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.
