Q$1$

Give me a full explaination and calculation on part b and d please

Figure Q$1$ shows a vertical pile which transfers the load $P$ from a building built on soft ground to the rigid bedrock below. The length of the pile is $L,$ its cross$-$sectional area is A and it is made of a material which has a modulus of elasticity of $E.$ The frictional resistance of the earth $($between the top surface and bedrock$)$ to the vertical displacement of the pile can be represented by $k($as an elastic stiffness per unit length of the pile$).$ The self$-$weight of the pile is $w$ per unit length.

$($a$)$ Use the principle of equilibrium of forces to derive the governing differential equation in terms of the primary variable $u($vertical displacement of the pile in the $x$ direction, as shown in Figure Q$1),$ including the complete boundary conditions.

$($b$)$ Neatly sketch a Tonti diagram representing this problem and correctly label all the boxes and the connecting links of the diagram and identify clearly the kinematic, constitutive and balance equations.

$($c$)$ Write down the total potential energy $($TPE$)$ functional for this problem.

$($d$)$ Derive the discretized finite element equation for a two$-$node pile element using the TPE variational principle, showing the key steps of the process clearly. Based on your derivation, write down the element stiffness matrix for a typical two$-$node pile element.

$($e$)$ Using two pile elements, solve for the maximum displacement at the top and mid$-$height of the pile under the action of the point load $(P)=100kN$ and the self$-$weight $(w)$ of the pile $=5k\frac{N}{m}.$ Assume the overall pile length $(L)=20m,$ elastic axial rigidity of the pile $(EA)=6000kN$ and stiffness of the earth's frictional resistance $(k)=30k\frac{N}{m^2}.$

$1$

Step $\frac{2}{3}$

Explanation

$($b$)$ Tonti Diagram

A Tonti diagram is a graphical representation that shows the relationships between different variables in a problem. It is commonly used in structural mechanics to represent the different equations and their interrelationships.

Kinematic: The primary variable is vertical displacement $($a$).$

Constitutive: The stress$-$strain relationship is given by Hooke's Law