General approach to analyzing such a beam:

1. Small Deflection Analysis:

- Determine the beam's geometry, material properties, and loading conditions (including the location and magnitude of the intermediate load).

- Apply the principle of superposition to break down the problem into simpler cases. Analyze the beam's response due to the intermediate load separately from other loads or boundary conditions.

- Apply the equations of linear elasticity to calculate the beam's deflection and slope under the given loading conditions. The most common equation used is the Euler-Bernoulli beam equation.

- Solve the governing differential equation using appropriate boundary conditions (e.g., zero deflection and zero slope at the fixed end of the cantilever).

- Determine the deflection and slope of the cantilever beam at the location of the intermediate load using the obtained solution.

1. Large Deflection Analysis:

- If the deflections are expected to be significant, a large deflection analysis may be necessary. In this case, linear elasticity assumptions are not valid, and nonlinear equations must be used.

- One approach is to use numerical methods such as the finite element method (FEM) to solve the governing equations. FEM software packages, such as ANSYS or Abaqus, can handle large deflection problems and provide accurate results.

- Define the beam's geometry, material properties, and loading conditions in the FEM software.

- Apply appropriate boundary conditions, including the fixed end of the cantilever and the load at the intermediate point.

- Solve the nonlinear equations iteratively to obtain the deflection and slope of the cantilever beam at the location of the intermediate load.

Good luck

Hi Mohammad Imam , Thanks for your answer; I asked for the intermediate load case. Could you please address that, if possible?

Cheers..

To proceed:

1. Define the variables:

- P: Magnitude of the load applied at point a (intermediate point)

- a: Distance from the left end of the beam to the point of load application

- x: Distance from the left end of the beam to the point where you want to calculate the deflection

- L: Length of the beam

2. Determine the equation for the bending moment (M) at any point x along the beam:

M = P(a - ux - x)

This equation represents the moment due to the load P at the point a. The term (a - ux - x) represents the moment arm, which is the perpendicular distance between the point of load application and the point of interest.

3. To calculate the deflection at point x, you need to integrate the bending moment equation twice:

d²y/dx² = M/EI

This equation relates the second derivative of the deflection (y) with respect to x to the bending moment (M), the modulus of elasticity (E), and the moment of inertia (I) of the beam's cross-section. The moment of inertia depends on the beam's geometry and is typically provided or calculated separately.

4. Solve the differential equation for deflection:

Integrate the equation d²y/dx² = M/EI twice with appropriate boundary conditions to find the equation for deflection y(x).

5. To calculate the end-slope angle Phi0 for the load at point a, you can still use equation :

Phi0 = -d²y/dx² at x = L

You can differentiate the deflection equation obtained in step 4 with respect to x and evaluate it at x = L to find the end-slope angle.

6. From the small angle deflection assumption, Phi0 should remain the same for a \< x ≤ L. Therefore, you can use equation to compute Phi0 for any point a \< x ≤ L.

Specific boundary conditions will affect the integration constants and the final form of the deflection equation.

Good luck!!

Mohammad Imam its good that you elaborated the process.

in step-3, I see that the curvature is assumed to be d²y/dx² = M/EI. This it self is an assumption for small angle deflection. Is it valid for large angle deflection?

Thanks..