Question

A 6 meter tall concrete column 40 cm square is supporting a 4 MN compressive load. The concrete has an elastic modulus of 30 GPa, compressive strength of 28 MPa, tensile strength of 3.5 MPa and Poisson’s Ratio 0.3. The concrete is reinforced by 16 #8 W40 axial rebar (200 GPa elastic modulus, 275 MPa compressive strength), spaced 8 cm apart on each side, 8 cm from the surface of the column.

What is the compressive stress on the column?

What is the axial modulus of the reinforced concrete column?

What is the axial deflection (change in length) of the column?

What is the transverse change in dimension (change in width)?

Will this column fail? Why?

Answer 1

A ,Compressive stress= compressive force/area

=4*10^6/400^2 =25N/mm^2

B, Axial modulus is the ratio of stress in a single direction to strain in same direction .Here we have compressive stress., to find axial modulus we need strain in the corresponding direction

Axial modulus=compressive stress/ compressive strain

For that first we need to find to find change in length , change in length, =PL/AE

Where p= applied load ,L =length of column, A = Area of column, E=Modulus of elasticity

Change in length=4×10^6×6×1000/400^2×30×10^3=5mm

Compressive strain = change in length/ orginal length =5/6000

=8.33×10^-4

Axial modulus=25/8.33×10^-4=30012Mpa

C, Axial deflection =change in length=PL/AE=5mm(alread find in b part )

D, we have to find change in transverse dimension..for that we have to use the equation of poissons ratio=lateral strain /linear strain

Here linear strain= compressive strain=8.33×10^-4

Poissons ratio is given=0.3

0.3=lateral strain / linear strain

0.3=lateral strain/8.33×10^-4

Lateral strain=0.3×8.33×10^-4

=2.5×10^-4

Here we have to find change in transverse dimension,

Lateral strain = change in transverse dimension/orginal transvere dimension

2.5×10^-4= change in transverse dimension/400

Change in transverse dimension=0.1mm= 1cm

D, columb will fail ,according to table 21 of Is 456:2000,permissible compressive stress in concrete for M30 grade is 8 Mpa and for M25 is 6Mpa...so that for M28 grade the permissible stress will be between 6 and 8 Mpa..here the internal stress developed is 25 Mpa..as the internal stress is greater than permissible stress , the column will fail