beam design theory

8/11/2019 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

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9 - 4

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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beam design theory

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