Massachusetts Institute of Technology Department of Mechanical Engineering Cambridge, MA 02139 2.002 Mechanics and Materials II Spring 2004 Laboratory Module No. 1 Elastic behavior in tension, bending, buckling, and vibration. 1 Objectives In this laboratory session we will review elementary concepts concerning the isotropic linear elastic behavior of materials and structures. We will consider elastic loading in simple tension, cantilever beam-bending, vibration and buckling. We also intro-duce experimental methods for quantifying elastic properties of materials and elastic response of structural components. 2 Lab Tasks In this laboratory module we will perform the following tasks: • review concepts of stress and strain, and the definition of the isotropic elastic constants: Young’s modulus (E) and Poisson’s ratio, (ν); • conduct an elastic-level tension test on a strip specimen of 6061-T6 aluminum alloy, mounted with two strain gages. The load and applied to the specimen will be monitored (load cell), and strain gauges mounted both parallel and perpendicular the bar axis will monitor the respective strain components (axial and transverse). • review elastic analysis of tip-loaded cantilever beams and use a cantilever beam as a leaf spring, inferring E-values from both local bending strain measurements and from tip deflection. • review elastic theory for lateral buckling of slender axially-compressed members (Euler buckling), and conduct simple buckling experiments to obtain estimates of E in slender members of different materials, cross-sections, and lengths. • review elastic beam vibration, and use the natural vibration frequency of can-tilever beams to estimate E of the material. 13 Lab Assignments: Specific Questions to Answer 1. Using the dimensions of the tensile specimen, recorded values of the axial load, and the axial and transverse strain gage readings, determine Young’s modulus (E) and the Poisson ratio (ν) for this material. 2. Record the specimen dimensions and strain gage locations on the instrumented cantilever. Record the values for axial strain at each gage, and the lateral tip deflection, for the tip loadings applied in class. How well do these measurements agree with theoretical values based on b eam theory? How well do they predict E? Discuss. 3. Use the elastic structural stiffnesses (load/tip-displacement) measured via can-tilever bending of the specimens, along with specimen dimensions, to infer an E-value for each material. How well do these values agree with other measure-ments of E? Discuss. 4. Use the critical elastic buckling load measured on the specimens you are given to estimate E for that material. How well do these values agree with other measurements of E? 5. Use the natural frequencies of the vibrating cantilever beams measured in the lab, along with the specimen dimensions and the appropriate mass values to estimate Young’s modulus, E. 6. Do the different test methods (tension test with strain gauges; instrumented cantilever and cantilever stiffness; natural frequency, buckling) provide consis-tent estimates of E? Discuss. 4 Review of Cantilever Beam Bending (See also Crandall, Dahl and Lardner, sections 3.5; 7.5; 8.3) Shear force and bending moment equations: The distributed loading (force/length) is q(x); transverse shear force is V (x); and bending moment is M (x). For the cantilever, q(x) = 0 in 0 < x < L. (See the CDL text for sign conventions on positive deflection, shear force, distributed load, and bending moment.) dV (x) dx + q(x) = 0; dV (x) dx = 0 ⇒ V (x) = constant = −P ; (4.1) dM (x) dx +V (x) = 0; dM (x) dx = −V (x) = +P ; ⇒ M (x) = P x+constant = −P (L−x). (4.2) 2bXYPL{$\nu$},{$\nu$},{$\nu$},{$\nu$},hZbHere the integration constants can be directly evaluated from free body diagrams of the end-portion of the cantilever in the interval [x, L]. Axial stress at a generic point (x,y,z): M(x)yσxx(x, y, z) = − = P (L − x)y (4.3)I I Note: in writing the previous expressions for −M(x) = P (L − x), we have assumed that the origin of the x-axis is at the base of the cantilever (x = 0), and that the tip where load is applied is at x = L; thus, the generic coordinate value “x” measures distance from the base of the cantilever. The drawing of the coordinate axes in the figure can give rise to confusion here. Axial strain at a generic point (x,y,z): σxx(x, y, z) M(x)y�xx(x, y, z) = = − = P (L − x)y, (4.4)E EI EI where I ≡ � y2 dA is the area moment of inertia of the cross-section, which, for rectangular cross-sections of this orientation, is equal to: bh3 I = , (4.5)12 with b the width and h the thickness of the beam. Assuming that axial strain �(surf )(x) at position x has been measured on the surface of the beam (at y = h/2), in the presence of cantilever load P , we can obtain a| |local-strain-based estimate of the elastic modulus E as .M(x) h/2 6P (L − x)E = E(gaged cantilever) = | | = . (4.6)I �(surf )(x)| bh2 �(surf )(x)|| |3Here we use absolute value of the bending moment and the surface strain (even if it is measured on the compression side) in order to get a positive modulus. Convince yourself that this is mathematically correct in either case, y = ±h/2. An axially-oriented strain gage, mounted on the face of a cantilever beam, gives a signal proportional to the bending moment at its location; from the differential equation of moment equilibrium, the difference in signal (∝ ΔM) between two such gages, mounted a distance Δx apart, provides a direct measure of shear force (V = −ΔM/Δx). Lateral displacement v(x) and rotation φ(x): Within the assumptions of traditional elastic beam theory, the lateral