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Contents | 1. Introduction to structural design | 2. Loads | 3. Wood |

Introduction to steel | Material properties | Sectional properties | Design approaches | Construction systems |

Unlike tension elements designed in timber, two modes of failure are considered when designing tension
members in steel. First, the element might become functionally useless if yielding occurs across
its gross area, at the yield stress, *F _{y}* . Since internal tensile forces are generally uniform throughout the entire length of the element, yielding would result in extremely large deformations. On the other hand, if yielding commenced on the net area (where bolt holes reduce the gross area), the part of
the element subjected to yield strains would be limited to the local area around the bolts, and excessive
deformations would not occur. However, a second mode of failure might occur at these bolt
holes: rupture of the element could occur if, after yielding, the stresses across the net area reached the ultimate stress,

Another difference in the design of wood and steel tension elements occurs because nonrectangular
cross sections are often used in steel. If connections are made through only certain parts of the
cross section, as illustrated in Figure 4.11, the net area in the vicinity of the connection will be effectively
reduced, depending on the geometry of the elements being joined, and the number of bolts being used. This effective net area, *A _{e}*, is obtained by multiplying the net area,

Figure 4.11: Shear lag in steel tension element showing unstressed or under-stressed areas

Where all parts (i.e., flanges, webs, etc.) of a cross section are connected, and the so-called *shear
lag* effect described above cannot occur, the coefficient *U* is taken as 1.0, and the effective net area equals the net area, just as in timber design. For short connection fittings like splice plates and gusset
plates, *U* is also taken as 1.0, but *A _{e}* =

Figure 4.12: The Whitmore section for connecting plates limits the effective width of the plate to 2*L*tan 30° + *W* for both (*a*) welded connections; and (*b*) bolted connections

Finally, the lengths of tension members, other than rods and cables, are recommended, but not required to have, a slenderness ratio — defined as the ratio of effective length to least radius of gyration
— of 300, to prevent excessive vibrations and protect against damage during transportation and erection. The radius of gyration, a property of the cross section, is equal to , where *I* is the moment of inertia and *A* is the cross-sectional area of the element.

From the preceding discussion, it can be seen that two values for available strength, or allowable stress, in tension need to be determined: one for yielding of the gross area and one for failure (rupture) of the effective net area. These two values are:

(4.1)

and

(4.2)

where *F _{t}^{gross}* and

(4.3)

When computing the capacity based on yielding, the full gross area is available to resist the internal forces:

(4.4)

When computing the capacity on the effective net area:

(4.5)

The "available strength" limit states listed in Appendix Table A-4.2 are equivalent to these formulations based on allowable stress.

The following example illustrates the application of these principles to a steel tension problem. Different procedures are used for cables, eyebars, threaded rods, and pin-connected plates.

** Problem definition.** Find the maximum tension load,

Figure 4.13: Connection detail at gusset plate, with (*a*) section at gross area, (*b*) section at net area, and (c) truss elevation with 3-dimensional view for Example 4.1

** Solution overview.** Find cross-sectional dimensions and material properties; find gross area capacity; find effective net area capacity; the governing capacity is the lower of these two values. For gusset plate thickness, find effective width based on Whitmore section; apply equations for gross and net area capacity to determine required plate thickness.

*Problem solution*

**1.** From Appendix Table A-4.3, find cross-sectional dimensions (Figure 4.14):

*A _{g}* = 7.08 in

*d* = 7.93 in.

*b _{f}* = 6.50 in.

*t _{f}* = 0.400 in.

**2.** From Appendix Table A-4.1, find *F _{y}* = 36 ksi and

Figure 4.14: Cross-sectional dimensions of W8 × 24 for Example 4.1

**3.** Gross area: find capacity, *P*:

a. Using Equation 4.1 (or Appendix Table A-4.2) find *F _{t}^{gross}* = 0.6

b. Using Equation 4.4, P = *F _{t}^{gross}* ×

**4.** Effective net area: find capacity, *P*:

a. From Appendix Table A-4.9, find the shear lag coefficient, *U*:

*U* = 0.90 since the following criteria are met:

- Bolts connect wide-flange (W) shape? Yes.
- Flange width,
*b*is no less than 0.67d? In other words, 6.5 ≥ 0.67(7.93) = 5.3? Yes._{f} - Flange is connected with at least 3 bolts per line? Yes.

b. Find the net area, *A _{n}* (same as Equation 3.1 for wood). As shown in Figure 4.15:

Figure 4.15: Net area diagram for Example 4.1

c. *A _{e}* =

d. Using Equation 4.2, find *F _{t}^{net}* = 0.5

e. Using Equation 4.5, find P = *F _{t}^{net}* ×

**5.** Conclusion: failure on the effective net area governs since 148 kips < 156 kips. The capacity (allowable load) is 148 kips.

**6.** We now can determine the thickness of the gusset plate, stressed in tension, with two lines of bolt holes per plate, using the Whitmore section to determine the effective width of the plate. As can be seen in Figure 4.12, the effective width, *l _{w}* = 2(6)(tan 30°) + 3 = 9.9 in. The tensile capacity of the gusset plates may be based on either yielding of the gross area or rupture of the net area. First, the capacity based on yielding of the gross area of both plates is

** Problem definition.** Select a W section bolted as shown in Figure 4.16 with 5/8 in. diameter bolts, and 3 bolts per line, to resist a tension force of 100 kips. Assume A36 steel. The effective bolt hole diameter = bolt diameter + 1/8 in. = 5/8 + 1/8 = 3/4 in. = 0.75 in.

Figure 4.16: Net area diagram for Example 4.2

** Solution overview.** Find the required area based on net area capacity, assuming values for shear lag
coefficient,

*Problem solution*

**1.** *Gross area:* find required gross area based on yielding. From Equation 4.4, the required gross area, *A _{g}* =

**2.** *Effective net area:* find required gross area after determining effective net area based on rupture through failure surface (assume *U* = 0.9 and *t _{f}* = 0.4 in.):

a. From Equation 4.5, the required effective net area, *A _{e}* =

b. Working backwards, the required net area, *A _{n}* =

c. Finally, the required gross area can be computed: *A _{g}* =

**3.** Since 5.03 in^{2} > 4.63 in^{2}, the calculation based on effective net area governs, and a W section must be selected with *A _{g}* ≥ 5.03 in

a. Check a W8 × 18 with *A _{g}* = 5.26 in

b. Check a W6 × 20 with *A _{g}* = 5.87 in

Both the W8 ×18 and the W6 × 20 would work, as would many other wide-flange shapes. Of the two sections considered, the W8 × 18 is lighter (based on the second number in the W-designation that refers to beam weight in pounds per linear foot), and therefore would be less expensive.

Threaded rods are designed using an allowable tensile stress, *F _{t}* = 0.375

Where plates are connected with a single pin, as shown in Figure 4.17, the net area, *A _{n}*, is defined, not by the length,

(4.6)

The plate capacity in tension is governed by either yielding on the gross area or rupture on the net area, whichever is smaller (there is no *effective* net area in this case), with *P _{gross}* = 0.6

Figure 4.17: (*a*) Overview of pin-connected plates; (*a*) rupture on net area and (*a*) shear failure

Aside from failure in tensile rupture or yielding, a third limit state for pin-connected plates is
*shear failure*, or the relative *sliding* of areas *A _{sf}* as illustrated in Figure 4.17

A fourth limit state for pin-connected plates is bearing, or compressive stress caused by the pin
itself in direct contact with the adjacent plate. Here, the allowable stress on the projected area (*A _{pb}*) of the pin that bears on the plate is 0.9

All four limit states must be checked, with the capacity of the pin-connected plate determined
by the *lowest* of the four limits. Values for yield and ultimate stress used in these calculations, *F _{y}* and

For such pin-connected plates, as well as for all other bolted connections, the fasteners themselves, and not only the stresses they produce on the elements being joined, must also be checked. This aspect of structural element design is discussed more thoroughly in the section of this chapter dealing with steel connections.

© 2020 Jonathan Ochshorn; all rights reserved. This section first posted November 15, 2020; last updated November 15, 2020.