Amending the Traditional ‘ACI Commentary 𝐽 𝑐 Method’ and Other Sources

This article aims to amend the traditional formulas for polar moment of inertia suggested in Section R8.4.4.2.3 of ACI 318-19 and other sources. The authors claim that these formulas have been incorrectly derived as far back as 1960s due to an incorrect implementation of Steiner’s theorem (parallel axis theorem) for sections spanning in three-dimensional space. To support the claim, a formal proof using elementary calculus is presented in order to obtain the correct formulas with mathematical rigor. Then, the implications of using the traditional formulas versus the correct ones are investigated with the solution of a design problem related to a combined footing.


Introduction
The traditional 'ACI Commentary   Method,' presented nowadays in Section R8.4.4.2.3 of ACI 318-19 [1], has been extensively used in design and other sources [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] to check the maximum shear stresses developed around a slabcolumn connection due to the effect produced by an unbalanced moment at the slab-column connection.In this type of connections, the critical (shear) section is usually located at /2 from the face of the column, as in Fig. 1 which depicts the critical section of an interior column with rectangular cross section  1 -by- 2 and effective slab depth .Because the actual distribution of shear stresses along the critical section is in most cases complex to determine, the ACI Code still permits the use of the theory of elasticity along with the further assumption that the 'connection's (horizontal) plane remains plane after deformation' to estimate the order of magnitude of such stresses (Section 8.4.4.2.3 of ACI 318-19).Hence, under the premise that shear stresses will remain essentially linear about the centroid of the critical section, the shear stress,   , developed at point  with coordinate  can readily be determined by: where   is the shear force acting on the critical section (positive if the force acts in the positive -direction),  0 is the perimeter of the critical section,  is the distance from the extreme compression fiber to the centroid of the longitudinal reinforcement on the tension side of the slab,   is the fraction of unbalanced moment that is to be transferred by shear stresses on the critical section,   is the unbalanced moment acting on the slab-column connection (positive if the moment acts in the positive -direction),     is the torsional moment acting on the critical section,  ̅  is the (centroidal) polar moment of inertia of the critical section, and  is the location of point  with coordinate  where the shear stress,   (), is to be computed.Furthermore, Sections 8.4.4.2.2 and 8.4.2.2.2 of ACI 318-19 define the fraction   as: where  1 is the dimension of the critical section measured in the direction perpendicular to the torsional moment     , and  2 is the dimension of the critical section perpendicular to  1 (Figure 1).It is worth mentioning that if   () is positive, the shear stress (at point  with coordinate ) acts in the positive -direction, based on the adopted sign convention defined above.The polar moment of inertia,  ̅  (also referred to as   ), of an interior slab-column connection has been historically presented in the ACI Code (e.g., as of ACI 318-63 [2]) as: However, this expression, first suggested by Di Stasio and Van Buren in 1960 [8], is not entirely correct since it does not consider the fact that the two side faces parallel to     have a definite height equal to , which also contributes to the total polar moment of inertia of the critical section.To account for this extra contribution, the formula for  ̅  should rather be written as: where the missing term in Equation 3 is: Although Section R8.4.4.2.3 of ACI 318-19 [1] does point out that   is a property 'analogous' to polar moment of inertia, none of the sources consulted [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] discusses the reason why the missing term can be neglected.For instance, Wight & MacGregor [17] tacitly states that the polar moment of inertia,  ̅  , of critical sections spanning in threedimensional space is ill-defined and that one way to consider the contribution of the side faces (i.e., those faces parallel to     ) is to reduce their geometry as points with area-an unnecessary simplification to make, nonetheless, as we will see later.In this case, the missing term is a consequence of the simplification.Moreover, neglecting the missing term does not make the formula any nicer or simpler to use but it does cause the formula to lose a certain amount of information when dealing with deep slab-column connections.
In the following section, a formal proof of Equation 4 is provided to demonstrate the correctness of authors' claim using elementary calculus.This is followed by the proofs of the   expressions required for an edge and a corner slabcolumn connection.The article ends with a practical example to investigate the numerical differences between the traditional and the correct   expressions in the solution of a design problem related to a combined footing.

Definition of Polar Moment of Inertia
In mechanics, the polar moment of inertia of a section about the -axis is defined by.
where d is the differential element of total area , and  is the (shortest) distance measured from the -axis to d.The quantity  2 is already given as a function of  and  because a set of Cartesian coordinate systems will be used in the following sections.

Proof for an Interior Slab-column Connection
The critical section of an interior slab-column connection is usually comprised by four mutually orthogonal faces, as in Figure 2-a.The centroid, , of such a section is located at  1 /2 from face 2, at  2 /2 from face 1, and at /2 from the bottom of the critical section.For convenience, a (standard) Cartesian coordinate system is defined at point , such that: (1) the -axis is oriented parallel to the torsional moment     , and (2) the plane generated by the  and  axes is parallel to the slab's plane.As observed, the quantities  2 and d for faces 1 and 3 are given by: ) and for faces 2 and 4 they are given by: This result should not be surprising.The first summand is the usual  ̅  for faces 1 and 3; i.e., the sum of the moments of inertia about the  and  axes for faces 1 and 3.The second summand is nothing but the result of implementing Steiner's theorem on faces 2 and 4.

Proof for an Edge Slab-column Connection
Let's now consider the edge slab-column connection depicted in Figure 2-b.The centroid, , of the critical section is located at  ̅ from face 2, at  2 /2 from face 1, and at /2 from the bottom of the critical section, where: after noticing that  1 =  3 =  1 ,  2 =  2 , d 1 = d 3 = dd, d 2 = dd, and  0 = 2 1 +  2 .
A Cartesian coordinate system is once again defined at point  using the same orientation as before.This time, the quantities  2 and d for faces 1, 2 and 3 are given by: 2 2 (, ) =  ̅ 2 +  2 (13) Hence, the  ̅  of an edge slab-column connection is: Again, both these summands are the result of implementing Steiner's theorem on each face.

Proof for a Corner Slab-column Connection
Lastly, let's consider the corner slab-column connection depicted in Figure 2-c.The centroid, , of the critical section is located at  ̅ from face 2, at  ̅ from face 1, and at /2 from the bottom of the critical section, where: and similarly, whence  1 =  1 ,  2 =  2 , d 1 = dd, d 2 = dd, and  0 =  1 +  2 .
The quantities  2 and d for faces 1 and 2 are given by: Hence, the  ̅  of a corner slab-column connection is: As before, this same result could have been inferred from Steiner's theorem applied on faces 1 and 2.

Example
The following example is taken from Wight & MacGregor [17] on reinforced concrete (Example 15-5, p. 845) to provide context of the situation discussed here.The example (Figure 3) considers a combined footing supporting a 16in-by-24-in exterior column ( 1 = 16 in and  2 = 24 in), and a 24-in-square interior column.The height of the footing is 36 in, giving an effective depth of approximately  = 32.5 in.The shear force and unbalanced moment acting on the critical section of the exterior column are   = 405 kips and   = −6,950 kips-in, following the notation and sign convention adopted in this work.
Other implications of having a large  are demonstrated below.
Ergo, as  tends to infinity, the ratio  , correct / , traditional tends to 21 13 ≈ 1.62, which means that numerical differences of up to 62% can be expected between both expressions.Furthermore, if   were sufficiently small compared to the effect produced by     , the ratio  , correct / , traditional would tend to 13  21 ≈ 0.62 as  tends to infinity.Hence, a design based on the traditional   expression can produce an overestimation of up to 62% in the worst-case load scenario.This is equivalent to adding an unwanted safety factor of 1.62 to the design of the slab-column connection.
Finally, it is observed that as  tends to zero, the ratio  , correct / , traditional tends to 1.This is why in shallow slabcolumn connections where  is sufficiently small (e.g., as in flat floor systems), no significant numerical differences are expected between  , traditional and  , correct .In those cases, both expressions produce the same order of magnitude.

Conclusions
This article has presented the correct formulas for polar moment of inertia needed in Section R8.4.4.2.3 of ACI 318-19 [1] and other sources (e.g., [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]).The formulas were derived for three different types of slab-column connections to show how to treat each case individually.For other types, a similar approach can be followed.The implications of using the traditional formula versus the correct formula in the design of an edge slab-column connection were presented in Section 3, from where it was shown that numerical differences of up to 62% can be expected if the slab-column connection is sufficiently thick.
The purpose of this article was twofold.First, to realize that Section R8.4.4.2.3 of ACI 318-19 needs to be amended so that the second paragraph of the commentary reads: "For an interior column,   may be calculated by:   = centroidal polar moment of inertia of assumed critical section, Second, to hope that other authors will eventually take note and adopt the correct   expressions for use in their new manuscript releases.The authors believe that using the correct expressions for   would lead to a more consistent implementation of the traditional 'ACI Commentary Jc Method' in the years to come, especially for deep slab-column connections resisting significant unbalanced moment.
It is now up to the research community to establish whether the proposed amendment to the Jc method produces a safe procedure to design deep slab-column connections as per ACI 318.It is suggested that more research is devoted to this topic in order to obtain more reliable designs in the future.
As a final remark, one may wonder why the ACI Code defines   as a property analogous to polar moment of inertia.One plausible reason for resorting to the analogy is that it is customary to think that the concept of polar moment of inertia is limited to sections that can be laid out on a flat sheet of paper.In mechanics, however, this concept is more general and can encompass three-dimensional space as well.The more accurate definition of   was given in Section 2.1.

Data Availability Statement
The data presented in this study are available in the article.

Figure 2 .
Figure 2. Critical sections for different types of slab-column connectionsSubstituting these quantities into Equation 5 yields the  ̅  of an interior slab-column connection: