Timber_Structural_Elements_Lecture_EN

1.

LECT URE 3
Timber Structural Elements
Calculating tension, compression and bending members — and the connections that join them
Chapter 2 · Design of Structural Elements · Chapter 3 (opening) · Connections
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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2.

ROAD MAP
From design rules to the calculation of every member
§2.1 General provisions
§2.2 Solid elements
§2.3 Composite elements
How members are modelled, which two
failure states govern every check, and
how loads are combined.
Tension, compression, bending and their
combinations — strength, stability and
stiffness, each checked in turn.
Built-up sections on nails, bolts or
dowels: why the connections' give
(податливость) must enter the
calculation.
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
Ch. 3 (opening) ·
Connections
How elements are joined — glued,
bearing, dowel-type or metal-plate —
and the allowable slip of each.
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3.

§2.1
General provisions for design and calculation
• Load-bearing and enclosing members of roofs, floors and frames are, as a rule, modelled as single-span, simply supported systems; only a few elements — purlins,
roof panels, wall panels, including glued ones sized for transport — are allowed to run as continuous, multi-span schemes.
• Forces and deformations are found by ordinary structural mechanics, assuming elastic material behaviour; where it matters, the calculation also accounts for the
give (податливость) in the connections themselves — trusses with continuous top chords are a case where that matters most.
• Every structure is checked against two distinct groups of limit states: loss of load-bearing capacity or fitness for service (strength, shape and position stability)
against design loads, and unfitness for normal service (deflection, settlement, displacement) against the lower, normative loads.
• Preliminary member weights are found from a self-referential formula: weight depends on the load, and the load includes the weight — so the design resolves it
with a dedicated coefficient, k_sw (self-weight), tabulated by structure type and span.
• Cross-section widths carry a constructional floor even before the strength check runs: from 12 cm at an 18 m span up to 21 cm at 33–36 m for continuous-topchord beams, arches and trusses.
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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NUMBERS T O REMEMBER · §2.1
The reliability coefficient γ_n, by building class
1.0
Class I buildings
0.95
Class II buildings
0.9
Class III buildings
0.8
Temporary structures, ≤ 5-year service life
• γ_n is the reliability-by-purpose coefficient: it scales how conservatively a member is designed according to the consequences of its failure, not its
material or span.
• Every limiting value used in a check — load-bearing capacity, design resistance, and deformation — is divided by γ_n, so a lower-class or temporary
building is allowed a smaller safety margin than a permanent Class I structure.
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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§2.2
Tension members: central and eccentric loading
• A centrally tensioned member — a truss bottom chord is the classic case — is checked on its net cross-section, F_net, which already has bolt holes,
notches and other local weakening subtracted out.
• An eccentrically tensioned member (tension combined with bending) adds a second term to the same check: the bending moment divided by the
calculated section modulus, scaled by the ratio of the tension and bending design resistances.
• Weakenings located within any 20 cm stretch of the member are treated as if they all fell in one section — the calculation does not let a designer split a
single hole pattern across a short length to dodge the net-area penalty.
• The net-area rule itself depends on whether the weakening is symmetric or not, and on whether it reaches the section's edge — the same logic that
governs the compression checks on the next slide.
EQ. 5 · CENTRAL TENSION
EQ. 6 · ECCENTRIC TENSION
σ = N / F_net ≤ R_t · m_0
σ = N/F_net + M·R_t /(W_calc·R_b) ≤ R_t
N — design axial force; F_net — net cross-section; R_t — design tensile resistance;
m_0 = 0.8, a bending-mode-of-failure factor.
M — design bending moment; W_calc — section modulus with weakenings accounted for;
R_b — design bending resistance.
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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FIG. 2.1 · §2.2
How a weakening — and an end condition — change the check
• (a) and (b) set the net-area rule: a
symmetric weakening that stays clear of the
edge keeps most of F_net; one that breaks
through the edge is penalised harder.
(c) and (d) are tapered members — their
calculated section properties are taken at
the largest height in the checked length, not
an average.
(e) is the detail every column check
depends on: pinned, fixed or free-end
conditions rescale the real length l into a
design length l0 — from 0.65l for a fixedfixed member up to 2.2l for a fixed-free
(cantilever) one.
Fig. 2.1. Schemes of compressed elements and end fixity: (a) weakened without reaching the edge; (b) weakened, reaching the edge; (c, d) tapered along their
length; (e) design length l0 as a function of end fixity.
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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§2.2
Centrally compressed elements: strength and stability
• A compressed member is checked twice, against two different failure modes, and both must pass — the strength check on the net section, and a separate
stability check that can govern for anything slender.
• The stability check divides the applied stress by the product of a calculated area, the buckling coefficient φ, and a coefficient k_zhN that corrects for a
tapered or stepped section — so a slender or tapered column is penalised on top of its raw area.
• Which area to use for F_calc depends on how the weakening sits in the cross-section: full gross area if the weakened fraction stays under 25% and clear of
the edge, a scaled-up fraction if it exceeds that, and the net area outright if the weakening reaches the edge.
• φ itself is not a constant — it falls sharply as the member's slenderness λ rises, which is exactly what Fig. 2.2 plots for four common structural materials.
EQ. 7 · STRENGTH
EQ. 8 · STABILITY
σ = N / F_net ≤ R_c
σ = N /(F_calc · φ · k_zhN) ≤ R_c
R_c — design compressive resistance; F_net as defined for tension members above.
φ — buckling coefficient (Fig. 2.2, Table 2.2); k_zhN — tapered/stepped-section correction
(Table 2.1).
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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FIG. 2.2 · §2.2
The buckling coefficient φ falls fast past λ ≈ 70
• All four curves start at φ = 1.0 for a stocky member and collapse toward zero as λ grows — past λ ≈ 70 the four
materials separate clearly, with solid timber (curve 1) always the most forgiving.
• Slenderness itself is simple: λ = l0/r, the design length from Fig. 2.1(e) divided by the section's radius of gyration — so
every stability check traces straight back to how the member's ends are actually held.
• Slenderness has ceilings, not just a formula: 120 for columns and the compressed chords or struts of trusses, 150 for
other compressed members of lattice structures and truss tension chords, 200 for bracing members and other
tension members of lattice structures.
Fig. 2.2. φ against slenderness λ for: (1) solid timber, (2)
plywood, (3) plywood tubes, (4) plywood angle sections —
timber keeps the highest φ at any given λ.
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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9.

§2.2
Bending elements: normal and shear stress
• The basic bending check divides the moment by the calculated section modulus — straightforward for a simple beam, but W_calc again depends on whether
the section being checked is weakened or not.
• Where a beam bends about both axes at once ("skew" bending — a purlin sitting on a sloped roof plane is the everyday case), the two moment components
are checked as a sum against the same bending resistance.
• A separate check covers shear: the beam's calculated width, its gross moment of inertia and the first moment of the sheared-off part of the section all enter
the formula together.
• Bending members also carry hard ceilings on where notches may be cut at supports — never on the tension edge if it can be avoided, and only within a
depth of 0.25h even where permitted.
EQ. 12 · BENDING STRESS
EQ. 14 · SHEAR STRESS
σ = M / W_calc ≤ R_b
τ = Q·S_gross /(J_gross · b_calc) ≤ R_shear
M — design bending moment; W_calc — calculated section modulus, allowing for any
weakening.
Q — design shear force; S_gross, J_gross — gross first moment and moment of inertia; b_calc
— calculated width.
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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10.

FIG. 2.3 · §2.2
Notching a beam at its support, safely
• Notching the tension zone is discouraged
everywhere, and only allowed at supports, in solid
members, to a depth a ≤ 0.25h, and only where the
bearing reaction stays under a modest stress limit
on the bearing length c.
Fig. 2.3. Notches at beam supports: (a) unreinforced; (b) reinforced with glued-in rods (1) — the notch itself follows the same a, c and c1 ≥ 2a
proportions in both cases.
• An unreinforced notch (a) relies entirely on that
geometric limit; a reinforced one (b) adds glued-in
steel or high-strength-plastic rods to anchor the
lower zone against splitting from the notch corner
outward.
• The run-out length c1 past the notch must be at
least twice the notch depth — a short, abrupt step
concentrates exactly the splitting stress the
reinforcement or the depth limit is there to control.
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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§2.2
Stiffness: checking the deflection, not just the stress
• A member that easily passes its strength check can still fail in service if it sags too far — so every bending element gets a separate deflection check
against the normative (not the design) loads.
• The calculation starts from f0, the deflection of a constant-depth member under pure bending, and corrects it upward for shear deformation (via a
coefficient c that grows with the depth-to-span ratio) and for any variation in the member's depth along its length (via k).
• For a member of constant cross-section, k = 1; tapered members take k and c from a dedicated table, keyed to how the section varies and how the
moment diagram is shaped along that length.
• Skew bending gets the same treatment as its stress check: the total deflection is the vector sum of the deflections from each moment component, f =
√(f_x² + f_y²).
EQ. 15 · DEFLECTION
f = (f0 / k) · [1 + c·(h/l)²] ≤ f_limit
f0 — bending-only deflection of a constant-depth member; k — variable-depth correction; c — shear-deformation correction; h/l — depth-to-span ratio.
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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12.

NUMBERS T O REMEMBER · §2.2
Deflection limits, as a fraction of the span
Floor beams (inter-storey)
1/250
Attic floor beams
1/200
Battens, decking
1/150
Roof deck panels
1/250
Purlins, rafter legs
1/200
Glued roof beams (except cantilevers), trusses
1/300
Cantilevered members
1/150
Panels and fachwerk (curtain-frame) elements
1/250
Load-bearing members of gable-end walls
1/400
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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§2.2
Bending elements can also buckle sideways
• A slender rectangular beam that easily passes its bending-stress check can still fail by twisting and deflecting sideways out of its loaded plane — plane-form
stability, checked separately whenever the unbraced length l0 exceeds about 7 times the compressed flange's width.
• The check reuses the buckling-coefficient logic from compression members, but through a dedicated coefficient φ_M that folds in the section's width and
depth, the unbraced length, the shape of the moment diagram along that length, and how many points along the tension edge are laterally restrained.
• Fewer than four restraint points on the tension edge over the unbraced length actively penalise φ_M; four or more restraint points and the correction factor
simply drops out (k_zhM = 1).
• Where the tension edge actually is braced within the unbraced length, a further amplification factor is applied — reflecting that a beam restrained on its
tension side resists lateral-torsional buckling noticeably better than an unbraced one.
EQ. 16 · LATERAL-TORSIONAL STABILITY
σ = M /(φ_M · W_gross) ≤ R_b
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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§2.2
Eccentrically compressed and compression-bending members
• A member under both axial compression and bending — a column carrying an off-centre beam reaction, or one resisting wind alongside gravity load — cannot
simply add the two stresses linearly, because the axial force amplifies the bending deflection as the member deforms.
• The design instead recalculates the moment on the deformed shape: the applied moment M is divided by a factor ξ·k_n, where ξ itself falls as the axial force N
approaches the member's buckling capacity — the closer to buckling, the more the moment is amplified.
• k_n depends on how the moment diagram is shaped: it is exactly 1 for simply-supported members under symmetric sinusoidal, parabolic or similar smooth
moment diagrams (and for cantilevers), but must be interpolated for triangular or rectangular diagrams instead.
• Once the amplified moment M_d is known, the member is checked on shear, on deflection (against an amplified allowable), and — for rectangular sections — on
lateral-torsional stability, all using the same M_d rather than the original, unamplified M.
EQ. 20 – 22 · COMPRESSION-BENDING
σ = N/F_calc + M_d/W_calc ≤ R_c , M_d = M/(ξ·k_n) , ξ = 1 − N/(φ·k_zhN·R_c·F_gross)
ξ falls toward 0 as N approaches the buckling load; M_d is always ≥ M, so second-order effects are never ignored, only quantified.
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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RECAP · §2.2
Every solid element answers the same three questions
Strength
Stability
Stiffness
Is the stress on the net section, under the design load,
within the material's design resistance? Checked for
tension, compression, bending and shear alike.
Could the member buckle — in-plane as a column, or
sideways as a slender beam — before the stress limit is
even reached? Governed by slenderness λ and the
coefficient φ.
Under the lower, normative loads, does the member
deflect within its service limit? Checked separately,
because a beam can be strong enough and still sag too
far.
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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§2.3
Composite elements on flexible connections
• To build a cross-section larger than a single board or timber, boards or beams are packed together with cylindrical or plate fasteners — bolts, nails, or
toothed-plate connectors — rather than glue.
• Unlike a glued joint, every one of these mechanical connections is flexible under load: it slips slightly as the structure deforms, which measurably
weakens the composite section compared with a solid one of the same overall size.
• Where the built-up section bears fully on all its layers and works in compression, strength and stability still use the same Eq. 7 and Eq. 8 as a solid
member — but with F taken over all the branches together, and φ evaluated at a slenderness that has been inflated to account for that connection slip.
• How much the slenderness is inflated depends on the connection type and count — a stiffer, more numerous set of fasteners barely changes the
calculation; a sparse or flexible one can significantly cut the section's effective stability.
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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FIG. 2.4 · §2.3
Three ways to pack a built-up section
(a), the simplest case, is a pack of boards bearing across their whole section
— its stability about the x-axis uses the ordinary slenderness formula; about
the y-axis, it uses a reduced slenderness that folds in the connection's give.
(b) adds short spacer blocks between the branches, changing how load
transfers through the section without changing which formula applies.
(c) is the special case: only some branches actually bear on each other, so the
calculated area and moment of inertia are taken only over the bearing
branches — not the full stack — for both the strength and the y-axis stability
checks.
Fig. 2.4. Composite elements on flexible connections: (a) a stack of boards bearing on their full section; (b) the same,
with short spacer blocks between branches; (c) the same, with only some branches actually bearing.
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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§2.3
Reduced slenderness: pricing in the connection's slip
• About the section's own axis (x on Fig. 2.4), the composite element's slenderness is the ordinary λ = l0/r — connection slip barely matters in that direction.
• About the perpendicular axis (y), the calculation instead uses a reduced slenderness that combines the whole section's slenderness (amplified by a coefficient
μ_y) with the slenderness of a single, unconnected branch acting alone.
• μ_y itself grows with the branch dimensions and the number of shear planes and fasteners, and shrinks with a tabulated compliance coefficient k_c that is
specific to the fastener type — steel dowels, nails, or oak dowels each get their own value.
• However the reduced slenderness comes out, it is capped at the value the branches would have acting completely independently — a composite section can
never be treated as less stable than the sum of its unconnected parts.
EQ. 28 · REDUCED SLENDERNESS (Y -AXIS)
λ_n = √[ (μ_y·λ_y)² + λ_1² ] ,
μ_y = √[ 1 + k_c · b·h·n_seam /(l0²·n_fasteners) ]
λ_y — whole-section slenderness ignoring slip; λ_1 — slenderness of one branch alone; k_c — connection compliance (Table 2.5).
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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19.

NUMBERS T O REMEMBER · §2.3
Not every fastener gives the same way
1/(5d²)
steel dowel, central compression, d ≤ a/7
1/d²
oak dowel, central compression
1/(10d²)
steel nail, central compression
• These are the compliance coefficient k_c from Table 2.5 — a smaller value means a stiffer connection, which pulls the reduced slenderness in Eq. 28 closer to
the whole-section value.
• The pattern is consistent across the whole table: steel dowels outperform steel nails at the same diameter, and a connection loaded in combined compressionand-bending is always more compliant than the same connection under pure central compression.
• A separate pair of coefficients, k_w and k_zh, scale a composite beam's effective section modulus and moment of inertia for bending — both improve (move
toward 1) as the span grows and as the number of layers in the built-up section drops.
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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§2.3
Composite elements under compression and bending
• A built-up section carrying compression together with bending — or loaded eccentrically — is checked with exactly the same Eq. 20–25 used for a solid
member, with one substitution: the section modulus W is replaced everywhere by k_w·W, the composite-corrected value.
• Where the amplification coefficient ξ (Eq. 22) is computed, the slenderness that goes into it is the reduced slenderness of Eq. 28 — so a flexibly-connected
section is amplified more than a stiff or solid one under the same load.
• The number of fasteners itself is sized from the shear flow between the built-up section's layers — the more the bending moment changes along a length, the
more shear must cross the joint, and the more connectors that length requires.
• Out-of-plane stability is still checked with the ordinary Eq. 8, using the full composite section; a further check verifies the single most heavily loaded branch on
its own, using its own individual buckling coefficient and design length.
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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21.

CHAPT ER 2 IN NUMBERS
The figures worth remembering
1.0 / 0.95 / 0.9 / 0.8
γ_n by building class I / II / III / temporary
150 / 200
slenderness limit, other lattice members /
bracing
0.25h
max. notch depth at a beam support, Fig. 2.3
1/300 / 1/150
deflection limit, glued roof beams / cantilevers
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
≥ 2a
minimum run-out length c1 past a support
notch
m_0 = 0.8
bending-failure-mode factor, central tension,
Eq. 5
120
slenderness limit, columns and truss
chords/struts
7·b_n
unbraced length above which lateral stability
governs
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CHAPT ER 3 (OPENING) · §3.1
Connections of structural elements
General provisions and classification — how joints carry load, and why every one but a glued joint is flexible
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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§3.1
Why elements are joined, and how
• Elements are connected for three distinct reasons: to extend a member's length (splicing), to build up a larger cross-section (packing several pieces
together), and to form joints where members cross or meet.
• The connection types in use are: glued joints, checked on shear; end-bearing joints and notched seats for elements meeting in compression; dowel-type
fasteners (bolts, nails, pins), which work in bending; glued-in steel rods, which resist pull-out, punching-through or shear; and metal connectors that resist
tension outright.
• Every connection type except a glued one is inherently flexible (податливое) — elastic and plastic deformations develop in it during service from the
material's own properties, and in most cases that flexibility is actually useful, because it lets the joint behave in a ductile, forgiving way rather than a
brittle one.
• Design practice favours more fasteners of lower individual capacity over fewer, heavier ones — spreading the load this way avoids concentrating force
onto whichever fastener happens to land on a timber defect, such as a knot, which would otherwise overload its neighbours.
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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NUMBERS T O REMEMBER · §3.1
How much a joint is allowed to slip
Perpendicular-to-grain bearing, solid timber elements
3 mm
Same, glued (laminated) elements
2 mm
On dowel-type fasteners (except glued-in rods)
2 mm
End-bearing joints and notched seats, end to end
1.5 mm
On metal connector plates of any type
1 mm
Glued-in rods resisting pull-out, across the grain
0.5 mm
Glued-in rods resisting pull-out, along the grain
0.25 mm
These allowances feed directly into the structure's overall deformation calculation — glued joints alone are excluded, since gluing is treated as effectively rigid.
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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CHAPT ER SUMMARY
§2.1 – §3.1 at a glance
§2.1
General provisions
Single-span modelling, two limit-state groups, γ_n by building class
§2.2
Tension & compression
Net-area rules, strength vs. stability, the buckling coefficient φ
§2.2
Bending elements
Normal and shear stress, support notches, deflection, lateral stability
§2.2
Combined loading
Compression-bending with second-order moment amplification
§2.3
Composite elements
Built-up sections on flexible fasteners; reduced slenderness
§3.1
Connections (opening)
Glued, bearing, dowel-type, glued-rod and metal-plate joints
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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26.

DESIGN OF BUILDINGS AND STRUCTURES
Every member, checked three ways
• Strength, stability and stiffness are checked separately for every tension, compression and bending member — and combined
loading simply layers these checks rather than replacing them.
• Composite, mechanically-fastened sections need one extra idea on top of the solid-member rules: the connection's own flexibility
has to be priced into the slenderness before stability can be checked at all.
• Next: Chapter 3 continues with the calculation of each connection type in detail — glued, bearing, dowel-type and metal-plate joints.
TIMBER STRUCTURES · DESIGN OF LOAD-BEARING ELEMENTS · CH. 2 – 3
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