Hancock G.J., Murray Th.M., Ellifritt D.S. Cold-Formed Steel Structures to the AISI Specification

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10.4 SYSTEM ANCHORAGE REQUIREMENTS

10.4.1 Z-Purlin Supported Systems

The principal axes of Z-purlins are inclined from the center-

line of the web at an angle y

, as shown in Figure 10.12a.

For Z-purlins commonly used in North America, y

between 20



and 30



. For an unrestrained purlin, the

component of the roof loading that is parallel to the web

causes a Z-purlin to de¯ect in the vertical and horizontal

directions, as shown in Figure 10.12b. The component of

the roof loading perpendicular to the purlin web decreases

the de¯ection perpendicular to the web. For steep roofs this

component can even cause movement in the opposite direc-

tion, as shown in Figure 10.12c. Since the bottom ¯ange of

the purlin is restrained at the rafter connection and the top

¯ange is at least partially restrained by the roof deck, a

purlin tends to twist or ``roll'' as shown in Figure 10.12d.

Devices such as antiroll clips at the rafters or intermediate

lateral braces are used to minimize purlin roll. The force

FIGURE 10.12 Z-purlin movement: (a) axes; (b) unrestrained

movement; (c) movement because of large downslope component;

(d) panel and anchorage restraints.

Chapter 10

318

induced in these devices is signi®cant and must be consid-

ered in design.

From basic principles (Ref. 10.16), the required

anchorage or restraint force for a single Z-purlin with

uniform load (W ) parallel to the web is

 0:5



W 10:7

where I

is the product moment of inertia and I

is the

moment of inertia with respect to the centroidal axis

perpendicular to the web of the Z-section as shown in

Figure 10.12a. For this formulation, the required anchor-

age force for a multi-purlin-line system is directly propor-

tional to the number of purlins (n

). However, Elhouar and

Murray (Ref. 10.17) showed that the anchorage force

predicted by Eq. (10.7) is conservative because of a system

effect. Figure 10.13 shows a typical result from their study.

The straight line is from Eq. (10.7), and the curved line is

from stiffness analyses of a multi-purlin-line system. The

difference in the anchorage force between the two results is

a consequence of the inherent restraint in the system due to

purlin web ¯exural stiffness and a Vierendeel truss action

FIGURE 10.13 Anchorage forces vs. number of purlin lines.

Roof and Wall Systems

319

caused by the interaction of purlin webs with the roof panel

and rafter ¯anges. This Vierendeel truss action explains the

relative decrease in anchorage force as the number of purlin

lines (n

) increases, as shown in the ®gure.

Section D3.2.1 of the AISI Speci®cation has provisions

that predict required anchorage forces in Z-purlin-

supported roof systems supporting gravity loading and

with all compression ¯anges facing in the same direction.

The provisions were developed by using elastic stiffness

models of ¯at roofs (Ref. 10.17) and were veri®ed by full-

scale and model testing (Ref. 10.18). For example, the

predicted anchorage force in each brace (P

) for single-

span systems with restraints only at the supports, Figure

10.14, is

 0:5bW; with b 

0:220b

1:5

0:72

0:90

0:60

10:8

where W  applied vertical load (parallel to the web) for

the set of restrained purlins

b  ¯ange width

d  depth of section

t  thickness

 number of restrained purlin lines

FIGURE 10.14 Single-span purlin system with anchorage at

rafters.

Chapter 10

320

The restraint force ratio, b, represents the system effect

and was developed from regression analysis of stiffness

model results.

Assuming that the top ¯ange of the Z-purlin is in the

upslope direction, the anchorage force P

for single-span

systems with restraints only at the rafter supports becomes

 0:5b cos y ÿ sin yW 10:9

with y roof slope measured from the horizontal as shown

in Figure 10.12c. The terms W b cos y and W sin y are the

gravity load components parallel and perpendicular to the

purlin web, respectively. The latter component is also

referred to as the downslope component. Equations of this

form are included in AISI Speci®cation Supplement No. 1.

Two important effects were not taken into account in

the development of Eq. (10.9). First, the internal system

effect b applies to the forces W cos y and W sin y. Second,

the internal system effect reverses when the net anchorage

force changes from tension to compression with increasing

slope angle. Further, the stiffness models used to develop

the AISI provisions assume a roof panel shear stiffness of

2500 lb=in., whereas the actual shear stiffness can be

between 1000 lb=in. (standing seam sheathing) to as high

as 60,000±100,000 lb=in. (through-fastened sheathing).

Neubert and Murray (Ref. 10.19) have recently

proposed the following model, which eliminates the de®-

ciencies inherent in Eq. (10.9). They postulate that the

predicted anchorage force in any given system is equal to

the anchorage force required for a single purlin (P

) multi-

plied by the total number of purlins (n

), a brace location

factor (C

), a reduction factor due to system effects (a), and

modi®ed by a factor for roof panel stiffness (g). Thus,

 P

n

*a  n

g10:10

Roof and Wall Systems

321

Figure 10.15a shows a Z-purlin with zero slope and

applied load W

, which is the total gravity load acting on

each purlin span:

 wL 10:11

where w distributed gravity load on each purlin

(force=length)

L span length

The ®ctitious force W

I

) is the ®ctitious lateral force

from basic principles (Ref. 10.16). The couple W

db results

from the assumed location of the load W

on the ¯angeÐ

that is, db from the face of the web as shown in Figure

10.15a. Figure 10.15b shows the set of real and ®ctitious

forces associated with a single purlin on a roof with slope y.

The purlin is assumed to be pinned at the rafter connection.

The set of forces accounts for the following effects: W

sin y

is the downslope component, W

cos yI

) is the down-

slope component of the ®ctitious lateral force, and

cos ydb) is the net torque induced by eccentric loading

of the top ¯ange. Summation of moments about the pinned

support results in







cos y ÿ sin y



10:12

For low slopes, P

is positive (tension) and for high

slopes P

is negative (compression). A positive value indi-

cates that the purlin is trying to twist in a clockwise

FIGURE 10.15 Modeling of gravity loads: (a) forces for a single

purlin on a ¯at roof; (b) forces for a single purlin on a sloping roof.

Chapter 10

322

direction; a negative value means twist is in the counter-

clockwise direction. The anchorage force is zero for

 tan

ÿ1





10:13

For y < y

; P

is in tension, and for y > y

; P

is in compres-

sion. The distance db has a signi®cant effect on the value of

the intercept angle y

. In Ref. 10.19, db was taken as b=3,

based on anchorage forces measured in tests of zero-slope

roof systems (Ref. 10.17).

Statistical analysis of stiffness model results was used

to develop a relationship for the system effect factor a:

a  1 ÿ C



n

* ÿ 110:14

with C

a constant factor that depends on the bracing

con®guration and n

* is described below. The factor a is

dimensionless and, when included in Eq. (10.10), models

the reversal of the system effect when P

changes from

tension to compression. The coef®cient C

was determined

from a set of regression analyses with values tabulated in

Table 10.1. The value of the coef®cient differs for each

bracing con®guration because bending resistance changes

depending on the distance between a brace and rafter

supports or other braces.

Equation (10.10) is quadratic with respect to n

because a is linear with n

. For some value of n

, denoted

as n

pmax

; P

will reach a maximum and then decrease as

is increased above n

pmax

. From basic calculus, n

pmax

determined as

pmax

 0:5 

10:15

However, the required anchorage force can never decrease

as the number of purlins is increased. Therefore n

* is used

in Eqs. (10.10) and (10.14) instead of n

, where n

* is the

Roof and Wall Systems

323

minimum of n

pmax

and n

. In other words, P

remains

constant when the number of purlin lines exceeds n

pmax

The brace location factor C

in Eq. (10.10) represents

the percentage of total restraint that is allocated to each

brace in the system. The sum of the C

coef®cients for the

braces in a span length is approximately equal to unity. The

values for C

were determined from a regression analysis

and are tabulated for each bracing con®guration in Table

10.1. For multiple span systems, the C

values are larger

for exterior restraints than the corresponding interior

restraints, as expected from structural mechanics.

Figure 10.16 shows a typical plot comparing the Eq.

(10.10) with, g  0, to the AISI Speci®cation Eq. (10.9) with

respect to slope angle y. Equation (10.10) predicts slightly

TABLE 10.1 Anchorage Force Coef®cients for Use in Eq. (10.10)

Bracing con®guration C

Support anchorages only

Single span 0.50 5.9 0.35

Multispan, exterior supports 0.50 5.9 0.35

Multispan, interior supports 1.00 9.2 0.45

Third-point anchorages

Single span 0.50 4.2 0.25

Multispan, exterior spans 0.50 4.2 0.25

Multispan, interior spans 0.45 4.2 0.35

Midspan anchorages

Single span 0.85 5.6 0.35

Multispan, exterior spans 0.80 5.6 0.35

Multispan, interior spans 0.75 5.6 0.45

Quarter-point anchorages

Single span, outside

points 0.25 5.0 0.35

Single span, midspan location 0.45 3.6 0.15

Multispan, exterior span

points 0.25 5.0 0.40

Multispan, interior span

points 0.22 5.0 0.40

Multispan, midspan locations 0.45 3.6 0.25

Third-point plus support anchorages

Single span, at supports 0.17 3.5 0.35

Single span, interior locations 0.35 3.0 0.05

Multispan, exterior support 0.17 3.5 0.35

Multispan, interior support 0.30 5.0 0.45

Multispan, third-point locations 0.35 3.0 0.10

Chapter 10

324

greater anchorage force for low-slope roofs and signi®cantly

less anchorage force for high-slope roofs. The latter is due

to the inclusion of system effects in the downslope direction

which are ignored in the AISI provisions. Figure 10.17

shows a similar plot with respect to the number of

restrained purlin lines.

FIGURE 10.16 Anchorage force vs. roof slope: Eqs. (10.9) and

(10.10).

FIGURE 10.17 Anchorage force vs. number of purlins: Eqs. (10.9)

and (10.10).

Roof and Wall Systems

325

The stiffness models used to develop the AISI provi-

sions, Eq. (10.9), included a roof deck diaphragm stiffness of

 2500 lb=in. with the assumption that the required

anchorage force for systems with roof decks stiffer than

2500 lb=in. is negligible. Roof deck diaphragm stiffness was

taken as



4aD

10:16

where P point load applied at midspan of a rectangular

roof panel,

L panel's span length

a panel width

and

D de¯ection of the panel at the location of the

point load.

In Ref. 10.19 stiffness models of roof systems with up

to eight restrained purlins were analyzed, and it was found

that deck stiffness above 2500 lb=in. caused signi®cant

increases in the anchorage forces for systems with four or

more purlin lines. The panel stiffness modi®er (g) in Eq.

(10.10) accounts for this effect. As shown in Figure 10.18,

FIGURE 10.18 Anchorage force vs. panel stiffness (stiffness

model).

Chapter 10

326

the anchorage force varies linearly with the common loga-

rithm of the roof panel shear stiffness over a ®nite range of

stiffnesses, which leads to the following equation for the

panel stiffness modi®er:

g  C

log

2500 lb/in.



10:17

where G

roof panel shear stiffness (lb=in.)

constant from regression analysis of stiffness

model

When G

 2500 lb=in., g  0 as necessary, and g is

positive for G

> 2500 lb=in. and negative for

< 2500 lb=in. Thus, when G

> 2500 lb=in., the ancho-

rage force is increased, and when G

< 2500 lb=in., the

required anchorage force is decreased. The values of C

which depend on the location of a brace relative to rafter

supports and other braces, are tabulated for each bracing

con®guration in Table 10.1.

The effect of g is to adjust the system effect factor, a.In

Eq. (10.10) g is multiplied by n

instead of n

*, because as

panel stiffness changes, change in anchorage force depends

on the total number of purlins in the system and n

pmax

longer applies. The panel stiffness modi®er g is valid only

for 1000 lb=in.  G

 100,000 lb=in. This is the range of

linear behavior and most roof decks have shear stiffness

within this limitation.

For very large panel stiffness, Eq. (10.10) can predict

an anchorage force greater than that obtained from basic

principles [Eq. (10.7)]. However, the maximum anchorage

force is the anchorage force ignoring system effects. Thus,







10:18

Figure 10.19 is a typical plot of anchorage force versus

panel stiffness for Eq. (10.10), compared with stiffness

model results.

Since the stiffness models used to develop Eq. (10.10)

had eight purlin lines or fewer, Eq. (10.10) must be used

Roof and Wall Systems

327