Tension Members

Tension Members

Learn the fundamental principles of designing structural steel tension members, focusing on the limit states of gross section yielding, net section fracture, block shear, and the effects of shear lag.

Tension Member

A structural element subjected to axial forces that attempt to elongate the member. Common examples include chords in trusses, bracing in frames, and cables in suspension bridges.

Design Philosophies: ASD and LRFD

The design of tension members follows two primary philosophies outlined by the AISC specification: Load and Resistance Factor Design (LRFD) and Allowable Strength Design (ASD). The required strength must be less than or equal to the available strength.

LRFD: $P_u \le \phi P_n$ (Required factored strength $\le$ Design strength)
ASD: $P_a \le P_n / \Omega$ (Required allowable strength $\le$ Allowable strength)

Where $P_n$ is the nominal strength of the member, determined by the controlling limit state.

Gross Section Yielding

The limit state where the entire cross-section yields before fracture occurs. This governs ductile failure along the member's length.

P_n = F_y A_g

Variables

Symbol	Description	Unit
$P_{n}$	Nominal yielding strength	N, kips
$F_{y}$	Specified minimum yield stress of the steel	MPa, ksi
$A_{g}$	Gross cross-sectional area of the member	$m m^{2}, i n^{2}$

Caution

Always ensure that the units for yield stress ( $F_y$ ) and gross area ( $A_g$ ) are consistent before calculating nominal strength. For example, multiplying $F_y$ in ksi by $A_g$ in $\text{in}^2$ directly yields kips. Mixing metric and US Customary units, or failing to convert correctly, is a common error.

Resistance Factors for Gross Yielding

LRFD: $\phi_t = 0.90$
ASD: $\Omega_t = 1.67$

Net Section Fracture (Rupture)

The limit state where the member fractures across the net area, typically at a connection with bolt holes, before the entire member can yield.

P_n = F_u A_e

Variables

Symbol	Description	Unit
$P_{n}$	Nominal fracture strength	N, kips
$F_{u}$	Specified minimum tensile strength (ultimate strength)	MPa, ksi
$A_{e}$	Effective net area	$m m^{2}, i n^{2}$

Resistance Factors for Net Fracture

LRFD: $\phi_t = 0.75$
ASD: $\Omega_t = 2.00$

Net Area Calculation ( $A_{n}$ )

The net area ( $A_n$ ) is the gross area minus the area lost due to holes.

Hole Diameter for Design ( $d_h$ ): The effective hole diameter is the bolt diameter plus an allowance for clearance and damage during punching/drilling. Typically, $d_h = d_{bolt} + 1/8 \text{ in}$ (or $+ 3 \text{ mm}$ ).
For a cross-section with multiple holes in a straight line across the load axis: $A_n = A_g - \Sigma(d_h \cdot t)$ .

Cochrane's Rule for Staggered Holes

Calculates the net width along a zigzag failure path when holes are staggered.

w_n = w_g - \Sigma d_h + \Sigma \frac{s^2}{4g}

Variables

Symbol	Description	Unit
$w_{n}$	Net width along the staggered path	mm, in
$w_{g}$	Gross width	mm, in
$d_{h}$	Effective hole diameter	mm, in
$s$	Longitudinal center-to-center spacing (pitch) of any two consecutive holes	mm, in
$g$	Transverse center-to-center spacing (gage) of the same two holes	mm, in

Effective Net Area and Shear Lag

Accounts for the uneven distribution of stress when a tension member is connected by only some of its elements (e.g., an angle connected by one leg).

A_e = U A_n

Variables

Symbol	Description	Unit
$A_{e}$	Effective net area	$m m^{2}, i n^{2}$
$A_{n}$	Net area (or gross area if welded)	$m m^{2}, i n^{2}$
$U$	Shear lag factor (dimensionless, $\le 1.0$ )	-

Calculating Shear Lag Factor ( $U$ )

The general formula for the shear lag factor, reflecting the distance the force must travel from the connection to the member centroid:

U = 1 - \frac{\bar{x}}{L}

$\bar{x}$ : Connection eccentricity (distance from the shear plane to the centroid of the cross-section).
$L$ : Length of the connection in the direction of loading.

Note: AISC Table D3.1 provides specific $U$ values for various standard cases (e.g., $U=1.0$ when all elements are connected).

Block Shear Strength

A tearing failure mode involving a combination of shear rupture or yielding along a longitudinal plane and tension rupture along a transverse plane.

R_n = 0.60 F_u A_{nv} + U_{bs} F_u A_{nt} \le 0.60 F_y A_{gv} + U_{bs} F_u A_{nt}

Variables

Symbol	Description	Unit
$R_{n}$	Nominal block shear strength	N, kips
$A_{nv}$	Net area subject to shear	$m m^{2}, i n^{2}$
$A_{gv}$	Gross area subject to shear	$m m^{2}, i n^{2}$
$A_{nt}$	Net area subject to tension	$m m^{2}, i n^{2}$
$U_{bs}$	Tension stress reduction factor ( $1.0$ for uniform stress, $0.5$ for non-uniform stress)	-

Resistance Factors for Block Shear

LRFD: $\phi_t = 0.75$
ASD: $\Omega_t = 2.00$

Tear-out Strength

Tear-out is a specific localized failure where a single bolt tears through the material ahead of it toward a free edge. It is checked as part of the bearing strength of the connection.

R_n = 1.2 L_c t F_u \le 2.4 d t F_u

Where $L_c$ is the clear distance from the edge of the hole to the edge of the material or the next hole, $t$ is thickness, and $d$ is bolt diameter. The limit state is governed by $\phi = 0.75$ (LRFD).

Built-up Tension Members

When single shapes are insufficient, multiple shapes (like double angles $\angle\angle$ ) are bolted or welded together.

Individual components must be connected at intervals (using stitch bolts or batten plates) to ensure composite action.
The slenderness ratio ( $L/r$ ) of individual components between connectors must not exceed 300 to prevent vibration.

Pin-Connected Members and Eyebars

Specialized tension members transferring force through a single large pin hole. They experience high localized stress concentrations.

Pin-Connected Failure Modes: Tensile rupture on the net section, shear rupture on the effective area beyond the hole, bearing on the projected area, and gross yielding.
Eyebars: Differ from generic pin-connected plates in that their heads are proportionally enlarged to strictly prevent failure in the head before yielding occurs in the main body.

Slenderness Limits

Excessive slenderness causes tension members to vibrate or sag under their own weight. This is a serviceability requirement.

\frac{L}{r} \le 300

Where $L$ is the unbraced length and $r$ is the minimum radius of gyration. Note: Tension rods and cables are explicitly exempt from this limit.

Tension Member Design Steps

Determine loads: Calculate the required tensile strength ( $P_u$ or $P_a$ ) based on factored load combinations.
Check gross yielding: Compute capacity based on $F_y$ and $A_g$ .
Check net fracture: Compute effective net area ( $A_e$ ) considering hole sizes, staggered paths, and shear lag ( $U$ ), then compute capacity based on $F_u$ .
Check block shear: Evaluate the governing combination of shear yielding/rupture and tension rupture along the connection block.
Check local tear-out/bearing: Verify the connection strength at individual bolt locations.
Verify slenderness: Ensure $L/r \le 300$ for standard structural shapes.
Select controlling capacity: The smallest nominal capacity ( $P_n$ ) among all applicable limit states dictates the design strength of the member.

Interactive Simulation

Note

Use the interactive simulation below to explore how different cross-sectional areas, steel grades, and limit states affect the capacity of a tension member.

Tension Member Capacity Analysis

Yield Stress,

F_y

(ksi)

Ultimate Stress,

F_u

(ksi)

Gross Area,

A_g

(in²)

Net Area,

A_n

(in²)

Shear Lag Factor,

U

1.00

Limit State Capacities ( $\\phi P_n$ )

Gross Yielding97.2 kips

Net Rupture108.8 kips

Block Shear75.9 kips

Governing Design Strength

75.9 kips

Key Takeaways

Tension member capacity is controlled by the lowest strength from multiple limit states: gross yielding, net fracture, block shear, and connection tear-out.
Yielding is a ductile failure over the member's length, while fracture and block shear are abrupt tearing failures at connections.
The net area calculation must account for the actual damage footprint of the hole ( $d_{bolt} + 1/8\text{"}$ ) and potential staggered failure paths using Cochrane's Rule ( $s^2/4g$ ).
Shear lag ( $U$ ) penalizes the connection efficiency when not all cross-sectional elements are directly fastened to the support.
A slenderness limit of $L/r \le 300$ prevents vibration and sagging in standard members, but rods and cables are exempt.

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Tension Member

Design Philosophies: ASD and LRFD

Gross Section Yielding

Caution

Resistance Factors for Gross Yielding

Net Section Fracture (Rupture)

Resistance Factors for Net Fracture

Net Area Calculation (AnA_nAn​)

Cochrane's Rule for Staggered Holes

Effective Net Area and Shear Lag

Calculating Shear Lag Factor (UUU)

Block Shear Strength

Resistance Factors for Block Shear

Tear-out Strength

Built-up Tension Members

Pin-Connected Members and Eyebars

Slenderness Limits

Tension Member Design Steps

Interactive Simulation

Note

Tension Member Capacity Analysis

Limit State Capacities (phiPn\\phi P_nphiPn​)

Net Area Calculation ( $A_{n}$ )

Calculating Shear Lag Factor ( $U$ )

Limit State Capacities ( $\\phi P_n$ )