Four Hollow Columns

solved YoungsModulus

Q. Four identical $F_{c} = \frac{m_{s} \cdot g}{4}$ hollow cylindrical columns $A_{c} = A_{o} - A_{i}$ of mild steel support a big structure of mass 50000 kg $m_{s} = 50000$ kg. The inner and outer radii of each column are 30 cm $r_{i} = 30$ cm and 60 cm $r_{o} = 60$ cm respectively. Assuming the load distribution to be uniform, and taking Young's modulus of steel as 2.0 × 10¹¹ N/m² $Y_{c} = 2.0 \times 10^{11}$ N/m² and g as 9.8 m/s² $g = 9.8$ m/s², what is the compressional strain of each column?

Inputs

mass

m_{s}

5e+04 kg

inner

r_{i}

30 cm

outer

r_{o}

60 cm

modulus

Y_{c}

2e+11 N/m²

gravity

g

9.8 m/s²

Outputs

strain

ε_{c}

How it's solved

Solved with YoungsModulus

Given

$m_{s} = 50000$ kg
$r_{i} = 30$ cm
$r_{o} = 60$ cm
$Y_{c} = 2.0 \times 10^{11}$ N/m²
$g = 9.8$ m/s²
$A_{c} = A_{o} - A_{i}$ from “hollow cylindrical columns”
$F_{c} = \frac{m_{s} \cdot g}{4}$ from “Four identical”

Find $ε_{c}$ .

Working

$r_{i} = 30$ cm $= 0.3$ m
$r_{o} = 60$ cm $= 0.6$ m
$Y_{c} = 2.0 \times 10^{11}$ N/m² $= 2 \times 10^{11}$ Pa
Four identical $F_{c} = \frac{m_{s} \cdot g}{4}$
$F_{c} = \frac{50000 \cdot 9.8}{4}$
$F_{c} = 122500$ N
a round rod s cross section(outer rim) $A_{o} = π \cdot {r_{o}}^{2}$
$A_{o} = 3.14159265 \cdot {0.6}^{2}$
$A_{o} = 1.13097336$ m²
a round rod s cross section(inner hole) $A_{i} = π \cdot {r_{i}}^{2}$
$A_{i} = 3.14159265 \cdot {0.3}^{2}$
$A_{i} = 0.282743339$ m²
hollow cylindrical columns $A_{c} = A_{o} - A_{i}$
$A_{c} = 1.13097336 - 0.282743339$
$A_{c} = 0.848230016$ m²
stress is load over area(column) $σ_{c} = \frac{F_{c}}{A_{c}}$
$σ_{c} = \frac{122500}{0.848230016}$
$σ_{c} = 144418.374$ Pa
hookes law(column) $Y_{c} = \frac{σ_{c}}{ε_{c}}$
$2 \times 10^{11} = \frac{144418.374}{ε_{c}}$
$ε_{c} = 7.22091871 \times 10^{−7}$
$ε_{c} \approx 7.22 \times 10^{−7}$ to 3 significant figures

Conclusion

Each column carries a quarter of the weight over the ring of steel between its two radii; steel is so stiff that it shortens by less than a millionth of its length.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :four_hollow_columns do4  kind :numerical5  describes "Four identical hollow cylindrical columns of mild steel support a big structure of mass %{mass}. The " \6            "inner and outer radii of each column are %{inner} and %{outer} respectively. Assuming the load " \7            "distribution to be uniform, and taking Young's modulus of steel as %{modulus} and g as %{gravity}, " \8            "%{strain}?"9  given mass: "50000 kg"10  given inner: "30 cm"11  given outer: "60 cm"12  given modulus: "2.0e11 N/m2"13  given gravity: "9.8 m/s2"14  asks strain: "what is the compressional strain of each column"15end

solution.rb

1require_relative "../questions/four_hollow_columns"2require_relative "../science/mechanical_properties_of_solids/youngs_modulus"34solve :four_hollow_columns, using: YoungsModulus do5  objects :structure, :column, :inner_hole, :outer_rim67  take mass: structure.mass8  take inner: inner_hole.thickness9  take outer: outer_rim.thickness10  take modulus: column.youngs_modulus11  take gravity: g1213  deduce_from "hollow cylindrical columns", column.cross_section => outer_rim.cross_section - inner_hole.cross_section14  deduce_from "Four identical", column.load => structure.mass * g / 41516  strain = column.strain17  answer strain:, figures: 31819  conclusion "Each column carries a quarter of the weight over the ring of steel between its two radii; " \20             "steel is so stiff that it shortens by less than a millionth of its length."21end

playground.rb

1require_relative "../solutions/four_hollow_columns"23play_with :four_hollow_columns do4  title "Four Hollow Columns"5end

The laws it drew on

quantity	written	units	within	from
pi	$π$	—	0 … ∞	YoungsModulus
mass	$m$	kg	0 … ∞	YoungsModulus
gravity	$g$	m/s²	—	YoungsModulus
load	$F$	N	—	YoungsModulus
thickness	$r$	m	0 … ∞	YoungsModulus
diameter	$D$	m	—	YoungsModulus
cross section	$A$	m²	—	YoungsModulus
original length	$L$	m	—	YoungsModulus
extension	$Δ L$	m	—	YoungsModulus
stress	$σ$	Pa	—	YoungsModulus
strain	$ε$	—	—	YoungsModulus
youngs modulus	$Y$	Pa	—	YoungsModulus
thickness of ground	$r_{g}$	m	0 … ∞
diameter of ground	$D_{g}$	m	—
thickness of structure	$r_{s}$	m	0 … ∞
diameter of structure	$D_{s}$	m	—
thickness of column	$r_{c}$	m	0 … ∞
diameter of column	$D_{c}$	m	—
thickness of inner hole	$r_{i}$	m	0 … ∞
diameter of inner hole	$D_{i}$	m	—
thickness of outer rim	$r_{o}$	m	0 … ∞
diameter of outer rim	$D_{o}$	m	—
cross section of ground	$A_{g}$	m²	—
cross section of structure	$A_{s}$	m²	—
cross section of column	$A_{c}$	m²	—
cross section of inner hole	$A_{i}$	m²	—
cross section of outer rim	$A_{o}$	m²	—
stress of ground	$σ_{g}$	Pa	—
load on ground	$F_{g}$	N	—
stress of structure	$σ_{s}$	Pa	—
load on structure	$F_{s}$	N	—
stress of column	$σ_{c}$	Pa	—
load on column	$F_{c}$	N	—
stress of inner hole	$σ_{i}$	Pa	—
load on inner hole	$F_{i}$	N	—
stress of outer rim	$σ_{o}$	Pa	—
load on outer rim	$F_{o}$	N	—
strain of ground	$ε_{g}$	—	—
extension of ground	$Δ L_{g}$	m	—
original length of ground	$L_{g}$	m	—
strain of structure	$ε_{s}$	—	—
extension of structure	$Δ L_{s}$	m	—
original length of structure	$L_{s}$	m	—
strain of column	$ε_{c}$	—	—
extension of column	$Δ L_{c}$	m	—
original length of column	$L_{c}$	m	—
strain of inner hole	$ε_{i}$	—	—
extension of inner hole	$Δ L_{i}$	m	—
original length of inner hole	$L_{i}$	m	—
strain of outer rim	$ε_{o}$	—	—
extension of outer rim	$Δ L_{o}$	m	—
original length of outer rim	$L_{o}$	m	—
youngs modulus of ground	$Y_{g}$	Pa	—
youngs modulus of structure	$Y_{s}$	Pa	—
youngs modulus of column	$Y_{c}$	Pa	—
youngs modulus of inner hole	$Y_{i}$	Pa	—
youngs modulus of outer rim	$Y_{o}$	Pa	—
mass of structure	$m_{s}$	kg	0 … ∞
mass of column	$m_{c}$	kg	0 … ∞
mass of inner hole	$m_{i}$	kg	0 … ∞
mass of outer rim	$m_{o}$	kg	0 … ∞

r_{r} = \frac{D_{r}}{2}

radius is half the diameter

A_{r} = π \cdot {r_{r}}^{2}

a round rod s cross section

σ_{r} = \frac{F_{r}}{A_{r}}

stress is load over area

ε_{r} = \frac{Δ L_{r}}{L_{r}}

strain is stretch over length

Y_{r} = \frac{σ_{r}}{ε_{r}}

hookes law

r_{g} = \frac{D_{g}}{2}

radius is half the diameter(ground)

r_{s} = \frac{D_{s}}{2}

radius is half the diameter(structure)

r_{c} = \frac{D_{c}}{2}

radius is half the diameter(column)

r_{i} = \frac{D_{i}}{2}

radius is half the diameter(inner hole)

r_{o} = \frac{D_{o}}{2}

radius is half the diameter(outer rim)

A_{g} = π \cdot {r_{g}}^{2}

a round rod s cross section(ground)

A_{s} = π \cdot {r_{s}}^{2}

a round rod s cross section(structure)

A_{c} = π \cdot {r_{c}}^{2}

a round rod s cross section(column)

A_{i} = π \cdot {r_{i}}^{2}

a round rod s cross section(inner hole)

A_{o} = π \cdot {r_{o}}^{2}

a round rod s cross section(outer rim)

σ_{g} = \frac{F_{g}}{A_{g}}

stress is load over area(ground)

σ_{s} = \frac{F_{s}}{A_{s}}

stress is load over area(structure)

σ_{c} = \frac{F_{c}}{A_{c}}

stress is load over area(column)

σ_{i} = \frac{F_{i}}{A_{i}}

stress is load over area(inner hole)

σ_{o} = \frac{F_{o}}{A_{o}}

stress is load over area(outer rim)

ε_{g} = \frac{Δ L_{g}}{L_{g}}

strain is stretch over length(ground)

ε_{s} = \frac{Δ L_{s}}{L_{s}}

strain is stretch over length(structure)

ε_{c} = \frac{Δ L_{c}}{L_{c}}

strain is stretch over length(column)

ε_{i} = \frac{Δ L_{i}}{L_{i}}

strain is stretch over length(inner hole)

ε_{o} = \frac{Δ L_{o}}{L_{o}}

strain is stretch over length(outer rim)

Y_{g} = \frac{σ_{g}}{ε_{g}}

hookes law(ground)

Y_{s} = \frac{σ_{s}}{ε_{s}}

hookes law(structure)

Y_{c} = \frac{σ_{c}}{ε_{c}}

hookes law(column)

Y_{i} = \frac{σ_{i}}{ε_{i}}

hookes law(inner hole)

Y_{o} = \frac{σ_{o}}{ε_{o}}

hookes law(outer rim)

A_{c} = A_{o} - A_{i}

hollow cylindrical columns

F_{c} = \frac{m_{s} \cdot g}{4}

Four identical