000_[Jonathan_Ochshorn]_Structural_Elements_for_Architecture_400

102 CHAPTER 6 Tension elements

Example 6.5 Design steel tension element

Problem definition

Select a W section bolted as shown in Figure 6.12 with 5 8-in.-diameter bolts, and three

bolts per line, to resist a tension force of 100 kips. Assume A36 steel. The effective bolt hole

diameter ϭ bolt diameter ϩ 1 in. ϭ 5 ϩ 1 ϭ 3 in. ϭ 0.75 in.
8 8 8 4

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

U, and flange thickness, tf; find required area based on gross area capacity; use the larger of
the two area values to provisionally select a W section; check using “analysis” method if either

U is smaller or tf is larger than assumed values. The area of the selected W section can be
somewhat smaller than the “required” area if either U is larger or tf is smaller than assumed
values—check using analysis method.

Problem solution

1. Gross area: Find required gross area based on yielding. From Equation 6.17, the required
gross area, Ag ϭ P/Ftgross ϭ 100/(0.6 ϫ 36) ϭ 4.63 in2 .

2. Effective net area: Find required gross area after determining effective net area based on

rupture through failure surface (assume U ϭ 0.9 and tf ϭ 0.4 in.): Ae ϭ P/Ftnet ϭ
a. From Equation 6.18, the required effective net area,

100/(0.5 ϫ 36) ϭ 3.44 in2

b. Working backward, the required net area, An ϭ Ae/U ϭ 3.45/0.9 ϭ 3.83 in2.

c. Finally, the required gross area can be computed: Ag ϭ An ϩ (bolt hole
area) ϭ 3.83 ϩ 4(0.75 ϫ 0.4) ϭ 5.03 in2.

3. Since 5.03 in2 Ͼ 4.63 in2, the calculation based on effective net area governs, and

a W section must be selected with Ag Ͼ 5.03 in2. Many wide-flange shapes could be

FIGURE 6.12
Net area diagram for Example 6.5

Steel 103

selected. From Table A-4.3, the following candidates are among those that could be

considered:
a. Check a W8 ϫ 18 with Ag ϭ 5.26in2. Two assumptions need to be tested: that U ϭ 0.9

and that tf ϭ 0.4. From Table A-4.3, bf ϭ 5.25in., d ϭ 8.14in., and tf ϭ 0.330in. From
Table A-6.1, the criteria for U ϭ 0.9 requires that bf ϭ 5.25 Ն 0.67d ϭ 0.67(8.14) ϭ 5.4
5in. Since this condition is not met, we must use U ϭ 0.85. Additionally, the flange thick-
ness is different from our assumed value of 0.40in., so that the calculation of net and effec-

tive area will change: An ϭ Ag Ϫ (bolt hole area) ϭ 5.26 Ϫ 4(0.75 ϫ 0.330) ϭ 4.27in. and
Ae ϭ U ϫ An ϭ 0.85(4.27) ϭ 3.63in2. The capacity based on rupture through the effec-
tive net area is P ϭ Ftnet ϫ Ae ϭ (0.5 ϫ 58)(3.63) ϭ 105 kips. The capacity based on
yielding on the gross area has already been found satisfactory (since the gross area of the

W8 ϫ 18 is greater or equal to the required gross area computed earlier). Therefore, the
W8 ϫ 18 is acceptable.
b. Check a W6 ϫ 20 with Ag ϭ 5.87 in2. The same two assumptions need to be tested:
that U ϭ 0.9 and that tf ϭ 0.4. From Table A-4.3, bf ϭ 6.02 in., d ϭ 6.20 in., and
tf ϭ 0.365 in. From Table A-6.1, the criteria for U ϭ 0.9 requires that bf ϭ 6.02 Ն
0.67d ϭ 0.67(6.20) ϭ 4.15 in. Since this condition is met, and since its net area is
greater than assumed (this is so because its flange thickness, tf, is less than the value
assumed, so that the bolt hole area is less than assumed, and therefore the net area is

greater than assumed), the W6 ϫ 20 is acceptable.

Both the W8 ϫ 18 and the W6 ϫ 20 would work, as would many other wide-flange
shapes. Of the two sections considered, the W8 ϫ 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.

Steel threaded rods

Threaded rods are designed using an allowable tensile stress, Ft ϭ 0.375Fu, which
is assumed to be resisted by the gross area of the unthreaded part of the rod. This

value for the allowable stress is found by dividing the nominal rod tensile strength of

0.75Fu by a safety factor, Ω ϭ 2.00. While there are no limits on slenderness, diame-

ters are normally at least 1 of the length, and the minimum diameter rod for struc-
500

tural applications is usually set at 5 iwn.itAhssaurmeai,ngAAϭ36πs(t5e1e6l),2w, citahnFsuuϭpp5o8rtksai (Table
A-3.12), the smallest acceptable 8 tensile

rod

load, P ϭ Ft ϫ A ϭ 0.375Fu ϫ π(516)2 ϭ 21.75 ϫ 0.3068 ϭ 6.67 kips.

Pin-connected plates

Where plates are connected with a single pin, as shown in Figure 6.13, the net area,
An, is defined, not by the length, b, on either side of the pin hole, but rather by an
effective length, beff ϭ 2t ϩ 0.63 Յ b, where t is the thickness of the plate:

An ϭ 2tbeff (6.19)

104 CHAPTER 6 Tension elements

FIGURE 6.13
Definition of net area for pin-connected plates

The plate capacity is then governed by either yielding on the gross area or rup-
ture on the net area, whichever is smaller (there is no effective net area in this case),
with Pgross. ϭ 0.6Fy ϫ Ag and Pnet. ϭ 0.5Fu ϫ An as before. It is possible to cut the
plate at a 45° angle as shown in Figure 6.13, as long as length c is greater or equal to
length a, which in turn must be greater or equal to 1.33beff.

For pin-connected plates, as well as for all other bolted connections, the fasten-
ers themselves, as well as the stresses they produce on the elements being joined,
must also be checked. This aspect of connection design is discussed more thor-
oughly in Chapter 9.

REINFORCED CONCRETE

Concrete, having very little tensile stress, is ordinarily not used for tension elements.
Where it is used, its strength in tension can be taken as approximately 10% of its
compressive strength, or 0.1fcЈ . The cylinder strength of concrete, fcЈ , is the ulti-
mate (highest) compressive stress reached by a 6 in. ϫ 12 in. cylinder of concrete
after 28 days of curing. Reinforced concrete, consisting of steel bars embedded
within a concrete element, would not normally be a good choice for a pure tension
element, since the steel reinforcement would be doing all the work. In this case,
one might wonder what would justify the added expense of casting concrete around
the steel. In fact, two justifications are possible: first, in a reinforced concrete