M15e Bending of beams

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Cuthbert Scott
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1 Fakultät für Physik und Geowissenschaften Physikalisches Grundpraktikum M5e Bending of beams Tasks. Determine Young s modulus E for two metal rods of different material but of the same crosssectional form by a bending experiment. Perform the measurements with an optical (cathetometer) or an electronic (inductive position sensor) technique.. Determine the area moment of inertia of a third metal rod with a different cross-sectional form using the value of Young s modulus determined in task. Compare to the value obtained by calculation using the geometrical dimensions.. Measure the bending stiffness of rods with more complex cross-sectional form, of folded paper or cardboard, etc. Literature Physics, P. A. Tipler, rd Edition, Vol., Chap. - Physikalisches Praktikum,. Auflage, Hrsg. W. Schenk, F. Kremer, Mechanik,.0,.,.. Accessories Mass pieces, blades with stirrup and bowl for mass pieces, mirror, cathetometer, micrometer screw gauge, vernier caliper, steel tape measure, metal rods, inductive position sensor Keywords for preparation - Hooke`s law, stress-strain-diagram, proportional limit, elastic limit, fracture point - Physical description of bending, Young s modulus, shear modulus, other elastic constants - Definition of the moment of area, moment of area of rods of various cross sections - Neutral fiber - Measurement of bending angle ϕ and bending vector s

2 Remarks The bending of the beams is measured for 0 different loads (mass m) up to a given maximum value, once with increasing load and once with decreasing load. The deflection is plotted as a function of the force of gravity; from the slope of the graphs the desired quantities can be calculated. Maximum values of the loads Material Cross section Mass m max / g Steel rectangular 000/000 Steel circular 000 Brass rectangular 000 Brass circular 000 Aluminum rectangular 600 Aluminum circular 00 Measurements with the inductive position sensor The inductive position sensor of the type D5/400 with amplifier and indicator panel uses an unguided ferromagnetic tracer pin and is designed for a measurement range of ±0 mm. In this range the deviations from linearity are specified as 0.5%. At the start of the measurement (without any load) the tracer pin should be positioned within the transducer such that it moves freely and without friction and such that the digital display shows a value near zero. Subsequently the transducer should be lowered down to a reading of about +0 mm. Then the deflection during increase and decrease of the load can be measured in the range +0 mm to -0 mm with a relative measurements accuracy of %. See also Measurements with the cathetometer The cathetometer is used for the measurement of vertical distances (e.g. level differences, liquid levels etc.). It consists of a massive vertical column with a millimeter scale. On the column a slider carrying a telescope or a diopter with horizontal optical axis can be adjusted. Similar to a caliper the slider has a nonius scale for reading off level differences with an accuracy of fractions of millimeters. With a screw the vertical fine adjustment of the telescope can be made. There is a cross line in the field of view of the telescope. The cathetometer is mounted on three knuckle feet allowing for a precise control of the vertical alignment. Before the measurement the cathetometer has to be adjusted.

3 Theoretical background Consider a homogeneous rod (density ρ, cross section A) suspended by two blades (distance l) as shown in Fig.. Each of the blades then carries a load of 0.5F 0. The weight of the rod is given by = gρal () under the assumption that the blade distance l is equal to the rod length. The rod is already bent by its own weight without an additional load. When an additional force F is applied at the center of the rod, see Fig., the bending is increased. In Fig. (b) an equivalent arrangement is shown that is used in the further discussion. In this equivalent arrangement the force F acts upwards. In the process of bending the upper layers of the rod are compressed and the lower layers are extended. Along the center of the rod there is a layer of unchanged length, called neutral fiber. In a two-dimensional model the shape of the neutral fiber y(x) can be calculated in a straightforward manner, if the following assumptions are made:. Hooke s law is valid.. A plane cross section of the rod remains plane under bending.. The bending is sufficiently small such that the derivative dy/dx is small compared to unity for all x. On the rod cross section at the position x, see Fig. (b), the following torque acts: l/ x. () M = gρa wdw+ ησ( η) da 0 A w denotes the distance of the mass element ρadw from the position x, see Fig. (b), η the distance of the surface element da from the neutral fiber, see Fig., and σ(η) the stress at position η. Fig. (a) Beam suspended by two blades and (b) equivalent arrangement.

4 Fig.. Volume element deformed by bending (exaggerated). For the stress σ(η) one obtains from Hooke s law and Fig. : dξ η ση ( ) = Eεη ( ) = E = E, () ξ r where E denotes Young s modulus, ε(η) the strain at a distance η from the neutral fiber, dξ the elongation/compression of the layer at a distance η from the neutral fiber and r the radius of curvature. Substituting Eq. () into Eq. () leads to an integral of the form I η A = η da (4) denoted as (axial) area moment of inertia. It has the unit m 4. The bending stiffness B of a rod is defined as the product of Young s modulus and the area moment of inertia B = EI η. Solving Eq. () with the substitution of Eq. (4) leads to l E M = gρa x + I r η. (5) The force 0.5(F+F 0 ) exerts at the position x a torque l M = x ( F ) +. (6) In equilibrium M = M yielding l η ( ) ρ = + EI F l x g A x r 4. (7) This relates the curvature /r of the neutral fiber to the external force F and the weight of the rod. The curvature of a curve can be calculated using Frenet s equations for the trihedron. This yields dy =± dx r dy + dx /. (8) 4

5 With the third assumption, dy/dx «, this can be simplified to /r ±d y/dx. Integration of the equation dy l EIη F( l x) gρa x ± = + dx 4 (9) with the boundary conditions dy/dx(x = 0) = y(0) = 0 leads to dy l η ρ ( ) ± EI = F lx x + g A x x dx 4 and (0) l l 4 η ρ ± EI y = F x x + g A x x. () Since in the present case y(x) is positive in the interval 0 < x < l/, in Eq. () the positive sign has to be used. At the position x = l/ both the displacement y(l/) and the slope dy/dx(l/) of the neutral fiber function is maximum. With the bending angle dy ϕ tan ϕ = ( x = l / ) () dx one obtains from Eq. (0) EIηϕ = Fl gρal 6 + or () l F+ ϕ =. (4) 6EI η Accordingly, the displacement or bending vector s = y(x = l/) of the rod is obtained from Eq. () as 5 l F+ 8 s =. (5) 48EI η The quantities ϕ and s are measured in the experiment by the cathetometer and the inductive position sensor, respectively, as a function of the force of gravity F. The area moment of inertia I η can be obtained from Eq. (4) by integration over the respective cross-sectional area. Young s modulus can be calculated from the slopes df/dϕ and df/ds, respectively. 5

Coupled pendulums and degree of coupling

Fakultät für Physik und Geowissenschaften Physikalisches Grundpraktikum Me Coupled pendulums and degree of coupling Tasks. Measure for three different positions of the coupling spring: a) the oscillation