Yu W., LaBoube R.A. Cold-Formed Steel Design

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SHELL ROOF STRUCTURES 335

Figure 9.26 Roof plan of superbay hangar. (Courtesy of Lev

Zetlin Associates, Inc.)

1.82

0.0934-in.- (2.4-mm-) thick sheet, 26 in. (660 mm) wide,

with two 9-in.- (229-mm-) wide by 7

-in.- (191-mm-) deep

0.0516-in.- (1.3-mm-) thick steel hat sections welded to the

ﬂat sheets. Figure 9.28 shows a typical cross section of the

steel deck used.

In order to be able to use this type of structural system in

any area of the world, prestressed cables are incorporated

into the shell structures (Fig. 9.26). Since the structural

strand cables induce a prestress in the shell, the system is

readily adaptable to any geographic site.

A comparison of various types of designs indicate

that this type of building with its hyperbolic paraboloids

weighs approximately 40% less than a conventional steel

construction.

9.4.3.2 Types of Hyperbolic Paraboloid Roofs The

surface of a hyperbolic paraboloid may be deﬁned by

two methods.

1.80

As shown in Fig. 9.29, with the x, y,

and z axes mutually perpendicular in space, the surface

is formulated by two straight lines called generators. One

line, parallel to the xz plane, rotates about and moves

along the y axis; the other, parallel to the yz plane, rotates

about and moves along the x axis. The intersection of the

generators is contained in the surface of the hyperbolic

paraboloid.

Figure 9.30 shows several types of hyperbolic paraboloid

roofs which may be modiﬁed or varied in other ways to

achieve a striking appearance.

In general, type I is the most pleasing of the shapes

available. The edge beams are in compression and are

usually tubular members. For this type of roof, the most

serious problem is the horizontal thrust at the supporting

columns. Usually the columns are kept short in order to

transfer the thrust down to the ﬂoor where tie rods can

be hidden. Four units of this type with a common center

column probably provide the most rigid roof structure, as

shown in Fig. 1.14.

Type II is an inverted umbrella, which is the easiest and

the least expensive to build. The edge members of this type

are in tension, and engineers usually use angles as edge

members.

Type III is the most useful type for canopy entrance

structures. The edge members connected to the columns are

in compression and are usually tubes, while the outside edge

members are in tension and could be angles or channels.

In some cases, one-half of the roof may be kept horizontal

and the other half tilts up.

Generally speaking, type IV is the most useful of all the

available shapes. The entire building can be covered with a

completely clear span. The horizontal ties between columns

on four sides can be incorporated in the wall construction.

9.4.3.3 Analysis and Design of Hyperbolic Paraboloid

Roofs The selection of the method of analysis of hyper-

bolic paraboloid roofs depends on the curvature of the shell

used. If the uniformly loaded shell is deep (i.e., when the

Figure 9.27 Construction of superbay hangar. (Courtesy of Lev Zetlin Associates, Inc.)

1.82

336 9 SHEAR DIAPHRAGMS AND ROOF STRUCTURES

Figure 9.28 Cross section of steel deck used in superbay hangar.

(Courtesy of Lev. Zetlin Associates, Inc.)

1.82

span–corner depression ratio a/h showninFig.9.24is

less than or equal to approximately 5.0), the membrane

theory may be used. For the cases of a deep shell subjected

to unsymmetrical loading and a shallow shell, the ﬁnite-

element method will provide accurate results.

9.80

In the membrane theory, the equation of the surface of

a hyperbolic paraboloid can be deﬁned from Fig. 9.31.

Assume

and

(9.63)

z =

hxy

= kxy (9.64)

where k = h/ab, in which h is the amount of corner

depression of the surface having the horizontal projections

a and b.

If we rotate coordinate axes x and y by 45

◦

, as shown

in Fig. 9.32, the equations for two sets of parabolas can

be obtained in terms of the new coordinate system using x



and y



9.81

Substituting

x = x



cos 45

◦

− y



sin 45

◦

(9.65)

y = y



cos 45

◦

+ x



sin 45

◦

(9.66)

into Eq. (9.61), one can obtain a new equation for z in

terms of x



and y



z =

k[x

2

− y

2

] (9.67)

In Eq. (9.67), if the value of x



remains constant, as

represented by line a



–a



in Fig. 9.33, the equation for

the parabolic curve can be written as follows, where the

negative sign indicates concave downward:



=−

2

(9.68)

When the value of y



remains constant, Eq. (9.66) can be

obtained for a concave upward parabola:



2

(9.69)

Figure 9.29 Prestressed cables used in hangar roof. (Courtesy of Lev Zetlin Associates, Inc.)

1.82

SHELL ROOF STRUCTURES 337

Figure 9.30 Types of hyperbolic paraboloid roofs.

Figure 9.31 Dimensions used for deﬁning the surface of hyper-

bolic paraboloid roof.

Figure 9.32 Coordinate system using x



and y



axes.

For a constant value of z, the hyperbolic curve can be

expressed by

= x

2

− y

2

(9.70)

Figure 9.34 shows a concave downward parabolic arch

subjected to a uniform load of w/2, where w is the roof

Figure 9.33 Sketch used for deriving equations for parabolic

and hyperbolic curves.

Figure 9.34 Concave downward parabolic arch.

load per square foot. Since the bending moment throughout

a parabolic arch supporting only a uniform load equals zero,

H(−h) =

(w/2)L

(9.71)

H =−

16h

(9.72)

Use Eq. (9.68) for y



= L/2 and z



= h.Then

h =

(9.73)

Substituting the value of h in Eq. (9.72), one obtains Eq.

(9.74) for the horizontal thrust H,

H =

wab

(9.74)

The above analogy can also be used for the concave

upward parabolic tie.

It can be seen that if the load is applied uniformly

over the horizontal projection of the surface, compres-

sive membrane stress results in the concave downward

parabolas, and tensile membrane stress results in the

concave upward parabolas. These tensile and compressive

membrane stresses are uniform throughout the surface and

are equal to wab/2h, in which w is the applied load per

unit surface area (Fig. 9.35). Since the compressive and

tensile membrane stresses are equal in magnitude and are

perpendicular to each other, a state of pure shear occurs in

planes of 45

◦

from the direction of either membrane stress.

Thus only shear stress need be transmitted along the joints

of the panels. The force in edge beams resulting from the

shear of q = wab/2h along the edge members is shown in

Fig. 9.35.

338 9 SHEAR DIAPHRAGMS AND ROOF STRUCTURES

Figure 9.35 Membrane and shear stresses in panels and forces

in framing members.

Figure 9.36 Example 9.4.

Example 9.4 illustrates the design of an inverted hyper-

bolic umbrella by using the membrane theory.

Example 9.4 Use the ASD method to design the inverted

steel hyperbolic paraboloid umbrella roof shown in

Fig. 9.36

9.76

for w = 30 psf on horizontal projection. Use

A36 steel for framing members.

SOLUTION

1. Determination of Panel Shear. The panel shear can be

determined as follows:

v =

30 × (15)

2 × 4

= 845 lb/ft

In accordance with Section D5 of the AISI speciﬁ-

cation (Table 9.1), the factor of safety for gravity load

alone is 2.45 for welded connections. The required

ultimate shear strength of the roof deck system is

therefore

= 845(2.45) = 2070 lb/ft

2. Selection of Steel Roof Deck. Select a type of steel roof

deck from catalogs to satisfy the following require-

ments:

a. Flexural strength

b. Shear strength

c. Deﬂection

3. Design of Framing Members

a. Tension Members AB and AC. Considering the

quadrant ABCD as shown in Fig. 9.34, the tensile

force is

T = va =





a = 845(15) = 12, 680 lb

Using the AISC ASD speciﬁcation with an

allowable stress of 22 ksi for A36 steel, the

required area is

A =

12.68

= 0.58 in.

Using L4 × 3 ×

in., the actual area is 1.69 in.

b. Compression Members BD and CD. Considering

section 1–1 of Fig. 9.37, the compressive force is

C = vc =





c = 845(15.55) = 13, 140 lb

Since the compression member will support two

quadrants, the total compressive force for design is



= 2C = 2(13,140) = 26,280 lb

Using the AISC ASD Speciﬁcation with an

allowable stress of 22 ksi, the required area is

A =



= 1.195 in.

Selecting two L4 × 3 ×

in., the actual area is 3.38 in.

The use of an allowable stress of 22 ksi is due to the fact

that the compressive force varies from zero at point B to

the maximum value at point D and that the member is

continuously braced by steel panels.

The above discussion is based on the requirement

of strength. In some cases the deﬂection of hyperbolic

paraboloid roofs at unsupported corners may be excessive

and therefore will control the design of this type of

structure. Research work

9.80

indicates that the corner

Figure 9.37 Forces in framing members.

SHELL ROOF STRUCTURES 339

deﬂection is affected by (1) the shear and bending stiffness

of steel panels, (2) the bending and axial stiffness of edge

members, (3) the type and arrangement of connections

used to join steel panels and to connect steel panels

to edge members, and (4) the eccentricity of the shear

force transmitted from the deck to edge members. The

ﬁnite-element method may be used to predict deﬂection.

If the corner deﬂection is found to be critical, the design

may be improved by increasing the bending stiffness of the

edge members and steel panels and the curvature of the

shell.

In the design of hyperbolic paraboloid roofs, considera-

tion should also be given to the possible buckling of the

steel deck and the overall buckling of edge members. The

determination of the buckling load can be made by either

the energy approach or the ﬁnite-element method.

9.80

9.4.3.4 Research on Cold-Formed Steel Hyperbolic

Paraboloid Roofs A full-scale hyperbolic paraboloid

shell constructed of one-layer standard steel roof deck

sections has been tested at Cornell University by Nilson.

9.79

The test setup is shown in Fig. 9.38. Note that the jack

loads are applied upward because of convenience of

testing. The specimen was 15 × 15 ft (4.6 × 4.6 m) with a

3-ft (0.92-m) rise, which represents one of the four quarter

surfaces of an inverted umbrella (type II), as shown in

Fig. 9.30. The panels used in the specimen were standard

0.0635-in.- (1.6-mm-) thick steel roof deck sections, 18 in.

(457 mm) wide, with ribs 6 in. (152 mm) on centers and 1

in. (38 mm) high. Panels were welded to each other along

the seam joints and to the perimeter frame members.

The structural feasibility of the hyperbolic paraboloid

shell structure has been demonstrated by Nilson’s test. The

results indicate that shear values obtained from horizontal

diaphragm testing can be used conservatively in the design

of curved shear surfaces.

The structural behavior of a hyperbolic paraboloid shell

has been studied further at Cornell University by Winter,

Gergely, Muskat, Parker, and Banavalkar under the spon-

sorship of the AISI. Findings of this study are reported

in Refs. 9.80 and 9.82–9.87. In addition, investigations of

hyperbolic paraboloid roofs have been conducted by indi-

vidual steel companies.

9.88

It is expected that additional

design information may be developed from the previous

research.

9.4.3.5 Curvilinear Grid Frame-Type Hyperbolic Para-

boloids

9.72,9.75

The above discussion is made on the basis

that the cold-formed panels are stressed as a membrane

in the hyperbolic paraboloid roof. This membrane type of

hyperbolic paraboloid structure is mostly used for rela-

tively small structures. For larger structures a hyperbolic

paraboloid roof may be constructed with structural members

as a curvilinear grid frame. The construction shown in Fig.

9.39 was developed by Hutton of Purdue University and

consists of struts and ties. The ties of the system can be

designed as cables. Either analytical or graphic solutions

are possible.

With regard to the design of ribs, the top ribs are designed

for compression and bending and the bottom ribs are

designed for compression only. The fascia is designed for

combined axial and bending stress. In addition, the ground

tie should be provided as a tension member.

In the curvilinear grid frame type, cold-formed steel

decks do not participate in shear membrane stresses; rather

a grid of structural steel members supports the loads.

Two grid frame methods are possible. They are the

double-arch and the orthogonal systems. For the double-

arch grid system, the positive arches are compressive

Figure 9.38 Testing of hyperbolic paraboloid shell.

9.79

340 9 SHEAR DIAPHRAGMS AND ROOF STRUCTURES

Figure 9.39 Curvilinear grid frame type hyperbolic para-

boloid.

9.76

elements and the negative arches are curved tensile

members. The analysis of this system for concentrated

load and unbalanced load can be complex.

For the orthogonal grid system, as shown in Fig. 9.39,

the ribs are straight members parallel to the edge members

and supported by a series of ties. Two layers of corrugated

steel panels may be used perpendicular to each other to

replace the ribs.

In addition to the types of shell roofs discussed above,

two structural systems (barrel shells and composite truss

and sheet panels) generally used for lightweight utility and

farm buildings have been discussed by Abdel-Sayed in

Ref. 9.89.

9.5 METAL ROOF SYSTEMS

For metal roof systems framed using cold-formed steel

C- or Z-section purlins the attached metal roof panel may

inﬂuence the structural performance of the C- or Z-section.

Sections 4.24 and 4.25 summarize the behavior of beams

having one ﬂange attached to deck or sheathing. Section

5.7 provides the AISI design provisions for a compression

member having one ﬂange attached to deck or sheathing.

Because the C- or Z-section is singly or point symmetric,

the application of gravity loads will cause torsion in the

purlin and lateral displacements of the roof system. This

behavior is due to the slope of the roof, the loading

of the member eccentric to its shear center, and for the

Z-section the inclination of the principal axes. The lateral

displacements may create instability of the roof system.

The capacity of the purlin roof system is dependent upon

the abilty of the roof system to restrain the purlins and

therefore minimize the torsion in the sections. To avoid

an instability of the roof system, lateral stability may be

provided by the roof sheathing and anchorage devices. The

anchorage device is designed to resist the lateral anchorage

force that is generated in the roof system

9.117–9.120

Research by Neubert and Murray

9.121

determined that

the anchorage force P

can be determined by calculating

the force introduced into the system by each purlin and

then multiplying by a reduction factor that accounts for

the system stiffness. Equation (9.75) deﬁnes the anchorage

force to be resisted by each anchorage device,



i=1



effi,j

total



(9.75)

where P

is the anchorage force introduced into the system

by each purlin and the reduction factor is the ratio of the

effective stiffness of the anchorage device, K

eff

to the total

stiffness of all system elements providing support to the

purlin, K

total

The anchorage force for each purlin anchored to a rigid

anchor is

= (C



1000



+ (C

)

(m + 0.25b)t



× α cos θ −(C

) sin θ



(9.76)

where C

= support restraint coefﬁcients

= total required vertical load

supported by the i

purlin in a

single bay

= product of inertia of full

unreduced section about

centroidal axes parallel and

perpendicular to the purlin web

= 0 for C-sections)

L = purlin span length

m = distance from shear center to

midplane of web (m = 0for

Z-sections)

b = top ﬂange width of purlin

t = purlin thickness

= moment of inertia of full

unreduced section about

centroidal axis perpendicular to

purlin web

d = depth of purlin

α =+1 for top ﬂange facing in

up-slope direction

= -1 for top ﬂange facing in

down-slope direction

θ = angle between vertical and plane

of purlin web

The stability of each purlin is provided by the total

effective stiffness of all elements providing support to the

purlin. This total effective stiffness is a combination of

METAL ROOF SYSTEMS 341

inherent system stiffness and the effective stiffness of each

anchorage device as deﬁned by

total,i



j=1



eff

i,j



+ K

sys

(9.77)

The inherent stiffness K

sys

isameasureoftheresis-

tance to lateral displacement of the purlin. This resistance is

from the ﬁxity of the purlin to the rafter connection and the

moment resistance at the panel-to-purlin interface connec-

tion. The inherent stiffness is

sys



1000



)

ELt

(9.78)

Where C

reﬂects the stiffness of the rotational spring

constant for the connections.

The stiffness K

eff

is the effective anchor stiffness which

consists of the anchor device stiffness and the axial behavior

of the roof panel acting in series,

eff



i,j

(C6)LA



−1

(9.79)

where K

is the lateral stiffness of the anchorage device, A

is the cross-sectional area of the roof panel per unit width,

and C

relates the cross-sectional area and the purlin span.

The previous discussion focused on the required strength

for an anchorage device; however, for an anchorage device

to function properly in a roof system it must provide a

minimum level of stiffness as deﬁned by

req





i=1



(9.80)

Additional background information as well as design

examples are given in Ref. 9.122.

CHAPTER 10

Corrugated Sheets

10.1 GENERAL REMARKS

Corrugated sheets (Fig. 10.1) have been used in building

construction since about 1784. This is one of the oldest

types of cold-formed steel products. At present, numerous

types of corrugated sheets with different coatings are being

produced by many manufacturers. Several standard corru-

gated steel sheets are generally available for building

construction and other usage.

In general, the design methods described in previous

chapters are also applicable to the design and use of corru-

gated steel sheets. However, certain simpliﬁed formulas for

computing the sectional properties of standard corrugated

steel sheets can be used in design. Following an investiga-

tion conducted by the AISI during 1955–1957, a publication

entitled “Sectional Properties of Corrugated Steel Sheets”

was issued by the Institute in 1964 to provide the necessary

design information for corrugated sheets.

1.87

This chapter is intended to discuss the a pplication

of arc-and-tangent-type corrugated steel sheets and

trapezoidal-type corrugated sheets (Fig. 10.2) and the

design of such cold-formed steel products. The information

included herein is based on AISI publications

1.87–1.89

and

other references.

Figure 10.1 Cross section of typical arc- and tangent-type corru-

gated sheets.

Figure 10.2 Cross section of typical trapezoidal-type corrugated

sheets.

10.2 APPLICATIONS

Corrugated steel sheets are frequently used for rooﬁng

and siding in buildings because the sheets a re strong,

lightweight, and easy to erect. In many cases, they are used

as shear diaphragms to replace conventional bracing and

to stabilize entire structures or individual members such

as columns and beams.

10.19,10.20

The shear diaphragms and

diaphragm-braced beams a nd columns were discussed in

Chapter 9.

Figure 1.18a shows the use of standard corrugated

sheets for exterior curtain wall panels. The application of

unusually large corrugated sections in frameless stressed-

skin construction is shown in Fig. 1.18b. In addition,

corrugated steel pipe of galvanized sheets has long been

used in drainage structures for railways, highways, and

airports.

1.18,1.88,1.89

Figure 10.3 shows a typical corrugated

metal pipe culvert used for highway systems. Other corru-

gated steel products have been used for retaining walls,

guardrails, conveyor covers, aerial conduits, and other

purposes.

1.18,1.88

During recent years, corrugated sheets have been

used in ﬂooring systems for buildings and bridge

construction.

1.88,10.1

These products have also been used

as web elements for built-up girders in order to increase

web stiffness instead of using a relatively thicker plate or

a thin web with stiffeners. The Macomber Panlweb girder

shown in Fig. 10.4 consists of 0.075–0.15 in. (1.9–3.8

mm) corrugated web for depths of 20–40 in. (0.51–1.02

m).

10.2

Reference 10.3 discusses the required connections

for beams with corrugated webs. The fatigue strengths of

girders with corrugated webs were reported in Refs. 10.4

and 10.25.

10.3 SECTIONAL PROPERTIES AND DESIGN

OF ARC- AND TANGENT- TYPE CORRUGATED

SHEETS

In 1934, Blodgett developed a method to compute the

sectional properties of arc- and tangent-type corrugated

sheets.

10.5

The computation of the moment of inertia and

the section modulus for standard corrugated sheets has been

simpliﬁed by Wolford.

10.6

In the computation, design curves

and tables can be used to determine factors C

and C

Eqs. 10.1 a nd 10.2

1.87

I = C

+ C

t (10.1)

343

344 10 CORRUGATED SHEETS

Figure 10.3 Typical corrugated metal pipe culvert installatio n on interstate highway system.

1.87

Figure 10.4 Macomber Panlweb girder.

10.2

S =

d + t

(10.2)

where I = moment of inertia, in.

S = section modulus, in.

b = width of sheet, in.

d = depth of corrugation, in.

t = thickness of sheet, in.

= factors depending on shape of

arc-and-tangent-type corrugation.

Using Wolford’s charts, as s hown in Figs. 10.5–10.9,

the values of the moment of inertia, section modulus, area,

radius of gyration, and length of tangent can be computed

by the following procedure:

1. Compute the midthickness radius R



= R +

2. Compute values of q and K,

q =



and K =

where p is the pitch.

3. From Fig. 10.5, determine the angle α for the

computed values of q and K .

4. From Figs. 10.6 and 10.7, determine C

and C

using K and angle α.

5. From Figs. 10.8 and 10.9, determine λ and the m/d

ratio.

6. Compute I and S by using Eqs. 10.1 and 10.2.

7. Compute

A = λbt

8. The radius of gyration is

r =



9. The length of the tangent is d × m/d.

Based on the method outlined above, the sectional prop-

erties of several types of corrugated sheets have been devel-

oped and published in Refs. 1.87 and 1.88. The accuracy

of Eqs. (10.1) and (10.2) has been veriﬁed by beam tests

conducted under the sponsorship of the AISI. Some of the