Tension members

Tension members are the simplest steel members to design, because tension straightens a member rather than buckling it. They are still worth careful study, because the connection at each end removes material and concentrates stress, and most tension members are governed by what happens at their ends. This lesson designs the bottom chord of the maintenance bay roof truss and introduces the three limit states of every tension member: yielding of the gross section, fracture of the net section and block shear.

Where tension members are used

Truss bottom chords and the diagonals of a Pratt truss under gravity load, hangers that carry a floor or a hoist from above, sag rods that support girts and purlins, and the diagonals of tension-only bracing are all tension members. Their cross-sections range from round bars and rods to angles, double angles, tees, channels, W shapes, HSS and plates. The choice usually depends on how the member is connected: a double angle or tee bolts or welds easily to a single gusset plate, and a W shape suits heavy truss chords with connections on both flanges.

Yielding of the gross section

Away from the connections, a tension member yields over its whole cross-section when the stress reaches \(F_y\). Yielding over a long length would stretch the member enough to distort the structure, so it is treated as an ultimate limit state:

Net area

At a bolted connection, the holes remove material and the stress across the remaining net section is higher than in the body of the member. Fracture across the net section is the second limit state. The net area is the gross area minus the area of the holes along the critical path:

Holes are normally 2 mm larger than the bolt. Punching damages a thin ring of material around the hole, so S16 deducts a further 2 mm when holes are punched. For an M20 bolt in a punched hole the allowance is 24 mm; for a drilled hole it is 22 mm. Fabricators decide how to make the holes, so designers usually assume punched holes unless the drawings require drilling.

When holes are staggered, the fracture path can zigzag from hole to hole. A diagonal step is less weakening than a straight cut through the same number of holes, because part of the path is inclined. The empirical correction, proposed by Cochrane in 1922 and still used, adds \(s^2/4g\) for each diagonal link in the path:

where \(s\) is the stagger (the pitch between holes along the member) and \(g\) is the gauge (the transverse spacing between lines of holes). Every reasonable path is checked and the one with the smallest net width governs. For an angle, the gauge between holes in the two legs is measured along the centreline of the thickness: the sum of the two gauges from the heel minus the thickness.

Shear lag and the effective net area

When only part of a cross-section is connected, such as one leg of an angle, the force enters the member through the connected part and spreads into the unconnected part over some length. Near the connection the unconnected part carries less than its share, and the net section fails at a lower load than its full area would suggest. This is shear lag. The longer the connection and the closer the centroid of the member is to the plane of the connection, the smaller the loss.

S16 accounts for shear lag with an effective net area \(A_{ne}\) that is smaller than \(A_n\). For angles connected by one leg, \(A_{ne} = 0.80A_n\) when there are four or more bolts in each line and \(0.60A_n\) when there are fewer. For W shapes and channels connected by the flanges only, and for welded connections, the standard gives other reductions that depend on the shape, the number of bolts and the length of weld. The fracture resistance is

Because the net section uses \(F_u\) with a lower resistance factor, and the gross section uses \(F_y\) with a higher one, a member can be governed by either. For ductile behaviour it is better if the gross section yields first, which needs a net-to-gross ratio of roughly \(F_y/F_u\) or more after the factors.

Block shear

A connection can also tear out as a block. The block separates along a shear plane parallel to the force, through the line of bolts, and a tension plane perpendicular to it. Block shear governs short connections with few bolts and thin material, and it must be checked in the member and in the gusset plate.

S16 combines fracture on the net tension area with yielding on the gross shear area, using an average of \(F_y\) and \(F_u\) for the shear plane:

The efficiency factor \(U_t\) is 1.0 when the block is symmetric and the tension plane is loaded uniformly, as in a gusset with two lines of bolts loaded through its centre. It is smaller when the tension is uneven: 0.6 for an angle connected by one leg and for a coped beam with one line of bolts, and lower still for a coped beam with two lines.

Slenderness

Tension members do not buckle, but a very slender member sags under its own weight, vibrates and is easily damaged in handling. S16 limits the slenderness ratio \(L/r\) of tension members to 300, except for rods and cables, which are usually installed with a little pretension.

Designing the bottom chord

The truss bottom chord L2L3 carries \(T_f = 705\) kN (lesson 1). Owen chooses a double angle, two L102x102x9.5 back to back with a 10 mm gap for the gusset plates. Each angle is connected to the gussets by one line of four M20 ASTM A325 bolts in punched holes, at 75 mm pitch with a 40 mm end distance, on a gauge line 64 mm from the heel.

Note that net fracture governs here, not gross yielding. The holes and the 0.80 shear lag factor remove more than a fifth of the strength. A longer connection does not help in S16 once there are four bolts in a line, but welding the angles to the gusset, as lesson 10 shows, avoids the hole deduction altogether.

Sofia's program makes the same checks and adds a staggered-hole example: a hanger plate for the hoist with four lines of holes, alternate lines staggered.

# Truss bottom chord L2L3: 2L102x102x9.5, 350W, one line of 4 M20 bolts per angle
Fy, Fu = 350.0, 450.0
phi, phiu = 0.90, 0.75
Tf = 705.0                      # kN, from lesson 1
A1, t, xbar = 1850.0, 9.5, 29.0 # one angle: area (mm2), thickness, centroid from back
leg, g = 102.0, 64.0            # leg length and gauge from the heel (mm)
ha = 20 + 4                     # hole allowance for M20 in punched holes (mm)
n_bolts, pitch, end = 4, 75.0, 40.0

Ag = 2 * A1
Tr_yield = phi * Ag * Fy / 1e3
An = 2 * (A1 - ha * t)
An_eff = (0.80 if n_bolts >= 4 else 0.60) * An
Tr_frac = phiu * An_eff * Fu / 1e3
print(f"Gross yielding:  Tr = 0.90 x {Ag:.0f} x {Fy:.0f} = {Tr_yield:.0f} kN")
print(f"Net fracture:    An = {An:.0f}, An' = {An_eff:.0f} mm2, Tr = {Tr_frac:.0f} kN")

# Block shear in each angle: shear along the bolt line, tension to the toe of the leg
Ut = 0.6
Agv = (end + (n_bolts - 1) * pitch) * t
Ant = (leg - g - ha / 2) * t
Tr_block = 2 * phiu * (Ut * Ant * Fu + 0.6 * Agv * (Fy + Fu) / 2) / 1e3
print(f"Block shear:     Ant = {Ant:.0f}, Agv = {Agv:.0f} mm2 per angle, Tr = {Tr_block:.0f} kN (two angles)")

Tr = min(Tr_yield, Tr_frac, Tr_block)
print(f"Governing Tr = {Tr:.0f} kN, Tf = {Tf:.0f} kN, utilization = {Tf/Tr:.2f}")
r = 31.2                        # mm, radius of gyration of the pair in the plane of the truss
print(f"Slenderness L/r = {3000/r:.0f} (limit 300)")

# Staggered holes in a 250 x 12 mm hanger plate: gauge 60 mm, stagger 50 mm
b, tp, gs, s = 250.0, 12.0, 60.0, 50.0
paths = {"straight, 2 holes": b - 2 * ha,
         "zigzag, 4 holes": b - 4 * ha + 3 * s**2 / (4 * gs)}
for name, wn in paths.items():
    print(f"Plate path {name:18s}: net width = {wn:.1f} mm")
wn = min(paths.values())
print(f"Critical net area = {wn*tp:.0f} mm2")

Gross yielding:  Tr = 0.90 x 3700 x 350 = 1166 kN
Net fracture:    An = 3244, An' = 2595 mm2, Tr = 876 kN
Block shear:     Ant = 247, Agv = 2518 mm2 per angle, Tr = 1006 kN (two angles)
Governing Tr = 876 kN, Tf = 705 kN, utilization = 0.80
Slenderness L/r = 96 (limit 300)
Plate path straight, 2 holes : net width = 202.0 mm
Plate path zigzag, 4 holes   : net width = 185.2 mm
Critical net area = 2223 mm2

Rods, cables and pin-connected members

Threaded rods are used for hangers, sag rods and light bracing. Their strength is governed by the area at the root of the threads, and S16 treats them like bolts in tension, with a resistance based on a fraction of the gross area times \(F_u\). Cables and strands have their own standards. Pin-connected plates and eyebars, once common in bridges, appear today in exposed structures; their proportions are limited so that the plate does not fracture or dish around the pin.

Practical points

• Size tension members by the connection, not by the body. Start from the bolt or weld pattern and check net fracture and block shear first.
• Keep gussets compact and connections concentric. Eccentric connections add bending that the simple checks do not cover.
• For double angles, add stitch bolts or fillers at intervals so the two angles act together and do not rattle.
• A truss chord also carries load reversal in some combinations: under wind uplift the bottom chord may go into compression and need bracing to the roof.