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Metal Structures

Lecture XIII

Steel trusses

Contents

Definition → #t / 3

Geometry and cross-sections → #t / 7

Types of truss structures → #t / 15

Calculations → #t / 29

Example → #t / 57

Results of calculations - verification → #t / 79

Death weight of truss → #t / 90

Examination issues → #t / 92

Truss – theory (idealization):

• Straight bars only;• Hinge joints;• Forces in joints only;

There are axial forces only

Schwedler-Żurawski formula for straight bar:

d M(x) / dx = Q(x)d Q(x) / dx = q(x)

Forces in joints only → no loads along bar (q(x) = 0):

q(x) = 0 → Q(x) = const = C → M(x) = C x + A

Hinges:

M(0) = 0 → A = 0 ; M(L) = 0 → C = 0

M(x) = 0 ; Q(x) = 0

Definition

Photo: Author

→ Des #2 / 2

Truss – real:

• Bars with imperfections;

• No ideal hinge joints;

• Gravity along bars;

It’s rather framePhoto: Author

Effort (calculation as a truss) ≈ Effort (calculation as a frame)

Time of calculations (truss) << Time of calculations (frame)

Because of these reasons, we calculate real truss as ideal truss.

Chord

Web members = brace members = cross bars

Chord

There is common name for every not-chord bars (web member or brace member or cross bar).

In Polish language exist two words: „słupki” (vertical bars) and „krzyżulce” (inclined bars).

Common mistake is to design different type of cross-section only because of different name or

direction - for web members important are only axial forces and lengths, not names.

Photo: waldenstructures.com

There are many different shapes of trusses.

Geometry and cross-sections

Photo: tridenttruss.com

Photo: steelconstruction.info

Photo: e-plytawarstwowa.pl

Photo: domgaz.com.plPhoto: konar.eu

Photo: i435.photobucket.com Photo: community.fansshare.net

Specific subtype are trusses with paralell chords. For these trusses, forces in top and

bottom chord are similar.

| NEd, top | / | NEd, bottom | ≈ 0,90 – 1,10

Photo: waldenstructures.com

Photo: gillmet.com.pl

max (h1; h2) ≤ 3,40 m

max (L1; L

2) ≤ 18,00 m

max (L3; L

4) ≤ 12,00 m

Initial drawings - geometry

h = L (1/10 ~ 1/15)

H = L (1/5 ~ 1/10)

a ≥ 5o

30o ≥ b ≥ 60o or b ≈ 90o

Photo: Author

→ Des #2 / 9

PN B 03200 EN 1993

Elements Each types of cross-sections are accepted

Joints No additional requirements Additional requirements → many

types of cross-sections are not

accepted

Cross-sections of truss members

Modern types of cross-sections

(EN)

Old types of cross-sections (PN-B)

Chords

Web

members

Photo: Author

Whenever we use hollow sections, we must

hermetically close the ends by welding to

prevent corrosion inside HS. Such type of

corrosion can develop without apparent

external symptoms and can cause damage

of structure without warning signs.

Photo: architectureau.com

Photo: Błędy wykonawcze podczas realizacji konstrukcji stalowych, Litwin M, Górecki M, Budownictwo i Architektura 4 (2009) 63-72

D

C

DD

Number of different cross-sections, designed in truss: 2 - 5:

Chords Brace members

Top and bottom - the same I-beam

or

Two different I-beams, first for top, second for bottom

1-3 different RHS

1-3 different CHS

Top and bottom - the same RHS

or

Two different RHS, first for top, second for bottom

1-3 different RHS

1-3 different CHS

Top and bottom - the same CHS

or

Two different CHS, first for top, second for bottom

1-3 different CHS

Types of truss structures

Truss purlins

„Classical” trusses

Multi-chords trusses

Truss-grates

Truss-frames

Spatial trusses

Laced columns

Span [m]

Photo: Author

Truss purlins

Photo: structural-steelbuilding.com

Photo: CoBouw Polska Sp. z o. o.

"Normal" truss - forces are applied in nodes; there

are axial forces in chords and cross-bars only.

Truss purlin - continuous load from roofing;

there are axial forces in chords and cross-

bars; in addition top chord is bending.

Photo: Author

Truss purlin

Photo: construdare.com

→ #7 / 37

I-beam purlin: bi-axial bending.

Wind

Dead weight

Snow

Imposed load

a

(D + S + I) cos a + W

(D + S + I) sin a

Wind

Dead weight

Snow

Imposed load

D + S + I + W cos a

W sin a

Truss purlin: horizontal force

W sin a

has very small value and can be neglected

(acts on roofing, not purlins). All loads act in

plane of truss. There is need wedge to install

truss purlin in vertical position.

a

Photo: Author

→ #7 / 38

„Classical” trusses

Photo: steelconstruction.infoPhoto: domgaz.com.pl

Photo: waldenstructures.com

2-chords flat trusses, used

as a single roof girders.

Multi-chords trusses

Photo: steelconstruction.info

Photo: multimetalgb.ca

3 or 4 trusses connected each other in triangle or square

cross-section. Often used for temporary structures

(bandstand) or for masts and towers.

Photo: rktruss.com

Photo: conference-truss-hire.co.uk

Photo: stretchtents.com.auPhoto: eioba.pl

Truss-grates

Photo: cdn8.muratorplus.smcloud.net

Photo: qdjinfei.en.made-in-china.com

Photo: Author

Complex of trusses of the same height, perpendicular each other.

Frame-truss

Photo: wikipedia

Photo: bbsc303.arch.school.nz

"Classical" portal frame (two columns and

roof girder), but made of trusses, not I-beams.

Spatial trusses

Photo: miripiri.co.in

Photo: urwishengineers.com

Photo: civiltech.ir Photo: shreeengineering.in/

2 or 3 layers of bars + cross-bars

between layers.

Layers can be flat, cylidrical or

spherical. More information about this

type of structures will be presented on

IInd step of studies.

Photo: cnxzlf.com

Photo: wikipedia

Photo: urwishengineers.com

Laced columns

Photo: zksgrzelak.eu

Structure looks like truss, but its behaviour and calculation model is completely different than for truss.

For laced columns, battened columns and closely spaced build-up compression members

we use specific algorithm of calculations → #t / 43

Calculations

NEd / Nb,Rd ≤ 1,0

Nb,Rd = χ A fy / γM0

NEd / Nt,Rd ≤ 1,0

Nt,Rd = A fy / γM1

χ - lecture #5:

Flexural buckling

Torsional buckling

Flexural-torsional buckling

Photo: Author

Critical length for compressed chord of truss is depend on plane of analysis. For vertical

plane, critical length is equal distance between cross-bars. For horizontal plane,

critical length is equal distance between bracings:

Photo: Author

Flexural buckling of chords:

Top chords in compression; buckling in plane: critical length = distance between nodes

Photo: Author

Critical length for compressed chord of truss is depend on plane of analysis and part of

structure. For web members, critical length = distance between nodes.

→ Des #2 / 34

Flexural buckling of chords:

Top chords in compression; buckling out of plane: critical length = distance between

horizontal bracings

Photo: Author

→ Des #2 / 35

Flexural buckling of chords:

Bottom chords in compression; buckling in plane: critical length = distance between

nodes

Photo: Author

→ Des #2 / 36

Flexural buckling of chords:

Bottom chords in compression; buckling out of plane: critical length = distance between

vertical bracings

Photo: Author

→ Des #2 / 37

Bracings are very important for behaviour of structures, including for trusses. For

calculations, there are used special algorithms and models.

More information about bracings will be presented on lecture #15.

Photo: Author

Buckling

Flexural Flexural, torsional,

flexural-torsional

c = cy = cz

(only if lcr, y = lcr, z )

c = min( cy ; cz) c = min( cy ; cz ; cT ; cz, T)

(hot rolled I)(welded I)

The result of calculations is buckling factor c.

It is calculated in different way for different cross-sections.

Photo: Author

→ #5 / 42

The same stiffness on each direction (Jy = Jz ):

Flexural buckling (y);

Flexural buckling (z) if lcr, y ≠ lcr, z (for example – distance between bracings is not the

same as between web members).

Photo: Author

Different stiffness on each direction (Jy ≠ Jz ):

Flexural buckling (y);

Flexural buckling (z);

Additionally, is possible that lcr, y ≠ lcr, z

Hot-rolled

Photo: Author

Welded cross-section or less than two axes of symmetry:

Flexural buckling (y) (axis u for L section);

Flexural buckling (z) (axis v for L section);

Torsional buckling;

Flexural-torsional buckling;

Additionally, is possible that lcr, y ≠ lcr, z ≠ lcr, T

Welded

Photo: Author

Buckling length ratio μ for truss members:

Element Direction Cross-section

I H pipe closely-spaced

build-up members

other

Chord In plane 0,9 0,9 1,0 1,0

Out of plane 1,0 0,9 1,0 1,0

Brace In plane 0,9 0,9; 1,0; 0,75 1,0; A 1,0; A

Out of plane 1,0 1,0; 0,75 1,0; A 1,0; A

Bolted connections

Chords parallel and dtruss bracing / dchord < 0,6

L sections:

leff, i = 0,5 + 0,7 li i = y, z

leff, n = 0,35 + 0,7 ln n = u, v

_ _

_ _

EN 1993-1-1 BB.1.1

Not recommended type of cross-sections

because of requirements for joints

Photo: EN 1993-1-1 fig 1.1

Main axes for differen cross-sections.

There is important, that axes are not

horizontal and vertical for L-sections.

Special way of calculations;

Resistance depends on distance between batten plates (a) and number of their

planes.

There were used special type of bars for old-type truss: closely spaced build-up

compression members.

Photo: img.drewno.pl

Photo: Author

Photo: EN 1993-1-1 fig 6.13

laced compression members,

battened compression members,

closely spaced build-up compression members

We must analysed initial deformations for members with two or more chords

(according to EN 1993-1-1 p.6.4). This means, these elements are always bent, even

if axial force acts only.

MEdII = eimperf NEd

More information will be presented on lecture #13

and # 19

→ #6 / 76

Photo: EN 1993-1-1 fig 6.7

Photo: EN 1993-1-1 fig 6.13

n = 29 n = 41

For these three types of members (laced

compression members, battened

compression members, closely spaced

build-up compression members) we

should use special way of calculations.

Of course, we can put full geometry

(chords and lacing system or batten

plates) into computer programm, but

each membes consist of many sub-

members

Photo: Author

Photo: Author

There is possible, that in structure we have more than 100 000 sub-members; it means very long term of calculations.

Photo: s9.flog.pl

laced compression members → lecture #19;

battened compression members → lecture #19;

closely spaced build-up compression members → #t;

Special way of calculations – we no analysed sub-members, but one equivalent uniform

element of effective geometry. Next, cross-sectional forces, calculated for this member,

are recalculated to new values of cross-sectional forces applied to chord and lacing

system or batten plates.

Photo: EN 1993-1-1 fig 6.13

Photo: EN 1993-1-1 fig 6.7

2 or 1 batten

planes:

Small distance: a ≤ 70 imin a ≤ 15 imin

Big distance: a > 70 imin a > 15 imin

imin = iv (for one L section)

EN 1993-1-1 tab. 6.9

There are four possibilities for closely spaced build-up compression members:

Photo: Author

For small distance: calculations as for uniform cross-sections, i.e.:

Ju1, Ju1, JW, JT – according to real cross-section;

Flexural buckling (u1);

Flexural buckling (v1);

Torsional buckling;

Flexural-torsional buckling;

Additionally, is possible that lcr, u1 ≠ lcr, v1 ≠ lcr, T

Photo: Author

No batten plates: calculation as for two independent L section

Axial force = 0,5 NEd

Flexural buckling (u);

Flexural buckling (v);

Torsional buckling;

Flexural-torsional buckling;

Additionally, is possible that lcr, u ≠ lcr, v ≠ lcr, T

Photo: Author

Batten plates exist, but on big distance:

Photo: Author

Buckling about y - y axis (material axis; axid goes through chords):

• No imperfections;

• Global axial force NEd only;

• Critical length = length of member;

• Moment of inertia for cross-section = 2 · moment of inertia for one

chord;

Photo: Author

Buckling about z - z axis (immaterial axis; axis goes out of chords):

• No imperfections;

• Global axial force NEd only;

• Critical length = length of member;

• Moment of inertia for cross-section = effective moment of inertia for

cross-section;

Photo: Author

Buckling about y1 - y1 axis (own axis of chord):

• Bow imperfections;

• Axial force Nch, Ed in one chord;

• Bending moment from imperfections Mch, Ed in one chord;

• Shear force from imperfections Vch, Ed in one chord;

• Critical length = distance of battens;

• Moment of inertia for cross-section = moment of inertia

for one chord about axis of one chord;

Photo: Author

Nch, Ed = NEd / 2 + 2 MEdII zs Ach / (2 Jeff)

MEd II = NEd e0 / [1 - (NEd / Ncr) - (NEd / SV)]

e0 = L / 500

Ncr = p2 E Jeff, / (m L)2

Numbers of modules between battens or lacings ≥ 3

Length of modules should be equal

Recommended is odd number of modules

EN 1993-1-1 6.4.1

l = m L / i0

i0 = √ [J / ( 2 Ach ) ]

l meff

0

2 - l / 75

1,0

≥ 150

≤ 75

75 ~ 150

SV = min { 24 X / [1 + 4 Jch, v zs / (n Jb a )] ; 2 p X }

Jeff = 2 zs2 Ach + 2 meff Jch

X = E Jch,v / a2

EN 1993-1-1 (6.73), (6.74)

EN 1993-1-1 tab. 6.8

n - number of battens planes

zs - distance between centre of

gravity for total cross-section and

centre of gravity for one chord

Xch - geometical characteristic of

one chord

Jb - moment of inertia for cross-

section of batten

J = 2 zs2 Ach + 2 Jch

VEd = p MEdII / (n L)

h0 = 2 zs

For chord:

Vch, Ed = VEd / 2

Mch, Ed = a VEd / 4

For batten:

Vb, Ed = VEd a / (2 h0)

Mb, Ed = a VEd / 2

Photo: EN 1993-1-1 fig 6.11

S 235

L 60x40x5

Ach = A (L 60x40x5) = 4,79 cm2

Ju1 (L 60x40x5) = 19,75 cm4 ; iu1 = 2,03 cm

Jv1 (L 60x40x5) = 3,50 cm4 ; iv1 = 0,85 cm

Jy1 (L 60x40x5) = 17,2 cm4 ; iy = 1,89 cm

Jz1 (L 60x40x5) = 6,11 cm4 ; iz = 1,13 cm

Ju (2L 60x40x5) = 93,29 cm4 ; iu = 3,14 cm

Jv (2L 60x40x5) = 24,44 cm4 ; iv = 1,60 cm

L = 4,000 m

d = 8 mm

a = 400 mm

NEd = 20,000 kN

L 60x40x5 → Ist class of cross-section

Example

Photo: Author

No batten plates → #t / 58 - 62

Batten plates in one plane → #t / 63 - 72

Batten plates in two planes → #t / 73 - 77

No batten plates: calculation for single L-section chord (according to example, #5 / 45):

NEd, 1 = NEd / 2 = 10,000 kN

NRd = A fy = 112,565 kN

mu = mv = mT = 1,00

L0u = L0v = L0T = 4,000 m

Ncr, u = p2 EJu / (mu L0u)2 = 25,584 kN

Ncr, v = p2 EJv / (mv L0v)2 = 4,534 kN

JW , JT – approximation according to #5 / 41

JW = 0,727 cm6

JT = 0,375 cm4

zs = 2,73 cm

i0 = √ (iu2 + iv

2) = 2,20 cm

is = √ (i02 + zs

2) = 3,50 cm

Ncr, T = [p2 EJw / (mT l0T)2 + GJT] / is2 = 248,036 kN

m = min[√ (mv / mT) ; √ (mT / mv)] = 1,000

x = 1 - (m zs2 / is

2) = 0,392

Ncr, zT = {Ncr, v + Ncr, T - √ [(Ncr, v + Ncr, T)2 - 4 Ncr, v Ncr, T x] } / (2 x) = 4,484 kN

A fy = NRd = 112,565 kN

lu = √(A fy / Ncr, u) = 2,098

lv = √(A fy / Ncr, u) = 4,983

lT = √(A fy / Ncr, T) = 0,674

lvT = √(A fy / Ncr, vT) = 5,010

L 60x40x5 → tab. 6.1, 6.2, 1993-1-1 → au = av = aT = avT = 0,49

Fu = [1 + au (lu - 0,2) + lu2] / 2 = 3,165

Fv = [1 + av (lv - 0,2) + lv2] / 2 = 14,085

FT = [1 + aT (lT - 0,2) + lT2] / 2 = 0,843

FvT = [1 + avT (lvT - 0,2) + lvT2] / 2 = 14,229

cu = min{1/[Fu + √ (Fu2 - lu

2)] ; 1,0} = 0,181

cv = min{1/[Fv + √ (Fv2 - lv

2)] ; 1,0} = 0,037

cT = min{1/[FT + √ (FT2 - lT

2)] ; 1,0} = 0,741

cvT = min{1/[FvT + √ (FvT2 - lvT

2)] ; 1,0} = 0,036

c = min(cu ; cz ; cT ; cvT) = 0,036

A fy = 112,565 kN

c A fy = 4,052 kN

NEd / A fy = 0,089

OK.

NEd / c A fy = 2,212 > 1,000

Wrong, buckling, destruction!

Batten plates in one plane;

Accordin to #t / 47, limit between small distance and long distance is equal

15 imin = 15 iv = 128 mm;

Distance betwen batten plates

a = 400 mm

a > 15 imin → big distance

There is even number of modules, but odd number is only recommended, not obligatory.

Ach = (L 60x40x5) = 4,79 cm2

Jch, u = Ju1 (L 60x40x5) = 19,75 cm4

Jch, v = Jv1 (L 60x40x5) = 3,50 cm4

Ju = Ju (2L 60x40x5) = 93,29 cm4

Jv = Jv (2L 60x40x5) = 24,44 cm4

e0 = L / 500 = 8 mm

zs = 2,73 cm

m = 1

i0 = √ [Jv / ( 2 Ach ) ] = 1,60 cm

l = m L / i0 = 250 → #t / 55 → meff = 0

Jeff = 2 zs2 Ach + 2 meff Jch, v = 71,40 cm4

X = E Jch,v / a2 = 45,938 kN

Jbatten = 103 ∙ 1 / 12 = 83,33 cm4

SV = min {24 X / [1 + 4 Jch, v zs / (n Jb a )] ; 2 p X } = 288,637 kN

Ncr = p2 E Jeff / (m L)2 = 92,491 kN

MEdII = NEd e0 / [1 - (NEd / Ncr) - (NEd / SV)] = 0,224 kNm

Nch, Ed = NEd / 2 + 2 MEdII zs Ach / Jeff = 18,203 kN

Vch, Ed = p MEdII / n L = 0,088 kN

Mch, v, Ed = Vch, Ed a / 4 = 0,022 kNm

Axis v goes through chords → material axis

Axis u goes out of chords → immaterial axis

Photo: Author

v - v flexural buckling → only axial force; moment of inertia Jv

NEd = 20 kN

NRd = A fy = 2 Ach fy = 225,130 kN

Lcr,u = L = 4,000 m

iv = √ (Jv / A) = 1,60 cm

lu1 = (Lcr,v / iv) (1 / l1) = 2,662

Buckling curve c → a = 0,49

Fv = 4,646

cv = 0,118

NEd / (cv NRd) = 0,753 < 1,000

OK

u - u flexural buckling → only axial force; moment of inertia = Jeff

NEd = 20 kN

NRd = A fy = 2 Ach fy = 225,130 kN

Lcr,u = L = 4,000 m

iu = √ (Jeff / A) = 2,73 cm

l = 1,560

Buckling curve c → a = 0,49

F = 2,050

c = 0,296

NEd / (c NRd) = 0,431 < 1,000

OK

v1 - v1 flexural buckling → interaction axial force - bending moment → interaction flexural-torsional buckling - lateral buckling

Nch, Ed = 18,203 kN

Mch, v, Ed = 0,022 kNm

Vch, Ed = 0,088 kN

Lcr, v1 = a = 0,800 m

Lcr, LT = a = 0,800 m

mz1 = mLT = 1,0

iv1 = √ (Jv1 / Ach) = 0,85 cm

Nch, Rd = Ach fy = 112,565 kN

Mch, Rd, v1 = 0,415 kN

Vch, Rd ≈ Ach fy / √3 = 64,989 kN

Ncr, v = p2 EJv / (mv a)2 = 36,079 kN

Ncr, T = [p2 EJw / (mT a)2 + GJT] / is2 = 249,881 kN

m = min[√ (mv / mT) ; √ (mT / mv)] = 1,000

x = 1 - (m zs2 / is

2) = 0,392

Ncr, vT = {Ncr, v + Ncr, T - √ [(Ncr, v + Ncr, T)2 - 4 Ncr, v Ncr, T x] } / (2 x) = 25,889 kN

lvT = √(A fy / Ncr, vT) = 2,085

avT = 0,49

FvT = 3,135

cvT = 0,183

Mcr = is √ (Ncr, z Ncr, T) = 2,815 kNm

lLT = √ (Wy fy / Mcr) = 0,384

FLT = [1 + aLT (lLT -0,2) + lLT2] / 2 = 0,605

cLT = 0,932

Interaction between flexural-torsional buckling and lateral buckling → lecture #18

Rough approximation (EN 1993-1-1 NA.20):

Nch, Ed / (c NRd) + Mch, Ed / (cLT MRd) ≤ 0,8

Nch, Ed / (c NRd) + Mch, Ed / (cLT MRd) = 0,884 + 0,057 = 0,941 > 0,800

Wrong, buckling, destruction!

Interaction between V and M → lecture #16

Battens resistance, welds between battens and chords → lecture #19

Batten plates in two planes;

Accordin to #t / 47, limit between small distance and long distance is equal

70 imin = 70 iv = 597 mm;

Distance betwen batten plates

a = 400 mm

a < 70 imin → small distance

Jb, u = Ju1 (2L 60x40x5) = 93,29 cm4

Jb, v = Jv1 (2L 60x40x5) = 24,44 cm4

Jw (2L 60x40x5) = 59,02 cm6 (calculations according to Mechanics of Materials, thin-walled

theory)

JT ≈ 2 JT (L 60x40x5) = 0,750 cm4

NRd = 2 A fy = 225,130 kN

mu = mv = mT = 1,00

L0u = L0v = L0T = 4,000 m

Ncr, u = p2 EJu1 / (mu L0u)2 = 120,847 kN

Ncr, v = p2 EJv1 / (mv L0v)2 = 31,659 kN

zs = 2,73 cm

i0 = √ (iu2 + iv

2) = 2,20 cm

is = √ (i02 + zs

2) = 3,50 cm

Ncr, T = [p2 EJw / (mT l0T)2 + GJT] / is2 = 495,740 kN

m = min[√ (mv / mT) ; √ (mT / mv)] = 1,000

x = 1 - (m zs2 / is

2) = 0,392

Ncr, zT = {Ncr, v + Ncr, T - √ [(Ncr, v + Ncr, T)2 - 4 Ncr, v Ncr, T x] } / (2 x) = 30,448 kN

lu = √(A fy / Ncr, u) = 1,365

lv = √(A fy / Ncr, u) = 2,667

lT = √(A fy / Ncr, T) = 0,674

lvT = √(A fy / Ncr, vT) = 2,706

L 60x40x5 → tab. 6.1, 6.2, 1993-1-1 → au = av = aT = avT = 0,49

Fu = [1 + au (lu - 0,2) + lu2] / 2 = 1,717

Fv = [1 + av (lv - 0,2) + lv2] / 2 = 4,661

FT = [1 + aT (lT - 0,2) + lT2] / 2 = 0,843

FvT = [1 + avT (lvT - 0,2) + lvT2] / 2 = 4,776

cu = min{1/[Fu + √ (Fu2 - lu

2)] ; 1,0} = 0,362

cv = min{1/[Fv + √ (Fv2 - lv

2)] ; 1,0} = 0,118

cT = min{1/[FT + √ (FT2 - lT

2)] ; 1,0} = 0,741

cvT = min{1/[FvT + √ (FvT2 - lvT

2)] ; 1,0} = 0,114

c = min(cu ; cz ; cT ; cvT) = 0,114

A fy = 225,130 kN

c A fy = 25,844 kN

NEd / A fy = 0,089

OK.

NEd / c A fy = 0,774 < 1,000

OK.

Summary:

No batten plates: effort 2,212 > 1,000

Battens in one plane (big distance): effort 0,941 > 0,800 ≈ 1,176 > 1,000

Battens in two planes (small distance): effort 0,774 < 1,000

Conclusions:

Batten plates in one planes is more effective than no batten plates, but the best are two

planes of batten plates

Results of calculations - verification

Fch-up

Fch-down

Fwm

Sx = 0 ; Fwm / Fch-down ≈ 0 →

|Fch-down| ≈ |Fch-up|

Fi

Photo: Author

q

There is possible to estimate force in top and bottom chords; we use beam-analogy

q = S Fi / L

Mmax = q L2 / 8 or q L2 / (8 cos2 a)

|Fch-down| ≈ |Fch-up| ≈ Mmax / H

Photo: Author

Similar shape of axial forces in chord and bending moment for single-span beam;

the same for axial forces in web members and shear force in single-span beam.

Photo: Author

But this estimation is not true for symmetrical supports - there are additional axial forces in chords.

Photo: Author

There is big difference between symmetrical supports and unsymmetrical supports (approximation)

Photo: Author

Ntop:

Nc, Ed, sym ≈ 1,5 Nc, Ed, unsym

Nbottom:

Nt, Ed, sym ≈ 0,5 Nt, Ed, unsym

Photo: Author

There is big horizontal reaction for symmetrical support. Because of this reaction, truss will

behave as for unsymmetrical supports after short time of exploatation (deformations of

columns, local destruction of masonry or concrete structure around anchor bolts).

Calculation as for symmetrical supports means, that forces taken under consideration in

design project are completely different than real forces in structure: for bottom chord real are

much more bigger than theoretical. This means big probability destruction of bottom chord

and collapse total structure.

Better way is to model the truss with unsymmetrical supports - this is closer its real

behaviour.

Photo: Author

There is difference between two directions of web members

Photo: Author

Deformations: elongation (tensile force) and abridgement (compressive force):

Transmission of forces form chords to supports and zero force members:

Photo: Author

Stifness of I-beam:

JI = Jtop flange + Jweb + Jbottom flange

For symmetrical cross-section:

Jtop flange = Jbottom flange ≈ 2 (h1, I / 2)2 Aflange

JI ≈ (h1, I2 / 2) Aflange + Jweb

Photo: Autor

Stifness of truss (Konstrukcje metalowe, M. Łubiński, A Filipiak, W. Żółtowski,

Arkady 2000):

Jtruss ≈ 0,7 [Atop chord Abottom chord / (Atop chord + Abottom chord)] (h1, truss / 2)2

For symmetrical cross-section:

Atop chord = Abottom chord = Achord

Jtruss ≈ 0,7 [Achord2 / (2 Achord)] (h1, truss / 2)2 = 0,7 (h1, truss

2 / 2) Achord

Stifness of I-beam:

JI ≈ (h1, I2 / 2) Aflange + Jweb

For different types of I-beams, Jweb = 7% - 25% of Jflange

Stifness of truss:

JI ≈ 0,7 (h1, truss2 / 2) Achord

For Jtruss = JI , we need Achord >> Aflange and h1, truss >> h1, I

h1, I = L / 20 - L / 25 ; h1, truss = L / 10 - L / 15 → h1, truss > h1, I

Achord - CHS, RHS, I-beam ; Aflange - plate → Achord >> Aflange

Death weight of truss

Ist proposal (PN B 02001):

gT

= [ 2 / a + 0,12 (g + q)] L / 100

gT, g (roofing + purlins), q (snow + wind) → [kN/m2], characteristic values

a (distance between trusses), L → [m]

IInd proposal:

|Fch-down| ≈ |Fch-up| ≈ Mmax / H →

Ach-down ≈ Ach-up = A = Mmax / (H fy) →

gch-down ≈ gch-up ≈ gweb members = A dsteel L →

gT

= 3 dsteel L Mmax / (H fy)

Types of truss structures

Algorithm of calculation for compressed closely spaced build-up compression members

Examination issues

Laced members - słup wielogałęziowy skratowany

Battened members - słup wielogałęziowy z przewiązkami

Closely spaced build-up members - pręt wielogałęziowy

Thank you for attention

© 2016 Tomasz Michałowski, PhD

tmichal@usk.pk.edu.pl