The Design of Beams: General Notes
4.15 DEFLECTION .1 Introduction
4.15.7 Creep and dynamic effects
The deflection limits given in BS 5268-2 are often criticized for being too sim- plistic but in reality they achieve the required purposes without being too obvious in their execution. For example, the limit of 0.003 ¥ span (1/333 of the span) is calculated under full load whereas in structural steelwork the limit of 1/360 span is under imposed load only. Taking the ratio of dead to imposed load for a domes- tic timber floor as 1 to 3, the dead load deflection limit is then 1/1332 and the imposed load 3/1332.
In simple terms, the deflection due to creep under a permanent load can cause the calculated elastic deflection to increase by about 50%. It is also accepted that the proportion of imposed load permanently on a domestic floor, e.g. due to fur- niture and fittings, is a small proportion of the design-imposed load. Taking this proportion as 33%, the total deflection (including creep) under full load becomes:
Assuming, in a steelwork design, a similar ratio of 1 to 3 for dead and imposed loads, the total deflection becomes
This demonstrates that creep has not been entirely ignored in the BS 5268-2 limit.
1 3 3
1 360
1 270
+ ¥ = span for a steel no creep construction under full load.( ) 1 5 1
1332
1 5 0 25 3 1332
0 75 3 1332
4 875 1332
1 273
. ¥ . . . .
+ ¥ ¥
+ ¥
= = span for a timber floor d
d
d d
t m
v m
= + = + Ê
Ë ˆ 1 1 15 36 ¯
2
. h
L
dt= 5 + ¥
384 1 2
8 4 3
3 2 2
pL EI
pL h EI .
GA EI
=3h 4 2
Creep is likely to be a major problem where a large part of the imposed load is in place for long periods of time, e.g. floors used for storage. In these circumstances BS 5268-2 recommends the use of Emineven where the construction is load-sharing.
The ratio of Emean/Emin=1.5 (section 2.2.5) can be seen to provide some allowance for creep in the deflection calculation.
Seviceability requires consideration of the ‘comfort’ of the user, and into this comes dynamic effects such as vibration as a person walks across a floor. This is not a simple topic to resolve for it relates to the mass of the floor construction and the actual arrangement of structural members in the floor providing lateral distri- bution of applied dynamic load and damping of the generated vibration. The problem became manifest in the 1970s as domestic floor spans tended to extend beyond the then typical 3.6 m. Analysis of the relationship between mass and stiff- ness showed that a finite deflection limit would provide an adequate control of the dynamic effects for domestic floors. Experience has shown that the threshold of human tolerance requires the natural frequency of the floor to be greater than 8 Hz.
A simple analogy of floor vibration to that of a spring gives
where n0 =natural frequency (Hz)
g =acceleration due to gravity (9810 mm/s2)
dG =dynamic deflection of the floor under its own weight.
The dynamic deflection is calculated using the dynamic Evalue which, for sim- plicity, can be taken as 1.33Emean, so taking 75% of the normal dead load deflec- tion, calculation gives the dynamic dead load deflection and the above expression becomes
After much discussion due to the range of possible values for mass, stiffness and damping characteristics a limit of 14 mm was set in BS 5268-2. Taking the range of typical dead loads for domestic floors as 0.25 kN/m2 (25% of the total) to 0.75 kN/m2(33%) then with the 14 mm total deflection limit, the natural frequencies are 9.6 and 8.4 Hz respectively. The success of this limit can also be judged by the very few complaints that have been recorded since its introduction! BS 5268-2 also requires the use of Eminfor floors subject to vibration such as gymnasia and dance floors. This provides an increase in both dead load mass and stiffness.
A development of the 14 mm philosophy is to say that an adequately stiff floor can usually be achieved where the dead load deflection does not exceed 3.5 mm (the 25% example above) irrespective of the imposed load category. Care must be exercised in using this approach where rythmic applications of load can occur, e.g.
a dance floor or marching soldiers on a bridge.
In present-day terminology these effects are grouped under the heading ‘Ser- viceability’. The approach given in EN 1990 ‘Basis of design’ (otherwise known as Eurocode 0) leads to a more transparent assessment of the serviceability crite- ria. Verification of these criteria should consider the following:
n0
1 2
9810 0 75
= = 18 2
p . d d d
.
G G G
or 18
in simple terms
n g
0
1
=2
p dG
The Design of Beams: General Notes 93
• deformations that affect the appearance, the comfort of users or the function- ing of the structure (including the functioning of machines or services) or that cause damage to finishes or non-structural elements
• vibrations that cause discomfort to people, or that limit the functional effec- tiveness of the structure
• damage that is likely to adversely affect appearance, durability or the func- tioning of the structure.
The applications of these serviceability principles are set out in Eurocode 5 DD ENV1995-1: 1994. The modulus of elasticity to be used in all deflection calcula- tions is the mean value. In particular the procedures give a more reasoned approach where more than one type of imposed load is applied to a member, e.g. a beam supporting both roof and floor loads.
For deflection there are three basic requirements:
(1) instantaneous deflection under imposed loads
(2) final deflection under total load, including creep
and for domestic floor joists (3) total instantaneous deflection
where uinst G,i=instantaneous deflection under dead load i uinst Q,j=instantaneous deflection under imposed load j uc =in-built camber
and ufin G =
ufin Q =
with y0,j =load combination factor (see Table 4.8) y2,i =load combination factor (see Table 4.8) kdef =factor for creep deflection (see Table 4.9)
Where a timber member having a moisture content greater than 20% is loaded and then dries out under load to a service class 2 condition, the kdefvalue for service class 3 should be increased by 1.00, e.g. the long-term value becomes 2.00 + 1.00 = 3.00. This reflects the well-recognized phenomenon when timber dries while under load and is a very good reason for ensuring that structural timbers intended for use in service class 2 are at the correct moisture content when installed.
Consider the domestic floor with dead load 25% of the total and the span less than 4.666 m. BS 5268-2 gives the deflection limit as span/333 using Emean. This
u k u j k
j
j j
inst Q,( + , def) + inst Q, ( , + , def)
>
Â
1 2 1
1
0 2
1 y y y
u i k
i
inst G,( + def)
>
Â
0
1
u i u
i
j j
inst G, inst Q, mm
> >
Â
+Â
£0 0
14 ufin G ufin Q uc
span + - £ 200
u u j j
j inst Q,1 inst Q,
span
+ £ 350
>
Â
y01
,
has been shown previously to give span/270 with creep. EC5 gives the long-term deflection, including creep, in service class 2 as:
There is therefore very little difference between the two methods for the simple condition of a single imposed load.