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