# Beam Bending and Superposition

Aerospace Engineering

Ashby Building

Stranmillis Road

Belfast

BT9 5AH

Stage 1 Laboratory Report

Beam Bending and Superposition

Author

TutorProf. Menary

Semester1

Date28/11/2011

Summary

An investigation into beam bending and superposition. Being able to analyse how beams bend is an essential tool for all engineers. By using mathematics and material properties, engineers are able to compute structural deformation thus verifying a structures fitness for use. In this experiment a simply supported beam of aluminium is loaded with point forces in three different cases. A clock gauge is positioned in the middle of the beam to measure the deflection. The results of a complex arrangement of forces can be deduced by the superposition of more simple cases. Superposition is possible only when the response of the structure is linear, e.g. when deflection is directly proportional to the applied load. Also the experimental and theoretical deflections of the beam will be compared and a percentage error obtained. There was a second test performed in this investigation demonstrating the influence the 2nd moment of area, also known as the second moment of inertia, had on the load carrying capacity of the beam. The results from test 1 show that it is possible to deduce the deflection of the beam when loaded with point forces by superposition. Results from test 2 show that the deflection of a beam is influenced greatly by its moment of inertia, i.e. with a greater value of inertia there is a smaller deflection.

Contents

Page

Nomenclatureii

Introduction1

Apparatus and Procedure5

Results6

Discussion10

Conclusions11

References11

Bibliography11

.

Nomenclature

Symbols:

EYoung’s Modulus (Pa)

LLength of beam (m)

bBreadth of beam (mm)

dDepth of beam (mm)

ISecond moment of inertia (m4)

FForce (N)

δ Deflection (mm)

Introduction

Due to the amount of physical applications where the total deflection that can be tolerated must be limited, the ability to predict such deflections is a necessity. An example of this is designing a steel frame to hold a glazed panel, during which only minimal deflection is allowed to prevent the glass fracturing [1]. The limiting factor in the design of beams loaded by weights is often either the permissible deflection or the bending moment. For given loads, the positions at which they are placed determine the deflection and bending moment along the beam; therefore deflection and bending moment can be controlled to some extent by judicious sequencing of the loads. Unfortunately it can be computationally difficult to determine the best sequence of loads on the beam [2].

The deflection of a beam depends on its length, its cross-sectional shape, the material, where the deflecting force is applied, and how the beam is supported [3]. According to the method of superposition, it is possible to break a complex loading problem down into a combination of simple loading conditions and construct the solution by adding the solutions to the simple loading conditions together. Superposition is only possible when all the load-deflection-slope reactions relations are linear [4]. In other words, the deflection at any point in a beam subject to multiple loading is equal to the sum of the deflections caused by each load acting separately [2]. It must be appreciated, however, that the principle of superposition is only valid whilst the beam material remains elastic [5]

In test 1 three different loading cases were set up. In each case a mass was loaded on to a single point on the beam, this acted as a point force. The deflection was measured for each mass, starting with 0 kg and ending with 80 kg. The same beam is used for each test; its properties are listed in table 1.

Table 1

Young’s Modulus (E)| 70| GN/m2|

Length (L)| 1.397| M|

Breadth (b)| 25.4| mm|

Depth (d)...

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