Introduction
The notes below show the principles used in calculating the
strength of axially loaded timber members. The principles used relate to columns , stanchions, and struts. For members loaded in compression the primary failure mode is buckling. It is important to note that
the notes below are outline notes. For detailed calculations it is necessary to refer to the identified codes.
The principles used are based on the requirements
of BS 5268;Part 2.
Axially loaded members are often also subject to other loads i.e. bending and torsional loads. The notes below
only relate to members subject to concentric axial loads.
Symbols / Units
a = distance (m)
α = angle of grain (deg /rads)
A = Area (m^{2})
b = breadth of beam/thickness (m)
E = Modulus of Elasticity (N/m^{2}
E_{mean} = mean value Modulus of Elasticity (N/m^{2}
E_{min} = min value Modulus of Elasticity (N/m^{2}
G = Modulus of Rigity (N/m^{2} /Pa
h = depth of section (m)
i =radius of gyration (m)
I = Second Moment of Area (m^{4}
L =Length /span/ (m)
L_{e} =Effective Length /Effective span (m)
m = mass (kg)
n = number
λ = slenderness ratio
Q = First moment of area(m^{3}
ρ_{average} = average density (kg / m^{3})
M = Moment (Nm)
σ_{m.a,ll} = Applied bending stress parallel to grain (N/m^{2})
σ_{m.g,ll} = Grade bending stress parallel to grain (N/m^{2})
σ_{m.adm,ll} = Permissible bending stress parallel to grain (N/m^{2})

F_{v} = Applied shear Force (N)
τ_{m.a,ll} = Applied shear stress parallel to grain (N/m^{2})
τ_{m.g,ll} = Grade shear stress parallel to grain (N/m^{2})
τ_{m.adm,ll} = Permissible shear stress parallel to grain (N/m^{2})
τ_{r.a,ll} = Applied rolling shear stress parallel to grain (N/m^{2})
τ_{r.adm,ll} = Permissible rolling shear stress parallel to grain (N/m^{2})
Δ_{m} = bending deflection (m)
Δ_{s} = shear deflection (m)
Δ_{total} = total deflection (shear + bending) (m)
Δ_{adm} = pemissible deflection (m)
σ_{c.a,ll} = Applied compressive stress parallel to grain (N/m^{2})
σ_{c.g,ll} = Grade compressive stress parallel to grain (N/m^{2})
σ_{c.adm,ll} = Permissible compressive stress parallel to grain (N/m^{2})
σ_{c.a,l} = Applied compressive stress normal to grain (N/m^{2})
σ_{c.g,l} = Grade compressive stress normal to grain (N/m^{2})
σ_{c.adm,l} = Permissible compressive normal parallel to grain (N/m^{2})
σ_{t.a,ll} = Applied tensile stress parallel to grain (N/m^{2})
σ_{t.g,ll} = Grade tensile stress parallel to grain (N/m^{2})
σ_{t.adm,ll} = Permissible tensil stress parallel to grain (N/m^{2})

Relevant Standards..For comprehensive list of standards Wood related Standards
BS 5268 2 ;2002 Structural use of timber � Part 2: Code of practice for permissible
stress design, materials and workmanship.
Design Methods
In respect to columns subject to compression it is importatant the note that limit for out of
straightness is L/300 for timber sections and L/500 for glued (Glulam) sections.
Failure due to buckling is caused by the following.
 Inherent eccentricity of loading to centroid of section
 Imperfections in timber cross section.
 Nonuniformity of timber material.

The effect of these variables is to introduce initial bending with consequent bending stresses
which result in failure much earlier than would result from compressive stresses alone.
Failure due to buckling is related to
 The cross section of the column
 The slenderness.
 The permissible stress of the material.

The slenderness is calculated using the equation
The effective length L_{e} is related to the actual length L depending on the end conditions as shown in the figure below.
The radius of gyration is calculated as follows..
The maximum slenderness ratio is, according to the BS EN 52682, as follows.
 The slenderness ratio has a maximum value of 180
 any compression member carrying dead and imposed loads other than loads resulting from winds
 any compression member , however loaded , which by its deformation will adversely affect the stress in another member carrying dead and imposed loads other than wind.
 The slenderness ratio has a maximum value of 250
 any member normally subject to tension or combined tension and bending arising from dead and imposed loads bu subject to a reversal of strees only from wind
 any compression member, carrying only self weight nad wind loads..

For compression members with slenderness ratios of less than 5, without undue eccentricity of loading, the
permissible stress should be taken as the grade compression parallel to the grain stress modified as
appropriate for moisture content, duration of loading and load sharing.
σ_{c,adm,ll} = σ_{c,g,ll}.K_{2}.K_{3}.K_{8}
For compression members with slenderness ratios equal to or greater than 5, the permissible stress should
be calculated as the product of the grade compression parallel to the grain stress, modified as appropriate
for moisture content, duration of loading and load sharing, and the modification factor, K_{12} as provided in table below
σ_{c,adm,ll} = σ_{c,g,ll}.K_{2}.K_{3}.K_{8}.K_{12}
E /σ_{c,ll}  Value of K_{12} 
Values of slenderness ratio =λ 
< 5 
5 
10 
20 
30 
40 
50 
60 
70 
80 
90 
100 
120 
140 
160 
180 
200 
220 
240 
250 
Equivalent Le /b for rectangular sections 
<1,4 
1,4 
2,9 
5,8 
8,7 
11,6 
14,5 
17,3 
20,2 
23,1 
26,0 
28,9 
34,7 
40,5 
46,2 
52,0 
57,8 
63,6 
69,4 
72,3 
400 
1000 
0,975 
0,951 
0,896 
0,827 
0,735 
0,621 
0,506 
0,408 
0,330 
0,271 
0,225 
0,162 
0,121 
0,094 
0,075 
0,061 
0,051 
0,043 
0,040 
500 
1000 
0,975 
0,951 
0,899 
0,837 
0,759 
0,664 
0,562 
0,466 
0,385 
0,320 
0,269 
0,195 
0,148 
0,115 
0,092 
0,076 
0,063 
0,053 
0,049 
600 
1000 
0,975 
0,951 
0,901 
0,843 
0,774 
0,692 
0,601 
0,511 
0,430 
0,363 
0,307 
0,226 
0,172 
0,135 
0,109 
0,089 
0,074 
0,063 
0,058 
700 
1000 
0,975 
0,951 
0,902 
0,848 
0,784 
0,711 
0,629 
0,545 
0,467 
0,399 
0,341 
0,254 
0,195 
0,154 
0,124 
0,102 
0,085 
0,072 
0,067 
800 
1000 
0,975 
0,952 
0,903 
0,851 
0,792 
0,724 
0,649 
0,572 
0,497 
0,430 
0,371 
0,280 
0,217 
0,172 
0,139 
0,115 
0,096 
0,082 
0,076 
900 
1000 
0,976 
0,952 
0,904 
0,853 
0,797 
0,734 
0,665 
0,593 
0,522 
0,456 
0,397 
0,304 
0,237 
0,188 
0,153 
0,127 
0,106 
0,091 
0,084 
1000 
1000 
0,976 
0,952 
0,904 
0,855 
0,801 
0,742 
0,677 
0,609 
0,542 
0,478 
0,420 
0,325 
0,255 
0,204 
0,167 
0,138 
0,116 
0,099 
0,092 
1100 
1000 
0,976 
0,952 
0,905 
0,856 
0,804 
0,748 
0,687 
0,623 
0,559 
0,497 
0,440 
0,344 
0,272 
0,219 
0,179 
0,149 
0,126 
0,107 
0,100 
1200 
1000 
0,976 
0,952 
0,905 
0,857 
0,807 
0,753 
0,695 
0,634 
0,573 
0,513 
0,457 
0,362 
0,288 
0,233 
0,192 
0,160 
0,135 
0,116 
0,107 
1300 
1000 
0,976 
0,952 
0,905 
0,858 
0,809 
0,757 
0,701 
0,643 
0,584 
0,527 
0,472 
0,378 
0,303 
0,247 
0,203 
0,170 
0,144 
0,123 
0,115 
1400 
1000 
0,976 
0,952 
0,906 
0,859 
0,811 
0,760 
0,707 
0,651 
0,595 
0,539 
0,486 
0,392 
0,317 
0,259 
0,214 
0,180 
0,153 
0,131 
0,122 
1500 
1000 
0,976 
0,952 
0,906 
0,860 
0,813 
0,763 
0,712 
0,658 
0,603 
0,550 
0,498 
0,405 
0,330 
0,271 
0,225 
0,189 
0,161 
0,138 
0,129 
1600 
1000 
0,976 
0,952 
0,906 
0,861 
0,814 
0,766 
0,716 
0,664 
0,611 
0,559 
0,508 
0,417 
0,342 
0,282 
0,235 
0,198 
0,169 
0,145 
0,135 
1700 
1000 
0,976 
0,952 
0,906 
0,861 
0,815 
0,768 
0,719 
0,669 
0,618 
0,567 
0,518 
0,428 
0,353 
0,292 
0,245 
0,207 
0,177 
0,152 
0,142 
1800 
1000 
0,976 
0,952 
0,906 
0,862 
0,816 
0,770 
0,722 
0,673 
0,624 
0,574 
0,526 
0,438 
0,363 
0,302 
0,254 
0,215 
0,184 
0,159 
0,148 
1900 
1000 
0,976 
0,952 
0,907 
0,862 
0,817 
0,772 
0,725 
0,677 
0,629 
0,581 
0,534 
0,447 
0,373 
0,312 
0,262 
0,223 
0,191 
0,165 
0,154 
2100 
1000 
0,976 
0,952 
0,907 
0,863 
0,818 
0,773 
0,728 
0,681 
0,634 
0,587 
0,541 
0,455 
0,382 
0,320 
0,271 
0,230 
0,198 
0,172 
0,160 
Timber beams subject to axial compression and bending
Timber members pinned at both ends which are subject to bending and axial
compression should be so proportioned that:
Spaced Columns
Spaced comumns are two or more equal parallel columns used together as supports and
spaced using packing blocks as shown below. These composite arrangements can be used individually or as component
parts of lattice girders.
L = overall length of columns (m)
l _{e} = thickness of end packing pieces (m)
l _{i} = thickness of intermediate packing pieces (m)
l = pitch packing pieces (m)
a = centre distance between component columns (m)
b = column thickness (m)
End packing.
The end packing length l_{e} should be sufficient to accommodate the nails, screws
or connectors required to transmit, between the abutting face of the packing and
one adjacent column, a shear force equal to:
A = Total cross section area of column (m^{2})
l _{e} = thickness of end packing pieces (m)
n = number of columns
The spacing of the columns should be less than 3 x the thickness of the thinner column.
Intermediate packing.
The length of the intermediate packs l _{i} must be at least 230mm . Also the connecting methods should be designed
to transmit at least 50% of that force required for an end pack.i.e the shear forc to be resisted =
The spacing of the columns should be less than 3 x the thickness of the thinner column.
For cases where L=< 30b then only intermediate packing is required.
In any event, sufficient packings should be provided to ensure that the slenderness ratio (L_{e}/ i )_{1C} of the local portion of an individual shaft between packings is limited to
either 70, or to 0.7 times the slenderness ratio of the whole column, whichever is the lesser. L_{e}for the local part should be based on the distance between the packing centroids
Load Capacity
The load capacity of a spaced column is calculated as the least of the following.
Bending about XX axis...Axial capacity = ( Total column Area.) x ( σ_{c,adm,ll})
Bending about YY axis.....Axial capacity = ( Total column Area.) x ( σ_{c,adm,ll})
where the effect length L_{eyy} is assessed in accordance with the diagram above and multiplied by
K_{13}
Individual buckling.....Axial capacity = (n x area of one column) x ( σ_{c,adm,ll})
where the effective length L_{e} is the length between packings and the rzdius of gyration i
iy for the yy axis of a single column.
Connection  K_{13} 
Ratio of space to thickness of thinner member 
0  1  2  3 
Nailed  1,8  2,6  3,1  3,5 
Scewed/bolted  1,7  2,4  2,8  3,1 
Connectored  1,4  1,8  2,2  2,4 
Glued  1,1  1,1  1,3  1,4 
Tension Members
The permissible stress for members subject to tensile loads provided in the tables are generally
based on solid timber with a width of 300mm. For members with width other than 300m a factor K_{14} must
be applied. ref K_{14}.
A timber member subject to direct tensile load is simple simply sized by ensuring the actual stress is less than the admissible stress
calculated using the appropriate grade stress and modifying factors where applicable
 K_{2} relates to the moisture content of the timber
 K_{3} relates to the duration of the load.
 K_{6} relates to the shape of the cross section.
 K_{8} relates load sharing factors.
 K_{14} relates to the depth of the section

Therefore
σ_{t.adm,ll} = σ_{t.g}. K_{2}.K_{3}.K_{6}.K_{8}.K_{14}
σ_{t} ≤ σ_{t.adm,ll}
σ_{t,ll} = Calculated tensile stress parallel to grain
σ_{t.adm,ll} = Admissible tensile stress parallel to grain
σ_{t.g,ll} = Grade tensile stress parallel to grain
For members subject to combined bending and axial tensile the following equation applies
