Joist and Beam Sizing ~ calculating overall dimensions alone is
insufficient, checks should also be made to satisfy: resistance to
deflection, adequate safe bearing and resistance to shear.
Deflection † should be minimal to prevent damage to plastered
ceilings. An allowance of up to 0„003 ¾ span is normally
acceptable; for the preceding example this will be:-
0„003 ¾ 5000 mm = 15 mm
The formula for calculating deflection due to a uniformly
distributed load (see page 546) is: ~
5WL^3
384EI where I =
bd^3
12I=^50 ¾ð^225 Þ
3
12 = 4„75¾ð^10 Þ7So, deflection =^5 ¾^4500 ¾ð^5000 Þ
3(^384) ¾ (^10800) ¾ 4 : (^75) ¾ð 10 Þ^7
= 14„27 mm
NB. This is only just within the calculated allowance of 15 mm,
therefore it would be prudent to specify slightly wider or deeper
joists to allow for unknown future use.
Safe Bearing ~
=compression perpendicular to grainload at the joist end, W/2
¾ breadth
=2„4^4500 =^2
¾^50
=19mm:
therefore full support from masonry (90 mm min.) or joist hangers
will be more than adequate.
Shear Strength ~
V=2bdv 3
where: V = vertical loading at the joist end, W/2
v = shear strength parallel to the grain, 0.71 N/mm^2
Transposing:-
bd =3V2v=^32 ¾^2250
¾^0 :^71
= 4753 mm^2 minimum
Actual bd = 50 mm¾225 mm = 11,250 mm^2
Resistance to shear is satisfied as actual is well above the minimum.
Timber Beam Design