Bending of an Aluminum beam
CIVE 3202 A6E – Mechanics of Solids II (Winter 2013)
Experiment 2: Bending of an aluminum I-beam
Introduction
“Beams are long straight members that are subjected to loads perpendicular to their longitudinal axis and are classified according to the way they are supported”[1]. When a beam is subjected to an external load there are unseen internal forces within the beam that one must be aware of when implementing it into any design or structure. These internal forces create stress and strain that could result in failure or deformation. This lab looked at how an aluminum cantilevered beam performed under symmetric and unsymmetrical bending as well as the stresses and strains developed as a result.
Objective “To study the stress and strain induced in an I-beam under symmetric and unsymmetrical bending” [2].
Theory: σ – Normal stress (Mpa) ε – Strain (mm/mm)
M – Moment (kN∙m)
I – Moment of inertia (mm^6)
E – Modulus of elasticity (Mpa)
G – Modulus of elasticity (Mpa) v – Poisson’s ratio.
L – Length (m)
*Subscripts x, y, z indicate plane of reference.
The strain rosettes are orientated so that θb = 0, θc = -45, and θa = 45.
The strain gauge equations then simplify to εx = εb, εy= εc+ εa- εb, and γxy = εc- εa
Using Hooke’s Law: σx= εxE, σy= -v σx, τxy=γxyG
This Experiment consisted of symmetric and unsymmetrical bending. For symmetric bending the relevant theory is as follows:
Because the moment about the z-axis here is zero the equation equates to: Where: My = PLA.
When rotated 45 degrees:
My = PLA Cos(45) and Mz = PLA Sin(45) there is compressive stress along the y-x axis
The moment of inertia about the y-axis is found by determining the inertia of the shape and subtracting the imaginary parts as shown:
The max normal stress with be at the furthest distance
Experiment 2: Bending of an aluminum I-beam
Introduction
“Beams are long straight members that are subjected to loads perpendicular to their longitudinal axis and are classified according to the way they are supported”[1]. When a beam is subjected to an external load there are unseen internal forces within the beam that one must be aware of when implementing it into any design or structure. These internal forces create stress and strain that could result in failure or deformation. This lab looked at how an aluminum cantilevered beam performed under symmetric and unsymmetrical bending as well as the stresses and strains developed as a result.
Objective “To study the stress and strain induced in an I-beam under symmetric and unsymmetrical bending” [2].
Theory: σ – Normal stress (Mpa) ε – Strain (mm/mm)
M – Moment (kN∙m)
I – Moment of inertia (mm^6)
E – Modulus of elasticity (Mpa)
G – Modulus of elasticity (Mpa) v – Poisson’s ratio.
L – Length (m)
*Subscripts x, y, z indicate plane of reference.
The strain rosettes are orientated so that θb = 0, θc = -45, and θa = 45.
The strain gauge equations then simplify to εx = εb, εy= εc+ εa- εb, and γxy = εc- εa
Using Hooke’s Law: σx= εxE, σy= -v σx, τxy=γxyG
This Experiment consisted of symmetric and unsymmetrical bending. For symmetric bending the relevant theory is as follows:
Because the moment about the z-axis here is zero the equation equates to: Where: My = PLA.
When rotated 45 degrees:
My = PLA Cos(45) and Mz = PLA Sin(45) there is compressive stress along the y-x axis
The moment of inertia about the y-axis is found by determining the inertia of the shape and subtracting the imaginary parts as shown:
The max normal stress with be at the furthest distance
References: [1] – Mechanics of Materials Eight Edition, R.C. HIBBELER, 2011 [2], [3] - CIVE 3202 – Mechanics of Solids II (Winter 2013) Experiment 2: Bending of an aluminum I-beam. Obtained from: http://webct6.carleton.ca/webct/cobaltMainFrame.dowebct