PORTAL FRAMES FOR INDUSTRIAL BUILDINGS

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Main structural and non-structural elements used for the current construction with portal frames

1)-Roof sheeting or decking (sandwich panels); 2)- Structural elements (columns and rafters); 3)- Beams for

sustaining the wall sheeting or decking; 4)-Purlins; 5)- Wall panels

Current structures used for various activities: workshops, deposits, retail centers; others.

Wide variety of spans: 12 m….60 m but the efficiency of the system is for 20 m…30 m;

Bays between 6 m…10(12) m

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Variation of the cross section of columns and rafters shows the static system adopted

COLUMNS

FIXED IN FOUNDATION

ARTICULATED

RAFTERS RIGIDLY INTERCONNECTED

ARTICULATED IN THE MIDDLE OF THE SPAN

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a)-medium span; b)- curved rafter

c)-with intermediary floor; d)- with crane girders on brackets

e)-portal frame with two even spans

f)-portal frame with two uneven spans

g)- double pitched rafter; h)- frame with variable sections of the

column and girder (build up sections, welded)

VARIOUS TYPES OF STRUCTURES FOR CURRENT PORTAL FRAMES

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STRUCTURAL SYSTEMS

•Usually smaller spans and bays (up to 20 m span; under 9 m bays);

•Arched or portal framed constructions, one or more bays.

•Three-hinged frame (statically determinate) the moments and crown deflections are greater than for the other types; not sensitive to ground settlement

• Rigid portal frames smallest values of moments of all three structures;

•Two-hinged or hinge-less portals (arch theory, which presupposes the existence of the appropriate horizontal resistance at the feet of the portals to provide the arch thrust) comparatively small horizontal displacements at the footings cause considerable redistribution of the moments.

Weak soils call for articulations in the foundations; if the horizontal thrusts are important (arch spread), then sag rods are placed in the transversal frame between the columns at the foundation level.

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•Structural elements: solid I sections (joist or welded plate) or compound: -The rafter may be a plate girder or a hot-rolled profile; - Joints between column and rafter and the apex area have greater dimension in the plane of the frame, reinforced with transversal stiffeners and wider flanges; – A rational solution is to use higher strength steels (supple webs). - Rafters must be insured against lateral buckling (flexural-torsional buckling), usually with the purlins. If the roof is sustained directly on the rafter (or skylights are placed in the roof plane), the top flange of the rafter must be reinforced (section 2-2). Rafters provided with sag rods (fig. 2.h) the girder has one support articulated and the other fixed. The effort in the sag-rod is X1, determined with the relationship: One support may shift the translation is equalized in both supports; the stiffness of the columns causes for a diminished value of the internal force in the sag-rod:

1111 X

11

111

1

1111 )(

EI

hKOpened or closed cross sections are used for the rafters

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CONNECTION BETWEEN THE COLUMN AND THE RAFTER

KNEES OF THE RIGID FRAMES

Rigid joint welded connections are preferred, by cutting the hot rolled profiles with flame and processing the edges after heating. Both butt welds and fillet welds are used ( on a steel plate as support). For build up sections almost all of the cases presented herein have transversal stiffeners 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 we have to strengthen the cross section in the shape of a haunch with a straight or curved line.

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APEX JOINT-CONNECTION BETWEEN RAFTERS

Bolted rigid connection between rafters with stiffened section modeled according to EN 1993-1-8 regulations:

1-end plate; 2)- stiffening part; 3)-rafter (current cross section); 4)-Welding of the flange to the end plate; 5)- welding of the web to the end plate; 6)- Bolts in shear; 7)- bolts in tension

A-Tension area; B-Shear area; C-compressed area

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JOINT BETWEEN COLUMN AND RAFTER

Bolted rigid connection between rafter (tappered beam) and column (EN 1993-1-8 regulations):

I- The rafter need an increased cross section for a greater stiffness (lower height of the column and greater span of the rafter);

II- In the tension area above the N.A. bolts will take the forces in the connection; in the compression area the web of the column is stiffened against local buckling

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RULES FOR THE DESIGN OF THE JOINTS BETWEEN COLUMN AND RAFTER

• After strength and stability verifications it may result a thicker web in the haunched area or an increased number of stiffeners;

• The bisetrix of the angle of intersection between the axes of the column and of the rafter put in evidence by a continuous stiffener (diagonally) which takes all the important values of shear stresses in the joint;

• Transversal stiffeners are developed all over the entire area in compression, up to the N. A. of bending in the cross section;

• The stiffeners 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;

• Bolted connections are used for the knees of rigid frames when tensions are not so important, or the conditions of transportation and execution impose.

I

zM

A

H

Vierendeel’s tappered beam formula:

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HA

AHH

A

AH

ATAT

ii

iii

;

;

00

000

Axial efforts resulted from the bending moment that act in the outside (T0) and inside (Ti) flange, respectively. Also the components of the horizontal or normal thrust may be determined, in the outside (H0) or inside (Hi) flanges, respectively:

The whole analysis is developed safely assuming that the flanges of the girder take the whole of the bending moment, M and transmit the whole of the thrusts, H while the web transmits only shear.

where: σ0, σi – average bending stresses in the outside and inside flanges; A0, Ai – cross sectional area of the outside and inside flanges

The girder will impose a tensile force of T0 – H0 in the outside flange and a compressive force of TI + HI in the inside flange at the boundaries of the knee.

HAA

AH

HAA

AH

d

MTT

i

i

i

i

0

0

0

00

0

In the case when shear is transferred from flanges to web through means of fillet welds (plate girders), welding is verified with the following relationship:

In the top edge of the web plate the thrust T is equal to T0-H0, while in the bottom edge it is equal to (TI + HI)-H . Since (TI + HI)-H = TI +HI – (HI +H0) =TI – H0=T0 – H0. Therefore the shear forces in the top and bottom edges of the web plate are equal. Similarly, those in outside and inside vertical edges are equal.

where: T- total thrust in the flange; L- length of the side of the web plate; t- thickness of the web; aw- throat of the fillet weld.

d,vw

ww

w fa)a2L(2

T:so;

tL2

T

C

.Te

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SPECIFIC RULES FOR THE DESIGN OF THE JOINTS BETWEEN COLUMN AND RAFTER-contin

• Experimental evidence shows that there is no tensile stress at the extreme corner of the knee, as the load takes a direct path across the web. The tensile force is assumed to vary uniformly in the outer flanges from a maximum value at the intersection with the vertical line that limits the depth of the column to zero at the outside corner.

• Each of the flange loads is transmitted into the knee web plate within the lengths of its sides, and this plate is the only means by which the bending moment is transferred from the girder to the column. Consequently, heavy shear forces appear in the knee.

• When shear is transferred from flanges to web through means of fillet welds (plate girders), welding

must be verified. • The shear forces in the top and bottom edges of the web plate are equal. Similarly, those in outside

and inside vertical edges are equal. • The web must have increased thickness or provided with suitable stiffeners (for the simple knee,

diagonal stiffeners are used). • The thickness of the diagonals is designed considering the necessity of taking the stresses that

remain uncovered by the web. For corners without haunch the area Ad is extracted from the equilibrium conditions expressed for the effort in the top flange V, which becomes a shear force taken by the web and the diagonal with their full capacities:

cos3

00

M

y

d

M

y

wwc

wr

fA

fth

h

M

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CORNERS WITH HAUNCHED KNEE

2

0'

21

0'

1d 45sinC45sinCF

0

00

Mydd

Myf2

'

2Myf1

'

1

/fAF

/fAC;/fAC

ff AA 21

fd AA 75.0

Usually:

and:

For θ1 and θ2 > 120 the haunched area is not stressed more than the adjacent areas in the column and in the rafter.

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Different solutions for haunched knees: a), b)- rafter and column with constant cross section: c), d) – rafter and respectively, both rafter and column, with tappered section

PRACTICAL DESIGN RELATIONSHIPS

For rectangular web plates Osgood / /considers the equilibrium stage in a rectangular plate of uniform thickness t, loaded with forces and couples, for which it may written:

xyxzyxyx FF2bMFF2aM

by;ax

by;ax

The normal stresses σx and σz along the boundaries: vary uniformly

along the boundaries: are equal with zero

Airy Stress Function f(x,z) is used for the determination of the stresses σx and σz

2

xy

2

yxyx

22

The principal in plane stresses in the knee web may be computed:

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ybxa

M3F

abt4

1

xayb

M3F

abt4

1

2

y

yy

2

xxx

where the normal stresses x and y are:

and the shear stresses xy are:

2

2

x2

2

yxyyxyxxyb

y1M

2

3

a

x1M

2

3yFxFFF2aM

abt4

1

In the relationships above the values of the forces and couples are:

b

M

jk1

n

jk1

mHjk1Fx

Vp21Fy

HbMp

rpVVF 0xy bjk1

nMjHHFyx

Mjk1

n

jk1

m1Mx

HbMr21M 0y

15

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The formulae are quoted in their original form, only the normal stresses are negative for compression and positive for tension. In the relationships p and r are the proportions of V and M respectively, which are taken by each of the flange of the column at the edge y=b. If a flange is not continuous all over the knee, as in the case of the riveted or bolted constructions, it will transfer stress only partially across the discontinuous section. Also, k and j are proportions of H and m and n are proportions of M, which are taken by the top and bottom flanges of the beam portion of the knee at the edge x=a. In the case when there is no discontinuity, k=j and m=n. The flanges carry no transverse shear. In the case of a welded frame:

VaMM

BbHAaVM

0

0

p21HbMM

n21MM

0y

x

Vp21F

Hj21F

y

x

b

nMjHHF

HbMa

rpVVF

yx

0xy

The greatest stress occurs in the inside corner of the knee: x=+a; y=+b. The greatest shear stress in the web occurs at the point with coordinates:

x

2x

y

2y

M3

bFy;

M3

aFx

The maximum stress is determined with the following relationship:

2

xy

2

yx

max2

Wright / /: The point where the greatest shear stress occurs is very near the centre of the web: where:

bbaa

c

I

tb

I

ta

abt

M

331

4

22

aVbHMMc 0

Mc is the moment of the inside corner of the frame; Ia -a– the section modulus of the knee along a horizontal axis, including the vertical flanges and the web plate; Ib-b- the corresponding section modulus along a vertical axis

NOTE

KNEES WITH CURVED FLANGES

cr

r

U

cM

Ar

M

A

N

Stresses on curved cross sections are determined based on Winkler-Resal formulae:

N- the normal thrust; A- the cross sectional area of the bar; M- the applied bending moment; r- the radius of curvature of the bar taken to the N.A. of the section; c- the distance from the N.A. to the fibre being considered; positive when measured outside the curvature and negative when measured inside; U-a figure analogous to the moment of inertia, I and which may be replaced by I when the value of r is greater than 2d, where d is the depth of the bar. It is expressed considering the polar coordinates of a curved line:

dAcr

crU

A

2

A

w

wbrrU

i2

12 log30258.2For r < 2d and a plate girder section:

o

exto

i

i

r

cA

r

cAIU

22

int

8.1...4.1d

rIU )05.1...15.1(

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Usually the flanges are not equal and:

More frequently the values are used :

o

o

i

i

cr

c

U

rM

Ar

M

A

N

cr

c

U

rM

Ar

M

A

N

Distribution of radial stress σr on the bottom flange of the curved bar and averaged value r

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tR

Cr

KNEES WITH CURVED FLANGES

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r

b

0t

'

'

dx

b

316.1k);k(f'b

'b

f

2'

rt

b

fr

dsrt'b

2

drt'b2

2

dsinrt'b2N

dsb

Np

'r

R

tp rr

2

)b(

R

tM

2'r

t

2t

f

tt

t1

M6

W

M

tr2t

2rechiv

Symbols and determination of the radial and tangential stresses in the bottom flange of the curved bar with I section: a)- determination of σr; b)- determination of σt

DETAILS FOR SPECIFIC SHAPES OF KNEES AND APEX

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CAPITALS UNDER ROOF RAFTER, FOUNDATIONS AND OTHER DETAILS FOR PORTAL FRAMES

Capital for a hinged connection between the column and the rafter Hinged connection between the foundation and the column

Fixed connection between the foundation and the column

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Position of bolts in the case of articulated base

Forces under the base plate of columns subjected to heavy compression and small moments

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Position of bolts in the case of fixed base (without or with gusset plates)

Forces under the base plate of columns subjected to moderate compression and heavy moments

The rafters are usually sensible to flexural-torsional buckling so adequate solutions must be provided for preventing. In the case of plastic design the length of the segment of the rafter situated in the proximity of the structural beam –to-column connection must be insured against lateral instability in order to be able to develop full plastic moments. Current solutions consist in: - placing braces between the bottom flange of the eave purlins and the bottom flange (in compression) of the rafter itself; - rigid support of the purlin on the rafter and transversal stiffeners on the web of the rafter in the cross section where the purlin is placed.

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Solutions providing lateral stability of the rafters

Bending moments diagram on a rigid frame connection and the purlins placed on the top flange of the rafter: solutions for preventing the lateral instability (1-1 and 2-2 ): a- purlins continuous; b-purlins fixed