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Transcript of beam design theory
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Adapted from:
Material selection in Mechanical Design
by Michael F. Ashby
&Strength and Stiffness of Engineering Systems
by Frederick A. Leckie & Dominic J. Bello
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Superposition
9 - 2
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21.34 m long, 2.3 m wide, 51.4
mm thick, all laminated glass
skywalk.
Slip resistant top surace.
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A 10 kg weight is hung on the oar 2.05 m from the collar and the deflection atthis point is measured. A soft oar will deflect nearly 50 mm; a hard oar willdeflect nearly 50 mm.
The oar itself is made of spruce wood, weighs 4 to 4.3 kg and costs around 150to 250 dollars. CFRP (carbon fiber reinforced polymer) weighs about 3.9 kg.
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Atomic Force Microscope
The AFM consists of a cantileverwith a sharp tip (probe) atits end that is used toscan the specimen surface. Thecantilever is typically siliconor silicon nitridewith a tipradius of curvatureon the order of nanometers.
Whenthe tip is brought into proximity of a sample surface,forcesbetween the tip and the sample lead to a deflectionof the cantilever according to Hooke's law. Depending onthe situation, forces that are measured in AFM include
mechanical contact force, van der Waals forces, capillaryforces, chemical bonding, electrostatic forcesetc.
http://en.wikipedia.org/wiki/Cantileverhttp://en.wikipedia.org/wiki/Siliconhttp://en.wikipedia.org/wiki/Silicon_nitridehttp://en.wikipedia.org/wiki/Radius_of_curvature_(applications)http://en.wikipedia.org/wiki/Siliconhttp://en.wikipedia.org/wiki/Silicon_nitridehttp://en.wikipedia.org/wiki/Radius_of_curvature_(applications)http://en.wikipedia.org/wiki/Forcehttp://en.wikipedia.org/wiki/Forcehttp://en.wikipedia.org/wiki/Hooke's_lawhttp://en.wikipedia.org/wiki/Hooke's_lawhttp://en.wikipedia.org/wiki/Van_der_Waals_forcehttp://en.wikipedia.org/wiki/Capillarityhttp://en.wikipedia.org/wiki/Capillarityhttp://en.wikipedia.org/wiki/Chemical_bondhttp://en.wikipedia.org/wiki/Coulomb's_lawhttp://en.wikipedia.org/wiki/Van_der_Waals_forcehttp://en.wikipedia.org/wiki/Capillarityhttp://en.wikipedia.org/wiki/Capillarityhttp://en.wikipedia.org/wiki/Chemical_bondhttp://en.wikipedia.org/wiki/Coulomb's_lawhttp://en.wikipedia.org/wiki/Coulomb's_lawhttp://en.wikipedia.org/wiki/Coulomb's_lawhttp://en.wikipedia.org/wiki/Chemical_bondhttp://en.wikipedia.org/wiki/Capillarityhttp://en.wikipedia.org/wiki/Capillarityhttp://en.wikipedia.org/wiki/Van_der_Waals_forcehttp://en.wikipedia.org/wiki/Hooke's_lawhttp://en.wikipedia.org/wiki/Hooke's_lawhttp://en.wikipedia.org/wiki/Forcehttp://en.wikipedia.org/wiki/Radius_of_curvature_(applications)http://en.wikipedia.org/wiki/Silicon_nitridehttp://en.wikipedia.org/wiki/Siliconhttp://en.wikipedia.org/wiki/Cantilever -
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The total load is 1200 kN
Load on each beam 300 kN
Distributed load w per unit length 300/10 =30kN/m
Ans vmax= -29.7 mm ( is 1/337 of span)
smax= 142 MPa.
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Strength and Stiffness
StrengthMaterial strength and Factor of Safety
Stiffness
Deflection index f =deflection/spanFor structural members f = 1/240
For plastered ceilings f=1/360
For automobile chassis f = 1/240
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Length and Allowable loads for Steel
and Al
Steel and Al have a similar strength 250 MPa
Esteel= 200 Gpa, EAl= 70 GPa
FOS = 2, f =1/240 Since strength is same for both metals
allowable load based on maximum stress is
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Based on deflection
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For the aluminum beam, the load is limited by strength for L< 7.2 m. Longer
aluminium beams are limited by deflection.
It is especially important to consider both strength and deflection criteria in
aluminium structures. While replacing steel with aluminum of the same cross-
section (here, S510-128) reduces the weight (the density of aluminum is about
one-third that of steel), deflection can become the failure mode.
For the s510x128 steel beam, the allowable load based on strength is lower than
allowable load based on deflection for L< 20.6 m.
Steel has high modulus so deflection is rarely major concern in practical systems.
i iff d i b f
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For a given stiffness design a beam of
minimum mass
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Maximize
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For a given load design a beam of
minimum mass
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Maximize
The ratio b/d is known but b and d have to be determined. After the
material has been chosen beam has to be sized depending on strength
and stiffness.
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Ashbys Bending Stiffness Elastic
Shape Factor
For a beam stiffness =C (EI/l3)
For a circular cross-section I =pr4/4 and A = pr2
I= A2/4p
For a circular section shape factor is 1.
Shape factor is a measure of stiffness of cross section I per
its A2.
h f
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Shape factor
Rectangular section
I-Section
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Minimum weight design using shape
factor
For a given stiffness (In earlier slide we had taken b=h)
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For the wooden beam
For a rectangle with d/b =2
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