Question

A simply supported wood beam with a span of L = 15 ft supports a uniformly distributed load of w0 = 270 lb/ft. The allowable bending stress of the wood is 1.95 ksi. If the aspect ratio of the solid rectangular wood beam is specified as h/b = 1.75, calculate the minimum width b that can be used for the beam.

Answer 1

The minimum width is
`b = 4.32 \ in`

Step-by-step explanation:

The free body diagram is shown on the first uploaded image

The Summation of the moment about A is equal to 0 from the diagram we can see its wedged

i.e
`\sum M_A = 0`

`C_y *L = 270 (2L)/(3) *(L)/(3)`

`C_y *L -270 (2L)/(3) *(L)/(3) = 0`

Given from the question that L = 15 ft

`C_y *15 -270 * (2*15)/(3)*(15)/(3) =0`

`C_y = 900lb`

=>
`C_y =900lb \ (upward)`

The net upward force experienced by the beam = 0

i.e
`\sum F_y = 0` because of the canceling downward force

`A_y +C_y = 270 * (2L)/(3)`

`A_y + 900 = 270 * 10`

`A_y = 1800lb`

The maximum bending always occurs where the shear force is zero

`A_y - 270*x =0`

`1800 - 270 * x = 0`

`x = (1800)/(270) =6.67ft`

To obtain the maximum bending moment

`M_(max) = A_y *6.67 - 270 *6.67 *(6.67)/(2)`

`M_(max) = 1800 *6.67 - 270*6.67*(6.67)/(2)`

`=6000\ lb.ft`

To obtain the width

`(M)/(I) =(\sigma)/(y)`

The equation above is the bending equation

Where M is the bending moment
`= 6000(12) \ lb in`

Note: the multiplication by 12 is to convert the value to inches

I is the moment of inertia
`=(bh^3)/(12)`

Note: the division by 12 is to convert the value to inches

and
`\sigma` is the bending stress = 1.95 ×1000 ksi

Then y is the distance from natural axis =
`=(h)/(2)`

Substituting this into the formula we have

`(6000(12))/((bh^3)/(12) ) = (1.95(1000))/((h)/(2) )`

`(6000(12))/((bh^2)/(6) ) =1.95 * 1000`

given that
`(h)/(b) = 1.75`

=>h = 1.75 b

Substituting into the equation

`(6000(12))/((b(1.75b)^2)/(6) ) = 1.95(1000)`

`(6000(12))/(1) (6)/(3.063b^3) = 1750`

`(432000)/(3.063b^3) =1750`

`3.06b^3 =(432000)/(1750)`

`3.063b^3 = 246.86`

`b^3 = 80.593`

`b = \sqrt[3]{80.593}`

`b = 4.32 \ in`