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vEd - longitudinal shear stress at the junction between one side

of a flange and the web

if the flange is in the compression zone:

if the flange is in the tension zone:- the part of tension reinforcement Asplaced in flange,

- the moment variation

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The flange in compression

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The flange in tension

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the transverse reinforcement per unit length is:

to prevent crushing of the compression struts in the flange:

IfvEd < kfctd , no extra reinforcement above that for flexure is required. (k=0.40)

fyd

fEd

f

sf

cotf

h

s

a

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1. Calculate the longitudinal design shear stresses at the web-flange interface:

xh

F

f

cdEd

=

The design stages - shear between web and flanges of T-sections

x -for uniformly distributed load, the maximum value allowed is half

distance between the section with zero moment and that where

maximum moment occurs,

-for points loads, the maximum value not exceed the distance

between the loads

eff

weffcd

b

bb

z

MF

=

2

1

M -the moment variation over the distance x

If : ctdEd f. 400 then no shear reinforcement is required and proceeddirectly to stage 4.

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2. Checking the shear stresses in inclined struts:

( )ffck

Ed tancot.

f

+

511

- the angle for the inclination of the concrete strut is recommended in EC2 to be:

-to prevent crushing of concrete in the compression struts, the longitudinal shear

stress is limited to:

where:

- the lower value of struts angle is first tried and if the shear stresses are too high,

the angle is calculated with:

o

ckck

Edf

ff

.

sin. 45

2501200

500 1

=

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3. Calculate the transverse shear reinforcement required:

fyd

fEd

f

sf

cotf

h

s

a

- the required transverse reinforcement per unit length may be calculated with:

4. The requirements of transverse steel:

-EC2 requires that the area of transverse steel should be greater of:

a) that given by the above equation, or

b) half that given by above equation plus the area of steel required

by transverse bending of the flange.

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Torsion of reinforced concrete beam

torsional moments produce shear stresses => principal tensile stresses are inclined atapproximately 45o,

diagonal cracking occurs when these tensile stresses exceed the tensile strength of theconcrete.

the cracks basically form a spiral, running up one face of the beam, across the top, downthe other side, and back across the bottom to connect with another crack on the first face.

in torsion, the crack on the far face will be at the opposite angle to that on the front facewhile in shear the cracks on both faces will be at the same angle.

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II. The torsion design model

the real element is replaced with a virtual spatial truss with reinforcements as tensionmembers and concrete as compressive struts between links.

failure will occur by reinforcement yielding, coupled with crushing of the concretealong line A-A as the cracks on the other faces open up.

the beam section is converted to a hollow box section:

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Torsional reinforcements

torsional reinforcement consists in closed links and longitudinal bars that will carry theforces from increasing torsional moment after cracking

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The equivalent section

TEd -the applied design torsion.A

k- the area enclosed by the centre-lines of the connecting walls, includinginner hollow areas.

t,i - the torsional shear stress in wall i.VEd,i-the shear force in a wall idue to torsiontef,i - the effective wall thickness.zi - the side length of wall idefined by the distance between the intersection

points with the adjacent walls

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Equivalent section in torsion design

A - the total area of the cross-section within theouter circumference, including inner hollowareas.

u - the outer circumference of the cross-section

The characteristics of the equivalent section

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Torsion of complex sections composed from rectangles

TEd,i the torsion moments of the section component elements

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Reinforcement design to torsion

- the longitudinal reinforcement for torsion:

- the transverse reinforcement for torsion:

- the longitudinal reinforcement from bending moment design is increased with thevalues obtained from the torsion design.- the same stirrups diameter for shear and torsion.- the transverse reinforcement for torsion: only closed stirrups.

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Reinforcements for torsion

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The torsion resistance of an element

for approximately rectangular solid sections only minimum reinforcement isrequired if the following condition is satisfied:

TRd,c is the torsional cracking moment, which may bedetermined by setting t,i = fctd

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Example 1: Design of torsional reinforcementsIt is required to be design the torsional reinforcement for a beam to an ultimate torsional

moment TEd=24kNm. The beam was previously designed to bending moment (2f 18 and2f 16) and shear force (stirrups f 8/200) . (C30/37, S500).

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