
### 1.0 Question

Consider the pile/ground-beam structure we saw previously, Fig 1. With the additional structural and loading information provided, confirm that:

- the ground beam does not exceed the EQU limit state.

- the rear pile does not exceed the STR limit state.

- the strata onto which the rear pile bears satisfies the GEO limit state.

<Image
	src="/images/courses/fundamentals-of-reinforced-concrete-design-to-eurocode-2/worked-example-1-determining-uls-actions/img1.jpg"
	alt="Foundation structure. | EngineeringSkills.com"
	width="1000"
	height="580"
	caption="Fig 1. Foundation structure."
/>

### 2.0 Solution

#### 2.1 Ground beam - EQU

The EQU limit state is concerned with instability arising from the structure behaving as a rigid body. So, as we look at this structure, it becomes apparent that we need to confirm that if the ground beam did behave as a rigid body (however unlikely that may be), it would not rotate about point $B$. In other words, under EQU loading, no uplift should occur at $A$.

The most onerous loading pattern for this case is when the maximum loading, $w_{max}$, is applied between $B$ and $C$ and the minimum, $w_{min}$, is applied between $A$ and $B$. For $w_{max}$, both the permanent and variable actions are unfavourable and therefore attract the highest possible partial factors from the Table in Fig. 2 below.

<Image
	src="/images/courses/fundamentals-of-reinforced-concrete-design-to-eurocode-2/worked-example-1-determining-uls-actions/img2.jpg"
	alt="Partial factors and combinations for ultimate limit states EQU, STR and GEO (repeated from lecture 4). | EngineeringSkills.com"
	width="1000"
	height="313"
	caption="Fig 2. Partial factors and combinations for ultimate limit states EQU, STR and GEO (repeated from lecture 4)."
/>

$$
\begin{align*}
w_{max} &= \gamma_G\:g_k + \gamma_Q\:q_k\\\\
w_{max} &= 1.1\times 70 + 1.5\times 40\\\\
w_{max} &= 137\:kN/m
\end{align*}
$$

In the case of $w_{min}$, both permanent and variable actions are favourable,

$$
\begin{align*}
w_{min} &= \gamma_G\:g_k + \gamma_Q\:q_k\\\\
w_{min} &= 0.9\times 70 + 0\times 40\\\\
w_{min} &= 63\:kN/m
\end{align*}
$$

Now we can simply evaluate the equilibrium of the ground beam under this EQU loading, Fig. 3 and confirm that uplift doesn't occur at $A$.

<Image
	src="/images/courses/fundamentals-of-reinforced-concrete-design-to-eurocode-2/worked-example-1-determining-uls-actions/img3.jpg"
	alt="Structure subject to EQU loading for the ground beam. | EngineeringSkills.com"
	width="800"
	height="339"
	caption="Fig 3. Structure subject to EQU loading for the ground beam."
/>

To identify the reaction at $A$, we can evaluate the sum of the moments about $B$,

$$
\begin{align*}
\sum M_B &= 0\:\: (\curvearrowright+)\\\\
(R_A\times 16) + (137\times 8\times 4)&= (63\times 16\times 8)\\\\
R_A &= 230\:kNm
\end{align*}
$$

Since we assumed upwards pointing forces were positive and $R_A>0$, the reaction is pointing upwards, and uplift is not predicted at A; the ground beam, therefore, does not exceed the EQU limit state.

#### 2.2 Rear pile - STR

To confirm that the ground beam does not exceed the STR limit state, a simple check would be to confirm that the maximum compressive stress in the pile remains within the limit specified. In this case the most onerous load pattern is when the span from $A$ to $B$ experiences $w_{max}$ and $B$ to $C$ experiences $w_{min}$.

<Image
	src="/images/courses/fundamentals-of-reinforced-concrete-design-to-eurocode-2/worked-example-1-determining-uls-actions/img4.jpg"
	alt="Structure beam subject to STR loading for the rear pile. | EngineeringSkills.com"
	width="800"
	height="339"
	caption="Fig 4. Structure beam subject to STR loading for the rear pile."
/>

Referencing the second row from the table in Fig. 2 above,

$$
\begin{align*}
w_{max} &= \gamma_G\:g_k + \gamma_Q\:q_k\\\\
w_{max} &= 1.35\times 70 + 1.5\times 40\\\\
w_{max} &= 154.5\:kN/m
\end{align*}
$$

When calculating $w_{min}$, we observe the single source principle, which states that all permanent loads from the same source attract the same partial factor. As a result, the partial factor applied to $g_k$ will be $1.35$, even though we seek to minimise $w_{min}$. Remember, the single source principle applied to STR and GEO limit states only.

$$
\begin{align*}
w_{min} &= \gamma_G\:g_k + \gamma_Q\:q_k\\\\
w_{min} &= 1.35\times 70 + 0\times 40\\\\
w_{min} &= 94.5\:kN/m
\end{align*}
$$

The vertical reaction at $A$ is determined again by taking moments about support $B$,

$$
\begin{align*}
\sum M_B &= 0\:\: (\curvearrowright+)\\\\
(R_A\times 16) + (94.5\times 8\times 4)&= (154.5\times 16\times 8)\\\\
R_A &= 1047\:kN
\end{align*}
$$

Finally, we can calculate the pile cross-sectional area, $A$, to determine the compressive stress, $\sigma_c$,

$$
\begin{align*}
a &= \frac{\pi\times 750^2}{4}\\\\
a &= 441786\:mm^2
\end{align*}
$$

$$
\begin{align*}
\sigma_c &= \frac{1047\times 10^3}{441786}\\\\
\sigma_c &= 2.4\:N/mm^2 \:<\sigma_{all}
\end{align*}
$$

Therefore the rear pile satisfies the STR limit state.

#### 2.3 Bearing strata - GEO

The loading pattern required for this design check is the same as for the STR check we just completed. However, the partial factors used to determine $w_{max}$ and $w_{min}$ will be those for the GEO limit state. Therefore $w_{max}$ is obtained as,

$$
\begin{align*}
w_{max} &= \gamma_G\:g_k + \gamma_Q\:q_k\\\\
w_{max} &= 1.0\times 70 + 1.3\times 40\\\\
w_{max} &= 122\:kN/m
\end{align*}
$$

while $w_{min}$ is obtained as,

$$
\begin{align*}
w_{min} &= \gamma_G\:g_k + \gamma_Q\:q_k\\\\
w_{min} &= 1.0\times 70 + 0\times 40\\\\
w_{min} &= 70\:kN/m
\end{align*}
$$

The reaction at $A$ is obtained the same way as shown above and evaluates to $R_A=836 \:kN$. To check if the bearing strata satisfies the GEO limit state, we will simply calculate the end bearing pressure, $q_{bear}$, by dividing $R_A$ by the pile cross-sectional area, $a_p$,

$$
\begin{align*}
q_{bear} &= \frac{836}{0.441786}\\\\
q_{bear} &=1892.3\:kN/m^2 \:>>q_{all}
\end{align*}
$$

Since the bearing pressure is larger than the stated allowable bearing pressure, the GEO limit state is not satisfied, and the pile must be redesigned to reduce the bearing pressure developed in the supporting strata.

Note that the design calculations performed as part of each limit state check above are somewhat superficial (_back of the envelope_) and don't represent detailed designs. For example, there is more to designing a reinforced concrete pile than simply checking the compressive stress. The important part of this example question was correctly determining the design loading.


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