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| Item | Symbol | Value | Units / Notes |
|---|---|---|---|
| Retained height | H | 4.00 | m |
| Base width | B | 2.60 | m |
| Toe length | L_toe | 0.80 | m |
| Heel length | L_heel | 1.40 | m |
| Base thickness | t_base | 0.45 | m |
| Stem thickness (top / bottom) | t_stem | 0.25 / 0.45 | m |
| Soil unit weight | \(\gamma\) | 18.00 | kN/m³ |
| Soil friction angle | \(\varphi'\) | 30.0 | deg |
| Cohesion (effective) | \(c'\) | 0.0 | kPa (often 0 for granular) |
| Surcharge | \(q\) | 10.0 | kPa |
| Water enabled | - | False | 0/1 |
| Water height behind wall | \(h_w\) | 0.00 | m (clamped to H) |
| Concrete unit weight | \(\gamma_c\) | 24.00 | kN/m³ |
| Base friction | \(\mu\) | 0.500 | [-] |
| Passive enabled | - | False | 0/1 |
| Embedment depth (front) | \(D_f\) | 0.80 | m |
| Passive reduction factor | - | 0.50 | [-] |
| Allowable bearing check | - | True | 0/1 |
| Allowable bearing pressure | \(q_{allow}\) | 200.0 | kPa |
| Factors on actions | - | γE=1.35, γQ=1.50, γW=1.35 | |
| Stabilizing weight factor | - | γG,stab=1.00 | |
| Resistance factors | - | γR,slide=1.00, γR,OT=1.00, γR,bear=1.00 | |
| Concrete strength | \(f_{ck}\) | 30 | MPa |
| Rebar yield strength | \(f_{yk}\) | 500 | MPa |
| Partial factors (EC2) | - | γc=1.50, γs=1.15 | |
| Nominal cover | - | 50 | mm |
| Assumed bar diameter | - | 16 | mm |
| Minimum steel ratio | \(\rho_{min}\) | 0.00130 | [-] |
Note Geometry check: toe + heel + t_stem_bot = 2.650 m, but B input = 2.600 m (Δ=-0.050 m). The calculation uses your B for bearing footprint.
\[K_a=\tan^2\left(45^\circ-\frac{\varphi'}{2}\right)\]
\[K_a=\tan^2\left(45^\circ-\frac{30.0^\circ}2\right)=0.3333\]
Horizontal soil pressure distribution (effective):
\[\sigma_h(z)=K_a\gamma z\]
Computed resultant (numerical integration): E_soil = 48.00 kN/m, acting at y = 1.33 m above base.
Surcharge lateral pressure:
\[\sigma_{h,q}=K_a q,\quad E_q=\sigma_{h,q}H\]
\[\sigma_{h,q}=0.3333\times 10.0=3.33\ \text{kPa},\quad E_q=3.33\times 4.00=13.33\ \text{kN/m}\]
Acts at y = H/2 = 2.00 m above base.
Water pressure not included.
| Component | Resultant (kN/m) | Lever arm y (m) | Moment about toe (kNm/m) |
|---|---|---|---|
| Soil (active) | 48.00 | 1.33 | 64.00 |
| Surcharge | 13.33 | 2.00 | 26.67 |
| Water | 0.00 | 0.00 | 0.00 |
| Total (characteristic) | 61.33 | - | 90.67 |
| Total (design / factored) | 84.80 | - | 126.40 |
Computed per meter run. Centroid lever arms measured from toe.
| Component | Area (m² per m) | Weight (kN/m) | Lever arm x from toe (m) | Moment about toe (kNm/m) |
|---|---|---|---|---|
| Stem (avg thickness) | 1.400 | 33.60 | 1.03 | 34.44 |
| Base slab | 1.170 | 28.08 | 1.30 | 36.50 |
| Soil over heel | 5.600 | 100.80 | 1.95 | 196.56 |
| Soil over toe (optional) | - | 0.00 | 0.40 | 0.00 |
| Total (characteristic) | - | 162.48 | - | 267.50 |
| Total (design / stabilizing) | - | 162.48 | - | 267.50 |
\[R_{fric,d}=\frac{\mu V_d}{\gamma_{R,slide}},\quad R_{slide,d}=R_{fric,d}+E_{p,d}\]
\[R_{fric,d}=\frac{0.500\times 162.48}{1.00}=81.24\ \text{kN/m}\]
\[E_{p,d}=\frac{E_{p,k}}{\gamma_{R,slide}}=\frac{0.00}{1.00}=0.00\ \text{kN/m}\]
\[R_{slide,d}=81.24\ \text{kN/m};\quad H_d=84.80\ \text{kN/m};\quad util=1.044\]
\[util_{OT}=\frac{M_{drive,d}}{M_{resist,d}} \le 1.0\]
\[M_{drive,d}=126.40\ \text{kNm/m},\quad M_{resist,d}=\frac{M_{stab,d}}{\gamma_{R,OT}}=\frac{267.50}{1.00}=267.50\ \text{kNm/m}\]
\[x_R=\frac{M_{net,d}}{V_d},\quad e=\frac{B}{2}-x_R,\quad |e|\le\frac{B}{6}\ \text{(no tension)}\]
\[M_{net,d}=M_{stab,d}-M_{drive,d}=267.50-126.40=141.10\ \text{kNm/m}\]
\[x_R=\frac{141.10}{162.48}=0.868\ \text{m from toe},\quad e=0.432\ \text{m}\]
\[q_{avg}=\frac{V_d}{B},\quad q_{max/min}=q_{avg}\left(1\pm\frac{6e}{B}\right)\]
\[q_{avg}=\frac{162.48}{2.60}=62.49\ \text{kPa}\]
\[q_{max}=124.73\ \text{kPa},\quad q_{min}=0.26\ \text{kPa}\]
\[util_{bear}=\frac{q_{max}}{q_{allow}/\gamma_{R,bear}}\]
\[util_{bear}=\frac{124.73}{200.0/1.00}=0.624\]
This section designs required reinforcement per meter run using a simplified bending model: As = M/(fyd·z), with z ≈ 0.9d. You can refine with full EC2 section analysis if needed.
\[f_{cd}=\frac{f_{ck}}{\gamma_c},\quad f_{yd}=\frac{f_{yk}}{\gamma_s}\]
\[f_{cd}=\frac{30}{1.50}=20.00\ \text{MPa},\quad f_{yd}=\frac{500}{1.15}=434.78\ \text{MPa}\]
Design moment at stem base from factored lateral pressures:
\[M_{Ed,stem}=126.40\ \text{kNm/m}\]
Effective depth (assumed):
\[d\approx t_{stem,bot}-cover-\phi/2=450-50-16/2=392.0\ \text{mm}\]
\[As_{req}=\frac{M_{Ed}}{f_{yd} z},\quad z\approx0.9d\]
\[As_{req}=\frac{126.40\times10^6}{434.78\times (0.9\times392.0)}=824.0\ \text{mm}^2/\text{m}\]
\[As_{min}=\rho_{min} b d=0.00130\times 1000\times 392.0=509.6\ \text{mm}^2/\text{m}\]
Provide: As,stem = 824.0 mm²/m (max of required and minimum).
Toe/heel are designed as cantilever strips from the stem face using net upward pressure (soil reaction minus downward surcharge/self-weight terms).
\[M_{Ed,toe}=30.33\ \text{kNm/m},\quad M_{Ed,heel}=0.00\ \text{kNm/m}\]
\[d\approx t_{base}-cover-\phi/2=450-50-16/2=392.0\ \text{mm}\]
\[As_{req}=\frac{M_{Ed}}{f_{yd} z},\quad z\approx0.9d\]
\[As_{req,toe}=197.7\ \text{mm}^2/\text{m},\quad As_{req,heel}=0.0\ \text{mm}^2/\text{m}\]
\[As_{min}=\rho_{min} b d=509.6\ \text{mm}^2/\text{m}\]
Provide toe main steel: As,toe = 509.6 mm²/m
Provide heel main steel: As,heel = 509.6 mm²/m
| Check / Output | Value |
|---|---|
| Ka (Rankine active) | 0.3333 |
| Design horizontal driving Hd (kN/m) | 84.80 |
| Sliding utilization | 1.044 |
| Overturning utilization | 0.473 |
| Eccentricity e (m) | 0.432 |
| No-tension condition |e| ≤ B/6 | OK |
| Design base pressure qmax / qmin (kPa) | 124.73 / 0.26 |
| Bearing check | OK (util=0.624) |
| Stem base moment MEd (kNm/m) | 126.40 |
| Stem steel As provided (mm²/m) | 824.0 |
| Toe steel As provided (mm²/m) | 509.6 |
| Heel steel As provided (mm²/m) | 509.6 |
Equations rendered using MathJax. This report is intended as a clear calculation record; you can tune partial factors, passive assumptions, and detailed EC7/EC2 clause settings per project/NA.