Beam Bending and Superposition
School of Mechanical and Aerospace Engineering Ashby Building Stranmillis Road Belfast BT9 5AH
Stage 1 Laboratory Report
Beam Bending and Superposition
Author Tutor Prof. Menary Semester 1 Date 28/11/2011
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 Nomenclature ii Introduction 1 Apparatus and Procedure 5 Results 6 Discussion 10 Conclusions 11 References 11 Bibliography 11 . Nomenclature
Symbols:
E Young’s Modulus (Pa)
L Length of beam (m) b Breadth of beam (mm) d Depth of beam (mm)
I Second moment of inertia (m4)
F Force (N) δ Deflection (mm)
Introduction
Due to
Stage 1 Laboratory Report
Beam Bending and Superposition
Author Tutor Prof. Menary Semester 1 Date 28/11/2011
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 Nomenclature ii Introduction 1 Apparatus and Procedure 5 Results 6 Discussion 10 Conclusions 11 References 11 Bibliography 11 . Nomenclature
Symbols:
E Young’s Modulus (Pa)
L Length of beam (m) b Breadth of beam (mm) d Depth of beam (mm)
I Second moment of inertia (m4)
F Force (N) δ Deflection (mm)
Introduction
Due to
References: 1. http://en.wikipedia.org/wiki/Deflection_(engineering) 28/11/2011 2. Samir V. Amiouny, John J. Bartholdi III, John H. Vande Vate, Minimizing Deflection and Bending Moment in a Beam with End Supports, 1991, School of Industrial and Systems Engineering, Georgia Institute of Technology 3. http://www.clag.org.uk/beam.html 29/11/2011 4. T J Lardner, R R Archer, Mechanics of Solids an Introduction, 1994, Mc Graw-Hill, pp. 370-375 5. E J Hearn, Mechanics of Materials 1, 3rd edition, 2001, Butterworth Heinemann, pp. 112 Bibliography 1. E J Hearn, Mechanics of Materials 1, 3rd edition, 2001, Butterworth Heinemann 2. B B Muvdi, J W Mc Nabb, Engineering Mechanics of Materials, 3rd edition, 1980, Springer-Verlag 3. T J Lardner, R R Archer, Mechanics of Solids an Introduction, 1994, Mc Graw-Hill