s law F AEL L=Δ, (1) where E is the modulus of elasticity of the material the r od is made of. The linear Hooke’s law is valid for small, elastic and reversible exte nsions, only. 1.2. Modulus of elasticity for bending Forces needed to bend (a rod) are usually much smal ler than those needed in stretching. Unfortunately the theory for bending is more cumber some. Let’s imagine that the other end of a rod is firmly fixed and the other end is loade d with some weight, the rod bends. Various layers of the rod bend differently dependin g of their position. In the middle there is a neutral layer which does neither stretch nor comp ress. Layers situated above or underneath either stretch or compress. The neutral layer is located at the centre of mass of a cross section A. In fig. 1 a rod is being bent. The distance between the stretched layer (DD’) and the neutral layer (CC’) is denoted with s, the radius of curvat ure with R and the angle of curvature with φ. From the figure it can be seen that the length CC ’ is Rφ and that of DD’is ( R+s )φ. The relative stretching of DD’ compared to CC’ is t hus sφ/R φ = s/R . Hooke’s law gives now dF dA Es R=, (2) where dF is the infinitesimal force tending to reverse an i nfinitesimal layer dA to its original, equilibrium length. Notice that the exter nal, bending force acts not along the rod, FYSP102_K2 – 3 – but with a relative stretching ΔL/L achieved, the internal stretching force F/A obeys Hooke’s law. Figure 1. A bending rod (thickness a, width b) 1.3. Surface momentum of inertia In static equilibrium the torques of external force s and the internal tensions cancel each other. The infinitesimal force dF causes a torque dM = sdF to the cross section dA . Eq. (2) gives now dF s Es RdA =2 (3) The total torque MS is thus M Es RdA E Rs dA E RIS A = = = ∫ ∫2 2 (4) IA is called as the surface momentum of inertia or th e square-momentum of inertia of the surface. IA depends of the shape of A and of the axis respect to which it is calculated. For a cross section of rectangular shape (fig. 2) FYSP102_K2 – 4 – I s dA s b ds bsba A aa aaa a= = = = − −− ∫ ∫2 22 2 222 233 3 12 // /// / / , (5) and for a cylindrical rod (derive!) Ir d A= =π π4 4 464 , (6) where r is the radius and d the diameter of the solid cylinder. Figure 2. Cross section of a rectangular rod. FYSP102_K2 – 5 – 1.4. Radius of Curvature Figure 3. A bent rod and radius of curvature R In order to calculate the modulus of elasticity E one needs to know the radius of curvature. For small displacements in x-y co-ordinates 1/R ≈ d 2y/dx 2 (see Pennala 1995). Eq. 4 reads now M EI d y dx S A=2 2. (7) In the experiment you’ll use rods suspended from bo th ends and bend with weights placed in middle (the origin for calculations). A force F in origin is compensated by suspending forces F/2 in the ends. The torque of the suspending forces m ust equal to the torque Mu of the internal tension (forces): EI d y dx F lxA2 222= −( ), (8) where l is the length of the rod. By separating variables x and y and integrating twice (do this!) you’ll get EI yF lx x A= −2462 3 ( ) , (9) from which function y(x) is FYSP102_K2 – 6 – y xF EI lx x A( ) ( )= −2 4 62 3 (10) At both support positions x = l / 2 i.e. yFl EI A=3 48 (11) This result gives the displacement at the middle of the rod when the force F acts to the middle point, when the distance between the suspens ion points is l. Home work: a. How much does a metal rod, suspended from two point s, bend, if the middle point is loaded with 1,0 kg. The distance between suspension points is 0,50 m and the cross section is of rectangular shape ( b = 1,0 cm and a = 0,50 cm, see figure 1). Compute the displacement of the middle point for aluminium and steel rods. b. Do the rods bend more or less if values of quantiti es a and b are interchanged? c. Which one bends more easily, a rod whose cross sect ion is rectangular (each side equal to s) or a rod with circular cross section (diameter eq ual to s)? The rods are made of same material. 2. Equipment and measurements Metal rods to be studied are suspended with two sta nds placed on the ends. Use several weights in the middle and measure the amount of ben d in the middle with a micrometer screw. Measure the dimensions of the rods carefully . Use at least three different rods made of two different materials and having different geo metries (rectangular and cylindrical). FYSP102_K2 – 7 – 4. Analysis of results Plot amount of bending Δy as a function of the load (force applied). Using t he method of least square sum define the linear coefficients (wi th error) of the straight lines fitting best to the measured points. In each case, calculate from t he slope the modulus of elasticity. Compare the measured values of E to values in literature. Error estimate for E can be obtained easily using t he max-min-method. Literature: Pennala, Erkki: ”Lujuusopin perusteet”. Helsinki, O tatieto OY 1995, ch. 4, p.’s. 91-