Rigid Frames

8/12/2019 Rigid Frames

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Fig. #. tatic systems adopted for the rigid frames

Three$hinged frame has the advantage of being statically determinate although the moments

and crown deflections are greater than for the other types. The fixed portal induces the smallest

values of moments of all three structures.

%ll these types of structures have the disadvantage of imposing horizontal thrusts of some

magnitude upon the foundations& hence the total loading must not be so significant.

The calculations for the two$hinged or hinge$less portals are based upon the arch theory,

which presupposes the existence of the appropriate horizontal resistance at the feet of the portals to

provide the arch thrust. 'ere it should be specified that comparatively small horizontal

displacements at the footings will cause considerable redistribution of the moments. These types of

framing should not therefore be used in cases where the existence of ade!uate horizontal resistance

is suspect unless the effect of arch spread is included in the design calculations or unless positive

means are ta en to determine the extent of the lateral movement of the portal feet, usually by the

insertion of ties at the base level.

Observation The calculations for a three$pinned frame also call provisions of an ade!uate

horizontal thrust, but the effect of footing spread in this case is of little importance. %ll forces and

moments in the frame are increased in inverse ratio to that of the reduced crown height compared

with the original crown height, but no redistribution effects occur. "t is generally found that the

weight of a portal frame designed in elastic theory is greater than that of the lightest comparable

construction that uses trusses and columns, but this effect may be offset by savings and advantages

in other directions.

The portal frames may be of solid " sections (*oist or welded plate) or of lattice type. The

formulae dealing with the indeterminate structures, the two$hinged and the hinge$less frames are

given in specific tables (annex +). These may be used to calculate moments, shear forces and

thrusts when a solid section is used.


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everal solutions may be used three hinged frame& frames with hinges between the rafter

and the column& frames with rigid connection between the rafter and the column.

"n the case of the hinge between the rafter and the column, the latter must be fixed in the

foundations. The frames with rigid *oints may have the column fixed in the foundation or, if the

foundation ground is wea , the column may be articulated in the foundation. henever there mayhappen important settlements of the ground, or the construction is designed as dismounting

structure, the frames have three hinges, being static determinate.

The rafter may be a plate girder or a hot$rolled profile and in this case the advantages are

simpler fabrication, smaller height, anti $ corrosion resistance. The *oints between the column and

the rafter and the apex area have greater dimension in the plane of the frame (named also portal

frames), the cross section being specifically reinforced with transversal stiffeners and wider flanges.

The solution is rational when higher strength steels are used, the cross section having supple

webs made of higher strength steels.

Fig. -. everal types of frames with steel rafters made of hot rolled section or build up and columns with

different types of cross sections (steel and reinforced concrete)


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Fig. . %doption of different cross section for the rafter 1$1$ when purlins are placed at the top flange& #$#$

when purlins are missing (under the longitudinal s ylight)

The build up sections of rafters must be insured against lateral buc ling (flexural$tortional

buc ling)& if the roof is sustained by purlins, these elements are able to prevent this phenomenon

from happening (fig. -. section 1$1). "f the roof is sustained directly on the rafter or s ylights are

placed in the roof plane, the top flange of the rafter must be conse!uently reinforced (section #$#).

"n some cases, the rafter is provided with tie rod (fig. #.h). /alculation in this case considers

the hypothesis that the girder has one support articulated and the other fixed. The effort in the tie$rod

is X 1, determined with the relationship

1111 = X (1#)where

1$ is the displacement in the direction of X 1 in the case of the girder without tie$rod, loaded

with external forces&

11$ translations on the direction of the force X 1=1.The girder is bloc ed in the position, double articulated practically, the supports having the

possibility of moving elastic, horizontally. 0!uilibrium conditions are written for the displacements

of the top of the columns, as to obtain e!ual forces at both ends of the columns. "f the stiffness of the

columns is different from one to the other (fig. #.a.), the displacements of their top part are inverted

proportional with the stiffness of the columns and the fixed section is placed at the distance l 1 from

the column S 1 and is given by the relationships

l

EI h K

EI h K

EI h K

l

+

=

1

11

#

##

#

##

1

--

- and 1# l l l = (1-)

where K 1 and K 2 are coefficients that ta e into account the variation of the cross section of the

column.


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"n the case when one support may shift, the displacement would be expressed as

1111 = and is e!ual for both supports. The stiffness of the columns is the reason for a

diminished value of the effort in the tie$rod, 11 < and it is obtained with the relationship

1111

1111 )( =

EI

h K (1 )

The transversal cross section may be designed with two frames or more than two, in this case

the central frame may be in many cases higher and with a bigger span than the frames placed aside.

ifferent lifting capacities and wor ing conditions of the cranes in these buildings are the

reason to vary the stiffness of the columns in the same transversal section. "n most of the cases, the

central span has two greater columns, fixed in the foundations and rigid *oint between the rafter and

the column& the side spans have smaller more flexible columns, even articulated in their foundations

and the rafter articulated to the column (fig. #.1.i,*).

8.". #esign of the $nee of the rigid frames

The general design of the frame (column and rafter) is the one that imposes upon the details

used in the connection area and which are specific for any of these cases.

Fig. 2. 3nees for rigid frames 1)$ machine drilling& #)$ cut off part with oxi$acetylene flame

%s an important degree of stiffness must be achieved, the welded connections are preferred.

"n fig. 2. a, b different details for the hot rolled shapes are presented. They are obtained from cutting

the profiles with flame and processing the edges after heating. % steel plate is used as support and


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fillet welds are used. For build up sections the details from the fig. 2. c+h are used& almost all of the

cases presented herein have ribs for stiffening the webs of the column and of the rafter.

The bending moments in the plane of the frame have in most of the cases, the biggest values

in these connections. % strengthened cross section in that area is necessary and this may be obtained

in the shape of a haunch with a straight line (fig. 2. *), or curved line (fig. 2. h).Typical strength and stability chec ing have as a result a thic er web in the haunched area or

an increased number of ribs. "n particular, the bisetrix of the angle of intersection between the axes

of the column and of the rafter, respectively, must be put into evidence by a continuous rib

(diagonally) which ta es all the important values of shear stresses in the *oint.

4ibs (or sometimes simple gussets) are developed all over the entire area in compression,

their length reaching the 5. %. of bending of the cross section (where stresses don6t exist anymore).

4ibs placed at the end of the haunched area are in fact diaphragms on which both the web of

column and rafter and the web in the corner are welded with fillet welds.

7olted connections are also used for the nees of rigid frames, whether because the efforts

are not so important, or because the conditions of transportation and execution impose. "n the figures

below two of the most usually preferred details are presented. The second version of bolted

connection (fig. .*) is made along the cross beam (the rafter) where the bending moment in the

plane of the frame is supposed to be annulled.

8.".1. %nees for re!tangular frames"n figure 8, the rectangular frame$portal frame has a simple nee between the cross$beam

and the column.

Fig. 8. !uare nee between the column and the cross beam for a doubly articulated frame


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%t any section of a member, the stresses may be found from the normal expression of

stresses

I

y M

A

P = (12)

where

9$longitudinal thrust (i.e. 5 or ' in this case)&

%$ cross sectional area of the member&

:$ bending moment at the section&

y$ distance from the neutral axis of the fibre considered&

"$ moment of inertia in the cross section considered.

/onsidering the design purposes, the bending moment is determined at the limits of the nee,

i.e. in line with the inside flange of the girder or of the column.

hear stress is determined, considering the distribution in figure ; by dividing the shear forceto the gross web area.

Fig. ;. tress distributions on the edges of the nee plate and on the ad*acent girder

"f the forces applied in the *oint are these mentioned in figure 8 then one may determine the

axial efforts resulted from the bending moment that act in the outside (T


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/onse!uently, the girder will impose a tensile force of T 0 H 0 in the outside flange and a

compressive force of T I + H I in the inside flange at the boundaries of the nee.

The whole analysis is developed safely assuming that the flanges of the girder ta e the whole

of the bending moment, M and transmit the whole of the thrusts, H while the web transmits only

shear.These assumptions lead to the following relationships

H A A

A H

H A A

A H

d

M T T

ii

i

i

+=

+=

==

, diagonals are placed on the web (fig. ?.a). The thic ness of thediagonals is designed considering the necessity of ta ing the stresses that remain uncovered by the

web. For corners without haunch the area % d is extracted from the e!uilibrium conditions expressed

for the effort in the top flange A, which becomes a shear force ta en by the web and the diagonal

with their full capacities

cos+= - A -t hh M

d shea% ""#"%

( ?)

8.".". Corners without a $nee or with haun!hed $nee

hen the corner needs to be haunched, the verifications and the design of the rib will be in

the same manner. The web in the corner has bigger height in the cross section, so it may buc le in

the area in compression. "t would be more cautiously to consider that it might not be able to ta e

shear stress and in this situation the diagonal will ta e all the stress (figure ;.b).

a b

Fig. ?. tiffened rectangular nees for rigid connections a)$ with diagonal& b)$ with diagonal and

haunched section in the connection

From the e!uation of e!uilibrium of the forces in the plane we shall obtain the force in the

plane of the diagonal, F d

( ) ( )#


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d s

& &

A - 2

A -( A -(

== #B#1B1 &

(2 the areas of the flanges in compression& usually & & A A #1 = &

Ad $ area of the diagonal (diaphragm)& in the worst case, & d A A = ;2.< , accepted as aconstructive preliminary condition.

"f the angles 3 1, respectively 3 2 are bigger than 1# < the haunched area will not be stressed

more than the ad*acent areas in the column and in the rafter.

-e#o endation4 local buc ling of the web and of the flange (flanges) in compression must

be avoided and verifications should be run accordingly.

8.".&. #esign relationships o'tained '( the )meri!an Studies

Dne of the more easily adaptable groups of formulae for rectangular web plates is attributed

to 5s$ood (ref. +). "f we consider a rectangular plate of uniform thic ness t , loaded with forces and

couples (fig. C), in the e!uilibrium stage we may write

6 y6 y y 6y 6 2 2 ! M 2 2 a M = ## (1C)

Fig. C. 0!uilibrium status on the edges of the steel plate

The stress conditions in the plate must be determined, considering that the normal stresses = x

and = y along the boundaries ! ya 6 == & are uniformly varying and along the boundaries

! ya 6 ==

& are everywhere e!ual with zero.The Ai%y St%ess 'n#tion is used for these conditions and

( )( ) ( )( ) #-#-# -81

-81

),( 6 y! 6!! y 6a yd # 6y! y 6 & ++++++= (#


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a!t 2

#a!t

2 ! 6 y & -- == (##)

t a! M

d !t a

M ! 6 y --

-&

-== (#-)

The normal stresses are then determined

( )

( ) y! 6a

M 2

a!t

6a y!

M 2

a!t

y y y

6 6 6

+

+=

+

+=

#

#

-1

-1

(# )

The shear stress has the following relationship

( )

+++= #

#

#

#

1#

-1

#

-#

1

!

y M

a

6 M y 2 6 2 2 2 a M

a!t 6 y 6 y y 6y 6 6y (#2)

"n the relationships above the values of the forces and couples are

( )!

M 7,

n 7,

H 7, 2 6

++= 111 (#8)

( )8 p 2 y #1 = (#;)

( ) H! M p%

p8 8 2 6y = < (#?)

( )! 7, nM

7H H 2 y6 +=

1 (#C)

M 7,

n 7,

M 6

++= 111 (-


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a8 ! H M M # = < $ the moment of the inside corner of the frame&

I a $a > the section modulus of the nee along a horizontal axis, including the vertical flanges

and the web plate&

I !/! $ the corresponding section modulus along a vertical axis

8.&. Haun!hed %nees for Rigid Frames with Pit!hed Roofs

"n the case of the pitched roofs, it is very common to haunch the nees (fig. C).

Fig. 1.A. o##'%s at the thi%d point alon$ the dia$onal &%o the inside &lan$e to the

o'tside #o%ne% o& the &%a e. %lternatively the force in the flange may be resolved from the triangle

of forces from the same figure, the force that is nown, being that one in the inside flange of the

rafter.

"n figure C.b it is assumed that only one$half of the force continues along the flange of the

main member, as the diagram shows. The stresses in the details of nees in figure C. a, c and d arefound with the help of 8e%endeel? s Tape%ed 9ea formulae or 5lande%?s formulae .

The nees in figure C shows what surface is Ecovered by the above mentioned formulae. "t

is not possible to analyze the stresses in the hatched areas. till, if no critical stresses were found in

the area covered by the formulae, it is not li ely to be so in the hatched areas.


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The forces in the stiffeners at the limits of the nee are found by a resolution of forces (see S

in the detail C.a.). The welds are designed accordingly. The remaining stiffeners inside the nee can

be of nominal size, their primary function being to prevent local buc ling of the web and lateral

failure of the inner flange.

8.*. Ridges in Pit!hed Roofs

"n practical design they are evaluated as obtuse$angled nees. The shape of the angle

between the two rafters determines an important diminish of the forces. "f *oists are used it may be

unnecessary to add brac ets. "n the case of lighter loaded frames, it may be sufficiently to butt weld

the ends of the rafters.

"f a bigger area is needed in figure 11 a typical detail are presented.

a b

Fig. 11. /ross beams at ridge a)$ with *oists (lighter loading)& b)$ area of web with increased stiffness

8.+. %nees with Cur,ed Flanges

Goints with curved flanges are the sub*ect of a continuous controversy and the latest news in

the matter are herein presented.

% theoretical curved bar with parallel flanges (see figure 11) is solved with the in,le%/

-esal formulae.

a b


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Fig. 1#. %nnotations and representation of sectional efforts for the determination of stresses in the curved bar

with parallel flanges a), b)$ symbols used in the in ler$4esal formulae

#%

%

@

# M

A%

M

A

>

+

+= ( $ the normal thrust&

A$ the cross sectional area of the bar&

M $ the applied bending moment&

% $ the radius of curvature of the bar ta en to the 5.%. of the section&

#$ the distance from the 5.%. to the fibre being considered& positive when measured outside the

curvature and negative when measured inside&

@ $a figure analogous to the moment of inertia, I and which may be replaced by I when the

value of % is greater than 2d , where d is the depth of the bar. "t is expressed considering the polar

coordinates of a curved line

dA#%

#% @

A +=

#

"n the case when % 2d and the cross section is a plate girder composed of rectangles

= A"

"!% % @ i

#

1# log-


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Fig. 1-. ymbols for the determination of the stiffness @ on the cross section of a curved bar

"n figure 11, the sign of the efforts : and 5 is positive. The coordinate # is also positive

from the 5.%. to the external fibre and negative to the internal fibre. The diagram from the figureshows the sign of stresses =, negative for compression and positive for tension. The values of normal

stresses in the extreme fibres, internal and external, are

o

o

i

i

#% #

@ % M

A% M

A >

#% #

@ % M

A% M

A >

++

+=

+=

The stress in any fibre both sides the 5.%. is (see figure 1-)

a b

Fig.1 . pecific stresses determined by the efforts in the curved nee a)$stress distribution in the cross

section of a curved bar& b)$ radial forces in the web

The relationships above show a different distribution of the normal stresses than that for

straight bars& thus, in the internal part of the curve the stresses grow sensibly, while in the external

part they are smaller than those obtained with 5avier6s formulae (fig. 1#. a).


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"n the web, stresses = may be neglected being relatively small. Their maximum value is

obtained in the 5. %., without going over #


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'. 7leich (ref. ;) investigated the effects of this phenomenon and produced two coefficients

C and D, the first being associated with the longitudinal stresses and the second to the transverse

stresses.

"t must be specified that, while radial stresses B % are important the normal stresses along the

curvature radius are greatest in the 5.%. but almost insignificant as value (under #


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( ) -t

!and , , &

!

!

====

#B

B

B

&-18.1)&( (( )

& % t !

=

#B

(28)

K and L have tabulated values.

The design of the cross section is made considering that the bottom flange has modified

dimensions ( ! instead of ! ) and the stresses % will be determined.The curvature determines the existence of a radial component % p of the stresses % in the

flange, according to fig. 12.b

&

% % % %

dst !d t !

d t ! >

===

BBB

##

#sin# (2;)

7utds!

> p%

= B and thus the result is -

t p % %

=

.

The value of % p is constant and it causes a bending moment in the flange (fig. 12.a), which

for an unitary strip of flange has the value#)( #B!

-t

M % t = , and the normal stresses

#1

8

t

M

=

M t

&

t t

== .

The values of C and D are tabulated and they both depend on the function ! 2 :-t (see tab.1).

& % t !

=#B


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5ormal stresses developed in the plane of the nee impose the verification of B e#hi (N has

very small values and it is neglected)

t % t % e#hi+ += ## (2?)

For stresses with different signs the value of B e#hi may be greater than the strength of the

steel.0xperimental researches O-CO show that local plastic areas are not dangerous as long they do

not develop near the web, but on the external flange (flange in tension), while on the opposite flange

(flange in compression) this will not happen because % and B t have the same sign.

Jocal buc ling of the top flange in plastic domain happens only when its whole section is

plasticized. This may happen only when B e#hi reaches the yield limit in all the points.

(onside%in$ the %eal plasti# %ese% es o& the #%oss se#tion e6pe%i ental %es'lts p%o e a

s'&&i#iently #o e%in$ #al#'lation ay !e de eloped only on the !asis o& the lon$it'dinal st%esses %

* "ith the #ondition that the e&&e#ti e no% al st%esses do not e6#eed the st%en$th.

Rules for practical design:

The bottom flange must be greater than the top flange& smaller values of stresses and a

bigger capacity of the whole nee area is obtained this way&

3nees, as intersection of two horizontal structural framing elements, may be chec ed

D.3. if both sides are considered separately, as in the figure 12.##.

eb is insured against local buc ling if, apart from the disposition of the ribs anddiaphragms ( t d A A ;2.


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Fig.1?. /urved nees for frames used in practical design

Penerally, nees are not shaped li e curved bars, the ma*ority of the type shown in fig. 1C. a

and b having the outside flange straight. The analysis of these nees is carried out with the shape of

the Aierendeel6s Tapered 7eam formulae for a nee with curved inner flange (fig. 1C.b) which is

presented below. "n any section %$% we may write

a b

Fig. 1C. 3nees with straight outside flange a)$ usual cases& b)$ ymbols for the tapered beam formula

( )

sin1

the distances of the centroids of the outside and inside flanges, respectively from

the axis shown&

E > the shear stress&

M , 8 and P $ bending moment, shear force and thrust at the section %$%&


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t $ the web thic ness&

d $ the web depth&

##


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The stresses derived from these values are determined as for an ordinary beam, excepting

that shear is obtained from M 0, that is S , the total shear is%

M S


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Capitals under roof rafter0 foundations and other details for rigid frames

Fig. ##. %rticulated connection of the cross beam to the column of the frame

Fig. #-. /olumns of rigid frames hinged in the foundations$different details

Fig. # . 7ase of columns for rigid frames that insure bending moments transfer to the foundations


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Fig. #2. olutions providing lateral stability against flexural$tortional buc ling of the rafters (cross beams) ofthe rigid frames (with braces between the bottom flange of the eave purlins or with rigid support between the

purlin and the rafter)

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