Module 4: Combined Stresses in Timber and Connections

Module 4: Combined Stresses in Timber and Connections

$Learning Objectives$

Evaluate timber members subjected to combined axial force and bending using the applicable NSCP interaction provisions.
Determine bearing resistance at an angle to grain using Hankinson-type interpolation where applicable.
Identify fastener, member, and connection-group failure modes before selecting a connection.
Detail bolts, nails, screws, plates, and hangers with appropriate spacing, edge/end distance, and grain-direction awareness.
Trace lateral forces through wood diaphragms, chords, collectors, shear walls, hold-downs, and foundations.
Coordinate structural connections with architectural exposure, moisture, fire, tolerances, and erection sequence.

$NSCP Code Basis$

Combined-stress member checks are governed principally by NSCP Section 616. Timber connectors and fasteners are governed by Section 619. Architectural timber systems also rely on the framing, sheathing, floor/deck, connection, diaphragm, and shear-wall provisions in Sections 607–614.

$Prepare the Adjusted Inputs Before the Interaction Equation$

Combined-stress equations are only as correct as the adjusted values placed in them.

For sawn lumber ASD, establish the required chains first:

tension: $F_t'=F_t C_D C_M C_t C_F C_i$ ;
compression pre-stability: $F_c^*=F_c C_D C_M C_t C_F C_i$ , then determine $C_P$ and $F_c'=F_c^*C_P$ ;
bending stability input: form the beam-stability $F_b^*$ using the factors required by NDS 3.3.3 before calculating $C_L$ ;
stability modulus: $E_{min}'=E_{min}C_M C_t C_i C_T$ .

Use the factor set for the actual product and design method. A combined-action equation must not silently mix an adjusted value from one load combination with demand from another.

$Combined Axial Force and Bending$

Columns in frames, posts supporting eccentric beams, truss members with connection eccentricity, and members subjected to wind can carry axial force and bending simultaneously. The interaction equation depends on whether the axial force is tension or compression and, for compression, must reflect stability and second-order effects as required by the governing provision.

A simple linear stress sum can be useful for intuition but must not replace the actual NSCP interaction equation when the code requires a more complete form.

Tension plus bending

For uniaxial bending with axial tension, the NDS-family interaction requires both checks:

\frac{f_t}{F_t'}+\frac{f_b}{F_b^*}\le1.0

and

\frac{f_b-f_t}{F_b^{**}}\le1.0.

In this combined-tension check, $F_b^*$ is the reference bending design value multiplied by all applicable adjustment factors except $C_L$ , while $F_b^{**}$ is the reference bending design value multiplied by all applicable adjustment factors except the glulam volume factor $C_V$ . This combined-loading definition must not be confused with the beam-stability intermediate value in NDS 3.3.3, where the stability calculation excludes $C_{fu}$ , $C_V$ , and $C_L$ from its $F_b^*$ input. For biaxial bending, include the bending contribution about both principal axes as required.

Compression plus bending

For bending about one or both principal axes with axial compression, use the full stability interaction:

\left(\frac{f_c}{F_c'}\right)^2 + \frac{f_{b1}} {F_{b1}'\left(1-f_c/F_{cE1}\right)} + \frac{f_{b2}} {F_{b2}'\left[1-f_c/F_{cE2}-\left(f_{b1}/F_{bE}\right)^2\right]} \le1.0.

The interaction equation is not the only stability condition. Apply the companion NDS 2015 requirements for the actual bending case:

f_c<F_{cE1}

for uniaxial edgewise bending or biaxial bending,

f_c<F_{cE2}

for uniaxial flatwise bending or biaxial bending, and

f_{b1}<F_{bE}

for biaxial bending. For biaxial bending, also satisfy the independent stability inequality

\frac{f_c}{F_{cE2}}+ \left(\frac{f_{b1}}{F_{bE}}\right)^2<1.0.

The denominators in the main interaction equation are stability/moment-amplification terms, so they must remain positive. Check both column axes, beam stability where applicable, and all eccentricity/second-order demand. Eccentric compression creates additional bending $M=Pe$ and must be included in the applied moment; where eccentric compression governs, follow the applicable eccentric-loading provisions rather than treating $Pe$ as the only required check.

$Combined-Action Visualization$

Change compression force, eccentricity, member length, section dimensions, and the problem-defined material limits. Use the Isometric, Elevation, Member axis, and Section cutaway presets to inspect the same deterministic model from different views. Toggle the deformed shape, stress contours, neutral axis, section cut, and failure overlay; the deformation is labeled with a fixed visual exaggeration while the reported deflection remains physical. With positive compression, an eccentricity near one-sixth of the depth moves the opposite edge through zero stress and then into tension. The elastic stress field is a mechanics explanation, not the NSCP interaction or second-order check. In the grain-angle explorer below, compare the curve at 0°, 45°, and 90° with its endpoint properties rather than assuming linear interpolation.

Controls

80 kN

60 mm

3000 mm

150 mm

250 mm

12.0 MPa

8.0 MPa

16.0 MPa

9000 MPa

View presets

Contours:tensionlow tensionlow compressionhigh compressionDashed centerline is the undeformed reference.

Area A375.00 cm²

Inertia I195.3 ×10⁶ mm⁴

Section modulus S1562.5 ×10³ mm³

Moment Pe4.80 kN·m

Uniform stress P/A2.13 MPa

Bending stress M/S3.07 MPa

Compression edge5.21 MPa

Opposite edge-0.94 MPa

Neutral axis z₀-86.8 mm

Actual δmax3.07 mm

Elastic interaction screen0.37

Compression utilization43%

Opposite-edge tension utilization12%

Bending utilization19%

Teaching-limit screenWithin displayed limits

The opposite edge is in tension because the eccentricity exceeds approximately d/6.

The neutral axis lies inside the section and is shown in purple when the overlay is enabled.

Stress distribution through depth

The chart is normalized only as a graph; its edge values come directly from the member field.

compression →

This deterministic first-order visual uses

\sigma(z)=\frac{P}{A}+\frac{Pe}{I}z

for a rectangular section, with positive compression. The result is a mechanics teaching aid: NSCP Section 616 interaction, stability and second-order effects, duration/moisture/fire adjustments, bearing, fastener limit states, and the complete connection load path still require separate design checks.

$Interaction Ratio Interpretation$

An interaction value exactly equal to the code limit only satisfies the mathematical boundary of the check; it does not represent extra reserve. Report the governing utilization and avoid describing a ratio of exactly 1.00 as having a safety margin.

$Bearing at an Angle to Grain$

Connection forces are often neither perfectly parallel nor perfectly perpendicular to grain. Hankinson-type interpolation provides a rational transition between the two directional strengths for applicable wood-bearing or dowel-bearing properties.

Hankinson Formula

Directional interpolation between parallel- and perpendicular-to-grain resistance for an applicable wood property.

N_\theta= \frac{PQ}{P\sin^2\theta+Q\cos^2\theta}

Variables

Symbol	Description	Unit
$P$	Property parallel to grain.	-
$Q$	Property perpendicular to grain.	-
$\theta$	Angle between load direction and the grain.	-
$N_\theta$	Interpolated property at angle theta.	-

$Interactive Exploration$

Rotate the force relative to grain and compare the interpolated property with the parallel- and perpendicular-to-grain anchors. Use the visual only for properties where the governing timber provision permits Hankinson interpolation.

Controls

35 MPa

6.0 MPa

30 °

Parallel property P35.0 MPa

Perpendicular property Q6.0 MPa

Hankinson result Nθ15.85 MPa

Fraction of parallel value45.3%

Use this only where the governing NSCP timber provision permits Hankinson-type interpolation for the property being evaluated. Connection capacity still requires fastener, spacing, edge/end distance, group, splitting, tear-out, and connected-member checks.

$Connection Design Is a System of Limit States$

A timber connection can be controlled by fastener yielding, wood bearing, net-section tension, row tear-out, group tear-out, splitting, tension perpendicular to grain, withdrawal, connection eccentricity, plate yielding, or local crushing. The exact set depends on connection type and loading.

Do not size a connection by dividing load by a single fastener capacity and stopping there.

Connection design values and adjustment factors

For a dowel-type fastener carrying lateral load in ASD, the 2015 NDS-family connection framework uses a reference lateral design value $Z$ and applies only the factors permitted for that fastener and condition. The governing applicability table includes factors such as:

Factor	What it represents
$C_D$	load duration for ASD; the impact-load duration factor does not apply to connections
$C_M$	wet-service condition for the connection
$C_t$	sustained temperature effect
$C_g$	group action for qualifying rows of multiple fasteners
$C_\Delta$	geometry reduction when end distance or spacing is below the distance for full reference value but still within the permitted reduced range
$C_{eg}$	end-grain condition where applicable
$C_{di}$	diaphragm factor where applicable
$C_{tn}$	toe-nail factor where applicable

The basic bookkeeping form is therefore connection-specific, for example

For ASD,

Z' = Z\,C_D C_M C_t C_g C_\Delta C_{eg} C_{di} C_{tn}.

For LRFD, the corresponding chain replaces $C_D$ with the connection format-conversion, resistance, and time-effect factors:

Z' = Z\,K_F\phi\lambda C_M C_t C_g C_\Delta C_{eg} C_{di} C_{tn}.

Use only the factors that actually apply to the selected dowel-type lateral connection. Withdrawal connections and split-ring, shear-plate, timber-rivet, or other connector families use different applicability chains.

For the NDS 2015 LRFD framework, the general format-conversion and resistance factors for connection design values are $K_F=3.32$ and $\phi_z=0.65$ , together with the applicable time-effect factor $\lambda$ . Do not substitute member resistance factors such as $\phi_b$ or $\phi_c$ into a connection calculation.

$C_g$ and $C_\Delta$ are particularly important in multi-fastener layouts: adding fasteners does not guarantee capacity proportional to fastener count, and reduced end distance/spacing can reduce the reference lateral value. Use the governing NSCP/NDS tables or equations rather than assigning these factors from appearance.

Quantitative connection workflow

Input/check	What must be established
Fastener geometry	Diameter, length, threaded/reduced shank where applicable, penetration
Wood basis	Product/species/grade and the property basis required by the adopted connection table
Loading direction	Parallel, perpendicular, or angle to grain; single/double shear; withdrawal if present
Lateral yield modes	Governing wood-bearing and fastener-yield mode
Group behavior	Number of fasteners, rows, eccentricity, and load distribution
Geometry	End distance, edge distance, spacing, row spacing, hole size
Member limit states	Net section, row/group tear-out, splitting, tension perpendicular to grain
Hardware	Plate/hanger/washer/fastener strength and approved proprietary ratings

Use NSCP Section 619 or an approved product report for the numerical values rather than inventing a generic capacity for a bolt, nail, screw, or hanger.

$Spacing and Edge Distance Are Capacity Inputs$

Minimum spacing, end distance, and edge distance are structural variables. Every worked connection should either provide these dimensions or state that they were verified from the adopted NSCP table.

$Connection and Lateral-Load Workflow$

The workflow separates member interaction, fastener action, surrounding-wood failure paths, hardware, and building-system force transfer. Do not approve a connection from a per-fastener capacity alone.

Timber Combined-Action and Connection Workflow

Member interaction, fastener, surrounding-wood, and lateral-system checks for timber connections.

Resolve force path, grain, member actions, and geometry → Combined axial force and bending?; Combined axial force and bending? — Tension + M → Tension + bending: form Ft′, Fb*, Fb** and check interaction; Combined axial force and bending? — Compression + M → Compression + bending: interaction + stability checks; Combined axial force and bending? — Single action → Single action: verify uncoupled tension, compression, or beam limit states; Tension + bending: form Ft′, Fb*, Fb** and check interaction → Connection action?; Compression + bending: interaction + stability checks → Connection action?; Single action: verify uncoupled tension, compression, or beam limit states → Connection action?; Connection action? — Lateral → Lateral fasteners: determine yield modes, group action, and geometry; Connection action? — Withdrawal → Withdrawal: verify model, penetration, and grain condition; Connection action? — Combined → Combined lateral + withdrawal: apply the governing interaction provision; Lateral fasteners: determine yield modes, group action, and geometry → Check net section, splitting, tear-out, bearing, and cross-grain tension; Withdrawal: verify model, penetration, and grain condition → Check net section, splitting, tear-out, bearing, and cross-grain tension; Combined lateral + withdrawal: apply the governing interaction provision → Check net section, splitting, tear-out, bearing, and cross-grain tension; Check net section, splitting, tear-out, bearing, and cross-grain tension → Check hardware, eccentricity, proprietary ratings, and constructability; Check hardware, eccentricity, proprietary ratings, and constructability → Part of diaphragm/shear-wall load path?; Part of diaphragm/shear-wall load path? — Yes → Trace sheathing, chords, collectors, hold-downs, anchors, and foundation; Part of diaphragm/shear-wall load path? — No → Complete member + connection + system load path adequate?; Trace sheathing, chords, collectors, hold-downs, anchors, and foundation → Complete member + connection + system load path adequate?; Complete member + connection + system load path adequate? — Yes → Document governing interaction and connection limit state; Complete member + connection + system load path adequate? — No → Revise member, fastener group, connection detail, or lateral system; Revise member, fastener group, connection detail, or lateral system → Resolve force path, grain, member actions, and geometry

Resolve force path, grain, member actions, and geometry: terminator
Combined axial force and bending?: decision
Tension + bending: form Ft′, Fb*, Fb** and check interaction: process
Compression + bending: interaction + stability checks: subprocess
Single action: verify uncoupled tension, compression, or beam limit states: process
Connection action?: decision
Lateral fasteners: determine yield modes, group action, and geometry: subprocess
Withdrawal: verify model, penetration, and grain condition: process
Combined lateral + withdrawal: apply the governing interaction provision: process
Check net section, splitting, tear-out, bearing, and cross-grain tension: subprocess
Check hardware, eccentricity, proprietary ratings, and constructability: process
Part of diaphragm/shear-wall load path?: decision
Trace sheathing, chords, collectors, hold-downs, anchors, and foundation: process
Complete member + connection + system load path adequate?: decision
Revise member, fastener group, connection detail, or lateral system: process
Document governing interaction and connection limit state: terminator

$Fastener Geometry$

Spacing, end distance, edge distance, row spacing, penetration, hole size, and grain direction are structural variables. Tight spacing may reduce group effectiveness or trigger splitting even when the sum of individual fastener capacities appears adequate.

For bolts and dowel-type fasteners, use the NSCP Section 619 geometry and strength provisions applicable to the fastener and connection configuration.

$Architectural Connection Detailing$

Exposed timber connections should answer these questions before they reach construction documents:

Where does the force enter and leave the timber?
Is there adequate wood beyond the fastener group to prevent splitting and tear-out?
Can bolts actually be installed and tightened?
Can hidden steel plates drain and dry if moisture enters?
Will concealed metal heat faster than the surrounding timber in fire?
Are tolerances realistic for prefabricated or CNC-cut members?
Does the detail allow shrinkage without unintentionally restraining cross-grain movement?

Connection aesthetics should emerge from a credible force-transfer mechanism.

$Diaphragm$

A horizontal or sloped structural system that collects and transfers in-plane lateral forces to vertical resisting elements such as shear walls or frames.

$Wood Diaphragm Load Path$

Roof and floor sheathing can act as a diaphragm only when the panels, fasteners, boundary elements, chords, collectors, and supporting vertical system create a continuous load path. Large atria, stairs, skylights, courtyards, and façade setbacks can interrupt this path and create collector or transfer demands.

Architecture students should identify diaphragm boundaries and major openings at schematic-design stage rather than after the plan is fixed.

$Wood shear wall$

A vertical lateral-force-resisting assembly in which sheathing and fasteners transfer story shear while boundary members and hold-downs resist overturning actions.

$Shear Walls, Chords, Collectors, and Hold-Downs$

A shear wall is not just a sheathed partition. Its load path requires adequate panel nailing/fastening, chords or boundary framing, collectors where forces must be dragged around openings, anchorage for sliding, and hold-down action for overturning where required.

Door and window openings, short wall segments, irregular plans, and stacked openings are therefore architectural decisions with structural consequences.

$Post-and-Beam and Floor/Roof Framing$

NSCP Chapter 6 also contains framing and sheathing provisions because member design and system detailing are inseparable. Joists, rafters, beams, posts, hangers, blocking, sheathing, and wall lines must work together. A beam that is adequate in bending may still create an incomplete load path if the diaphragm, hanger, collector, or support detail is unresolved.

Key Takeaways

Combined axial and bending checks must use the applicable NSCP interaction provisions, not a convenient stress sum when the code requires more.
The Hankinson formula provides the required directional interpolation for applicable properties at an angle to grain.
Timber connections are governed by multiple fastener, member, and group limit states; spacing and grain direction are structural variables.
Sections 607–614 make wood framing, sheathing, diaphragms, and shear walls essential parts of an architecture-focused timber course.
Architectural openings and exposed connection aesthetics must be coordinated with a complete structural load path.