Tension Members

Tension Members

Tension members are structural elements subjected to axial tensile forces. They are common in roof trusses, building cross-bracing, bridge cables, and hangers. The primary design considerations involve the cross-sectional area, material strength, connection details, and potential slenderness issues.

Limit States

The design strength of tension members (

\phi_t P_n

) is the minimum of two primary strength limit states, and potentially a third serviceability limit state:

Tensile Yielding (Gross Section)

Occurs in the gross section away from connections. The member elongates significantly before failure, providing ample warning. This is a ductile failure mode.

Formula:

P_n = F_y A_g

Resistance Factor ( $\phi_t$ ): 0.90 (LRFD)
Safety Factor ( $\Omega_t$ ): 1.67 (ASD)

Tensile Rupture (Net Section)

Occurs at the connection where holes reduce the cross-sectional area and cause stress concentrations. The member fractures abruptly across the net area. This is a brittle failure mode.

Formula:

P_n = F_u A_e

Resistance Factor ( $\phi_t$ ): 0.75 (LRFD)
Safety Factor ( $\Omega_t$ ): 2.00 (ASD)

Where:

$F_y$ = Yield stress (e.g., 50 ksi for A992)
$F_u$ = Ultimate tensile stress (e.g., 65 ksi for A992)
$A_g$ = Gross cross-sectional area (from the AISC manual)
$A_e$ = Effective net area ( $A_n \times U$ )

Area Calculations

A_g

The total, uncut cross-sectional area of the member, found in the section properties table of the AISC Manual.

A_n

The area remaining after accounting for bolt holes and potential failure paths.

When fasteners (bolts or rivets) are placed in a straight line across the member width, the net area is simply the gross area minus the area of the holes.

A_n = A_g - \sum (d_h \times t)

Where:

$d_h$ = Effective hole diameter. This is the nominal bolt diameter plus a standard hole clearance ( $1/16"$ ) plus an allowance for damage to the edge of the hole during punching or drilling ( $1/16"$ ). Total $d_h = \text{bolt } d + 1/8"$ .
$t$ = Thickness of the connected part.

Staggered Fasteners

Accounting for diagonal failure paths when holes are not aligned transversely.

When bolts are staggered, the failure path might not be straight across. It might zigzag between holes. The critical net area is the smallest area along any possible failure path. Cochrane's formula adds a term (

s^2/4g

) for each diagonal segment of the path to account for the longer, inclined length.

A_n = A_g - \sum (d_h \times t) + \sum \frac{s^2}{4g} t

Where:

$s$ = Pitch (longitudinal spacing between any two successive holes).
$g$ = Gage (transverse spacing between the same two holes).

Interactive Net Area Calculator

Click on the bolt holes to simulate a potential fracture path. The calculator will automatically apply Cochrane's rule ($s^2/4g$) for staggered bolts.

Assume 3/4" bolts. Hole diameter = 3/4" + 1/8" = 0.875".
Select a valid logical path (e.g., A-B is invalid for tension rupture as it's a vertical tear, choose A-C-D instead).

Path Calculation

Select holes on the plate to generate a fracture path.

Governing Path

In actual design, you must calculate the Net Area ($A_n$) for all possible paths and select the smallest value to determine the governing design capacity for tension rupture.

A_e

Accounts for Shear Lag when not all elements of the cross-section are connected to the gusset plate or support.

A_e = A_n U

Where

U

is the shear lag factor. For welded connections where the member is attached over an area,

A_n = A_g

, so

A_e = A_g U

Understanding Shear Lag ( $U$ )

Shear lag occurs when a tension force is transferred to a member through some, but not all, of its cross-sectional elements (e.g., an angle connected by only one leg, or a W-shape connected only by its flanges).

The disconnected part of the cross-section (the unconnected leg or the web) is not fully effective at the immediate location of the connection. The force must "lag" or transfer via shear from the connected part to the unconnected part over the length of the connection (

L

General Formula for U (Case 2):

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

Where

\bar{x}

is the distance from the shear plane (the face of the connection) to the centroid of the cross-section of the tension member.

L

is the length of the connection in the direction of loading.

Other Common Cases for U (AISC Table D3.1):

Case 1: All elements connected. $U = 1.0$ .
Case 7 (W, M, S, HP Shapes): Connected by flanges with $\ge 3$ fasteners per line in direction of load. If $b_f \ge 2/3 d$ , then $U = 0.90$ . If $b_f < 2/3 d$ , then $U = 0.85$ .
Case 8 (Angles): Connected by one leg. If $\ge 4$ fasteners per line, $U = 0.80$ . If 2 or 3 fasteners per line, $U = 0.60$ .

Block Shear Strength

A tearing failure mode involving a combination of shear and tension along orthogonal planes.

Block shear can occur when a chunk of the connected material (like the corner of a gusset plate or the end of a beam web) tears out. It involves shear failure along a longitudinal plane (parallel to the load) and tension failure along a transverse plane (perpendicular to the load).

Block Shear Formula (AISC Eq. J4-5)

The nominal block shear strength

R_n

is the sum of the shear strength and the tension strength.

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}

Where:

$A_{nv}$ = Net area subject to shear.
$A_{nt}$ = Net area subject to tension.
$A_{gv}$ = Gross area subject to shear.
$U_{bs}$ = Tension stress reduction factor ( $1.0$ for uniform stress, $0.5$ for non-uniform stress like coped beams).
$\phi = 0.75$ (LRFD).

The formula checks two possibilities: shear rupture + tension rupture, and shear yielding + tension rupture. The lesser of the two governs.

Built-up Tension Members

Connecting multiple shapes to act as a single unit.

When a single standard rolled shape (like an angle or a W-shape) does not provide sufficient cross-sectional area or an adequate radius of gyration (

r

), a built-up member is fabricated by bolting or welding two or more shapes together. The most common configuration is double angles (

\angle\angle

), often separated by a small gap to accommodate a gusset plate.

Detailing Built-up Members

Interconnection Spacing: The individual components of a built-up tension member must be periodically connected to each other along their length. If they are not connected, each shape would act independently as an extremely slender member and could vibrate or rattle.
AISC Spacing Requirement: AISC dictates that the spacing ( $a$ ) between these intermittent connections (e.g., stitch bolts or small weld plates called "batten plates") must be such that the slenderness ratio of the individual component between the connections ( $a/r_i$ ) does not exceed 300.
Continuous Action: These connections ensure the separate shapes share the load equally and act compositely.

Pin-Connected Members and Eyebars

Specialized tension members used in bridges and historic structures.

Eyebars and pin-connected plates use a single large pin to transfer the tensile force. They are designed using different limit states because the stress concentration around the single pin hole is severe.

Failure Modes for Pin-Connected Members

Tensile Rupture on the Net Section: Checked across the hole diameter. $\phi = 0.75$ .
Shear Rupture on the Effective Area: Checked on the planes parallel to the load beyond the pin hole. $\phi = 0.75$ .
Bearing on the Projected Area: Checked on the projected area of the pin hole ( $d_{pin} \times t$ ). $\phi = 0.75$ .
Yielding on the Gross Section: Checked on the full member body. $\phi = 0.90$ .

AISC provides specific dimensional requirements for the "head" of an eyebar or pin-connected plate (e.g., edge distances and net widths) to ensure these failure modes do not occur prematurely.

Threaded Rods

Used for bracing, hangers, and tie-downs.

Threaded rods are designed based on their tensile stress area, which is less than the nominal gross area because of the material removed to cut the threads.

P_n = 0.75 F_u A_b

Where:

$A_b$ = Nominal (unthreaded) area of the rod.
$F_u$ = Ultimate tensile strength of the rod material.
The $0.75$ factor inherently accounts for the reduction in area due to threading (typically $\sim 75-80\%$ of $A_b$ ).
$\phi_t = 0.75$ (LRFD).

Fatigue and Cyclic Loading

Considerations for tension members subjected to repeated stress fluctuations.

While most building tension members carry relatively static loads, tension members in bridges, crane runways, and structures supporting heavy machinery are subjected to cyclic loading. Repeated fluctuations in stress can cause a progressive, localized structural damage known as fatigue.

Fatigue failure occurs abruptly at stress levels well below the material's yield strength (

F_y

). In tension members, fatigue cracks almost exclusively initiate at stress concentrations, such as bolt holes, welds, or sudden changes in cross-section.

AISC Fatigue Provisions (Appendix 3)

AISC Appendix 3 governs fatigue design. The design is based on the stress range (

F_{SR}

), which is the difference between the maximum and minimum stress in a load cycle, not the absolute peak stress.

F_{SR} = |f_{max} - f_{min}|

The allowable stress range (

F_{TH}

, the threshold stress) depends on:

The Detail Category (A through E'): This classifies the severity of the stress concentration. A plain, unpunched steel bar is Category A (high fatigue life). A member with a severe welded attachment might be Category E' (very low fatigue life).
The Number of Load Cycles ( $N$ ): Structures expected to undergo millions of cycles (e.g., a busy highway bridge) have much lower allowable stress ranges than structures experiencing only a few thousand cycles.

Slenderness Ratio

A serviceability recommendation to prevent vibration or sag.

While tension members do not buckle like compression members, excessive slenderness can cause vibration under wind loads, sagging under self-weight, or damage during transportation and erection.

Recommended limit: $L/r \le 300$
This is a serviceability recommendation, not a strict strength limit.
Exceptions: Tension rods, cables, and members acting purely as hangers are entirely exempt from this slenderness limit because they cannot resist significant bending or compression anyway.

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 (φP_n)

Gross Yielding0.0 kips

Net Rupture0.0 kips

Block Shear0.0 kips

Governing Design Strength

0.0 kips

The Whitmore Section

A method for evaluating the strength of gusset plates connecting tension members.

When a tension member connects to a gusset plate, the tensile stress spreads out from the fasteners into the plate. The Whitmore section is an effective width of the gusset plate assumed to carry the tensile force uniformly.

Whitmore Section Calculation

The force is assumed to spread out at a $30^\circ$ angle from the first row of fasteners to the last row of fasteners.
The effective width ( $L_w$ ) is measured perpendicular to the load axis at the last row of fasteners.
The tension capacity of the gusset plate is then checked using $P_n = F_y \times L_w \times t_{plate}$ (Yielding) and $P_n = F_u \times L_{w,net} \times t_{plate}$ (Rupture).

Key Takeaways

The capacity of a tension member is governed by the lesser of yielding on the gross section ( $\phi=0.90$ ) or rupture on the net section ( $\phi=0.75$ ).
Net section calculations must account for the actual hole size (bolt diameter + 1/8") and staggering (using Cochrane's $s^2/4g$ term).
Shear lag ( $U$ ) reduces the effective net area ( $A_e$ ) when a member is connected by only some of its elements, reflecting the inefficiency of force transfer. The formula $U = 1 - \bar{x}/L$ is standard.
Block shear is a critical tearing failure mode involving simultaneous shear and tension along orthogonal planes at the connection.
Built-up tension members (like double angles) require intermittent connections to ensure composite behavior and prevent vibration.
Special provisions apply to threaded rods (using $0.75A_b$ ), pin-connected members, and the gusset plates (Whitmore section) that connect them.
A slenderness limit of $L/r \le 300$ is recommended for serviceability to prevent sag and vibration, but rods are exempt.

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Limit States

Tensile Yielding (Gross Section)

Tensile Rupture (Net Section)

Area Calculations

Gross Area ( $A_g$ )

Net Area ( $A_n$ )

Staggered Fasteners

Interactive Net Area Calculator

Path Calculation

Governing Path

Effective Net Area ( $A_e$ )

Understanding Shear Lag ( $U$ )

Block Shear Strength

Block Shear Formula (AISC Eq. J4-5)

Built-up Tension Members

Detailing Built-up Members

Pin-Connected Members and Eyebars

Failure Modes for Pin-Connected Members

Threaded Rods

Fatigue and Cyclic Loading

AISC Fatigue Provisions (Appendix 3)

Slenderness Ratio

Tension Member Capacity Analysis

Limit State Capacities (φP_n)

The Whitmore Section

Whitmore Section Calculation

Tension Members

Limit States

Tensile Yielding (Gross Section)

Tensile Rupture (Net Section)

Area Calculations

Gross Area (AgA_gAg​)

Net Area (AnA_nAn​)

Staggered Fasteners

Interactive Net Area Calculator

Path Calculation

Governing Path

Effective Net Area (AeA_eAe​)

Understanding Shear Lag (UUU)

Block Shear Strength

Block Shear Formula (AISC Eq. J4-5)

Built-up Tension Members

Detailing Built-up Members

Pin-Connected Members and Eyebars

Failure Modes for Pin-Connected Members

Threaded Rods

Fatigue and Cyclic Loading

AISC Fatigue Provisions (Appendix 3)

Slenderness Ratio

Tension Member Capacity Analysis

Limit State Capacities (φP_n)

The Whitmore Section

Whitmore Section Calculation

Gross Area ( $A_g$ )

Net Area ( $A_n$ )

Effective Net Area ( $A_e$ )

Understanding Shear Lag ( $U$ )