Thin Web Beams
8.2 PRIMARY DESIGN CONSIDERATIONS
To illustrate the design concepts solid timber flanges and plywood webs will be adopted.
8.2.1 Bending deflection
Thin web beams are a composite section with flanges and webs of different ma- terials, and consequently, differing E values. On a few occasions, the E values of 164
Thin Web Beams 165 the flange and web may be close enough to justify the designer taking the section as monolithic and therefore calculating geometrical properties without any modi- fication for differing Evalues. However, the more normal situation involves dif- fering E values, and therefore the use of straightforward geometrical properties may not lead to a correct or sufficiently accurate design.
It will be seen from clause 2.9 of BS 5268-2 that the load-sharing K8factor of 1.1 together with the use of Emeanfor deflection calculations do not apply to built- up beams. Special provisions are given in clause 2.10.10 of BS 5268-2. For deflec- tion the total number of pieces of solid timber in both flanges should be taken to determine factor K28to be applied to the minimum modulus of elasticity. These factors are summarized in Table 8.1
The full cross section of the plywood should be adopted and as the bending of the beam is about an axis perpendicular to the plane of the board (i.e. edge loaded) the E value for plywood webs should be taken as the modulus of elasticity in tension or compression. Stresses and moduli may be modified for duration of load using the factor K36given in Table 39 of BS 5268-2 and summarized here in Table 8.2 for service classes 1 and 2.
The EIcapacity is determined as follows:
Referring to Fig. 8.1(a) for a full depth web,
(8.1) (8.2) (8.3) EIx =Eflange flangeI +Eweb webI
I b h h
flange
w3
= ( 3- ) 12 I th
web=
3
12
Table 8.1 Modification factor K28for deflection
Number of pieces Value of K28
2 1.14
4 1.24
6 1.29
8 or more 1.32
Table 8.2 Modification factor K36by which the grade stresses and moduli for long-term duration should be multiplied to obtain values for other durations
Value of K36
Duration of loading Stress Modulus
Long term 1 1
Medium term 1.33 1.54
Short/very-short term 1.5 2
Referring to Fig. 8.1(b) for a rebated web,
(8.4) (8.5) (8.6)
8.2.2 Shear deflection
The shear deflection will be a significant proportion of the total deflection (fre- quently 20% or more), and must be taken into account in determining deflection of thin web beams.
For simple spans the shear deflection at midspan for a section of constant EIx
is given in Section 4.15.5 as:
where Kform=a form factor
M0 =the midspan bending moment AG =the shear rigidity.
For an I or box section having flanges and webs of uniform thickness through- out the span, Roark’s Formulas for Stress & Straingives the formula for the section constant Kform(the form factor) as:
where D1=distance from neutral axis to the nearest surface of the flange D2=distance from neutral axis to extreme fibre
K D D D
D t t
D
form = + ( - ) r
Ê - Ë
ˆ
¯ È
ÎÍ
˘ 1 3 ˚˙
2 1 4
10
22 12
1 2 3
2 1
22 2
dV
=Kform ¥M AG
0
EIx=Eflange flangeI +Eweb webI
I B h h
flange I
w
=È ( - ) web
ÎÍ
˘
˚˙-
3 3
12 I thr
web =
3
12 Fig. 8.1
Thin Web Beams 167 t1 =thickness of web (or webs, in box beams)
t2 =width of flange (including web thickness)
r =radius of gyration of section with respect to neutral axis = ÷I—x/A . When transposed into the terms of Fig. 8.1(a) this becomes:
where A=the area of the full section
b =the total flange width (excluding thickness of web).
If ais made equal to hf/hso that hW=h(1 -2a) then:
or
where Kv=1 +6a(1 - a)(1 -2a)(b/t).
Hence shear deflection is:
For a solid section a =0.5, hence Kv=1.0 and
which agrees with the form factor Kform = 1.2 given in section 4.15.5 for solid sections.
There is little, if any, inaccuracy in adopting Roark’s recommended approxima- tion that Kformmay be taken as unity if Ais taken as the area of the web or webs only.
The shear deflection may then be simplified to:
(8.7) where M0 =the bending moment at midspan
Aw =the area of the webs
GW=the modulus of rigidity of the webs.
8.2.3 Bending
The method most usually adopted to determine the maximum bending stress at the extreme fibre of a section assumes that the web makes no contribution to the bend- ing strength. This method has much to commend it, although it will underestimate the bending strength of the section.
dv
w w
= M A G
0
dv=
( )=
h M G bh
M GA
2 0 3
0
10 12
1 2. dv
form v
x x
= K M = 10 =
GA
Ah K M I GA
h K M GI
v
0 2
0 2
0
10
K Ah K
form I
v x
=
2
10
K Ah
I
b
form t
x
= È + ( - )( - )ÊË ˆ¯ ÎÍ
˘
˚˙
2
10 1 6 1a a 1 2a
K Ah
I
h h h h
b
form t
x
w w
= + ( - ) Ê
Ë ˆ
¯ È
ÎÍ
˘
˚˙
2 2 2
10 1 3 3
2
Apart from the simplicity of this method, disregarding the contribution made by the web to bending strength reduces the design of web splices, if these should be required, to a consideration of shear forces only.
The applied bending stress is calculated as:
(8.8) where M0 =applied bending moment
Zflange=section modulus of flange only.
The section modulus is
(8.9) where Iflange =inertia of flange only
yh =the distance from the neutral axis to the extreme fibre being con- sidered =h/2.
The permissible bending stress is:
sm, adm= sgrade¥K3¥K27 (8.10)
where K3 =factor for duration of load
K27=factor for number of pieces of solid timber in one flange as shown in Table 8.3.
8.2.4 Panel shear
Panel shear stress is the traditional term used for horizontal shear stress in a thin web beam, not to be confused with ‘rolling shear stress’ as discussed in section 8.2.5.
The maximum panel shear stress (vp) occurs at the x–xaxis and is determined as:
(8.11) v VQ
p tI
g
=
Z I
yh flange
flange
= sm
flange
= M Z
0
Table 8.3 Modification factor K27for bending Number of pieces in
one flange Value of K27
1 1
2 1.11
3 1.16
4 1.19
5 1.21
6 1.23
7 1.24
8 or more 1.25
Thin Web Beams 169
where V =applied shear
Q=first moment of area of flanges and web or webs above the x–xaxis
=Qflange+Qweb
t =web thickness
Ig =second moment of area of the full section about the x–xaxis.
Referring to Fig. 8.2(a) for a full depth web:
(8.12)
(8.13) Referring to Fig. 8.2(b) for a rebated web:
(8.14)
(8.15) The applied panel shear stress should not exceed the value tabulated in Tables 40 to 56 of BS 5268-2 according to the type of plywood adopted. The listed grade stresses apply to long-term loading. For other durations of loading the stresses should be modified by K36given in Table 39 of BS 5268-2, and repeated in this manual as Table 8.2.
Q th
web r
= 8
2
Q Q Q
bh h h
t h h h h h
bh h h t
h h
f flange
f r r
f f
r
flange each side of web flange over web
= ( )+ ( )
= ( )Ê - Ë
ˆ
¯ È
ÎÍ
˘
˚˙+ Ê - Ë
ˆ
¯ -Ê - Ë
ˆ
¯ È
ÎÍ
˘
˚˙
ÏÌ Ó
¸˝
˛
= ( )Ê - Ë
ˆ
¯ È
ÎÍ
˘
˚˙+ÏÌ ( - ) Ó
¸˝
˛
2 2 2 2
2 8
2 2
Q th
web=
2
8
Q bh h h
flange f
=( )ÊË - fˆ¯
2 Fig. 8.2
8.2.5 Web–flange interface shear
‘Rolling shear stress’ (as discussed in section 8.2.6) is the traditional term used for the web–flange interface stress in a thin web beam and relates more specifically to plywood webs although BS 5268-2 retains the rolling shear definition for other web materials such as tempered hardboard and chipboard.
The applied rolling shear stress (vr) is determined as:
(8.16) where V =applied shear
Qf=first moment of area of flange only above the x–xaxis Ig =second moment of area of the full section about the x–xaxis T =total contact depth between webs and flanges above the x–xaxis.
Referring to Fig. 8.3(a) for a full depth web:
Qflange=Qflange as eq. (8.12) T=2 ¥hf
Referring to Fig. 8.3(b) for a rebated web:
Qflange=Qflange as eq. (8.14) T=2 ¥rd
The applied rolling shear stress should not exceed the value tabulated in Tables 40 to 56 of BS 5268-2 according to the type of plywood adopted. The listed grade stresses apply to long-term loading. For other durations of loading the stresses should be modified by K36given in Table 39 of BS 5268-2, and repeated in this manual as Table 8.2.
8.2.6 Rolling shear stress
The web–flange shear in a glued ply web beam is frequently referred to as ‘rolling shear’. Figures 8.4 and 8.5 represent a plan on the top flange of an I and box beam, and Fig. 8.6 is an idealized magnification of the junction between web and flange.
The term ‘rolling shear’ is frequently used because it best describes the appear- ance of the failure which can result if the ultimate stress at the junction between
v VQ
r TI
f g
= Fig. 8.3
Thin Web Beams 171
web and flange is exceeded. In transferring horizontal shear forces from web to flange, a ‘rolling action’ takes place. If the face grain of the plywood runs per- pendicular to the general grain direction of the timber in the flange, the rolling takes place at this interface. If the face grain of the plywood runs parallel to the general grain direction of the timber in the flange, the rolling will take place between the face veneer of the plywood and the next veneer into the plywood. To avoid this rolling action, the rolling shear must be limited by providing sufficient glued contact depth between flange and web.
Rolling shear in ply web beams applies only to the plywood, not to the solid timber. Any glue lines to the solid timber flanges in ply web beams are stressed parallel to grain.
Clause 4.7 of BS 5268-2 requires the permissible rolling shear stress to be multiplied by a stress concentration modification factor K37which has a value of 0.5.
Clause 6.10.1.5 of BS 5268-2 requires the permissible rolling shear stress to be multiplied by modification factor K70=0.9 if the bonding pressure is provided by nails or staples.
Fig. 8.4 Fig. 8.5
Fig. 8.6