Fig. 2 shows the critical sections for the bending moments according to the code are [26]: For the axes parallel to the X axis are: a ₁, a ₂, a ₃ … a _n−1, a _n. For the axes parallel to the Y axis are: b ₁, b ₂, b ₃ … b _n−1, b _n, and d ₁, d ₂, d ₃ … d _n−1, d _n.

Critical sections of the bending shear are presented at a distance d from the junction of the column with the footing, and these sections appear on en, fn and gn axes (see Fig. 3).

Figure 3 Source: Prepared by the author.

Critical section for the punching shear appears at a distance d/2 from the junction of the column with the footing in the two directions (see Fig. 4).

Figure 4 Source: Prepared by the author.

The design of a rectangular footing (rectangular isolated footing) that supports a square column with the following information are presented: the column is of 40x40 cm; H = 2.0 m; P _D1 = 600 kN; P _L1 = 300 kN; M _Dx1 = 200 kN-m; M _Lx1 = 100 kN-m; M _Dy1 = 100 kN-m; M _Ly1 = 50 kN-m; f’ _c = 35 MPa; f _y = 420 MPa; q _a = 250 kN/m ² ; γ _ppz = 24 kN/m ³ ; γ _pps = 15 kN/m ³ . Where: H is the depth of the footing, P _D1 is the dead load, P _L1 is the live load, M _Dx1 is the moment around the X axis of the dead load, M _Lx1 is the moment around the X axis of the live load, M _Dy1 is the moment around the Y axis of the dead load, M _Ly1 is the moment around the Y axis of the live load.

The load and moments that act on soil are: P ₁ = 900 kN; M _x1 = 300 kN-m; M _y1 = 150 kN-m.

The thickness that meets the bending moments, bending shear, and punching shear is of 45 cm.

Substituting the values of σ _p = 215.95 kN/m ² , P ₁ = 900 kN, M _x1 = 300 kN-m, M _y1 = 150 kN-m, L _a = L _x /2 into eq. (15)-(23) of part I and using the MAPLE-15 software are obtained: S _tmin = 8.27 m ² , L _x = 2.03 m, L _y = 4.07 m, σ ₁ = 215.95 kN/m ² , σ ₂ = 108.86 kN/m ² , σ ₃ = 108.86 kN/m ² , σ ₄ = 1.77 kN/m ² .

Now the practical dimensions of the rectangular footing that supports a square column are: L _x = 2.05 m, L _y = 4.10 m.

Substituting the values of L _x = 2.05 m, L _y = 4.10 m in the same MAPLE-15 software are obtained: S _tmin = 8.40 m ² , L _x = 2.05 m, L _y = 4.10 m, σ ₁ = 211.55 kN/m ² , σ ₂ = 107.08 kN/m ² , σ ₃ = 107.08 kN/m ² , σ ₄ = 2.61 kN/m ² .

The load and the bending moments around of the X and Y axes (factored mechanical elements) are [26]: P _u1 = 1200 kN; M _ux1 = 400 kN-m; M _uy1 = 200 kN-m.

The design of a rectangular footing (rectangular combined footing) that supports two square columns with the following information are presented: the columns are of 40x40 cm; H = 2.0 m; L ₁ = 6.00 m; P _D1 = 600 kN; P _L1 = 300 kN; M _Dx1 = 200 kN-m; M _Lx1 = 100 kN-m; M _Dy1 = 100 kN-m; M _Ly1 = 50 kN-m; P _D2 = 1200 kN; P _L2 = 600 kN; M _Dx2 = 400 kN-m; M _Lx2 = 200 kN-m; M _Dy2 = 200 kN-m; M _Ly2 = 100 kN-m; f’ _c = 35 MPa; f _y = 420 MPa; q _a = 250 kN/m ² ; γ _ppz = 24 kN/m ³ ; γ _pps = 15 kN/m ³ .

The loads and moments that act on soil are: P ₁ = 900 kN; M _x1 = 300 kN-m; M _y1 = 150 kN-m; P ₂ = 1800 kN; M _x2 = 600 kN-m; M _y2 = 300 kN-m.

The thickness that meets the bending moments, bending shear, and punching shear is of 80 cm.

Substituting the values of σ _p = 212.80 kN/m ² , P ₁ = 900 kN, M _x1 = 300 kN-m, M _y1 = 150 kN-m, P ₂ = 1800 kN, M _x2 = 600 kN-m, M _y2 = 300 kN-m into eq. (15)-(23) of part I and using the MAPLE-15 software are obtained: S _tmin = 21.99 m ² , L _a = 0.20 m, L _x = 8.07 m, L _y = 2.73 m, M _xT = 900.00 kN-m, M _yT = 0.00 kN-m, R = 2700 kN, σ ₁ = 212.80 kN/m ² , σ ₂ = 212.80 kN/m ² , σ ₃ = 32.71 kN/m ² , σ ₄ = 32.71 kN/m ² .

Now the practical dimensions of the rectangular footing that supports two square columns are: L _x = 8.10 m, L _y = 2.75 m.

Substituting the values of L _x = 8.10 m, L _y = 2.75 m in the same MAPLE-15 software are obtained: S _tmin = 22.27 m ² , L _a = 0.22 m, L _x = 8.10 m, L _y = 2.75 m, M _xT = 900.00 kN-m, M _yT = 1.00 kN-m, R = 2700 kN, σ ₁ = 209.40 kN/m ² , σ ₂ = 209.33 kN/m ² , σ ₃ = 33.09 kN/m ² , σ ₄ = 33.02 kN/m ² .

The loads and the bending moments around of the X and Y axes (factored mechanical elements) are [26]: P _u1 = 1200 kN; M _ux1 = 400 kN-m; M _uy1 = 200 kN-m, P _u2 = 2400 kN; M _ux2 = 800 kN-m; M _uy2 = 400 kN-m.

The factored resultant loads and the factored resultant bending moments by Eqs. (16) to (18) of part I are obtained: R _u = 3600 kN, M _uxT = 1200 kN-m, M _uyT = 60 kN-m.

The design of a rectangular footing (rectangular combined footing) that supports three square columns with the following information are presented: the columns are of 40x40 cm; H = 2.0 m; L ₁ = 6.00 m; P _D1 = 600 kN; P _L1 = 300 kN; M _Dx1 = 200 kN-m; M _Lx1 = 100 kN-m; M _Dy1 = 100 kN-m; M _Ly1 = 50 kN-m; P _D2 = 1200 kN; P _L2 = 600 kN; M _Dx2 = 400 kN-m; M _Lx2 = 200 kN-m; M _Dy2 = 200 kN-m; M _Ly2 = 100 kN-m; P _D3 = 1200 kN; P _L3 = 600 kN; M _Dx3 = 400 kN-m; M _Lx3 = 200 kN-m; M _Dy3 = 200 kN-m; M _Ly3 = 100 kN-m; f’ _c = 35 MPa; f _y = 420 MPa; q _a = 250 kN/m ² ; γ _ppz = 24 kN/m ³ ; γ _pps = 15 kN/m ³ .

The loads and moments that act on soil are: P ₁ = 900 kN; M _x1 = 300 kN-m; M _y1 = 150 kN-m; P ₂ = 1800 kN; M _x2 = 600 kN-m; M _y2 = 300 kN-m; P ₃ = 1800 kN; M _x3 = 600 kN-m; M _y3 = 300 kN-m.

The thickness that meets the bending moments, bending shear, and punching shear is of 80 cm.

Substituting the values of σ _p = 212.80 kN/m ² , P ₁ = 900 kN, M _x1 = 300 kN-m, M _y1 = 150 kN-m, P ₂ = 1800 kN, M _x2 = 600 kN-m, M _y2 = 300 kN-m, P ₃ = 1800 kN, M _x3 = 600 kN-m, M _y3 = 300 kN-m into eq. (15)-(23) of part I and using the MAPLE-15 software are obtained: S _tmin = 37.47 m ² , L _a = 0.20 m, L _x = 14.47 m, L _y = 2.59 m, M _xT = 1500.00 kN-m, M _yT = 0.00 kN-m, R = 4500 kN, σ ₁ = 212.80 kN/m ² , σ ₂ = 212.80 kN/m ² , σ ₃ = 27.37 kN/m ² , σ ₄ = 27.37 kN/m ² .

Now the practical dimensions of the rectangular footing that supports two square columns are: L _x = 14.50 m, L _y = 2.60 m.

Substituting the values of L _x = 14.50 m, L _y = 2.60 m in the same MAPLE-15 software are obtained: S _tmin = 37.70 m ² , L _a = 0.22 m, L _x = 14.50 m, L _y = 2.60 m, M _xT = 1500.00 kN-m, M _yT = 1.00 kN-m, R = 4500 kN, σ ₁ = 211.19 kN/m ² , σ ₂ = 211.17 kN/m ² , σ ₃ = 27.56 kN/m ² , σ ₄ = 27.53 kN/m ² .

The loads and the bending moments around of the X and Y axes (factored mechanical elements) are [26]: P _u1 = 1200 kN; M _ux1 = 400 kN-m; M _uy1 = 200 kN-m, P _u2 = 2400 kN; M _ux2 = 800 kN-m; M _uy2 = 400 kN-m, P _u3 = 2400 kN; M _ux3 = 800 kN-m; M _uy3 = 400 kN-m.

The factored resultant loads and the factored resultant bending moments by Eqs. (16) to (18) of part I are obtained: R _u = 6000 kN, M _uxT = 2000 kN-m, M _{uyT = -} 100 kN-m.

Substituting the corresponding values into eq. (9)-(15) to obtain the bending moments that act on the critical sections of the rectangular isolated footing are shown below: M _a1 = 671.68 kN-m; M _b1 = 270.31 kN-m.

After making different proposals, the effective depth is d = 37.00 cm, r = 8.00 cm, t = 45.00 cm.

Substituting the corresponding values into eq. (19)-(20) to obtain the bending shear forces that act on the critical sections of the rectangular isolated footing are shown below: V _{e1 = -} 568.20 kN; V _{f1 = - 367.43} kN. Now, the allowable bending shear forces by the concrete are: ∅ _v V _ce1 = 648.42 kN; ∅ _v V _cf1 = 1296.84 kN. Then, the two bending shear forces satisfy with the code [26].

Now, substituting the corresponding values into eq. (22) to obtain the punching shear force that acts on the critical section of the rectangular isolated footing is shown below: V _p1 = 1115.35 kN. Now, the allowable punching shear forces by the concrete are: ∅ _v V _cp1 = 2922.64 kN, ∅ _v V _cp1 = 3236.86 kN, ∅ _v V _cp1 = 1891.12 kN. Then, the punching shear force satisfies with the code [26].

The reinforcement steel areas for the rectangular isolated footing that result of the bending moments are shown below: A _sy = 50.39 cm ²; A _sx = 19.50 cm ², and the minimum steel areas according to the code are [26]: A _symin = 25.28 cm ²; A _sxmin = 50.57 cm ². Hence, the reinforcement steel areas for the design of the rectangular isolated footing are: A _sy = 50.39 cm ²; A _sx = 50.57 cm ². The proposed steel areas are: A _sy = 50.70 cm ² (10Ø1”) spaced at 20.56 cm; A _sx = 50.70 cm ² (10Ø1”) spaced at 43.33 cm.

Substituting the corresponding values into eq. (9), (15) and (17) to obtain the bending moments that act on the critical sections of the rectangular combined footing are shown below: M _a1 = 457.908 kN-m; M _a2 = 915.80 kN-m; M _b1 = 0.09 kN-m; M _d1= - 400.33kN-m; M _b2 = 943.58 kN-m; M _d2 = 692.53 kN-m; M _max12 = - 1540.55kN-m (maximum bending moment between the column 1 and column 2) in x _max12 = 1.37 m (position of the maximum bending moment between the column 1 and column 2).

The maximum bending moment between the column 1 and column 2 is obtained as follows: First, the position of the maximum bending moment is located from the equation of the shear forces, i.e., when the shear force is zero, the maximum moment is obtained.

After making different proposals, the effective depth is d = 72.00 cm, r = 8.00 cm, t = 80.00 cm.

The bending shear forces that act on the critical sections of the rectangular combined footing by eq. (19)-(11), and the allowable bending shear forces for the concrete by the code are obtained [26] (see Table 1).

Table 1

Axis	Shear force that acts (kN)	Allowable shear force (kN)	Width of analysis (m)
e 1	−319.05	467.79	0.76
e 2	−638.10	689.37	1.12
f 1	0*	1692.65	2.75
g 1	687.96	1692.65	2.75
f 2	−1165.61	1692.65	2.75
g 2	422.02	1692.65	2.75

* The axis falls outside of the footing

Then, the bending shear forces satisfy with the ACI code [26].

The punching shear force that act on the critical sections of the rectangular combined footing by eq. (22) and (24), and the allowable punching shear forces for the concrete by the code are obtained [26] (see Table 2).

Table 2

Analysis section	Shear force that acts (kN)	Allowable shear force (kN)	Critical perimeter (m)
Column 1 (boundary column)	1057.24	4948.69	2.68
		8101.84
		3202.09
Column 2 (inner column)	2198.61	8272.44	4.48
		11347.38
		5352.76

Source: Prepared by the author.

Then, the two punching shear forces satisfy with the ACI code [26].

The reinforcement steel areas for the rectangular combined footing that result of the bending moments are shown below (see Table 3).

Table 3

Direction of the steel	Steel location	Steel area (cm2)
Y axis	Under column 1 (width of c 1 + s + d/2)	Main steel	17.21
		Minimum steel	18.24
		Proposed steel	20.28(4Ø1”)
	Under column 2 (width of c 3 + d)	Main steel	34.70
		Minimum steel	26.88
		Proposed steel	35.49(7Ø1”)
	Steel at the bottom (width of L x − c 1 − c 3 − s − 3d/2)	Temperature steel	89.57
		Proposed steel	91.20(32Ø3/4”)
	Steel at the top (width of L x )	Temperature steel	116.64
		Proposed steel	116.85(41Ø3/4”)
X axis	Steel at the Bottom (width of L y )	Main steel	35.11
		Minimum steel	66.00
		Proposed steel	70.98(14Ø1”)
	Steel at the Top (width of L y )	Main steel	57.80
		Minimum steel	66.00
		Proposed steel	70.98(14Ø1”)

Source: Prepared by the author.

Substituting the corresponding values into eq. (9), (15) and (17) to obtain the bending moments that act on the critical sections of the rectangular combined footing are shown below: Ma1 = 433.44 kN-m; Ma2 = 866.88 kN-m; Ma3 = 866.88 kN-m; Mb1 = 0.08 kN-m; Md1 = − 406.22 kN-m; Mb2 = − 73.36 kN-m; Md2 = 448.43 kN-m; Mb3 = 198.02 kN-m; Md3 = − 123.47 kN-m; Mmax12 = − 1797.11 kN-m (maximum bending moment between the column 1 and column 2) in xmax12 = 4.17 m (position of the maximum bending moment between the column 1 and column 2); Mmax23 = − 1794.99 kN-m (maximum bending moment between the column 2 and column 3) in xmax23 = − 1.71 m (position of the maximum bending moment between the column 2 and column 3).

The maximum bending moment between the two columns is obtained as follows: First, the position of the maximum bending moment is located from the equation of the shear forces, i.e., when the shear force is zero, the maximum moment is obtained.

After making different proposals, the effective depth is d = 72.00 cm, r = 8.00 cm, t = 80.00 cm.

The bending shear forces that act on the critical sections of the rectangular combined footing by eq. (19), (20) and (11), and the allowable bending shear forces for the concrete by the code are obtained [26] (see Table 4).

Table 4

Axis	Shear force that acts (kN)	Allowable shear force (kN)	Width of analysis (m)
e1	−290.58	467.79	0.76
e2	−581.16	689.37	1.12
e3	−581.16	689.37	1.12
f1	0*	1600.32	2.60
g1	768.99	1600.32	2.60
f2	−879.44	1600.32	2.60
g2	759.28	1600.32	2.60
f3	−988.95	1600.32	2.60
g3	601.45	1600.32	2.60

* The axis falls outside of the footing

Then, the bending shear forces satisfy with the code [26].

The punching shear forces acting on the critical section of the rectangular combined footing by eq. (22) and (24) are obtained, and the allowable punching shear forces for the concrete by the code are obtained [26] (see Table 5).

Table 5

Analysis section	Shear force that acts (kN)	Allowable shear force (kN)	Critical perimeter (m)
Column 1 (boundary column)	1070.95	4948.69	2.68
		8101.84
		3202.09
Column 2 (inner column)	2202.51	8272.44	4.48
		11347.38
		5352.76
Column 3 (inner column)	2189.98	8272.44	4.48
		11347.38
		5352.76

Source: Prepared by the author.

Then, the two punching shear forces satisfy with the ACI code [26].

The reinforcement steel areas for the rectangular combined footing that result of the bending moments are shown below (see Table 6).

Table 6

Direction of the steel	Steel location	Steel area (cm2)
Y axis	Under column 1 (width of c 1 + s + d/2)	Main steel	16.27
		Minimum steel	18.24
		Proposed steel	20.28(4Ø1”)
	Under column 2 (width of c 3 + d)	Main steel	32.79
		Minimum steel	26.88
		Proposed steel	35.49(7Ø1”)
	Under column 3 (width of c 5 + d)	Main steel	32.79
		Minimum steel	26.88
		Proposed steel	35.49(7Ø1”)
	Steel at the bottom (width of L x − c 1 − c 3 - c 5 − s − 5d/2)	Temperature steel	165.60
		Proposed steel	168.15(59Ø3/4”)
	Steel at the top (width of L x )	Temperature steel	208.80
		Proposed steel	210.90(74Ø3/4”)
X axis	Steel at the Bottom (width of L y )	Main steel	16.58
		Minimum steel	62.40
		Proposed steel	65.91(13Ø1”)
	Steel at the Top (width of L y )	Main steel	67.76
		Minimum steel	62.40
		Proposed steel	70.98(14Ø1”)

Source: Prepared by the author.