Sample of calculation using deflection method??

Sure, here is an example of a calculation using the deflection method:

Let's say we have a simply supported beam with a span of 5 meters and a uniformly distributed load of 10 kN/m. We want to calculate the maximum deflection at the center of the beam.

First, we need to determine the reaction forces at the supports. Since the beam is simply supported, the reactions at each support will be half of the total load:

Reaction force = (10 kN/m * 5 m) / 2 = 25 kN

Next, we can calculate the maximum deflection at the center of the beam using the formula for deflection of a simply supported beam with a uniformly distributed load:

δ = (5 * 10^3 * 5^4) / (384 * 200 * 10^3) = 0.0651 meters

Therefore, the maximum deflection at the center of the beam is 0.0651 meters.

Problem:

Determine the deflection at point B of the beam shown below using the deflection method.

[Image of a beam with point loads and a support at the left end]

Beam Data:

- Length: L
- Modulus of Elasticity: E
- Area Moment of Inertia: I
- Point Loads: P1 at L/4 and P2 at 3L/4

Steps:

1. Calculate the reactions:

Using equilibrium equations, determine the reactions at the left and right supports.

2. Draw the bending moment diagram:

Plot the bending moment diagram for the beam considering the reactions and point loads.

3. Determine the deflection equation:

For a beam with uniformly distributed load, the deflection equation is:

```
v = (PL^3) / (3EI)
```

where:

- v is the deflection
- P is the concentrated load
- L is the length of the beam
- E is the modulus of elasticity
- I is the area moment of inertia

4. Apply the boundary conditions:

At the left end (A), the deflection is zero (v = 0). At point B, the deflection is unknown.

5. Superposition:

The total deflection is the sum of deflections due to each load. In this case, we have two loads: P1 and P2.

6. Calculate the deflection at B:

Due to P1:

```
v1 = (P1-(L/4)^3) / (3EI)
```

Due to P2:

```
v2 = (P2-(3L/4)^3) / (3EI)
```

Total deflection at B:

```
v_B = v1 + v2
```

7. Substitute the values:

Plug in the given values for P1, P2, L, E, and I into the deflection equations and calculate v_B.

Example Calculation:

Given:

- P1 = 10 kN
- P2 = 5 kN
- L = 6 m
- E = 200 GPa
- I = 2000 cm^4

```
v1 = (10 kN-(6 m/4)^3) / (3-200 GPa-2000 cm^4)
v1 = 0.0028 m

v2 = (5 kN-(3-6 m/4)^3) / (3-200 GPa-2000 cm^4)
v2 = 0.0014 m

v_B = 0.0028 m + 0.0014 m
v_B = 0.0042 m
```

Therefore, the deflection at point B is 0.0042 m.