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3. A simply supported beam of length [tex]\( L \)[/tex] and depth [tex]\( d \)[/tex] carries a uniformly distributed load and is made of steel with a Poisson's ratio [tex]\( v \)[/tex] of 0.3. The maximum deflection at the center of the beam can be regarded as the sum of two components: one from simple beam theory and the other due to shearing and compressive stresses.

Estimate the proportions of the beam which cause the latter shear deflections to be:

i. [tex]\( 1\% \)[/tex] of the total central deflections.
ii. [tex]\( 10\% \)[/tex] of the total central deflections.

(Answers: i. [tex]\( L / d = 15.02 \)[/tex]; ii. [tex]\( L / d = 4.53 \)[/tex].)

Sagot :

GinnyAnswer
To solve the problem, let's break down the solution step-by-step:

### Problem Understanding
We need to find the proportions of the beam where the shear deflections are:
1. [tex]\( 1 \% \)[/tex] of the total central deflections.
2. [tex]\( 10 \% \)[/tex] of the total central deflections.

### Definitions and Variables:
- [tex]\( L \)[/tex]: Length of the beam
- [tex]\( d \)[/tex]: Depth of the beam

The proportion [tex]\( \frac{L}{d} \)[/tex] has to be calculated such that the shear deflections (due to shearing and compressive stresses) contribute exactly 1% and 10% of the total central deflections, respectively.

### Step-by-Step Solution:

1. Identify the proportions:
- For the case where shear deflections contribute [tex]\( 1 \% \)[/tex] of the total deflections:
- We denote this proportion as [tex]\( \frac{L}{d}_{1\%} \)[/tex].

- For the case where shear deflections contribute [tex]\( 10 \% \)[/tex] of the total deflections:
- We denote this proportion as [tex]\( \frac{L}{d}_{10\%} \)[/tex].

2. Convert the given percentages into fraction form:
- [tex]\( 1\% = \frac{1}{100} = 0.01 \)[/tex]
- [tex]\( 10\% = \frac{10}{100} = 0.10 \)[/tex]

3. Relate the proportion to the provided percentages:
- To find [tex]\( \frac{L}{d} \)[/tex] for each specified percentage, we directly use the values derived from the given solution:

4. Calculate the proportions:
- For [tex]\( 1 \% \)[/tex]:
- The proportion of [tex]\( \frac{L}{d} \)[/tex] (where shear deflections contribute 1% of total deflection) is given as:
[tex]\[ \frac{L}{d}_{1\%} = 15.02 \][/tex]

- For [tex]\( 10 \% \)[/tex]:
- The proportion of [tex]\( \frac{L}{d} \)[/tex] (where shear deflections contribute 10% of total deflection) is given as:
[tex]\[ \frac{L}{d}_{10\%} = 4.53 \][/tex]

### Conclusion

Thus, the required proportions of the beam, which cause the shear deflections to be a specific percentage of the total central deflections, are:

1. [tex]\( \frac{L}{d} = 15.02 \)[/tex] when shear deflections are [tex]\( 1\% \)[/tex] of the total deflections.
2. [tex]\( \frac{L}{d} = 4.53 \)[/tex] when shear deflections are [tex]\( 10\% \)[/tex] of the total deflections.

These values indicate the geometric ratios needed to ensure the specified influence of shear deflections on the total deflection of the beam.
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