Module 6: Steel Tension Members
Steel Tension Members
Tension members are structural elements subjected to axial loads that tend to elongate the member (e.g., truss chords, bracing, cables). Because steel is incredibly strong in tension and not susceptible to buckling, tension members are highly efficient. However, their capacity is often limited by the connections, where the cross-section is reduced by bolt holes.
Tensile Yielding in Gross Section
The most basic limit state is the uniform yielding of the entire unperforated
(gross) cross-section (). If the member yields extensively, excessive
elongation occurs, leading to serviceability failures or redistribution of
loads that the structure cannot handle.
Checklist
- Nominal strength:
$$
P_n = F_y A_g
$$Checklist
- LRFD:
- ASD:
Key Takeaways
- Tensile yielding prevents massive, uncontrolled elongation of the entire member.
- It is based entirely on the gross cross-sectional area () and the yield stress ().
Tensile Rupture in Net Section
At connections (like bolted splices), the cross-sectional area is reduced by
holes (), and the stress distribution is uneven. The member can tear
(rupture) across this reduced net section. Because the failure is localized
and sudden, the capacity is based on the ultimate tensile strength ().
Checklist
- Nominal strength:
$$
P_n = F_u A_e
$$Checklist
- LRFD: (lower than yielding to reflect the brittle nature of rupture)
- ASD:
Key Takeaways
- Tensile rupture is a sudden, catastrophic failure at the connection.
- It occurs on the effective net area () and is governed by the ultimate strength ().
- The safety/resistance factors () reflect the brittle, dangerous nature of this failure compared to ductile yielding ().
Effective Net Area
Calculating the effective net area () involves two steps: finding the net
area () by subtracting hole areas, and then accounting for shear lag ().
Checklist
- Net Area (): The gross area minus the area of the holes. The standard hole diameter () used for calculations is typically larger than the nominal bolt diameter () to account for hole clearance () and damage during drilling/punching ().
- Adjusting for staggered holes along a diagonal failure path using the rule:
$$
A_n = A_g - \\sum (d_{hole} \\times t) + \\sum \\left(\\frac{s^2}{4g} \\times t\\right)
$$Staggered Fastener Paths
The s²/4g Rule (Cochrane's Rule)
When bolts are arranged in a staggered pattern, failure might not occur straight across the cross-section. The member might tear diagonally between holes if that "zig-zag" path provides less net area than the straight transverse path.
To account for the diagonal tear, the code uses Cochrane's rule. For any diagonal segment in a failure path, the net area is calculated by taking the gross area, subtracting the holes along the path, and then adding a quantity to compensate for the longer diagonal distance (which requires more force than a straight transverse tear):
- Evaluate all possible failure paths (e.g., straight A-B-C, zig-zag A-B-D-E).
- For every diagonal segment between two holes in the path, add the term multiplied by the member thickness ().
- = pitch (longitudinal center-to-center spacing of the two holes).
- = gauge (transverse center-to-center spacing of the two holes).
- The governing Net Area () is the minimum area produced by all valid path calculations.
$$
A_n = A_g - \sum d_h t + \sum \frac{s^2 t}{4g}
$$Shear Lag Factor ()
Shear lag occurs when some elements of the cross-section are not connected to the gusset plate (e.g., an angle bolted only by one leg). The stress must "flow" from the unconnected leg through the connected leg into the bolt, creating an uneven stress distribution. The unconnected part of the section is not fully effective in resisting tension.
The factor reduces the net area () or gross area () to the effective net area ().
(or for welded connections)
$$
U = 1 - \\frac{\\bar{x}}{L}
$$Key Takeaways
- Hole calculations require adding 1/8" to the bolt diameter to account for clearance and hole damage.
- The rule accounts for staggered bolt arrangements by increasing the net area along diagonal tear paths.
- Shear lag () mathematically discounts the unconnected portions of a cross-section, penalizing short, eccentric connections.
Specialized Tension Members
Beyond standard rolled shapes (like angles and W-shapes) bolted or welded to gusset plates, steel structures frequently utilize specialized tension-only elements.
Pin-Connected Members and Threaded Rods
- Threaded Rods: Commonly used for sag rods, tie downs, and bracing. The critical area is the root of the thread. For design, the nominal tensile area () is taken as (gross area of the unthreaded rod). The nominal tensile capacity is .
- Pin-Connected Members: Uses a single large pin to transfer load (like a hinge). Failure modes include tension on the effective net area across the pin hole, longitudinal shear behind the pin hole, and bearing failure of the plate against the pin.
- Eyebars: A specific type of pin-connected member where the ends are forged into a circular "eye". They are dimensioned specifically such that the body of the bar will yield before the eye fails in rupture or shear.
Slenderness Limit for Tension Members
Although tension members are not subject to buckling failures, the NSCP provides a recommended maximum slenderness ratio to prevent excessive vibration, sagging, and damage during transportation and erection.
Checklist
- Recommended Maximum Slenderness:
- This limit does not apply to tension rods or cables.
Block Shear Strength
Block shear is a complex failure mode at connections where a "block" of steel
tears out. It involves a combination of tension rupture on one plane and shear
yielding or rupture on perpendicular planes along the fastener group.
Checklist
- The nominal block shear strength () is the sum of the shear strength on the failure path parallel to the load and the tensile strength on the failure path perpendicular to the load.
- The equation utilizes a tension stress distribution factor, , which equals 1.0 when tension stress is uniform (e.g., angles, gusset plates, coped beams with single row of bolts) and 0.5 when tension stress is non-uniform (e.g., coped beams with double rows of bolts).
$$
R_n = 0.60 F_u A_{nv} + U_{bs} F_u A_{nt} \\le 0.60 F_y A_{gv} + U_{bs} F_u A_{nt}
$$Shear Lag Factor () Application
Shear lag significantly reduces the effective area when only part of a member is connected (e.g., an angle bolted by only one leg).
Checklist
- W-shapes connected by flanges: If and there are fasteners per line, .
- W-shapes connected by web only: If there are fasteners per line, .
- Single angles: Depending on the number of fasteners and connection length, typically ranges from to . The un-connected leg experiences shear lag, concentrating stress near the connection.
Key Takeaways
- Tension member capacity is governed by the lesser of Yielding in the gross section () and Rupture in the net section ().
- The Effective Net Area () accounts for the physical holes subtracted from the cross-section () and the uneven stress distribution caused by unconnected elements (Shear Lag Factor, ).
- Block Shear is an independent connection limit state involving simultaneous shear tearing and tension rupture along the perimeter of the bolt group. The capacity must exceed the applied tension load.
- The factor accounts for the uniformity of the tension stress distribution across the tearing block.