1. Structures and Stiffness
B. Furman
K. Youssefi
20SEP2007
K. Youssefi and B. Furman
Engineering 10, SJSU

2. Outline Newton’s 3rd Law Hooke’s Law Stiffness Area moment of Inertia
Orientation of cross section and stiffness
Comparison of cross sections
Materials and stiffness
K. Youssefi and B. Furman
Engineering 10, SJSU

3. Newton’s 3rd Law
Lex III: Actioni contrariam semper et æqualem esse reactionem: sive corporum duorum actiones in se mutuo semper esse æquales et in partes contrarias dirigi.
To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.
K. Youssefi and B. Furman
Engineering 10, SJSU

4. Newton’s 3rd Law - example
T, tension
Free body diagram
T, tension
Isolate the body of interest
Put back the forces that are acting
M*g
K. Youssefi and B. Furman
Engineering 10, SJSU

5. Hooke’s Law Robert Hooke (1635-1702)
Materials resist loads (push or pull back) in response to applied loads
This ‘resistance’ is accomplished by deformation of the material (changing its shape)
Tension (stretching)
Compression (shortening)
Stretching or shortening of chemical bonds in atoms
The science of Elasticity concerns forces and deformations in materials
K. Youssefi and B. Furman
Engineering 10, SJSU

6. Hooke’s Law, cont.
Hooke found that deflection was proportional to load
Load, N
Slope of Load-Deflection curve:
slope, k
Deflection, mm
The “Stiffness”
K. Youssefi and B. Furman
Engineering 10, SJSU

7. Stiffness Stiffness in tension and compression F F L
Forces F applied, length L, cross-sectional area, A, and material property, E (Young’s modulus) F L A F
K. Youssefi and B. Furman
Engineering 10, SJSU

8. Stiffness, cont. Stiffness in bendingA Ri B Ro F
How does the material resist the applied load?
Think about what happens to the material as the beam bends
Inner “fibers” (A) are in compression (radius of curvature, Ri)
Outer “fibers” (B) are in tension (radius of curvature, Ro)
K. Youssefi and B. Furman
Engineering 10, SJSU

9. Review Question 1 Stiffness is defined as: Force/Area Deflection/Force
Force/Deflection
Force x Deflection
Mass/area
K. Youssefi and B. Furman
Engineering 10, SJSU

10. Concept of Area Moment of Inertia
The Area Moment of Inertia is an important parameter in determine the state of stress in a part (component, structure), the resistance to buckling, and the amount of deflection in a beam.
The area moment of inertia allows you to tell how stiff a structure is.
The Area Moment of Inertia, I, is a term used to describe the capacity of a cross-section (profile) to resist bending. It is always considered with respect to a reference axis, in the X or Y direction. It is a mathematical property of a section concerned with an area and how that area is distributed about the reference axis. The reference axis is usually a centroidal axis.
The higher the area moment of inertia, the less a structure deflects (higher stiffness)
Mathematically, the area moment of inertia appears in the denominator of the deflection equation, therefore;
K. Youssefi and B. Furman
Engineering 10, SJSU

11. Mathematical Equation for Area Moment of Inertia
Ixx = ∑ (Ai) (yi)2 = A1(y1)2 + A2(y2)2 + …..An(yn)2
A (total area) = A1 + A2 + ……..An
Area, A A2 A1 y2 y1 X X
K. Youssefi and B. Furman
Engineering 10, SJSU

12. Moment of Inertia – Comparison1
Load
Maximum distance of 4 inch to the centroid I2 2
2 x 8 beam
Load
2 x 8 beam I1
Maximum distance of 1 inch to the centroid
I2 > I1 , orientation 2 deflects less
K. Youssefi and B. Furman
Engineering 10, SJSU

13. Moment of Inertia Equations for Selected Profiles
Round hollow section π 64
I =
[(do)4 – (di)4] do di
Round solid section d
π (d)4 64
I =
Rectangular solid section b h
bh3 1
I = 12
BH3 - 1
I = 12
bh3
Rectangular hollow section H B h b b h 1
I = 12
hb3
K. Youssefi and B. Furman
Engineering 10, SJSU

14. Example – Optimization for Weight & Stiffness
Consider a solid rectangular section 2.0 inch wide by 1.0 high.
1.0
I = (1/12)bh3 = (1/12)(2)(1)3 = , Area = 2
2.0
Now, consider a hollow rectangular section 2.25 inch wide by 1.25 high by .125 thick. H B h b
B = 2.25, H = 1.25
b = 2.0, h = 1.0
I = (1/12)bh3 = (1/12)(2.25)(1.25)3 – (1/12)(2)(1)3= = .1995
Area = 2.25x1.25 – 2x1 = .8125
( )/(.1167) = .20 = 20% less deflection
( )/(2) = .6 = 60% lighter
Compare the weight of the two parts (same material and length), compare areas. Material and length is the same for both profiles.
So, for a slightly larger outside dimension section, 2.25x1.25 instead of 2 x 1, you can design a beam that is 20% stiffer and 60 % lighter
K. Youssefi and B. Furman
Engineering 10, SJSU

15. Rectangular Horizontal
Review Question 2
Which cross section has the larger I?
Rectangular Horizontal
Rectangular Vertical
K. Youssefi and B. Furman
Engineering 10, SJSU

16. Stiffness Comparisons for Different sections
Square
Box
Rectangular Horizontal
Rectangular Vertical
K. Youssefi and B. Furman
Engineering 10, SJSU

17. Material and Stiffness
E = Elasticity Modulus, a measure of material deformation under a load.
Deflection of a Cantilever Beam
Fixed end
Support
F = force
L = length
Y = deflection = FL3 / 3EI
The higher the value of E, the less a structure deflects (higher stiffness)
K. Youssefi and B. Furman
Engineering 10, SJSU

18. Modulus of Elasticity (E) of Materials
Steel is 3 times stiffer than Aluminum and 100 times stiffer than Plastics.
K. Youssefi and B. Furman
Engineering 10, SJSU

19. Density of Materials
Plastic is 7 times lighter than steel and 3 times lighter than aluminum.
K. Youssefi and B. Furman
Engineering 10, SJSU

20. Wind Turbine Structure
The support structure should be optimized for weight and stiffness.
Support Structure
K. Youssefi and B. Furman
Engineering 10, SJSU

21. Review Question 3 Which material has the higher stiffness? Steel
Aluminum
Alumina ceramic
Nylon
Unobtanium
K. Youssefi and B. Furman
Engineering 10, SJSU

22. Examples of Achieving Structural Stiffness
‘Boss’
‘Gusset’
‘Ribbing’
K. Youssefi and B. Furman
Engineering 10, SJSU

23. Examples of Achieving Structural Stiffness, cont.
K. Youssefi and B. Furman
Engineering 10, SJSU

24. Examples of Achieving Structural Stiffness, cont.
Welded ‘box’ construction
K. Youssefi and B. Furman
Engineering 10, SJSU

25. Examples of Achieving Structural Stiffness, cont.
K. Youssefi and B. Furman
Engineering 10, SJSU

26. Examples of Achieving Structural Stiffness, cont.
‘Flange’
K. Youssefi and B. Furman
Engineering 10, SJSU

27. Examples of Achieving Structural Stiffness, cont.
K. Youssefi and B. Furman
Engineering 10, SJSU