Calculating the Dimensions of a Reinforced Concrete Column Under Stress

[anthonyk2013]

A concrete column having a modulus of elasticity of 20GN/m² is reinforced by two steel bars of 25mm diameter having modulus of 200GN/m².
Calculate the dimensions of a square section strut if the stress in the concrete is not to exceed 7MN/m² and the load is to be 400KN.

σ in conctete not to exceed 7*10⁶
Load =400KN
E of steel=200GN/m²
E of concrete=20GN/m²

Area of steel=3.14*25²/4*2=982*10^-3m²
Area of concrete=?

σ_c/E_c=σ_s/E_s

σ_c=σ_s*E_c/E_s

σ_c=σ_s*20/200

σ_c=σ_s*.1


7*10⁶=σ_s*.1

7/.1*10⁶=σ_s

σ_s=70*10⁶MN/M²

Wondering if I m on the right track?

 

[SteamKing]

You could indicate the units of your calculations.

 

[anthonyk2013]

You could indicate the units of your calculations.

Well do. Bad habit

 

[anthonyk2013]

Any help would be appreciated

 

[Chestermiller]

Any help would be appreciated

Let S be the side of the square cross section column. What is the total area of the cross section. What is the cross sectional area of the steel rods (this time calculate it correctly, taking into account that 1 mm = 0.001 m)? In terms of S, what is the cross sectional area of the concrete?

 

[anthonyk2013]

Let S be the side of the square cross section column. What is the total area of the cross section. What is the cross sectional area of the steel rods (this time calculate it correctly, taking into account that 1 mm = 0.001 m)? In terms of S, what is the cross sectional area of the concrete?

Area of the steel is 3.14*.025²/4=.0004908
Because there is two rods I multiply .0004908*2=.000981747m²

S=L*B

 

[Chestermiller]

Area of the steel is 3.14*.025²/4=.0004908
Because there is two rods I multiply .0004908*2=.000981747m²

S=L*B

What are L and B?

For the cross sectional area of the concrete, I get (S - 0.000981747) m².

 

[anthonyk2013]

What are L and B? Length x breath

For the cross sectional area of the concrete, I get (S - 0.000981747) m².

Yes S-0.000981747

 

[Chestermiller]

Yes S-0.000981747

Actually, S²-0.000981747. What is your recommendation as to what to do next in this problem?

 

[anthonyk2013]

I need the get area using the values I have.

do I use σ_c/E_c=σ_s/E_s

σ_c/σ_s=E_c/E_s

σ_c=F_c/A_c and σ_s=F_s/A_s

THEN F_c/A_c*A_s/F_s=E_c/E_s

 

[Chestermiller]

I need the get area using the values I have.

do I use σ_c/E_c=σ_s/E_s

σ_c/σ_s=E_c/E_s

σ_c=F_c/A_c and σ_s=F_s/A_s

THEN F_c/A_c*F_s/A_s=E_c/E_s

Let ε be the compressive strain in the steel and concrete. What value of ε causes the stress in the concrete to equal 7MN/m²? What is the stress in the bars? In terms of S, what is the compressive force in the bars and in the concrete? What are the sum total of the compressive forces? What value of S causes this total compressive force to equal 400 kN.

 

[anthonyk2013]

ε of concrete=ε in steel?

σ=E*ε

ε=σ/E ε=7MN/m₂/20N/m₂

Would the correct way the write this be ε=7*10⁶/20*10⁹=.35*10^-3

σ in steel=.35*200=70MN

Total force is 400KN so I need to find what load is on the steel.

 

[Chestermiller]

ε of concrete=ε in steel?

σ=E*ε

ε=σ/E ε=7MN/m₂/20N/m₂

Would the correct way the write this be ε=7*10⁶/20*10⁹=.35*10^-3

σ in steel=.35*200=70MN

Total force is 400KN so I need to find what load is on the steel.

So far, so good. Now, you know the stress in the steel. What is the load carried by the two steel bars?

 

[anthonyk2013]

So far, so good. Now, you know the stress in the steel. What is the load carried by the two steel bars?

σ=F/A→F=σ*A→F=70*10⁶*0.000981747m²=.06872MN/m²

or 69KN/m²

 

[anthonyk2013]

Total force is 400KN, force in steel is 69KN so 400KN-69KN=331KN

Force in concrete=331KN
σ in concrete=7MN/m²

Area=f/σ A=331*10³/7*10⁶=47.285*10³ or 47285

$\sqrt{}47285$=217.45mm²

Answer in my notes is 220mm²

 

[Chestermiller]

Total force is 400KN, force in steel is 69KN so 400KN-69KN=331KN

Force in concrete=331KN
σ in concrete=7MN/m²

Area=f/σ A=331*10³/7*10⁶=47.285*10³ or 47285

$\sqrt{}47285$=217.45mm²

Answer in my notes is 220mm²

You're still not being careful about units. You made two errors that canceled each other out. Incidentally, the length of the side of the square cannot be in mm². It has to be mm.

 

[anthonyk2013]

You're still not being careful about units. You made two errors that canceled each other out. Incidentally, the length of the side of the square cannot be in mm². It has to be mm.

217.45mm is what should have wrote?

 

[Chestermiller]

217.45mm is what should have wrote?

Not exactly. You had to add the area of the bars back to get the total area. Then take the square root. That's probably why you got 217 and the book had 220.

 

[anthonyk2013]

To get the dimensions I need to find the total area. Concrete +steel and square root that.

Thanks for your help Chester, big help again.