Structural design of timber to BS 5268–2:1996 The structural design of timber elements is based on permissible stresses and deflections derived from elastic theory. su bjec ectt to bend be ndin ing g Flexural members – me mbe rs subj Members subject to bending (i.e. beams) are assumed to behave in accordance with elastic bend be ndin ing g theo th eory ry provi pr ovi ded de d that th at the th e perm pe rmis issi sible ble mate ma teri rial al stre st ress sses es are ar e not no t exce ex ce eded ed ed.. The bend be ndin ing g expression can be applied to timber design: M! "fy "#$ %t any point across a section of a beam which is located a distance y from the neutral axis of a section& a stress f will will be developed as a conse'uence of applying a bending moment M to the section. The magnitude of the stress developed will vary with the second moment of area of the section I . !n the timber design code& σ is the designation for stress& hence the above e'uation may be written as: M/I M/ I = σ /y In timber design generall rectangular sections are used! t"erefore t"e maximum com#ressi$e and tensile bending stress %ill occur at t"e extreme fibres& '"us is e(ual to "alf t"e de#t" of t"e section&
%s both ! and y are geometric properties of the section it is convenient to combine the two terms in a single property which is referred to as the elastic modulus and denoted by the symbol ) . " !y urther& as rectangular sections are being considered& if b is the width of the section and " the depth then I and may be expressed as: ! " bh * +, and y "h, -ence elastic modulus& ) " (bh * +,)(h,) " b" 2 *6 +onsidering! from t"e bending ex#ression! M*I , elastic modulus t"en: M,
)
* and combining %it" t"e definition of
'"is can be rearranged to determine t"e maximum bending stress in t"e beam and t"en com#ared %it" t"e #ermissible stress t"at t"e beam ma carr& -t"er design re(uirements '"e ot"er c"ec.s re(uired to timber beams are s"ear! bearing! deflection and t"e maximum de#t" to breadt" ratio&
%s the beams are simply supported& and generally carry uniformly distributed loads& shear will be a maxi mum at the supports.
–
* ,
×
oad /ross0sectional area
Maximum shear stress " The end bearing area is dependent on the contact area with the beam support.
oad /ontact area
"
oad width of beam
×
bearing length
Maximum bearing stress " Bot" of t"ese $alues can t"en be com#ared %it" t"e #ermissible $alues obtained from t"e design code& /eflection & as with other structural materials& is a serviceabilit y re'uirement. Maximum deflections are determined using the standard deflection formulae and compared with the deflection limits given in the design code. 'imber as a structural material
1nli2e other construction materials& timber cannot be mixed to a pre0determined formula. The cut wood has to be inspected and graded by visual or mechanical means. The design code allows for a number of strength classes based on the inspection of the timber& or alternatively& if the species of timber is 2nown it may be classified as given in 'able 8 of the code according to its standard name. %ppropriate grade stresses are assigned to the graded timber. or flexure the appropriate grade stresses are: 3 3 3
4ending parallel to the grain /ompression perpendicular to the grain 5 he ar pa ra ll el to th e gr ai n
0ccount must also be ta.en of t"e loading and ex#osure conditions t"at t"e timber %ill be subect to& '"e design code lists almost t"irt factors t"at can be a##lied to t"e grade stresses& -nl a fe% %ill be of concern in t"is course& Modification factors Moisture content of timber related to service class. !t is difficult to artificially dry solid timber more than +66 mm thic2& unless it is specially dried. 45 7,89 recognises three services classes that are related to the conditions of end use. 5ervice classes + and , generally re'uire the timber to be artificially dried and the dimensions and properties of the dried timber can be ta2en as the grade values. 5ervice class * timber is used when the finished structure is fully exposed or if the timber is more than +66 mm thic2. !n this case the grade values must be modified by a factor 2 found in Table +* which allows for
the differing load carrying mechanisms of wet and dry timber.
Ser$ice
3xam#les of end use of
0$erage
Moisture content in
class
timber
moisture
eac" #iece at time
content
eac" #iece at time
*
#xternal fully exposed
; ,6 <
,
/overed and unheated
+9 <
,= <
,
/overed and heated
+7 <
,6 <
+
!nternal use and
+9<
,=<
continuously heated
+, <
,6 <
5 er vi ce cl as se s + an d , us e u nmo di fi ed str es se s a nd mo du li .
i .e . > , "+
5ervice class * timber uses modified stresses and moduli.
i.e. > , ?+
/uration of loading 4
The grade stresses based on the strength classes of the timber apply to long0term loading on the structural element. 'able 1 gives a modification factor for various load durations and list values of > * varying from +.6 to +.@7. > * is applied to the grade stresses only and does not apply to the modulus of elasticity. oad7s"aring sstems 8 % load0sharing system may be considered as being& for example& a series of four or more floor joists connected by fl oor ing in such a wa y that ac t together – a sta ndar d timbe r floor. Ar ovided that the joists are no farther apart than 8+6 mm centres then the grade stresses should be modified by the modification factor > 9 "+.+.
or all other systems > 9 may be ta2en as being e'ual to +.6. or load0sharing systems the mean modulus of elasticity should be used to calculate any deflections except in circumstances where dynamic loads may occur& e.g. gymnasia& where the minimum value should be used. /e#t" factor
The grade stresses based on the strength classes of the timber apply to materials having a depth (h) of *66 mm. % modification factor > @ is applied to the grade bending stress of beams having a depth other than *66 mm. or solid timber beams: Bepth of beam (h) mm
> @ value
@, mm or less @, ; h ? *66 h ; *66
+.+@ (*66h) 6.++ 6.9+(h, C D,*66) (h , C 78,66)
otc"ing of beams 5
Figure 2
Eotching the end of a beam for construction purpose s causes stre ss concentrations that must be allowed for in the shear calculation. The shear stress should be calculated by using the effective depth (h e ) shown in igure ,. The grade shear stress should be multiplied by a modification factor > 7 to obtain the permissible stress. or beams notched on the underside > 7 " h e h Eote: 4eams with notches on the top edge ar e not conside re d in this unit. /eflection The deflection is acceptable if the deflection of the fully loaded beam does not exceed 6.66* times the span of the member or +=mm whichever is the lesser. 'imber flexural members design exam#les Boarding
/hec2 the suitability of ,6mm tongued and grooved floor boarding spanning between 76mm × ,76mm timber joists at 866mm centres. The boards are of strength class /+=. Eote that boarding is nor mall y provided in lengt hs up to *m long. #ach board spans over a number of joists and for analysis purposes may be treated as a continuous beam. The maximum moment occurs at an internal support and may be found using M " w , +6. The maximum shear force (reaction) occurs at the outside support and may be ta2en as F" 6.=w.
%dditional data: Bead load inclusive of self0weight of boards !mposed load
6.+7 2Em , +.7 2Em ,
Sol uti on /onsider a width of boarding (b) " +m b"+666 mm (%ctual width of floor is immaterial if width of one metre is assumed) en gt h b et we en su pp or ts " 86 6m m " 6. 8m oad on boarding w "dead C imposed " 6.+7 C +.7 " +.87 2Em , /onsidering a typical +m width of board b"+.87 × + " +.87 2Em Bending
Maximum moment M " w , +6 " +.87 × 6.8 , +6 " 6.68 2Em #lastic modulus of board " bh , 8 " +666 × ,6 , 8 " 8888@ mm * %ctual bending stress& s " M " 6.68 × +6 8 8888@ " 6.D Emm , ermissible stress , grade bending stress #arallel to t"e grain × 2 × 4 × × 8
Grade stress from Table @ – /+= " =.+ Emm , > , – wet stresses modification factor – material ,6 mm thic2 – service class +
2 , 1
> * – duration of loading – on floor this may be ta2en as long term
4 , 1
> @ – depth factor – less than @, mm
, 1&1
> 9 – load0sharing – boards are load0sharing
8 , 1&1
Aermissible stress " =.+ × +.6 × +.6 × +.+@ × +.+ " 7.,9 Emm , ; 6.D Emm , boards suitable in bending S"ear
Maximum shear force F " 6.=w " 6.= × +.87 × 6.8 " 6.= 2E
v" Maximum shear stress
*F ,bh
"
* ,
× ×
6.=
×
+666
+6* ×
,6
" 6.6* Emm ,
ermissible stress , grade s"ear stress #arallel to t"e grain × 2 × 4 × 8
Grade stress from Table @ – /+= " 6.8 Emm , The modification factors used for bending are still applicable – except > @ that is applied to bending only. Aermissible stress " 6.8 × +.6 × +.6 × +.+ " 6.88 Emm , ; 6.6* Emm , boards suitable in shear /eflection
/onsidering the beam as continuous& ∆ " w = *9=#! # mean
from Table @ – /+= (one board cannot act on its own) # " 8966 Emm ,
! "bh * +, " +666 × ,6 * +, " 88888@ mm =
∆ " w = *9=#! " +.87 × 866 = (*9= × 8966 × 88888@) " 6.+* mm Aermissible deflection (clause ,.+6.@) " 6.66* × span " 6.66* × 866 " +.9 mm %ctual deflection less than permissible – beam is suitable. Floor oists
The floor joists for the boarding example above also re'uire to be chec2ed. !t may be assumed that the joists are simply supported over a span of *.8 m and bear on to bloc2wor2 supports +66 mm wide. The revised dead load to include for the self0weight of the beam may be ta2en as 6.*= 2Em , . The joists are strength class /+8. Sol uti on /entres of joists
oadjoist
866mm " 6.8m w " (dead C imposed) × centres " (6.*= C +.7) × 6.8 " +.+ 2Em
Bending
or a simply supported beam Maximum moment M " w , 9 " +.+ × *.8 , 9 " +.@9 2Em
#lastic modulus of board " bh , 8 " 76 × ,76 , 8 " 7,69** mm * %ctual bending stress& σ " M " +.@9 × +6 8 7,69** " *.=, Emm , ermissible stress , grade bending stress #arallel to t"e grain × 2 × 4 × × 8
Grade stress from Table @ – /+8 " 7.* Emm , > , – wet stresses modification factor – material ,6 mm thic2 – service class +
2 ,1
> * – duration of loading – on domestic floor this may be ta2en as long term
4 ,1
> @ – depth factor (clause ,.+6.7) > @ " (*66h) 6.++ " *66,76) 6.+ " +.6,
,1&;2
> 9 – load0sharing – boards are load0sharing
8 ,1&1
The assumption is that the floor boards are of sufficient length to distribute the load over at least four joists. " 7. * × +.6 × +.6 × +.6, × +.+ " 7.D7 Emm , ? *.=, Emm ,
A er mi ss ib le be nd in g s tr es s 4eam satisfactory in bending. S"ear
Maximum shear force F " w, " +.+ × *.8, " +.D9 2E
v"
*F ,bh
"
* ,
× ×
+.D9 76
×
×
+6* ,76
" 6.,= Emm,
Maximum shear stress ermissible stress , grade s"ear stress #arallel to t"e grain × 2 × 4 × 8
Grade stress from Table @ – /+8 " 6.8@ Emm , The modification factors used for bending are still applicable (> @ is only applicable to bending) Aermissible stress " 6.8@ × +.6 × +.6 × +.+ " 6.@= Emm , ; 6.,= Emm ,
joist suitable in shear
Bearing
Falue of react ion " w, " +. + × *.8, " +.D9 2E Hoist bears on to a +66mm wide support and width of joist is 76mm $eaction +.D9 × +6* " " 6.*D Emm, 4earing length × width +66 × 76 %ctual bearing stress " ermissible stress , com#ression #er#endicular to t"e grain × 2 × 4 × 8
Grade stress from Table @ – compression perpendicular to the grain – ,., Emm , Two values of compression perpendicular to the grain are given in Table @. Ihich value should be usedJ $eference should be made to Eote + of the table. The modification factors used for shear are still applicable Aermissible stress " ,., × +.6 × +.6 × +.+ " ,.=, Emm , ; 6.*D Emm , bearing length is suitable /eflection
%s the beam is simply supported& ∆ " 7w = *9=#! # mean from Table @ – /+= (one board cannot act on its own) # " 9966 Emm , ! "bh * +, " 76 × ,76 * +, " 87.+ × +6 8 m m = ∆ " 7w = *9=#! " 7 × +.+ × *866 = (*9= × 9966 × 87.+ × + 6 8 ) " =., mm Aermissible deflection (clause ,.+6.@) " 6.66* × span K += mm " 6.66* × *866 " +6.9 mm %ctual deflection less than permissible – beam suitable. otc"es
!f the beam is notched at the support& then the shear cross0sectional area is reduced and the modification factor > 7 applies (see clause ,.+6.=). /onsider the above beam with a @7mm notch on the underside.
Bimension h e " h – @7 " ,76 –@7 " +@7 mm
v" Maximum shear stress
*F ,bh
"
* ,
× ×
+.D9 76
×
×
+6* +@=
" 6.*= Emm,
>7 " heh " +@7,76 " 6.@ ermissible stress , grade s"ear stress #arallel to t"e grain × 2 × 4 × 8 × 5
Aermissible stress " 6.8@ × +.6 × +.6 × +.+ × 6.@ " 6.7, Emm , ; 6.*= Emm , Hoist is still suitable in shear. 'imber com#ression members
%s with all structural materials& the design of compression members is dependent on the slenderness ratio. Ihere the slenderness ratio& λ " e i e " effective length is found using 'able 18! which lists for conditions of end restraint& the ratio of e & where is the actual length. Falues given for e are 6.@& 6.97& +.6& +.7 and , . i is the radius of gyration of the section. %s only solid rectangular sections will be dealt with& there are two possible axes of buc2ling& x–x and the y–y. -ence there are two values of slenderness ratio:
The radius of gyration
λ x " ex i x
λ y " ey i y
i x " √ ! x %
i y " √ ! y %
Ihere ! (for a rectangular section) " bh * +, ! x "bh * +, and ! y "hb * +,
%rea % " bh
considering x–x axis
considering y–y axis
The critical slenderness ratio is the larger of the two !n no case should the slenderness ratio exceed +96 (see clause 2&11& ) The permissible stress is based on the comments of clause 2&11&5 which gives two design procedures: +. ,.
/ompression members with slenderness ratios less than 7 (short columns) /ompression members with slenderness ratios greater than 7 (slender columns)
!n both cases the permissible stress is ta2en as the grade compression stress parallel to the grain multiplied by the modification factors for moisture content& duration of loading and load sharing. ermissible stress , grade stress #arallel to t"e grain × 2 × 4 × 8
!n addition for members with a slenderness greater than 7& the above formula is multiplied by > +, given in Table +D. actor > +, varies with slenderness ratio as calculated above and with # σ c & ″ where # " minimum modulus of elasticity of the material& and
σ c & ″ " compression parallel to the grain.
Exa mple Single column
% timber column ,66mm × ,66mm is re'uired to carry a load of ,+6 2E. The load has been transferred to the column by timber joists such that the end restraint conditions top and bottom may be ta2en as restrained in position but not in direction. The height of column is ,.9 m and the timber may be ta2en as strength class /,@. The load may be considered as short term. Sol uti on %s timber is greater than +66mm thic2 it would be difficult to dry the section& so use wet stresses. Falues found in Table @ are modified by factor > , found in Table +*
rom Table @ σ c & ″ " compression parallel to the grain " 9., Emm ,
> , " 6.8
# mi n " 9,66 Emm ,
> , " 6.9
e "+.6 " ,966 mm ! " bh * +, " ,66 × ,66 * +, " +.*** × +6 9 m m = % " bh " ,66 L ,66 " =6666 mm , i" √ !% " 7@.@ mm l " e i " ,9667@.@ " =9.7 (for both axes) ? +96 suitable $atio # σ c & ″ " (9,66 × 6.9)(9., × 6.8) " +***.* modified by factor > , N rom Table +D: +*66 +=66
=6 6.96D 6.9++
76 6.@7@ 6.@86
Modification factor > +, for λ " =9.7 and # σ c & ″ "+***.*
> +, " 6.@8@
%lternatively for Table +D an e'uivalent slenderness e b may be used for rectangular sections& in this example ,966,66 " += rom Table +D: +*66 +=66 as before.
++.8 6.96D 6.9++
+=.7 6.@7@ 6.@86
> * for short term loading " +.7 > 9 for non load0sharing member " +.6 Aermissible stress " grade stress parallel to the grain × > , × > * × > 9 × > +, " 9., × 6.8 × +.7 × + × 6.@8@ " 7.88 Emm , %ctual compressive stress " oad%rea " ,+6 × + 6 * =6666 " 7.,7 Emm , %s this is less than 7.88 Emm , the column is suitable. Exa mple +olumn forming #art of a #artition %all
% timber column of @,mm × +89mm cross0section supports a medium term axial load of ,= 2E. The column forms part of a partition wall that is *.D m high and the columns are arranged such that there is no load sharing. The column is restrained in position only top and bottom and is provided wit h restraining side rails at the third points about the wea2er axis. /hec2 the suitability of strength class /,, to carry the load. Sol uti on %s there are two differing effective lengths and hence two different slenderness ratios& the critical axis must be identified
e " *.Dm !x "
e "+.*m
@, × +89* +,
!y "
+89
×
@,*
+,
*
! " bh +,
i" √ !%
,9.=7 × +6 8 mm =
7.,* × +6 8 mm =
√ ,9.=7 × +6 8 (+89 × @ ,)
√ 7.,* × +6 8 (+89 × @,)
=9.7 mm
,6.9 mm
*D66
+*66
=9.7
" 96.=
,6.9
" 8,.7
l " e I /ritical axis for buc2ling is the x–x axis 5ection is less than +66mm thic2 so service class + or , applies (> , " +.6) rom Table @
σ & ″ " compression parallel to the grain " @.7 Emm , c # mi n
" 8766 Emm ,
$atio # σ c & ″ " 8766@.7 " 98@ rom Table +D for the ratio value of 98@ and l " 96.=
> +, " 6.7+
> * for medium term loading " +.,7 > 9 for non0load sharing member " +.6 Aermissible stress " grade stress parallel to the grain × > , × > * × > 9 × > +, " @.7 × +.6 × +.,7 × + × 6.7+ " =.@9 Emm , %ctual compressive stress
" oad%rea " ,= × +6 * (@, × +89) " +.D9 Emm ,
%s this is less than =.@9 Emm , the column is suitable.
