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Design of Columns
Design of Columns
Column- a member loaded in compression such that either its length or eccentric loading causes it to experience more than pure compression
Can be classified as
a. Long columns
b. Short Columns
Design of Columns
Long columns- fail due to structurally(buckling or elasticity failure)
Short columns- fail to materially (yielding failure)
When P reaches critical load, column becomes unstable and bending develops rapidly
Critical load depends on end conditions
Design of Columns
Parameters needed in determining the type of column
Radius of Gyration,r: 𝑟 =𝐼
𝐴
Effective Length,Le: 𝐿𝑒 =𝐿
𝐶,where C=End Condition Constant
Slenderness Ratio,λ:𝜆 =𝐿𝑒
𝑟
Design of ColumnsEnd Conditions and Effective Length of Columns
C=1/4 C=1 C=2 C=4 C=1
Design of Columns
Compare 𝐿𝑒
𝑟with
2𝜋2𝐶𝐸
𝑆𝑦
1
2
if 𝐿𝑒
𝑟>
2𝜋2𝐶𝐸
𝑆𝑦
1
2, long columns, use Euler’s Formula
if 𝐿𝑒
𝑟<
2𝜋2𝐶𝐸
𝑆𝑦
1
2, short columns, use J.B. Johnson’s Formula
Design of Columns
Central Loading:
Euler’s Formula:
J.B. Johnson’s Formula:
Factor of safety:
Where P=maximum allowable load𝑃𝑐𝑟 = 𝑁𝑃
Design of Columns
Eccentric Loading: For both ColumnsMaximum deflection:
Factor of safety:Where P=maximum allowable load e=eccentricity, c=distnace of NA to extreme fiber
𝜎𝑚𝑎𝑥 = 𝜎𝑐𝑟 =𝑃𝑐𝑟𝐴
1 +𝑒𝑐
𝑟2𝑠𝑒𝑐
1
2
𝑃𝑐𝑟𝐸𝐴
𝐿𝑒𝑟
𝑃𝑐𝑟 = 𝑁𝑃
Design of Columns
Develop a specific Euler’s and J.B. Johnson’s Eqautions for the sizes of columns having
a) Round cross sections, diameter d is required
b) Rectangular cross section, h<b, base b is required
Design of Columns
Specify the diameter of a round column 1.5m long that is to carry a maximum load estimated to be 22kN. Use a design factor of 4 and consider the ends as pinned rounded. The column material selected has a minimum yiled strength of 500 Mpa at a modulus of elasticity of 207 Gpa.
Design of Columns
A round steel rod made of structural steel,with yield strength of 48ksi and E=30Mpsi, is to be used as a column, centrally loaded with 10 kips; N = 3 . Determine the diameter for L = 25 in.
Design of Columns
The column above is subjected to an axial compressive load of 15 kips. Made of AISI C1030, as rolled(Sy=51 ksi and E=30Mpsi), it has sectional length of 20 in. Assume a loose fit with the pins. What is
a. the critical load for this column
b. the design factor
c. the equivalent stress under a load of 15 kips
d. Maximum stress for the foregoing load, when eccentricity ratio=0.25 and eccentricity is Le/400