Prestressed Concrete - 2 Beam in Bending at Working Load

2. Prestressed Concrete:

Beam in bending at

working load

Introduction

Bending stresses in an uncracked section

Jacking force Po, initial force Pi and

effective force Pe.

Two important properties of a prestressed

section

Stress distribution across an uncracked section

University of Western AustraliaSchool of Civil and Resource Engineering 2004

INTRODUCTION

Basic assumptions:

Bernoulli/Navier postulate ( plane sections remain plane)

Linear performance for concrete

Tendon stress unaltered through range from transfer to working load

sc

ec

f c

Approx.

linear to

0.5 f c

NB Demonstrating that stresses are

acceptable at working load does not

ensure that section has adequate

safety. This is treated later in the

course.

Lecture 5

So how to estimate the bending stresses at working load? . . .

BENDING STRESSES IN AN

UNCRACKED SECTION

Start by assuming that the section is uncracked and linear, then demonstrate

that each is true.

Limiting stresses:

Concrete is linear up to about 0.5 fc where f c = current strength

Two conditions of concern: At transfer, when f c = f cp and

At maturity, when f c = f c

At transfer to ensure linearity : sb = - 0.25 (f cp)0.5

At maturity to ensure linearity : sa = - 0.25 (f c)0.5

sc

ec

f c

0.5 f cCONCRETE

Consider this example . . .

centroidal

axis

Tendon stressed

to force P

applied loading

Simple Beam - stresses in working load range

Examine this typical section, where

the applied bending moment = M

This section may be anywhere along the beam,

EXCEPT very close to the supports . . .

applied loading

axis

Examine this section, whereTendon stressed

to force P

centroidal

applied moment = M.

Simple Beam - actions at a typical section

Stresses due to prestress only:

e

qP R

Components of

force on concrete

Resultants at

section

P Pe

Pq

From these actions we can work out the stresses . . .

C = Rcosq = P

V = Rsinq= Pq

Prestress force

sa

sb

P

Pe

P/A Pe / Z

Pe / Z

= P/A - Pe/Z

= P/A + Pe/Z

(Pq does not cause

any bending at the

section)

So top fibre stress sa = P/A - Pe/Z and

bottom fibre stress sb = P/A + Pe/Z

NOTE:

Position of neutral axis is high in the section.

Some tension may occur high in the section.

Stresses depend on the eccentricity of prestress at section.

Now consider stresses due to applied loading . . .


Stresses due to prestress only:

applied loading

reaction

MM / Z

M / Z

These stresses are now added

to stresses due to prestress to

give the total stresses . . . .


= P/A + Pe/Z - M/Z

applied loading

reaction

P

M

V

M / Z

M / Z

sa

sb

Stresses due to: Prestress Applied loading Combined

So top fibre stress

s

a = P/A - Pe/Z + M/Z and

bottom fibre stress

s

b

Provided that section is not cracked in tension, AND . . .

. . . concrete is not too highly stressed in compression.


JACKING FORCE PoINITIAL FORCE PiEFFECTIVE FORCE Pe Jack applies force to

tendon at the live end.

The maximum value

of this force is the

Jacking Force Po

Immediately after jacking

and lock-off are completed,

tendon force at typical

location is less than Po due to

friction losses and anchorage

losses. This tendon force is

called Initial Force Pi

Tendon force progressively

diminishes with time due to

shrinkage and creep of concrete, and

relaxation of tendon. Ultimately the

force settles at a value called the

Effective Force Pe

Important for

anchorage design

Important at transfer

Important for maximum loads

Pe = 0.75 - 0.85 Pi approx.

Now we know what were doing!

Ok!

Lets try an example!

10.000

Study of stresses at mid-span of beam :

Applied working live load w = 25 kN/m (+ s/wt )

parabolically draped

tendon, stressed to

Pi of 1250 kN,

Pe 1000 kN, at mid-span.

e = 0 e = 0e = 225

BMD due to LL of 25 kN/m

312.5 kNm

400

750

e varies 0 to 225 mm

Note that the maximum eccentricity e maxis governed by the need to provide

enough space below the duct to place

concrete, and any secondary

reinforcement which may be needed.

Example 2.1

400

750

Stresses due to prestress only :

sa = P/A - Pe/Z

-3.33

sb = P/A + Pe/Z +11.67

NOTE:

Neutral

axis . . .

Pi = 1250 kN

. . . does not coincide

with centroidal axis

This is fictitious case, since self-weight of beam is engaged

when the beam is stressed and curves upwards . . .

So . . . .

Stresses due to p/s + s/wt of beam :

sa = P/A - Pe/Z + Mswt / Z400

750

-0.83

sb = P/A + Pe/Z - Mswt / Z

+9.17

NOTE:

both top and bottom fibre stresses are reduced by self-weight.

neutral axis is even higher.

Now what about stresses due to applied load? . . . .

Bending stresses

prestress

only

Stresses due to p/s, s/wt and applied load :

Mtot = Mswt + Mapplied = 93.75 + 312.5 = 406.25 kNm

sa = P/A - Pe/Z + Mtot /Z+8.16

sb = P/A + Pe/Z - Mtot /Z

-1.50

400

750

NOTE:

stress reversals at extreme fibres, so

movement of neutral axis.

Now check stresses to ensure our assumptions are o.k. . .

Bending stresses

prestress only

p/s and s/wt

Our stress estimates will not be valid if, at any time j,

Compressive stress anywhere exceeds 0.5 f cj , since above that stress the stress /strain curve for concrete is not linear, or

Tensile stress anywhere exceeds 0.25 (fcj ) 0.5, since above that tensile

stress the section may crack.

where fcj is compressive strength at that time j.

So we must check stresses at

Time of transfer, when f cp is probably less than f c, and

At full load, when concrete strength = f c.

If excessive stress is indicated anywhere,

then a different approach will be required

Lecture 4

Now consider stresses across the entire section . . .

Bending stresses Checking of stresses :

STRESS DISTRIBUTION ACROSS

AN UNCRACKED SECTION

Consider again Example 2.1. The applied load is all live load, (except for the

self-weight). Remember that live load may or may not be present. So in the

mature life of the beam, there are a range of extreme fibre stresses to be

considered.

The bending stress at any distance y below the

centroidal axis is:

sy = P/A + Pey/I - My/I

So the stress varies linearly with depth y

The range of stresses to be considered (for a simply

supported beam) is:

From MQ = 0 to MQ = MQ max

400

750

-0.17 +8.16

-1.50 +6.83

Life-time range of stresses :

Live load may, or may not, be present. So stresses will vary

throughout the beams life. The full range is displayed by a

diagram which pivots about the centroidal axis.

Note that there is a particular value of w, somewhere

between 0 and 25 kN/m for which the stress will be

uniformly compressive. More on this next lecture.

Example 2.1

total

p/s + s/wt

TWO IMPORTANT PROPERTIES

OF A PRESTRESSED SECTION

Decompression moment Mo or Mdec :

The total moment which is just enough to

eliminate tension in the bottom fibre:

sa

sb = 0

sb = 0 so P/A + Pe/Z - Mdec/Z = 0

then M dec = Z [P/A + Pe/Z]

Cracking moment Mcr :

The total moment which is just enough to

cause tensile cracking in the bottom fibre

sa

sb = - f cf

sb = - f cf so P/A + Pe/Z - Mcr/Z = - f cf

then Mcr = Z [P/A + Pe/Z + f cf ]

Rectangular section T section

I sectionBox section

Some Typical Prestressed Beam Sections :

y

Top fibre stress : sa = P/A - Pe/Ztop + M/Ztop , and

Bottom fibre stress : sb = P/A + Pe/Zbott - M/Zbott

Z is section modulus.

For sections which

are not symmetrical

about the horizontal

axis, Z is different

for top and bottom.

Z top = I / y top, and

Z bott = I / y bott.

Also, stress at any level y below centroidal axis :

sy = P/A + Pey/I - My/I

Enough for

today !

Bending stresses

y

y

Bending stresses must be checked for the transfer condition, and the maximum load condition.

Stresses must be checked to ensure assumptions of linearity and non-cracking are valid.

Simple beam theory is appropriate, noting that the action caused by prestress may be modelled in terms

of an axial load applied at the centroidal axis, and a

bending action.

Mdec and Mcr are important properties of a prestessed section.

This theory may be applied to any prismatic member.

SUMMARY

Documents

Prestressed Concrete - 2 Beam in Bending at Working Load