Restrained Beams in Bending

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Course 4Restrained beams in bending

Resistance of cross-section

Beams are perhaps the most basic of structural components.

A variety of section shapes and beam types may be used dependingon the magnitude of loading and the span

Loading

uniaxial bendingz

y

My

y

z

biaxial bending

z

y

My

z

y

M Mz

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Types of cross sections for members in bending:

Fig.1 Hot rolled beams

Fig. 2. Castellated beams

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Fig.3 Plate girders, box girders

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Fig.4 Riveted beams (old constructions)

Fig.5 Compound sections

Fig.6 Cold formed sections simple or connected using welding or screws

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beams may be with constant or variable inertia to adapt for the bending momentalong the beam

1:5>2< 2

Haunched or tapered beams

Lv LvL

Fig.7 Beams with variable inertia

Haunched segment length: Lv = (0.10.12 L)

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truss girders or vierendeel trusses may also be used to to resist in bending!

talpa comprimata

talpa intinsa

montant

Fig.8 Vierendeel truss

placa beton armat

grinda din otel

conectori

Fig.9. Composite steel - concrete beams

Generally, beams are members loaded mainly in bending; therefore they may berealized with any shape from simple sections or as compound sections.

o The ideal beam, loaded in pure bending, could be realized from flanges,only flanges will resist the bending, while web is design to assure thetwo flanges act together

o Pure bending is seldom met in practice

M

o Bending is usually accompanied by shearing. In this case, the web willcarry most of the shear forces.

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Selection of beam type according to loading and span

Nr. Type Max. span[m]

Observations

1. Simple angles

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As rails or purlins

Light loading2. Cold formed

sections 6-7- roof purlins

- wall rails- light deck girders

- light to moderate loading

3. Hot rolled channelsections 6-8

- secondary beams- roof purlins

- wall rails- moderate loading

4. I Hot rolledprofiles

IPE, HEB, HEA

12-35- main beams

- secondary beams for heavy

structures(ex. bridges)

5. Castellatedbeams 5-35

- large spans and moderate loading- mainly to optimize the steel

consumption

6. Plate girders withhigh depth and

stiffenersConstant or

variable depth


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Presizing the plate girders

h

c

b

ti

t

hi

t

The following formula may be used for presizing the plate girders:

Wh k

t (1)

W section modulus for bending about one axis

maxnecesar

MW

R= (2)

R = steel design resistance

0

y

M

fR

= (3)

0 = 1 - 1.10

k = 1,15 plate girders with constant depthk= 1.10 plate girders with variable depth

Percent of the flanges aria from the total cross area

(0.5 0.6)flange totalA=

Recommended values

h(m)


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For class 3

/ 14

235

y

c t

f

=

For plastic design (global analisys), beam cross section should be entirely class

1. For plastic design (at element), may be used also a class 2 cross section.

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Moment resistance of a beam (cross section)

M

R

x

z

z

y y

b

M

Mel

Mpl

el pl curbura

raspuns idealizat(elastic-plastic)

1R

=

Fig. 10 Curvature bending moment relation for a beam in pure bending

See Strength of Materials, Navier formula !!!!

Longitudinal strain of a fiber located at the distance z from the neutral axis may be giveby:

/z R = (7)

yz

yI=

(Navier) (8)

E = (Hook) =1 y

yR E I

= =

(9)

Relation M- is linear in the elastic range, for p <

p

pE

= (10)

where p is the limit of proportionality of steel for a elastic perfect plastic model

( p yf = ) and y p =

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M

fy

y

y

f

E = (11)

When elastic strain is reached in a fiber, y , the corresponding stress will be yf and it

cannot increase beyond this value. In the post elastic stage, fibers will successively

plastify, from the exterior to the neutral axis, until the cross section is fully plastified.

For rectangular cross section:

1.5pl

el

W

W=

=> Moment resistance in the post elastic range is doubled.

Efficiency of cross section in bending

Cross sectiontype

Properties

Rectangular(bxh)b=5.2cmh=10.4cm

HEHEA200

IPEIPE300

rectangularhollow section200x100x10mm

A(cm2) 53.8 53.8 53.8 51.7

Wel 93.7 388.6 557.1 235.5

Wpl 140.6 429.5 628.4 309.3

Wpl/Wel 1.5 1.11 1.13 1.31

Check for bendingSections class 1 and 2

,pl Rd SdM (14)

,/( / )

opl necesar Sd y MW M f (15)

Sections class 3

,el Rd Sd M (16)

, /( / )oel necesar Sd y M W M f (17)

Sections class 4

,el Rd Sd M

(18)/( / )

oeff Sd y M W M f (19)

el y

E=tg

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Shear force

It is based on formula of Juravski (see Strength of Materials) which is employed to

determine the shear stress Rd

z y

Rd Sd

i y

V S

t I = > (20)

zV - shear force on z direction

yS statical moment of area for y axis, at cross section level

Iy = moment of inertiati - web thickness

In principle, in order to simplify the verification, it may be neglected the contribution ofthe flanges.

Checking formula is (EN 1993-1-1):

0

,3

v y

sd pl Rd

M

fV V

=

(21)

Vsd shear force in the cross section, results form the statical analysisVpl,Rd rezistenta la taiere

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Restrained Beams in Bending

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