2-Lectures LEC 14

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Chapter 5 Deflection and StiffnessDeflection and Stiffness

A. Aziz Bazoune

2/5/2012 Chapter 5 1



OutlineOutline1. Spring Rates

2. Deflection in Tension, Compression & Torsion

3. Deflection due to Bending

4. Strain Energy

5. Castigliano·s Theorem

6. Statically Indeterminate Problems

7. Compression MembersLong Columns with Central Loading

Intermediate Length Columns with Central Loading

2/5/2012 Chapter 5 2



Solved ProblemsSolved Problems

LecLec. 14. 14

2/5/2012 Chapter 5 3



2/5/2012 Chapter 5 4



2/5/2012 Chapter 5 5



2/5/2012 Chapter 5 6



2/5/2012 Chapter 5 7



2/5/2012 Chapter 5 8

A column is pinned at both ends has a length of 4.50 ft and a solid circularcross-section of 1.25 in diameter. If it is made of steel AISI 1020 CD whatwould be a safe column loading?

Material AISI 1020 CD Steel = Sy

= 57 ksi (Table A-20)

E = 30 Mpsi (Table A-5)

Ends: Pinned = Rounded-rounded case = C=1 (Table 5-2)

Eq. (5-48)

Calculate the slenderness ratio

ProblemProblem

SolutionSolution

1/2 1/22 2 6

31

2 2 1 30 10

101.957 10 y

l CE

k S

T T¨ ¸ ¨ ¸v v v v¨ ¸

! ! !© ¹ © ¹© ¹ © ¹ vª º ª ºª º

l

k

¨ ¸© ¹ª º



2/5/2012 Chapter 5 9

Calculate the slenderness ratio

Since == Euler·s Column

Therefore

ProblemProblem

l k

4 2

2

64 4

4.5 12172.8

4 1.25 / 416

f t in

in

l l l k

I A d d

l l

d d

T T! !

v! ! ! !

1

l k l k "

2 2 6

2 2

30 109906

( / ) (172.8

1

)

cr P E C

A L k

T Tv v v¨ ¸! ! !

© ¹ª º

2 21.25

9906 9906 9906 121504 4

lbcr

D P A T T! v ! v v ! v v !



2/5/2012 Chapter 5 10

Determine the critical load on a steel column having a rectangular cross-

section, 12 mm X 18 mm and a length of 280 mm? It is proposed to use AISI1040 HR Steel. The lower end of the column is inserted into a close-fittingsocket and welded securely. The upper end is pinned.

Material AISI 1020 HR Steel

Sy = 290 MPa (Table A-20)

E = 207 GPa (Table A-5)

Fixed-rounded end Condition

C=1.2 (Table 5-2)

Radii of gyration

ProblemProblem

SolutionSolution

1 12 xx

xx

b I k

A

v! !

3h

b

v 185.196

12 12mm

h

h! ! !

v



2/5/2012 Chapter 5 12



2/5/2012 Chapter 5 13

Therefore

ProblemProblem

2

26

6 6

9

1

2

290 10 1290 10 80.8 234 10

2 21 07 1.2 0

2N/m

cr y

y

S P LS

k E C A T

T

¨ ¸! © ¹ª º

¨ ¸v! v ! v© ¹

vª º v

6 6 6234 10 234 10 18 12 10 50,544Ncr P A

! v v ! v v v v !



2/5/2012 Chapter 5 14

Design a rectangular section steel column having the following

specifications:

Width of the column (w ) is three times its thickness (t)

Maximum load that could be carried by the column is 100 kN

Material has a yield strength (Sy = 500 MPa) and a modulus of elasticity(E=207 GPa)

Assume end conditions constant C=1 in the weakest direction. Column length l=150 mm and 400 mm)

Cross-section area is

or

ProblemProblem

SolutionSolution

23 3 A w t t t t ! v ! v !

3t A!



2/5/2012 Chapter 5 15

Ixx

Iyy

or

ProblemProblem

SolutionSolution

23 3 A w t t t t ! v ! v !

3t A!

3 3 4 41 12 1 12 3 4 0.25 xx

I bh t t t t ! ! ! !

3 3 4 41 12 1 12 3 27 12 2.25 yy I hb t t t t ! ! ! !



2/5/2012 Chapter 5 16

ProblemProblem



2/5/2012 Chapter 5 17

Leonhard Euler

Portrait by Johann Georg Brucker

BornApril 15, 1707(1707-04-15)Basel, Switzerland

DiedSeptember 18 [O.S. September 7] 1783St Petersburg, R ussia

ResidencePrussiaR ussia

Switzerland

Nationality Swiss

Field Mathematics and physics

Institutions Imperial R ussian Academy of SciencesBerlin Academy

Alma mater University of Basel

Religion Calvinist[1]

Leonhard Paul Euler (pronounced Oiler ; IPA [l]) (April15, 1707 ² September 18 [O.S. September 7] 1783) was a

pioneering Swiss mathematician and physicist, who spentmost of his life in R ussia andGermany. He published more papers than any othermathematician of his time.[2]

Euler made important discoveries in fields as diverse ascalculus and graph theory. He also introduced much of themodern mathematical terminology and notation, particularlyfor mathematical analysis, such as the notion of a mathematical f

unction.

[3]

He is alsorenowned for his work in mechanics, optics, and astronomy.Euler is considered to be the preeminent mathematician of the 18th century and one of the greatest of all time. He isalsoone of the most prolific; his collected works fill 60²80 quartovolumes.[4] A statement attributed to Pierre-Simon Laplaceexpresses Euler's influence on mathematics: "Read Euler,

read Euler, he is the teacher (master) of

us all".

[5]

Euler was featured on the sixth series of the Swiss 10-francbanknote[6] and on numerous Swiss, German, and R ussianpostage stamps. The asteroid 2002 Euler was named in hishonor. He is also commemorated by the Lutheran Church ontheir Calendar of Saints on May 24.

http://en.wikipedia.org/wiki/Leonhard_Euler



2/5/2012 Chapter 5 18

Q UESTIONS ?Q UESTIONS ?

2-Lectures LEC 14

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