(Koerner, 1999) Design a 7m high geogrid-reinforced wall when the reinforcement vertical maximum spacing must be 1.0 m. The coverage ratio is 0.80 (Refer to Fig. Ex. 19.5). Given: Tu = 156 kN/m, Cr = 0.80, Ci = 0.75. The other details are given in the figure.

Question

(Koerner, 1999) Design a 7m high geogrid-reinforced wall when the reinforcement vertical maximum spacing must be 1.0 m. The coverage ratio is 0.80 (Refer to Fig. Ex. 19.5). Given: [latex]T_{u}=156 kN / m , C_{r}=0.80[/latex], [latex]C_{i}=0.75[/latex]. The other details are given in the figure.

Accepted Answer

Internal Stability

From Eq. (19.14)

[latex]p_{h}=p_{a}+q_{h}=\left(\gamma z K_{A}+q_{h} ight)[/latex] (9.14)

[latex]\begin{aligned}&p_{h}=\left(\gamma z K_{A}+q_{h} ight)=\gamma z K_{A}+q_{s} K_{A} \&K_{A}= an ^{2}\left(45^{\circ}-\phi / 2 ight)= an ^{2}\left(45^{\circ}-32 / 2 ight)=0.31 \&p_{h}=(18 imes z imes 0.31)+(15 imes 0.31)=5.58 z+4.65\end{aligned}[/latex]

1. For geogrid vertical spacing.

Given [latex]T_{u}=156[/latex] kN/m

From Eq. (19.10) and Table 19.5, we have

[latex]T_{a}=T_{u}\left(\frac{1}{R F_{I D} imes R F_{C R} imes R F_{B D} imes R F_{C D}} ight)[/latex] (19.10)

[latex]\begin{aligned}&T_{a}=T_{u}\left[\frac{1}{R F_{I D} imes R F_{C R} imes R F_{B D} imes R F_{C D}} ight] \&T_{a}=156 \frac{1}{1.2 imes 2.5 imes 1.3 imes 1.0}=40 kN / m\end{aligned}[/latex]

But use [latex]T_{ design }=28.6 kN / m ext { with } F_{ s }=1.4 ext { on } T_{a}[/latex]

From Eq. (19.28)

[latex]h=\frac{T_{a} C_{r}}{p_{h}}[/latex] (19.28)

[latex]T_{ design }=\frac{h p_{h}}{C_{r}}[/latex]

[latex]28.6=h \frac{5.58 z+4.65}{0.8}[/latex]

or [latex]h=\frac{22.9}{5.58 z+4.65}[/latex]

Maximum depth for h = 1 m is

[latex]1.0=\frac{22.9}{5.58 z+4.65} ext { or } z=3.27 m[/latex]

Maximum depth for h = 0.5m

[latex]0.5=\frac{22.9}{5.58 z+4.65} ext { or } z=7.37 m[/latex]

The distribution of geogrid layers is shown in Fig. Ex. 19.5.

2. Embedment length of geogrid layers.

From Eqs (19.27) and (19.24)

[latex]T_{a}=T F_{s}=\left(\gamma z K_{A}+q_{h} ight) h F_{s}[/latex] (19.24)

[latex]F_{R}=2 C_{i} C_{r} L_{e} p_{o} an \phi \geq T F_{s}[/latex] (19.27)

[latex]2 C_{1} C_{r} L_{e} p_{o} an \phi=T_{H} F_{s}=p_{h} h F_{s}[/latex]

Substituting known values

[latex]2 imes 0.75 imes 0.8 imes\left(L_{e} ight) imes 18 imes(z) an 32^{\circ}=h(5.58 z+4.65) 1.5[/latex]

Simplfying [latex]L_{e}=\frac{(0.62 z+0.516) h}{z}[/latex]

The equation for [latex]L_{R}[/latex] is

[latex]\begin{aligned}L_{R} &=(H-z) an \left(45^{\circ}-\phi / 2 ight)=(7-z) an \left(45^{\circ}-32 / 2 ight) \&=3.88-0.554(z)\end{aligned}[/latex]

From the above relationships the spacing of geogrid layers and their lengths are given below.

Layer	Depth	Spacing	[latex]L_{e}[/latex]	[latex]L_{e}[/latex] (min)	[latex]L_{R}[/latex]	L (cal)	L (required)
No.	(m)	h (m)	(m)	(m)	(m)	(m)	(m)
1	0.75	0.75	0.98	1	3.46	4.46	5
2	1.75	1	0.92	1	2.91	3.91	5
3	2.75	1	0.81	1	2.36	3.36	5
4	3.25	0.5	0.39	1	2.08	3.08	5
5	3.75	0.5	0.38	1	1.8	2.8	5
6	4.25	0.5	0.37	1	1.52	2.52	5
7	4.75	0.5	0.36	1	1.25	2.25	5
8	5.25	0.5	0.36	1	0.97	1.97	5
9	5.75	0.5	0.36	1	0.69	1.69	5
10	6.25	0.5	0.35	1	0.42	1.42	5
11	6.75	0.5	0.35	1	0.14	1.14	5

External Stability

(a) Pressure distribution

[latex]\begin{aligned}&P_{a}=\frac{1}{2} \gamma H^{2} K_{A}=\frac{1}{2} imes 17 imes 7^{2} an ^{2}\left(45^{\circ}-30 / 2 ight)=138.8 kN / m \&P_{q}=q_{s} K_{A} H=15 imes 0.33 imes 7=34.7 kN / m \& ext { Total } \approx 173.5 kN / m\end{aligned}[/latex]

1. Check for sliding (neglecting effect of surcharge)

[latex]\begin{aligned}&F_{R}=W an \delta=\gamma imes H imes L an 25^{\circ}=18 imes 7 imes 5.0 imes 0.47=293.8 kN / m \&P=P_{a}+P_{q}=173.5 kN / m \&F_{s}=\frac{293.8}{173.5}=1.69>1.5 OK\end{aligned}[/latex]

2. Check for overturning

Resisting moment [latex]M_{R}=W imes \frac{L}{2}=18 imes 7 imes 5 imes \frac{5}{2}=1575 kN - m[/latex]

Overturning moment [latex]M_{O}=P_{a} imes \frac{H}{3}+P_{q} imes \frac{H}{2}[/latex]

or [latex]M_{O}=138.8 imes \frac{7}{3}+34.7 imes \frac{7}{2}=445.3 kN - m[/latex]

[latex]F_{s}=\frac{1575}{445.3}=3.54>2.0 \quad OK[/latex]

3. Check for bearing capacity

[latex] ext { Eccentricity } e=\frac{M_{O}}{W+q_{s} L}=\frac{445.3}{18 imes 7 imes 5+15 imes 5}=0.63[/latex]

[latex]e=0.63<\frac{L}{6}=\frac{5}{6}=0.83 Ok[/latex]

Effective length [latex]=L-2 e=5-2 imes 0.63=3.74 m[/latex]

Bearing pressure [latex]=[18 imes 7+15]\left(\frac{5}{3.74} ight)=189 kN / m ^{2}[/latex]

[latex]F_{s}=\frac{600}{189}=3.17>3.0 OK[/latex]

Table 19.5 Recommended reduction factor values for use in Eq. (19.10) for determining allowable tensile strength of geogrids
Application Area	[latex]R F_{ID}[/latex]	[latex]R F_{CR}[/latex]	[latex]R F_{C D}[/latex]	[latex]RF _{B D}[/latex]
Unpaved roads	1.1 to1.6	1.5 to 2.5	1.0 to 1.5	1.0 to 1.1
Paved roads	1.2 to 1.5	1.5 to 2.5	1.1 to 1.6	1.0 to 1.1
Embankments	1.1 to 1.4	2.0 to 3.0	1.1 to 1.4	1.0 to 1.2
Slopes	1.1 to 1.4	2.0 to 3.0	1.1 to 1.4	1.0 to 1.2
Walls	1.1 to 1.4	2.0 to 3.0	1.1 to 1.4	1.0 to 1.2
Bearing capacity	1.2 to 1.5	2.0 to 3.0	1.1 to 1.6	1.0 to 1.2

Question 19.5: (Koerner, 1999) Design a 7m high geogrid-reinforced wall whe...

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