Tension Member Design -- AISC 360 Chapter D Reference

Tension members are structural elements that carry axial tensile force: bracing members, truss bottom chords, hangers, tie rods, and sag rods. They are among the simplest members to design, but the limit states -- yielding on the gross section, rupture through the net section, and block shear rupture -- must all be checked independently per AISC 360-22 Chapter D.

This reference covers every equation in Chapter D, the shear lag factor U table from Table D3.1, staggered hole net area calculations, slenderness limits, a full worked example, and multi-code comparisons.

Tension member behavior

When a steel member is loaded in axial tension, three limit states can occur, listed in order of increasing danger:

1. Yielding on the gross section -- the entire cross-section reaches the yield stress Fy. The member elongates permanently but does not fracture. This is a ductile failure with visible warning (elongation, necking). AISC assigns phi = 0.90 because the ductility provides inherent safety.

2. Rupture on the net section -- the member fractures through the weakest cross-section, typically through bolt holes or at a reduced section. This is a brittle, sudden failure with no warning. AISC assigns phi = 0.75 to provide additional safety margin for this dangerous mode.

3. Block shear rupture -- a block of material tears out of the connected element along a combination of shear and tension planes. This is also a sudden failure and must be checked for all bolted and welded connections.

The design strength is the minimum of all applicable limit states: phiPn = min(yielding, rupture, block shear).

AISC 360-22 Chapter D equations

Yielding on gross section (AISC Eq. D2-1)

phiPn = phi * Fy * Ag    (phi = 0.90)

Where:

• phi = resistance factor for yielding = 0.90
• Fy = specified minimum yield stress of the steel (ksi)
• Ag = gross area of the member (in^2)

This limit state ensures the member does not undergo excessive elongation under service loads. Yielding is ductile, so the higher phi factor reflects the inherent safety of a failure mode that gives warning.

Rupture on effective net section (AISC Eq. D2-2)

phiPn = phi * Fu * Ae    (phi = 0.75)

Where:

• phi = resistance factor for rupture = 0.75
• Fu = specified minimum tensile strength of the steel (ksi)
• Ae = effective net area = U * An (in^2)
• U = shear lag factor (see Table D3.1 below)
• An = net area = Ag - sum of hole deductions (in^2)

Rupture is brittle and sudden -- the lower phi factor compensates for the lack of warning and the consequences of fracture.

Which limit state governs?

Quick check: Rupture governs when Ae/Ag < 1.20 * Fy / Fu.

For A992 steel (Fy = 50 ksi, Fu = 65 ksi): threshold = 1.20 _ 50/65 = 0.923. Rupture governs whenever Ae < 0.923 _ Ag, which is almost always true for bolted connections where U < 1.0 and holes reduce the net area.

For A36 steel (Fy = 36 ksi, Fu = 58 ksi): threshold = 1.20 _ 36/58 = 0.745. Rupture governs when Ae < 0.745 _ Ag.

Net area calculation (An)

The net area accounts for material removed by bolt holes:

An = Ag - sum(t * dh)

Where:

• t = thickness of the material at the hole
• dh = hole diameter for calculation purposes

Hole diameter for calculation

AISC Section B4.3b specifies that the design hole diameter is the nominal bolt diameter plus 1/16 inch for standard holes (accounting for punching damage):

For oversize and slotted holes, use the actual hole dimensions from AISC Table J3.3.

Staggered holes -- the s-squared / 4g rule

When bolt holes are arranged in a zigzag (staggered) pattern, the potential failure path is not straight across the section. The stagger adds length (and therefore area) to the failure path. AISC Section B4.3b provides the s-squared / 4g correction:

Net width = Gross width - sum(dh) + sum(s^2 / (4g))

Where:

• s = longitudinal (pitch) distance between staggered holes (in.)
• g = transverse (gage) distance between staggered holes (in.)
• dh = design hole diameter (in.)

The s-squared / 4g term is added for each diagonal segment in the failure path. You must check all possible failure paths through the bolt holes; the path with the smallest net area governs.

Example: A plate 10 inches wide with four 3/4" bolts in two staggered rows. Row 1 holes at gage 2" from left edge, Row 2 holes at gage 4" from left edge. Pitch s = 3" between staggered holes.

• Straight path through Row 1: An = (10 - 2*0.8125) * t = 8.375t
• Staggered path (Row 1 to Row 2): An = (10 - 20.8125 + 3^2 / (42)) _ t = (10 - 1.625 + 1.125) _ t = 9.500t
• Straight path through Row 2: same as Row 1 = 8.375t

The straight paths govern (8.375t). The stagger correction makes the diagonal path less critical in this case.

Shear lag factor U -- AISC Table D3.1

When tension is transferred to only part of the cross-section (e.g., through the flange of a W-shape), not all elements are fully effective. The shear lag factor U accounts for this uneven stress distribution.

Ae = U * An

For members where the tension load is transmitted by transverse welds: Ae = area of the directly connected elements only.

Table D3.1 -- Shear lag factors

Where bf = flange width, d = member depth.

Shear lag for HSS -- alternate calculation

For round and rectangular HSS connected by a gusset plate through a slot:

U = 1 - (x-bar / L)

Where x-bar = distance from outer face of gusset to centroid of member cross-section, L = length of connection (distance between outermost fasteners or weld length).

Alternatively, for round HSS with single gusset through a slot:

For rectangular HSS with single gusset through a slot:

Where D = outside diameter of round HSS, B = width of rectangular HSS perpendicular to the gusset.

Block shear rupture

Block shear is a combined shear-and-tension failure where a block of material tears out of the connected element. It must be checked for every bolted and welded tension connection.

AISC Eq. J4-5

phiRn = phi * [0.60 * Fu * Anv + Ubs * Fu * Ant]     <= phi * [0.60 * Fy * Agv + Ubs * Fu * Ant]

Where:

• phi = 0.75
• Anv = net area along the shear failure plane(s) = Agv - sum(t * dh) along shear line
• Agv = gross area along the shear failure plane(s)
• Ant = net area along the tension failure plane = Agt - sum(t * dh) along tension line
• Ubs = 1.0 when tension stress is uniform (typical), 0.5 when tension stress is non-uniform (e.g., coped beams with eccentricity)

The upper bound expression (with Fu on shear plane) governs when the net shear area is much less than the gross shear area. The lower bound expression (with Fy on shear plane) can govern when holes are few or small.

When block shear governs: Short connections (few bolts in a single line), thin connected elements, and coped beam ends are especially susceptible to block shear.

Slenderness limits

AISC Section D1 recommends a maximum slenderness ratio for tension members:

L / r <= 300

Where:

• L = unbraced length of the member
• r = governing radius of gyration (least r of the cross-section)

This is a serviceability recommendation, not a mandatory strength requirement. Members with L/r > 300 are prone to excessive sagging under self-weight, vibration, and visual appearance issues. The AISC Commentary on Chapter D notes that slenderness limits for tension members address:

• Gravity sag -- slack in horizontal tension members
• Vibration -- flexible members may vibrate under wind or seismic loads
• Handling and erection -- very slender members are difficult to transport and install without damage
• Appearance -- members that visibly sag reduce confidence in the structure

For rods and cables, which have very low r values, L/r is often impractical to limit to 300. In those cases, the engineer should address serviceability through other means (prestressing, sag rods, etc.).

Worked example -- WT8x25 tension member

Given:

• WT8x25 (ASTM A992: Fy = 50 ksi, Fu = 65 ksi)
• Connected through flange with four 3/4" A325 bolts in a 2x2 pattern
• Bolt gage = 4" (transverse), bolt pitch = 3" (longitudinal)
• Edge distance = 1.5" in direction of force, edge distance transverse = 1.5"
• Member length between connections = 18 ft
• Unbraced length for slenderness = 18 ft

Section properties from AISC Manual Table 1-8:

• Ag = 7.37 in^2
• bf = 7.075 in (flange width)
• tf = 0.630 in (flange thickness)
• d = 8.13 in (overall depth)
• rx = 2.18 in, ry = 1.47 in (radii of gyration)

Step 1: Check yielding (Eq. D2-1)

phiPn(yielding) = 0.90 * Fy * Ag
                = 0.90 * 50 * 7.37
                = 331.7 kips

Step 2: Calculate net area (An)

Two bolt holes through the flange:

dh = 13/16" = 0.8125 in (standard hole for 3/4" bolt)

An = Ag - n * dh * tf
   = 7.37 - 2 * 0.8125 * 0.630
   = 7.37 - 1.024
   = 6.346 in^2

Note: Only 2 holes are subtracted because the failure line crosses only two holes in the transverse direction through the flange.

Step 3: Determine shear lag factor U

The WT is connected through its flange. From Table D3.1, Case 3:

bf / d = 7.075 / 8.13 = 0.870 > 2/3

With 4 fasteners (2 per line in 2 lines), Case 3 applies:

U = 0.90

(For 3+ fasteners per line, Case 3 values apply.)

Step 4: Calculate effective net area (Ae)

Ae = U * An = 0.90 * 6.346 = 5.711 in^2

Step 5: Check rupture (Eq. D2-2)

phiPn(rupture) = 0.75 * Fu * Ae
               = 0.75 * 65 * 5.711
               = 278.4 kips

Step 6: Check block shear

Shear planes along each line of 2 bolts (length = 1.5" + 3" = 4.5" per shear line):

Agv = 2 * (4.5 * 0.630) = 5.670 in^2 (two shear planes)

Anv = Agv - 2 * (2 * 0.8125 * 0.630)
    = 5.670 - 2.048
    = 3.622 in^2

Ant = (gage - 2 * dh/2) * tf
    = (4.0 - 2 * 0.8125/2) * 0.630 ... but this is the tension strip between shear lines.

Actually, for a WT flange connected with 2 lines of 2 bolts, the block shear tension area is the strip between the two lines of bolts:

Ant = (gage - 2 * dh) * tf = (4.0 - 2 * 0.8125) * 0.630 = 2.375 * 0.630 = 1.496 in^2

Ubs = 1.0 (uniform tension distribution).

Upper bound (Fu on shear plane):

phiRn = 0.75 * (0.60 * 65 * 3.622 + 1.0 * 65 * 1.496)
      = 0.75 * (141.26 + 97.24)
      = 0.75 * 238.50
      = 178.9 kips

Lower bound (Fy on shear plane):

phiRn = 0.75 * (0.60 * 50 * 5.670 + 1.0 * 65 * 1.496)
      = 0.75 * (170.10 + 97.24)
      = 0.75 * 267.34
      = 200.5 kips

Block shear = min(178.9, 200.5) = 178.9 kips.

Step 7: Check slenderness

L / r(max) = (18 * 12) / 1.47 = 216 / 1.47 = 147 < 300   OK

Step 8: Summary

Design strength = 178.9 kips (block shear governs).

This result illustrates an important lesson: for short connections with few bolts, block shear often governs over the yielding and rupture checks.

Capacity table -- common tension members

Tensile strength for common shapes (A992 steel, Fy = 50 ksi, Fu = 65 ksi):

Note: Ae values are illustrative and depend on the specific connection configuration. Always calculate for the actual connection.

Shear lag factor quick reference -- by shape

Multi-code comparison

AS 4100-2020 (Australia)

AS 4100 Clause 7.2 covers tension members. The nominal section capacity is:

phiNt = phi * Ag * fy     (yielding)
phiNt = phi * An * 0.85 * fu   (rupture through holes)

Where phi = 0.90 for yielding and phi = 0.90 for rupture (single phi factor, but 0.85 reduction on fu for rupture).

Key differences from AISC:

• Single phi = 0.90 for both limit states (AISC uses 0.90/0.75)
• 0.85 factor applied to fu for rupture (AISC applies full Fu but with lower phi)
• No explicit shear lag factor U in the same formulation; instead AS 4100 requires a capacity reduction for eccentric connections
• Net area uses hole diameter + 2mm (metric) or bolt diameter + 1/16" (similar to AISC)
• Block shear: Clause 7.3 with Ubs concept similar to AISC

EN 1993-1-1 / Eurocode 3

Eurocode 3 covers tension members in Clause 6.2.3:

Nt,Rd = Ag * fy / gamma_M0     (yielding, gamma_M0 = 1.00)
Nt,Rd = An * fu / gamma_M2     (rupture, gamma_M2 = 1.25)

Key differences from AISC:

• Uses partial safety factors gamma_M0 and gamma_M2 instead of LRFD phi factors
• gamma_M2 = 1.25 provides the extra safety margin for rupture (equivalent to AISC phi = 0.75 * 1/1.25 = 0.60 relative to yielding)
• Effective area Ae = An * beta (beta is similar to shear lag factor U, from EN 1993-1-8)
• Net area deduction uses hole diameter + 2mm (standard holes) or hole diameter + allowance for oversize
• Block shear per EN 1993-1-8 Clause 3.10 with combined shear and tension resistance

CSA S16-19 (Canada)

CSA S16 Clause 13.2 covers tension members:

Tr = phi * Ag * Fy     (yielding, phi = 0.90)
Tr = phi * Ae * Fu     (rupture, phi = 0.75)
Tr = phi * (Ubs * Fu * Ant + 0.60 * Fu * Anv)   (block shear)

Key differences from AISC:

• Nearly identical formulation to AISC 360
• Same phi factors (0.90 yielding, 0.75 rupture)
• Shear lag factor U per Clause 12.3.3 with similar but not identical values
• Slenderness limit L/r <= 300 (same as AISC recommendation)
• Block shear formulation very similar; Ubs = 0.8 or 0.6 depending on tension uniformity

Common mistakes

1. Checking only yielding and forgetting rupture. Both must always be checked. Rupture typically governs for bolted connections. Always compute phiPn = min(yielding, rupture, block shear).

2. Using bolt diameter instead of design hole diameter for An. The net area deduction uses dh = bolt diameter + 1/16" for standard punched holes, not the nominal bolt diameter. For 3/4" bolts, dh = 13/16" not 3/4".

3. Forgetting the shear lag factor U. Using Ae = An (i.e., U = 1.0) when the connection does not engage the full cross-section overestimates capacity. Most W, WT, and angle connections have U < 1.0.

4. Not checking all failure paths for staggered holes. A straight path across the fewest holes may not be critical. The staggered path with the s-squared/4g correction could be smaller if it crosses more holes.

5. Applying L/r <= 300 as a mandatory strength limit. AISC Section D1 states this is a "suggested maximum" -- a serviceability recommendation. It must be checked but does not limit the design strength of the member.

6. Using Ubs = 1.0 when tension stress is non-uniform. For coped beams or connections with significant eccentricity, Ubs should be 0.5, not 1.0. This significantly reduces block shear capacity.

7. Not checking block shear for short connections. Block shear often governs when there are only 1-2 bolts per line. Many engineers only check yielding and rupture and miss this critical limit state.

8. Ignoring the upper and lower bounds of block shear. Both expressions must be evaluated and the minimum governs. Using only the Fu-based expression may be unconservative.

Frequently asked questions

What is the difference between yielding and rupture in tension members? Yielding occurs across the full gross section (Ag) at the yield stress Fy. The member elongates permanently but does not break -- it is a ductile failure with visible warning. Rupture occurs through the reduced net section (An) at the tensile strength Fu. The member fractures suddenly with no warning -- it is a brittle failure. AISC requires checking both, with different phi factors (0.90 for yielding vs 0.75 for rupture) to reflect the different consequences.

How do I calculate the net area for staggered bolt holes? Use the formula: An = Ag - sum(dh _ t) + sum(s^2 / (4g) _ t), where s is the longitudinal pitch between staggered holes and g is the transverse gage. The s-squared / 4g term is added for each diagonal segment. Check all possible failure paths through the bolt holes; the path giving the smallest net area governs. This is covered in AISC Section B4.3b.

When does block shear govern over yielding and rupture? Block shear often governs when: (a) the connection has few bolts (1-2 per line), creating short shear planes; (b) the connected element is thin; (c) there is a coped beam end with eccentricity. In the worked example above, block shear (178.9 kips) governed over rupture (278.4 kips) for a 2x2 bolt pattern.

What is the shear lag factor and when is it less than 1.0? The shear lag factor U accounts for the uneven stress distribution when tension is transferred through only part of the cross-section. U < 1.0 whenever the connection does not engage all elements uniformly -- for example, bolting through only the flange of a W-shape leaves the web partially ineffective. Values range from 0.60 (single angle with 2-3 bolts) to 1.0 (plate with bolts across full width).

Can a tension member have L/r > 300? Yes. The L/r <= 300 limit in AISC Section D1 is a serviceability recommendation, not a mandatory strength requirement. Engineers can exceed it with justification, but should address the resulting serviceability concerns (sag, vibration, handling, appearance). Rods and cables routinely exceed this limit and are addressed through prestressing or support.

What steel grades are most common for tension members? ASTM A992 (Fy = 50 ksi, Fu = 65 ksi) is the standard for W-shapes, WT, and S-shapes. ASTM A36 (Fy = 36 ksi, Fu = 58 ksi) is common for plates and angles. ASTM A500 Grade B or C (Fy = 46 or 50 ksi) is used for HSS. High-strength A572 Grade 65 (Fy = 65 ksi) is used for heavier truss chords and special applications.

Do welds affect the tension member design differently than bolts? Yes. For welded connections, there are no bolt holes so An = Ag for the member itself (but the weld effective throat may govern at the connection). Shear lag still applies when only part of the section is welded. The effective area for transverse welds is limited to the area of directly connected elements (U = area of connected elements / Ag per Table D3.1 Case 2).

How does pin-connected tension member design differ? AISC Chapter D3 covers pin-connected members (eye bars, clevises, etc.) with special requirements for the pin hole geometry, including minimum edge distance from the pin hole to the member edge, minimum width at the pin hole, and checking for tear-out and splitting behind the pin. These follow different equations than standard bolted connections.

Run this calculation

• Bolted Connections Calculator -- check yielding, rupture, block shear, bolt shear, and bearing/tearout in one tool
• Section Properties Database -- look up Ag, rx, ry for any AISC shape
• Splice Connection Calculator -- tension splice design with full Chapter D checks

Related references

• Shear Lag Factor U -- complete Table D3.1 values with worked examples
• Bolt Hole Sizes -- standard, oversize, and slotted hole dimensions
• Bolt Capacity Table -- A325 and A490 single/double shear values
• Bolt Bearing and Tearout -- AISC J3.10 bearing and tearout equations
• Steel Fy and Fu -- material properties for all common steel grades
• Block Shear Rupture -- detailed block shear calculation with examples
• Bolt Spacing Requirements -- minimum and preferred bolt spacing and edge distances
• Steel Code Comparison -- AISC 360 vs Eurocode 3 vs AS 4100 vs CSA S16
• Hanger Design -- tension hangers, threaded rods, and prying action
• Connection Design Workflow -- step-by-step connection design process

Disclaimer

This page is for educational and reference use only. It does not constitute professional engineering advice. All design values must be verified against AISC 360-22 Chapters D and J and the governing project specification. The site operator disclaims liability for any loss arising from the use of this information.