Effects of Change in Foundation Thickness

Under same loading applied on the same column spacing, same modulus of elasticity of the underlying soil, for different foundation thicknesses it is seen that contact stresses under column areas are decreasing to the stress levels of span locations, and distribution is getting more uniform. In other words, since as thickness increases the foundation system becomes more rigid than the underlying soil, so the stress concentration does not occur under the columns. Moreover, settlement tends to decrease as foundaton thickness increases since flexural stiffness (EI) increases, so that rotations and deformations of the mat foundation decrease.

Furthermore, modulus of subgrade reaction decreases as foundation thickness increases where other variables are kept constant.

Under same loading applied on the foundation over soil with same properties, for different column spacing cases over various thickness of foundation contact stresses under inner columns and under mid-spans are compared and shown in Figure 4.15.

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Figure 4.15 σyy vs t for various column spacings under the columns and mid-spans

Figure 4.15 demonstrates that as foundation thickness increases, contact stresses under column locations decrease rapidly and the trend is marginal for small foundation thicknesses. On the other hand, contact stresses under mid-span locations decreases at a smaller rate with respect to contact stresses under column locations, and may be considered as constant.

The variation of foundation displacements as a function of foundation thickness for both mid-span and column locations are shown in Figure 4.16 for different column spacings.

0 100 200 300 400 500 600 700

0.00 0.50 1.00 1.50 2.00 2.50

σyy(kN/m2)

t (m)

s=5m (inner column) s=5m (mid‐span) s=8m (inner column) s=8m (mid‐span) s=10m (inner column) s=10m (mid‐span)

q=100kPa E=50MPa

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Figure 4.16 δyy vs t for various column spacings under the columns and mid-spans

As shown in Figure 4.16, there is not a significant difference between settlement under column locations and span locations for a specific column spacing over the foundation thickness larger than 1.00 m. The difference becomes obvious as foundation thickness decreases. Moreover, the difference in settlement under column locations between different column spacings is more pronounced as foundation thickness decreases. In addition, settlement is not sensitive to the variations in foundation thickness as much as the one affected by the variation in modulus of elasticity of soil.

0.03 0.04 0.05 0.06 0.07

0.00 0.50 1.00 1.50 2.00 2.50

δyy(m)

t (m)

s=5m (inner column) s=5m (mid‐span) s=8m (inner column) s=8m (mid‐span) s=10m (inner column) s=10m (mid‐span)

q=100kPa E=50MPa

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For different column spacing cases over various thickness of foundation developed modulus of subgrade reactions under columns and under mid-spans are compared as given in the Figure 4.17.

Figure 4.17 t vs k for various column spacings under the columns and mid-spans

Figure 4.17 shows that as foundation thickness increases, modulus of subgrade reaction decreases for all cases. This trend is more obvious in thinner foundations, but becomes more marginal as foundation becomes thicker. Decrement of modulus of subgrade reaction at column locations are more considerable than the ones at mid-span locations. General trend is similar to behavior of contact stress, since settlement

2000 3000 4000 5000 6000 7000 8000

0.00 0.50 1.00 1.50 2.00 2.50

k(kN/m2)

t (m)

s=5m (inner column) s=5m (mid‐span) s=8m (inner column) s=8m (mid‐span) s=10m (inner column) s=10m (mid‐span)

q=100kPa E=50MPa

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depends less on foundation thickness but highly depends on the modulus of elasticity of soil.

For different column spacing cases over various thickness of foundation developed shear forces (Q) under columns and under mid-spans are compared as given in the Figure 4.18.

Figure 4.18 Q vs t for various column spacings under the columns and mid-spans

Shear forces under mid-span locations are said to be constant under foundations having different thicknesses. On the other hand, under columns shear forces increase as foundation becomes thicker irrespective to the column spacing.

Moreover, shear forces under column locations increase as column spacing increases

0

0.00 0.50 1.00 1.50 2.00 2.50

Q (kN/m)

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over specific foundation thickness due to the increase of applied loading through the columns. Whereas, average shear force under mid-span locations is said to be constant for irrespective of the column spacing and the foundation thickness.

Under same loading applied on the foundation over soil with same properties, for different column spacing cases over various thickness of foundation developed bending moment (M) under inner columns and under mid-spans are compared as given in the Figure 4.19.

Figure 4.19 M vs t for various column spacings under the columns and mid-spans

Increase in foundation thickness leads to increase in bending moment under both inner column locations and mid span locations. This is expected since according to

0

0.00 0.50 1.00 1.50 2.00 2.50

M (kN.m/m)

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simple bending theory, as foundation thickness increases, bending moment increases at same unit rotation.

Contrary to shear forces, bending moments increase as foundation thickness increases both under column and mid-span locations. Opposite to the effect of variation in deformation modulus, variation in foundation thickness greatly affects the bending moment beneath both the column and mid-span locations.

The ratio of contact stress between column and span locations for various foundation thicknesses are calculated. The relation is given in Figure 4.20.

Figure 4.20 σcolumn/σspan vs t for various column spacings

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

0.00 0.50 1.00 1.50 2.00 2.50

σcolumn/σspan

t (m)

s=5m s=8m s=10m

q=100kPa E=50MPa

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As it is illustrated in Figure 4.20, the σcolumn/σspan ratio decreases as foundation thickness increases. In addition this trend is more obvious for larger column spacings. Since, more rigid foundation leads to a more uniform distribution of contact stress, the differences between column locations and span locations decreases.

The ratio of foundation settlement between column and span locations for various foundation thicknesses are calculated. The relation is given in Figure 4.21.

Figure 4.21 δcolumn/δspan vs t for various column spacings

As demonstrated in Figure 4.21, settlement ratio between column and span locations is parallel to the previous stress that the ratio decreases as foundation thickness increases for same column spacing and the decaying curve gets much steeper as

0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60

0.00 0.50 1.00 1.50 2.00 2.50

δcolumn/δspan

t (m)

s=5m s=8m s=10m

q=100kPa E=50MPa

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column spacing increases, since foundation is less rigid so that the same amount of increase in rigidity is more effective on larger column spacing.

The ratio of modulus of subgrade reaction between column and span locations for various foundation thicknesses are calculated. The relation is given in Figure 4.22.

Figure 4.22 kcolumn/kspan vs t for various column spacings

Since decrease in contact stress is dominant than the decrease in foundation settlement through the entire cross-section, consequently modulus of subgrade reaction under column to the span decreases by the increase of foundation thickness.

Moreover, as column spacing increases this decay is more rapid. Furthermore, as

1.00 1.25 1.50 1.75 2.00 2.25 2.50

0.00 0.50 1.00 1.50 2.00 2.50

kcolumn/kspan

t (m)

s=5m s=8m s=10m

q=100kPa E=50MPa

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foundation rigidity inceases the ratio approaches to 1, that at infinite rigidity the contact stresses developed under the column locations are just same with ones developed under span locations. In other words, the modulus of subgrade reaction distribution would be uniform through the entire cross section as foundation thickness increases.

Several cases are studied for constant deformation modulus of soil, E = 50 MPa, and constant load pressure, q = 100 kPa, to generalize the contact stress distribution under the mat foundation having different thicknesses and loaded by the columns having different column spacings. Those comparisons are illustrated in Figure 4.23, Figure 4.24, Figure 4.25 and Figure 4.26 by means of normalized contact stress with respect to applied pressure.

Figure 4.23 Normalized contact stress distribution for various column spacings for t = 0.30 m

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Figure 4.24 Normalized contact stress distribution for various column spacings for t = 0.50 m

Figure 4.25 Normalized contact stress distribution for various column spacings for t = 1.00 m

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Figure 4.26 Normalized contact stress distribution for various column spacings for t = 2.00 m

All normalized stress distributions demonstrated in Figure 4.23, Figure 4.24, Figure 4.25 and Figure 4.26 are idealized and the general stress distribution is summarized as in Figure 4.27 and Table 4.3. Note that for s = 5 m and s = 8 m, contact stress zones are similar to the given case of s = 10 m in Figure 4.27.

0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15

‐21 ‐15 ‐9 ‐3 3 9 15 21

σyy/qapplied

x (m)

s=5m s=8m s=10m Uniform

q=100kPa E=50MPa t=2.00m

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Figure 4.27 Zones for contact stress distribution for variation in foundation thickness of Pattern B – s = 10 m

Table 4.3 Summary of normalized contact stresses at zones shown in Figure 4.27 for different columns spacings over soil having different “t”

η Zones

t (m) (E = 50 MPa; q = 100 kPa)

s = 5 m s = 8 m s = 10 m

0.30 0.50 1.00 2.00 0.30 0.50 1.00 2.00 0.30 0.50 1.00 2.00 A 2.10 1.30 1.05 0.88 4.60 2.20 1.18 0.88 6.70 3.10 1.35 0.90 B 1.00 1.00 1.03 0.85 0.95 1.00 1.00 0.85 0.91 0.95 1.00 0.84

s/3 s/3 s/3 2s/3

2s/3

Zone A Zone B s/3 2s/3 s/3 2s/3 s/3

86 Where;

100 (Equation 4.5)

η: Ratio of average contact stress in zones defined in Figure 4.27 to average applied load (i.e. 100 kPa) for variable foundation thickness.

From Figures 4.28 to 4.31, it is obvious that as column spacing increases the individual effect of a column is increasing so that the increase in contact stress occurs at larger area around the columns.

Moreover, Table 4.3 shows that, as foundation rigidity increases contact stress distribution becomes uniformer that the stresses under columns decrease and stress difference between mid-span and column locations decrease, irrespective of the column spacing.

From Table 4.2 and Table 4.3 it is understood that as deformation modulus decreases and/or foundation thickness increases, contact stress distribution under the foundation becomes uniform. This means the differences between the stresses beneath the column locations (Zone A) and stresses beneath the span locations (Zone B) decrease and approaches to the applied load pressure. Thus, in order to obtain uniform contact stress pressure under the foundation, the combined effect of deformation modulus of soil and foundation rigidity should be considered.

Table 4.3 shows that there is no significant change in the stresses beneath the span locations (Zone B) opposite to the column locations (Zone A). Thus, other than increasing the entire foundation thickness, only increasing the foundation thickness at Zone A is also studied.

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