## Analysis of RC circular sections – theory and code

The basic tasks involved in analysing the stresses and strains in a reinforced concrete section under combined bending and axial load are:

Assuming a horizontal beam with the top of the beam in compression and the bottom in tension

1. Find the depth of the Neutral Axis below the top face, that is the depth of the line along which the strain is zero
2. Find the stress in the concrete at the top face
3. Find stresses at other levels by simple proportion
4. Find forces and bending moments
5. Confirm that the calculated total force and moment in the section are equal to the applied load

The depth of the Neutral Axis is controlled by the eccentricity of the reaction force, which must be equal to the eccentricity of the applied load:

 Mr/Pr = M/P = E

where the bending moment is taken about any convenient axis, which for the purposes of this analysis will be taken as the top fibre of the section.

It can be shown that:

Pr= KQx
and
Mr = KIx

where Qx and Ix are the first and second moments of area about the Neutral Axis, at depth X below the top (most compressive) fibre of the section, and K = Top fibre stress / X

Hence  KIx / KQx = Ix / Qx = E + X

Polynomial equations to determine the value of X are derived for any section made up of a series of trapezoidal layers in a previous article, but if we wish to apply this equation to a circular section it is more convenient to solve by iteration.  The actual equation used in the analysis presented here is:

Qx(E + X) / Ix – 1 = 0

which behaves nicely as Q approaches zero.

The procedure to find X for a circular section is then:

1. Assuming that the reinforcement is distributed evenly around the circle, find the depth of each reinforcement layer from the top face.
2. Find the transformed area of a reinforcing bar in tension and compression.  That is the bar area x Steel Elastic Modulus / Concrete Elastic Modulus in tension, (since the bar displaces an equal area of concrete) bar area x (Steel Elastic Modulus / Concrete Elastic Modulus – 1) in compression.
3. Find the transformed area properties (area, level of centroid, first moment of area about layer depth,   of area about centroid depth) for the neutral axis at each steel layer, adjusting the transformed area for the numbers of bars in compression and tension.
4. Find the composite (concrete + steel) Ix and Qx for the Neutral Axis at each reinforcement layer, and hence find which layers it lies between.
5. Use Brent’s Method to find the Neutral Axis position that satisfies the governing equation given above.
6. Find the composite area properties for the Neutral Axis at this layer.
7. Hence find the stresses, strains, forces and bending moments in the section.
8. Check that the calculated total axial force and bending moment are equal to the applied loads.

Excerpts of the VBA code are shown below.  Full open source code is included in the download file: RC design functions5.zip

Composite area properties about an axis at depth X below the top fibre:

```Function CircleQxoI(SectData As Variant, X As Double) As Variant
Dim R As Double, H As Double, Theta As Double, A As Double, Xcr As Double
Dim Icc As Double, Iccent As Double, Icb As Double, Qcb As Double
Dim Ast As Double, Istc As Double, Xsc As Double, Ecc As Double
Dim Qsb As Double, Isb As Double, outA(1 To 8, 1 To 1), CallFunc As Long

If TypeName(SectData) = "Range" Then SectData = SectData.Value2
R = SectData(1)
Ast = SectData(2)
Xsc = SectData(3)
Istc = SectData(4)
Ecc = SectData(5)
CallFunc = SectData(6)

' Concrete properties
If X > 0 Then
If X < 2 * R Then
H = (R ^ 2 - (R - X) ^ 2) ^ 0.5
Theta = WorksheetFunction.Atan2((R - X), H)
A = R ^ 2 * (Theta - (Sin(2 * Theta) / 2))
Xcr = (2 * R / 3) * ((Sin(Theta) ^ 3) / (Theta - Sin(Theta) * Cos(Theta)))
Qcb = A * (X - (R - Xcr))
Icc = A * R ^ 2 / 4 * (1 + (2 * (Sin(Theta)) ^ 3 * Cos(Theta)) / ((Theta - Sin(Theta) * Cos(Theta))))
Iccent = Icc - A * Xcr ^ 2
Else
A = Pi * R ^ 2
Qcb = A * (X - R)
Iccent = A * R ^ 2 / 4
End If
Icb = Iccent + A * (Xcr - (R - X)) ^ 2

Else
Qcb = 0
Icb = 0
End If
' Reinforcement properties
Qsb = Ast * (X - (R - Xsc))
Isb = Istc + Ast * ((R - Xsc) - X) ^ 2

If CallFunc = 1 Then
CircleQxoI = ((Qcb + Qsb) * (Ecc + X) / (Icb + Isb)) - 1

Else
outA(1, 1) = X
outA(2, 1) = Icb
outA(3, 1) = Isb
outA(4, 1) = A
outA(5, 1) = Ast
outA(6, 1) = A + Ast
outA(7, 1) = Qcb
outA(8, 1) = Qsb
CircleQxoI = outA
End If

End Function
```

Use Brent’s Method to find the depth of the Neutral Axis within a specified layer

```Function QuadBrent(Func As String, ByVal Ak As Double, ByVal Bk As Double, Coefficients As Variant, _
Optional YTol As Double = 0.00000000000001, Optional MaxIts As Long = gMaxIts, _
Optional Subr As Long = 1, Optional XTol As Double = 0.0000000001) As Variant
Dim Ck As Double, Sk As Double, Dk As Double, XRange As Double
Dim Yak As Double, Ybk As Double, Yck As Double, Ysk As Double, Ydk As Double
Dim i As Long, LastStepB As Boolean, Temp As Double, ResA(1 To 3, 1 To 1) As Double, SubName As String

If TypeName(Coefficients) = "Range" Then Coefficients = Coefficients.Value2

Yak = Application.Run(Func, Coefficients, Ak)
Ybk = Application.Run(Func, Coefficients, Bk)

If (Yak < 0 And Ybk < 0) Or (Yak > 0 And Ybk > 0) Then
QuadBrent = "Yak and Ybk must have different signs"
Exit Function
End If

If Abs(Yak) < Abs(Ybk) Then
Temp = Ak
Ak = Bk
Bk = Temp
Temp = Yak
Yak = Ybk
Ybk = Temp
End If

Ck = Ak
Yck = Yak
LastStepB = True

For i = 1 To MaxIts

XRange = (Bk - Ak)

If (Abs(Ybk) < YTol) Then Exit For

If Yak <> Yck And Ybk <> Yck Then

If Subr = 1 Then SubName = "solveInvQuad" Else SubName = "solveQuad"
Sk = Application.Run(SubName, Ak, Yak, Bk, Ybk, Ck, Yck) '

Else
Sk = Bk - (Ybk) * (Bk - Ak) / (Ybk - Yak)
End If

If ((Sk - (3 * Ak + Bk) / 4) * (Sk - Bk)) > 0 Then
Sk = (Ak + Bk) / 2
LastStepB = True
ElseIf LastStepB = True Then
If Abs(Sk - Bk) >= Abs(Bk - Ck) / 2 Or (Abs(Bk - Ck) < Abs(XTol)) Then
Sk = (Ak + Bk) / 2
LastStepB = True
Else
LastStepB = False
End If
ElseIf Abs(Sk - Bk) >= Abs(Ck - Dk) / 2 Or Abs(Ck - Dk) < Abs(XTol) Then
Sk = (Ak + Bk) / 2
LastStepB = True
Else
LastStepB = False
End If

' Use Application.run to evaluate Func for the value Sk
Ysk = Application.Run(Func, Coefficients, Sk)
'----------------------------------------
Dk = Ck
Ydk = Yck
Ck = Bk
Yck = Ybk

If Yak * Ysk < 0 Then
Bk = Sk
Ybk = Ysk
Else
Ak = Sk
Yak = Ysk
End If
If Abs(Yak) < Abs(Ybk) Then
Temp = Ak
Ak = Bk
Bk = Temp
Temp = Yak
Yak = Ybk
Ybk = Temp
End If

Next i

If Abs(Ybk) < Abs(Ysk) Then
ResA(1, 1) = Bk
ResA(3, 1) = Ybk
Else:
ResA(1, 1) = Sk
ResA(3, 1) = Ysk
End If
ResA(2, 1) = i - 1
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