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
- Find the depth of the Neutral Axis below the top face, that is the depth of the line along which the strain is zero
- Find the stress in the concrete at the top face
- Find stresses at other levels by simple proportion
- Find forces and bending moments
- 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:
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:
- Assuming that the reinforcement is distributed evenly around the circle, find the depth of each reinforcement layer from the top face.
- 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.
- 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.
- Find the composite (concrete + steel) Ix and Qx for the Neutral Axis at each reinforcement layer, and hence find which layers it lies between.
- Use Brent’s Method to find the Neutral Axis position that satisfies the governing equation given above.
- Find the composite area properties for the Neutral Axis at this layer.
- Hence find the stresses, strains, forces and bending moments in the section.
- 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 ' SolveInvQuad uses inverse quadratic interpolation, SolveQuad uses Mullers method of quadratic interpolation 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 QuadBrent = ResA End Function