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The second part of the lab will focus on using the ISM 04 Beam Apparatus to determine the deflection f point-loaded simply-supported beams made of steel, brass or aluminum. Using the deflection measurements, an examination of the relationship between deflection and material properties will be shown along with a comparison of the materials based on their strengths and deflections, both theoretical and experimental. The last objective of this laboratory is to verify the theory of pure bending using the SMASH Beam Apparatus.

Introduction Engineers use beams to support loads over a span length.

These beams are structural members that are only loaded non-axially causing them to be objected to bending. “A piece is said to be in bending if the forces act on a piece of material in such a way that they tend to induce compressive stresses over one part of a cross section of the piece and tensile stresses over the remaining part” (Ref. 1). This definition of bending is illustrated below in Figure 1.

Figure 1 – Bending on a Cross Section 5 It can be seen from Figure 1 that the compressive force, C, and the tensile force, T, acting on the member are equal in magnitude because of equilibrium.

Therefore, the compressive force and the tensile force form a force couple whose moment is equal to either the tensile force multiplied by the moment arm or the compressive force multiplied by the moment arm. The moment arm is denoted, e, in Figure 1. Figure 2 – Bending Action caused by Transverse Loads Figure 2, shown above, is an illustration of bending action in a beam acted upon by transverse loads.

Bending may be accompanied by direct stress, transverse shear or torsion’s shear, however for convenience; bending stresses may be considered separately (Ref. 1).

In order to separate the stresses it is assumed that the loads are applied in the following manner: loads act in a plane of symmetry, o twisting occurs, deflections are parallel to the plane of the loads, and no longitudinal forces are induced by the loads or by the supports (Ref. 1). A beam or part of a beam that is only acted on by the bending stresses is said to be in a condition of “pure bending. ” However for many circumstances bending is accompany by transverse shear. The term “flexure” is used to refer to bending tests of beams subjected to transverse loading (Ref. ). A visual illustration of the transverse shear and bending moment can 6 be seen in the shear and bending moment diagrams of the beam. It is important o note that in a symmetrical 2-point loading scenario, the center portion of the beam will be in a condition of pure bending as such the bending stresses may be considered separately. “Deflection” of a beam is the displacement of a point on the neutral surface of a beam from its original position under the action of applied loads (Ref. 1).

Before the proportional limit of the material, the deflection, A, can be calculated using the moment of inertia, modulus of elasticity along with other section properties that will depend on the given situation imposed on the beam. The position of the load, the type of load applied on the beam, and the Engel of beam are examples of section properties that depend on the situation. The deflection equations for two common cases are listed below in equations (1) and (2). Case 1: Center deflection of a simple beam with freely supported ends and concentrated load, P, at the mid-span (Ref. ). Equation (1) where: A = deflection, (mm) P = load, (N) L = length of beam, (mm) E = modulus of elasticity (N/mm) = moment of inertia of section about the neutral axis, (mm) 7 Case 2: Center deflection of a simple beam with concentrated loads, each equal to P, at third points of span (Ref. 1). Equation (2) P load, (N) moment of inertia of section about the neutral axis, (mm) Deflection is a measure of overall stiffness of a given beam and can be seen to be a function of the stiffness of the material and proportions of the piece (Ref. 1).

Deflection measurements give the engineer a way to calculate the modulus of elasticity for a material in flexure. The stiffness of a given material is calculated using the following equation: Equation (3) p = load, (N) Stiffness (N/m) 8 A beam may fail in any of the following ways: A beam may fail by yielding of extreme fibers, in long span beams compression fivers act like those of a column ND fail by buckling, in webbed members excessive shear stress may occur and stress concentrations may build up in parts of beam adjacent to bearing blocks (Ref. 1).

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