Calculating the updated neutral axis and cross section properties

When nonlinear materials are present in the cross section, the neutral axis of the cross section shifts from its original position due to the difference of stiffness over the cross section. When concrete cracks, for example, the neutral axis will shift to the stiffer, uncracked side of the cross section so that equilibrium is maintained. The shift in the neutral axis needs to be taken into account by calculating the updated cross section bending stiffness about the shifted neutral axis. A study done by Walls (2016) demonstrated that if the shift of the neutral axis is not taken into account, the error in deflection calculations could be significant. The updated neutral axis position is calculated using the following expression:

𝑐′ = 𝑖+1 ∫ 𝐸(𝑦)𝑦𝑑𝐴𝑖 ∫ 𝐸(𝑦)𝑑𝐴𝑖 = ∑𝑛𝑘=1 𝑖𝐸𝑘𝐴𝑘𝑦𝑘 𝑖 ∑𝑛𝑘=1 𝑖𝐸𝑘𝐴𝑘 𝑖 (4.2) Where,

 𝑐′ refers to the updated neutral axis position, measured relative to the datum position;  𝑖 refers to the current iteration number;

 𝐸_𝑘 is modulus of elasticity for the 𝑘𝑡ℎ _element;

 𝐴_𝑘 is the area of a specific element; and

 𝑦_𝑘 is the distance measured from the point about which the internal moment is being calculated, to the neutral axis of the element being investigated.

The bending stiffness about the shifted neutral axis is calculated using the following expression:

(𝐸𝐼) 𝑖+1 _{= ∫ 𝐸}𝑖 _(𝑦)𝑦2_{𝑑𝐴 +}𝑃𝑐 ′ 𝑘 = ∑ 𝐸𝑘(𝐴𝑘𝑦𝑘 2_{+ 𝐼} 𝑘) 𝑖 𝑛 𝑘=1 𝑖 +𝑁𝑥𝑐′ 𝑘 (4.3) Where:

 𝐼𝑘 is the second moment of inertia of the element being investigated;

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 𝑘 is the curvature.

It is recommended that the datum axis be chosen to coincide with the original, unshifted neutral axis. When this assumption is made, the applied axial force acts on the datum axis throughout the Model Column Method procedure. As the neutral axis shifts, the axial force will have an effect on the bending stiffness calculation and will have to be taken into account. This effect is taken into account by the second term in Equation (4.3). It is important to note that the curvature in this term is the total curvature, resulting from the total internal moment on the cross section. The total internal moment is comprised of the moment resulting from the external wind force and the P-delta effects resulting from large lateral deflections. The total moment also includes the moment caused by the eccentricity of the axial force, which is a result of the shifted neutral axis. When there is no moment caused by the lateral wind forces or P-delta effects, the last term of Equation (4.3) can be discarded, since the neutral axis will not shift in this case.

An alternative to calculating the bending stiffness of the cross section is to make use of the Euler- Bernoulli beam theory. From this, a relationship between moment, bending stiffness and curvature is obtained. When using the Model Column Method to analyse a cross section, the curvature is assumed for which the moment needs to be found. The moment and the curvature is thus known after the completion of the analysis. The bending stiffness is then simply found by dividing the moment by the curvature. Once again, the total moment is used in this relationship. The alternative relationship that is used to calculate the bending stiffness can be expressed as follows:

𝐸𝐼 = 𝑀𝑡𝑜𝑡𝑎𝑙 𝑘𝑡𝑜𝑡𝑎𝑙 =(𝑀𝑤𝑖𝑛𝑑+ 𝑀𝑃−𝑑𝑒𝑙𝑡𝑎− 𝑁𝑥𝑐′) 𝑘𝑡𝑜𝑡𝑎𝑙 (4.4) Where:

 𝑀𝑤𝑖𝑛𝑑 is the moment caused by the external lateral wind force;

 𝑀_{𝑃−𝑑𝑒𝑙𝑡𝑎} is the moment caused by the own weight of the TWG when subject to lateral displacements; and

 𝑁𝑥𝑐′ is the term that takes account of the eccentricity of the axial force due to the shift of the

neutral axis.

The 𝑁𝑥𝑐′-term in Equation (4.4) above is subtracted from the other two moment terms because an

upward shift of the neutral axis would cause the axial force to decrease the total internal moment. Figure 4.2 is provided below to demonstrate this situation.

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Figure 4.2: Typical cross section subject to a moment and axial load

To clarify the process of how the neutral axis migrates when a cross section is subject to an axial load and moment, consider Figure 4.2. The moment acting on the cross section in Figure 4.2 comprises the moment caused by lateral loads and P-delta effects. This moment causes tension strains at the bottom of the cross section to occur, which results in cracking of the concrete. The reduction of stiffness at the bottom of the cross section results in the neutral axis to migrate upward, to the stiffer part of the cross section. Due to the axial force, 𝑁𝑥, still acting through the original neutral

axis position, a restoring moment is generated. This moment is equal to the axial force multiplied by the distance that the neutral axis shifts, which is denoted as c’. The effect of the restoring moment on the bending stiffness is considered by the inclusion of the last term in Equation (4.3) and Equation (4.4) above.