Design of slender reinforced concrete frames

Working Paper

Design of slender reinforced concrete frames

Author(s):

Aas-Jakobsen, Knut Publication Date:

1973

Permanent Link:

https://doi.org/10.3929/ethz-a-000674501

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ETH Library

Design of Slender Rein¬

forced Concrete Frames

^Y

KnutAas-Jakobsen

November 1973 Bericht Nr. 48

BirkhauserVerlag ^{Basel und} Stuttgart ^Institutfür Baustatik ETH Zürich

Design ^{of Slender Reinforced} Concrete Frames

von

Dr.sc.techn. Knut Aas-Jakobsen

Institut für Baustatik

Eidgenössische

Technische Hochschule Zürich

Zürich November 1973

FOREWORD

The two main

problems

^{in the}

analysis

^{of slender reinforced concrete frames}

are the influence of the

displacements

^{on the}

equilibrium

(second ^order

theory)

^{and the} non-linear material behavior of the concrete and steel.

The author has made ^a

general study

^{of the}

analysis

^and

design problem.

^He

developed

^a

practical design

^{method and tested the} aecuracy

against

^a pre-

viously developed general Computer

program for the

analysis

^{of such frames} (Bericht Nr. 45, Institut für Baustatik, ETH-Zürich, ^Januar 1973).

In its

present

^{form the}

study

constitutes a basis for the

development

^of

practically applicable Computer

programs. Furthermore it can serve in the formulation of

simplified design

^{rules for routine}

problems

^{in the}

design

of reinforced concrete columns.

The author has

prepared

^this

study

ⁱⁿ

partial

^{fulfilment of his} doctoral work at the Institute of Structural

Engineering.

Zürich, August

^{1973 Prof. Dr. B. Thürlimann}

CONTENTS

Page

1. Introduction 7

1.1

Objectives

⁷

1.2

Glossary

⁷

1.3 Outline 8

1.4

Background

⁹

2. Basis of the Methods 16

2.1 Determination of a

point

^{on the real}

load-displacement

^{curve 16}

2.2 Elastic frame

analysis

¹⁶

2.3 Cross sectional

analysis

¹⁷

3. Frame

Analysis

¹⁹

3.1 Procedure 19

3.2 Assumed material

properties

²³

4.

Design

^{Method 27}

4.1

Introductory

^{remarks 27}

4.2 General

design

^{method 27}

4.3

Simplified design

^{method 32}

4.4 Procedure 34

4.5

Design

^{criterion 36}

4.6

Special

^{cases 40}

5.

Aecuracy

^{of the}

simplified design

^{method 43}

5.1

Types

^{of errors 43}

5.2 Errors due to lack of "middle" strain

equality

⁴³

5.3 Errors due to lack of "curvature"

equality

⁴⁵

5.4 Errors in the estimated creep effect 54

6.

Design Examples

⁵⁸

6.1 Tail

bridge pier

⁶⁰

6.2 Arch 67

6.3 Tail

building

⁶⁷

Appendix

^A2 82

Appendix

^A3 88

Summary

91

Zusammenfassung

⁹²

Resume 92

Notation 95

References 98

INTRODUCTION

1.1

Objectives

The

present investigation

^{is concerned with the}

analysis

^and

design

^of

slender reinforced concrete frames

subjected

^{to short-time} and sustained

loading.

^The

study

^{is limited to}

plane

^frames.

The

objectives

^{of this}

investigation

^are

- the

development

^{of a method which enables the} determination of the maximum load

capacity

^{of a slender frame with}

given

^{cross sections}

and reinforcements. This method will be termed a frame

analysis.

- the

development

^{of a method}

whereby

^{cross sections and} reinforcements

can be determined for a frame

subjected

^to

given

^loads. This method will be termed a

design

^method.

Although

^{the basic}

assumptions

^are

essentially

^{the same for both the ana¬}

lysis

^{and the}

design,

^some differences do exist in the number of

simpli¬

fying assumptions

^{made. The} former will be formulated as

accurately

^as

possible

^{while in the} latter exactness is sacrificed for the sake of sim¬

plieity.

1.2

Glossary

The

key

^{terms and}

expressions

^{used in this}

investigation

^{are defined as}

follows:

§!§§tic §D§^y§i§

^{^s used to determine elastic}

forces,

^{M and N, and the}

corresponding

^strain distribution for all sections of the frame. The

given properties

^are

geometry

^{of the} frame, ^{loads and}

rigidities

^{for all members.}

The

equilibrium equations

^{in the elastic}

analysis

^{are formulated for the}

deformed frame. The

resulting

^{elastic forces, therefore, include second}

order effects.

Cross_sectional analysis

^{is used to determine inelastic}

forces,

^{M. and N.,} which are the resultants of the stresses over a cross section. Given pro¬

perties

^{are cross} section,

reinforcement,

material

properties

^{and strain} distribution. In the cross sectional

analysis

^{the real} behavior of the materials ^are considered.

M ⁼ M.

l N ⁼ N.

(1.1)

§train_eguality

^{is satisfied for a section when the strain e calculated in}

the elastic

analysis

^is

^equal

^{to the strain e. used in the cross sectional}

analysis.

^{The strain is}

given by

^(see

Fig.

^3, page 18)

1

e ⁼ e ^- (—) ^• z for the elastic analysis

m r ^J

1

e.= e .-(—)• ^z for the cross sectional ^analysis

l mi r. ^J

l

where

e , e ^. m mi

are the axial strain at the level of the

system

^line

Vr, Vr. ^are the curvatures

is the distance from the

system

^line

Strain

equality

^{is defined}

by

e ⁼ e .

m mi

1=1

r r.

(1.2)

1.3 Outline

The two main difficulties in the

analysis

^and

design

^{of slender reinforced} concrete frames are due to

- the influence of the

displacements

^{on the}

equilibrium

^{of the frame}

producing

^a

"geometrical non-linearity".

- the non-linear stress-strain-time relations for the materials

causing

a "material

non-linearity".

The two non-linearities are treated

separately

^{of each other. The}

geometri¬

cal

non-linearity

^{is considered in an elastic}

analysis,

^{and the material non-}

linearity

^{is taken into account in a cross sectional}

analysis.

^{The two non-}

- 9

linearities are

coupled through

^the

requirements

^{of force}

equality

(Eq.

^{1.1) and strain}

equality (Eq.

^{1.2). These}

aspects

^{are considered in}

Chapter

^2.

The elastic

analysis,

^the pross sectional

analysis

^{and the force- and} strain

equalities

^{outiined in}

Chapter

2, are the basis for both frame ana¬

lysis

^and

design. Chapter

^{3 shows how these can be}

applied

^{in a frame ana¬}

lysis.

^The

procedure

^{used in the}

design

^{method is discussed in}

Chapter

^4.

A

general design

^{method which has the same} aecuracy ^as the frame

analysis,

is outiined. The

procedure

^{is then}

simplified

^to

give

^{a more}

practical design

^method.

The aecuracy of the

simplified design

^{method is}

investigated

ⁱⁿ

Chapter

^5.

Three realistic

design problems

ⁱⁿ

Chapter

⁶ demonstrate the

efficiency

^of

the

proposed design

^method.

1.4

Background

The foundation for the

understanding

^{of the}

load-carrying capacity

^{of com¬}

pression

^{members and} the associated

phenomenon

^of

instability

^{is the}

theory

of Euler (1759)

[1]*

for the bifurcation load P of ^a concentrically loaded

e ^J

elastic column:

P

tt2

^• EI

*2

Euler was the first to realize that a column could reach failure not neces-

sarily by crushing,

^{but also}

by instability arising

^from deformations.

At the

beginning

^{of our}

Century,

^{the Euler}

theory

^{was extended to} columns with non-elastic materials. Two basic

hypothesis

^were formulated, ^known as the

tangent

^modulus

(Engesser)

^{and the reduced modulus}

(Engesser-

^{von Kar¬}

man) theories. The former

gives

^{the load where} bifurcation starts, ^and the latter a theoretical maximum value which can never be reached.

A most

signifieant

contribution to the

study

^of

eccentrically

^{loaded columns}

with non-elastic material

properties

^{was the work of von Karman}

^[2]

^{in 1910.}

He

presented

^a

general

^non-linear

analysis

^{based on the actual} stress-strain

relationship

^{of the material.}

Assuming

^that

plane

^{sections remain}

plane,

*) Numbers in braekets refer to references listed at the end.

ture

relationship.

^{From numerical}

integration

^{of the curvature}

along

^the

column

length,

^{the deflection}

shape

^{of the column was} determined.

Some

representative investigations

^{about slender reinforced concrete col¬}

umns and frames are summarized in

Fig.

^{1. The} survey is not

complete,

^but

it is believed to

represent

^{some of the more}

important developments.

^The

studies have been classified

according

^{to the}

type

^of structure, the

shape

of the deflection curve, the load

history

^{and the}

applied

creep law.

Most

investigations

^have concentrated ^on the

hinged

^column

subjected

^{to an}

axial load with constant end eccentricities. Broms

[4]

and

Pfrang

^{and Siess}

[5]

introduced elastic

Springs

^{with infinite moment}

capacity

^{at the ends}

of the column in order to simulate the effect of

connecting

^{beams. Breen}

^[6]

and Manuel and

MacGregor [10]

^studied

simple

^{braced frames where the real} behavior of the beams ^was taken into account. Aas-Jakobsen and Grenacher

[14]

developed

^{a method to}

study

any

type

^of

plane

^frames.

The

geometrical non-linearity

^{has either been taken into account}

by assuming

the form of the deflection curve (for instance ^a cosine wave) ^or

by

^deter¬

mining

^{the real curve. The former}

procedure requires considerably

^{less nume¬}

rical work.

Three

types

^{of load histories were considered in the}

past.

^{These were} (see

Fig.

^{2) :}

1. Short-time

loading

^{- the load increases}

instantaneously

^from

zero until maximum load

capacity

^is

reached.

2.

Long-time loading

^{- the load is increased}

instantaneously

to a desired load level and then sus-

tained until

instability

^occures.

3.

Long-short-time loading

^{- the load is}

kept

^{constant over a defined} time

period

^{and then raised} instantane¬

ously

^{to maximum load}

capacity.

The first studies of the influence of creep treated the

long-time loading

case

[7], [8], [9].

In later

investigations

^the

long-short-time loading

has

caught increasing

^{interest [10], [13],}

^[15].

^{This latter}

representa¬

tion is believed to be a more realistic Simulation of the actual

loading

conditions than the

long-time loading.

^The

long-short-time loading implies

that the effect of creep is taken into account at the

working

^{load level,}

structure

deflection

curveload

history

creep lawcomments

Baumann,1934 [3] hinged

and

clamped

columnreal

short-

time Broms, 1956 [4] restrained

column

cosine

wave

long- time

reduced modulus Pfrang and Siess,

1961

[5] restrained

columnreal

short- time Breen, 1962 [e] braced frame

real

short- time Warner

and

Thürlimann, 1963 [7J hinged

column

cosine

wave

long- time

rate

of

creep

I-section with

zeroweb

thickness Mauch

and

Holley, 1963 [8] hinged

column

cosine

wave

long- time

rate

of

creep

Green, 1966 [9] hinged column

real

long- time

reduced modulus Manuel

andMac

Gregor, 1966 [lü]

braced frame

real

long-short- time

rate

of

creep

Aroni,1967 [ll] hinged column

real

short- time prestressed column Oelhafen, 1970 [12] hinged

columnreal

long- time

~rate

of

creep

studied influence of

the

variance of material and sectional properties

onthecolumn

behavior Hellesland, 1970 [13] hinged column

cosine

wave

long-short- time

*•

general rheological. model for

concrete

Aas-Jakobsen and Grenacher, 1972 [14j general frame

real

long-short- time

rate

of

creep

Fig. 1: Review of previous investigations

long- ^time long-short-time

J

/

w

p-

\

\

T,

W

Fig. ^2: Types of load histories

and that the failure load is an instantaneous overload (such ^as wind and

earthquake).

The assumed stress-strain relations for concrete under short-time load dif- fer

only slightly

^{in the}

surveyed

papers. On the other hand the creep of concrete has been considered in several different ways.

Broms

[4]

and Green

[9]

^used the reduced modulus method. In this method the increase of strain under sustained load is taken into account

by increasing

all strains in the short-time stress-strain

diagram by

^{the same factor (see}

Fig.

^14, page 32). This

implies

^{that the Variation of concrete stress under}

the

long-time period

^is

neglected.

The rate of creep method (which is

equivalent

^{to the}

Dischinger

^{law) is} per-

haps

^{the method which has been used most}

frequently

^{to estimate the effect}

of variable stresses. The most

complete description

^{of the}

rheological

^be¬

havior of concrete seems to be the one

by

^Hellesland

^[13].

^{He also consi-}

ders the increase in

strength

^{due to continued}

hydration

^{of the}

^cement,

^and

the decrease in

strength resulting

^{at stresses above a certain} "critical"

stress.

All

investigations

^reviewed

report

^{a close}

agreement

^between

analysis

^and

tests. Oelhafen [12]

compared

^the theoretical confidence limits of the load-

displacement

^{curves with test results [15], and concluded that the Statisti¬}

cal

analysis

^{confirmed the}

reliability

^{of the}

computation

^method.

13

Most studies in the field of slender reinforced concrete columns and frames have been concerned with the behavior and maximum load

capacity

^{of such}

structures. The

design problem,

^{which is the}

prime

^{motive of this} research, has attracted

relatively

^{little interest.}

Thus,

^a gap ^seems to exist between the

present knowledge

^{about the} behavior of slender concrete structures on one hand, ^and the

practical

^{Solution to the}

design problem

^{on the other.}

With few

exceptions

^{the first}

design

^{methods for slender reinforced concrete}

columns were based on either the "u-method"

[3], [16]

or the American re¬

duetion factor method based on the

investigation by

^Broms

^[4].

Both methods reduee the

design

^{of a slender column to}

design

^{of a section. In the "id-}

method" the axial force and the first order moment are

multiplied by

^{a w-}

factor

greater

^than 1,0. ^In the American method the allowable axial force for the section is

multiplied by

^{a R-factor less than 1,0.}

A severe

shortcoming

^{of these earlier}

design

^{methods is that the same ec-}

centricity

^{is maintained for the section and the column. This is contradic-}

tory

^{to the actual behavior of slender columns where the reduetion in load}

carrying capacity

^{is caused}

by

^{the increased}

eccentricity

^{due to the de-} flections.

The Swiss SIA code

[17]

(1955) introduced an increased

eccentricity.

^Re¬

written in the form for ultimate load

design

ei 1 1 ^-

*

N

er

e is the total

eccentricity

^{for the section}

ei is the first order

eccentricity

^{from external loads}

N is the ultimate axial force

N is the maximum force for the

centrically

^{load column.}

er

About 1960 the CEB-recommendations

[18]

introduced ultimate load

design

and an additional moment

approach

^{for slender columns.}

In the last _years the German DIN and the American ACI codes have been re-

vised, both

introducing

^{ultimate load}

design. According

^{to the DIN code}

slender columns are

designed

^{for an additional moment, and the ACI code in¬}

creases the first order moment

using

^{a similar}

expression

^{as the SIA code}

referred to above.

The

design procedure

^{for the codes}

just

mentioned may be summarized ^as follows:

1. Determine the moments and axial forces from a first order elastic

analysis.

2. Determine the elastic

buckling length

^{of each column.}

3. Account for the slenderness effect

by increasing

^{the first}

order moments.

4.

Design

^{the cross section.}

The sources of errors in this

procedure

^are:

- The first and second

step imply

^an

assumption

^{of the El-values. These}

are often calculated on the basis of the net concrete cross section

although

^{it is well known that this is a rather}

rough approximation.

^The

rigidity

^also

depends

^{on the} reinforcement, the moments and axial

loads,

and a

satisfactory

^{El-value must take this into account.}

- The third

step

^{is based on the}

assumption

^{that the increase in moment}

due to the deflection of the structure ^can be related to the elastic

buckling length

^{of each individual column. This is an}

assumption

^often

used in the

theory

^{of elastic}

stability.

- In the third and fourth

step

^the

stability problem

^{is reduced to the} pro¬

blem of a short column associated with material failure under excessive strains in steel and concrete. Slender structures

usually

^{fail under mueh}

smaller strains, i.e. before the material failure state is reached. Gen-

erally,

^a

disagreement

^{between actual failure state and}

design

^state

exists in the

approximate

^{methods reviewed.}

In addition to the

simplified design

^{methods outiined above, most codes en-}

courage the ^use of more

comprehensive

^{methods which consider the}

secondary

moments and the actual material response.

A

design

^method

attempting

^{a better}

agreement

^{with the real} behavior is the so called "model column" method

presented

^{in the}

CEB-buckling

^manual

^[19].

The method is based ^{on an} assumed form of the deflection curve and the ac¬

tual stress-strain relations, ^and is restricted to cantilever columns.

Beck and Bubenheim

[20] developed

^a

design

^{method for}

plane

^{frames under}

short-time loads. Their method is based on an elastic second order ana-

15

lysis

^{and a cross sectional}

analysis

^{which determines the} "secant

rigidi-

ties" (the secant

rigidity

^{is the moment determined in a cross} sectional

analysis

^{divided with the}

corresponding

curvature). Cross sections and re¬

inforcements must be known in advance. The maximum load

capacity

^{is deter¬}

mined in a

"step by step" procedure

^{where the} secant

rigidities

^{are deter¬}

mined

iteratively

^{at each}

step. By

^{means of a}

special algorithm

^{the rein¬}

forcement is

changed

^{until the} calculated maximum load and the

given design

load coincide.

2. BASIS OF THE METHODS

2.1 Determination of a

point

^on the real

load-displacement

curve

A

point

^{on the}

load-displacement

^{curve is defined}

by equilibrium

^{and com¬}

patibility.

^{In the case of slender} concrete frames both the

geometrical

and material

non-linearity

^{must be} considered. These two

non-linearities,

which combine ^to affect the overall structural _response, can

conveniently

be considered in two

steps:

1. The

geometrical non-linearity

^{is accounted for in an elastic}

second order frame

analysis.

^This

analysis

^{is based on}

given

^loads

and assumed

rigidities

^and

yields

^elastic forces, M and N, and

corresponding strains,

^e_m and

Vr,

^{for all sections.}

Equilibrium

and

compatibility

^{are satisfied in the elastic}

analysis.

2. The material

non-linearity

^{is taken into account in the cross sec¬}

tional

analysis through

^{the use of the real} stress-strain-time relations. For a given° strain

distribution,

^{e .} and Vr., the in-

mi l

elastic forces, ^M. and N., can be determined.

A

point

^{on the real}

load-displacement

^{curve is now defined}

by

^force

equali¬

tyJ (M ⁼ M.; ^{N =} N.) and,

similarly,

^strain

equality

^{(Vr =} Vr.; ^{e = £} .)

ii im mi

for all sections. It should be

recognized

^{that the main}

problem

^involved

in the frame

analysis

^and

design

^{method is to}

satisfy

^{the force and strain}

equalities.

2.2 Elastic frame

analysis

The elastic ^analysis_J yields_J elastic forces, M and N, ^and

strains,

em and

m

Vr, for

given geometry

^{of the frame, loads and}

rigidities.

^The

analysis

^is

performed by

^{means of the finite element method. A detailed}

description

^of this method is

given

ⁱⁿ

Appendix

^{A1. A short review is}

presented

^{in the}

following.

The frame is divided into beams and columns which are termed members. The members are subdivided into elements. The elements are connected at their ends, the so-called nodal

points.

^The

displacements

^{of these nodal}

points

are the unknowns. Based ^on the

load-displacement

^{relation for each element,}

the

load-displacement

^{relation for the whole structure is}

developed,

^and may

symbolically

^{be written}

17 ^-

([KJ

⁺

[K2])-{w>

⁼

^{P}

^(2.1)

[Ki]

^{is the first order} stiffness matrix

[K2]

^{is the}

geometrical

^{stiffness matrix}

{w}

is the unknown

displacement

^vector

{P}

is the load vector

An elastic

analysis

may be of first or second order. In a first order ana¬

lysis

^the

equilibrium equations

^{are formulated for the undeformed}

^structure,

and the

geometrical

^{stiffness matrix}

[K2]

ⁱⁿ

Eq.

^{2.1 vanishes.}

In a second order

analysis

^the

equilibrium equations

^{are formulated for the}

deformed structure. The additional stiffness matrix

[K2]

^is

dependant

^{on the}

axial forces N in the elements. The axial forces

depend

^{in turn of the ex¬}

ternal loads. A non-linear relation between

displacements

^{and loads results.}

Due to the fact that the

geometrical

^{stiffness matrix}

[K2] depends

^{on the}

axial forces in the elements, ^an iterative

procedure

^{results. Based on}

assumed values of N, ^{new axial forces are} determined. The

procedure

^{is re¬}

peated

^{until assumed and calculated axial forces coincide.}

2.3 Cross sectional

analysis

The resultant inelastic forces, ^M. and N., ^are determined from

given

^strain distribution, cross section, reinforcement and material

properties

(see

Fig.

³⁾

M. ⁼ E ^{o • z •} AA 1

N. ⁼ E o ^• AA (2.2)

1

The strain e. is assumed to be

positive

^{when in tension and to be}

linearly

distributed over the section. Then

e. ⁼ e .-(—)• ^z (2.3)

1 mi r.

e . is the axial strain at the level of the

system

^line

mi

Vr. is the curvature

is the distance from the

system

^line

cross

section

/ ^AA \

²

strain internal forces

Ni

Fig. ^{3: Cross section,} strain distribution and forces

The stress a

depends

^{on the strain e.}

a ⁼ f(e.) (2.4)

The

adopted

stress-strain relations are different in the frame

analysis

^and

design ^method,

^{and are outiined in their}

respective chapters.

19

FRAME ANALYSIS

3.1 Procedure

The purpose of the outiined frame

analysis

^{is to determine} the maximum load for a

plane

^{frame with}

given

^{cross sections and} reinforcements. To be able to account for an

abritrary

^load

history,

^a

step by step procedure

^{is used.}

The increase of stress is related to the increase of strain and time, and loads and time are increased in small increments. A

Computer

program ^was

developed

ⁱⁿ

conjunetion

^{with this}

study,

^{and method and} program ^are des¬

cribed in detail in

[14].

However, as it forms an essential base for this

study,

^{a brief}

description

^{will be}

given

^below.

The maximum load

capacity

^{of a frame with}

given

^{cross sections and rein¬}

forcements is considered in two

steps.

^{First, a}

point

^{on the}

load-displace¬

ment curve is determined, ^and second, it is checked whether this

point

^re¬

presents

^{the maximum load}

capacity.

Fig.

^{4 shows the flow-chart used for}

determining

^a

point

^{on the load-dis¬}

placement

^{curve. The}

procedure

^{starts with assumed}

rigidities

^{for all ele¬}

ments. In a second order elastic

analysis,

^{the elastic forces M and N, and}

the strain distribution

expressed

^byJ ^{middle strain e and curvature (Vr)} m

are determined for all elements. The inelastic forces M. and N. are deter-

l l

mined in a cross sectional

analysis

^{based on the strain} distributions found in the elastic

analysis.

^{Thus strain}

equality

^is

automatically

^satisfied.

Force

equality

^{then becomes the iteration} criterium of the

procedure.*

If

equality

^{is not satisfied,}

improved

^secant

rigidities

^{are determined}

from the inelastic forces, ^and the

procedure

^is

repeated.

The maximum load

capacity

^of slender reinforced concrete frames is asso¬

ciated with

instability

^{as indicated in}

Fig.

^{5. In load controlled} proce¬

dure where the external load is increased

step by step,

^{a poor} convergence

can be

expected

^{near to the maximum load. In a}

displacement

^controlled pro¬

cedure, where a characteristic

displacement

^{is increased}

step by step,

and the

corresponding

^{load is} calculated, ^no

problem

^of convergence is encountered.

*) A

slight

^different

procedure

^{was used in the} program

[14]

in order to ensure convergence. In the cross sectional

analysis

^{the middle strain e .}

mi was determined

iteratively

^to

satisfy

^{the axial force}

equilibrium.

^The curvature was

kept

^{constant. Moment and middle strain}

equality

^became

the iteration criteria in this case.

geometry ^of frame, loads, materials,

cross

sections, reinforcements

EI.EA

[k]

⁼

[k,]+[k2]

M N

(1/r)

> f ({«})

M, N,

f Um,1/r)

no

El

⁼

EA

⁼

M; ^/ (1/r) N-, ^/ em

given

assumed

2

nd

order elastic analysis

cross

sectional analysis

ye^.

^one

^point

^on

^{the load-}

displacement

^curve

determined

secant

rigidities

Fig. ^4: Flow-Chart for frame analysis

21

maximum load capacity, instability ^failure

material failu're

*-

displacement

Fig. ^5: Load-displacement ^{curve for a slender} reinforced concrete frame

In the

present study,

^{external loads on a frame are diveded into constant}

and

proportional

^loads

(Fig.

^{6). The latters are}

proportional

^{to a load}

factor X, the^maximum value of which is searched. A

displacement

controlled

procedure

^{will be used. The}

displacement

^{w is} increased in

steps

^{until the}

maximum value of X has been found.

For each value of the

speeified displacement

w, the

corresponding

^{load fac¬}

tor X is found

iteratively

^{as outiined in the}

following.

^First

rigidities

are assumed for all elements. Then, for the same

rigidities,

^{the load fac¬}

tor X is increased in

steps

^{until calculated and}

speeified displacement

eoineide. New

rigidities

^{can now be determined in the cross sectional ana¬}

lysis.

^The

procedure

^is

repeated

^{until assumed and calculated}

rigidities

agree. The outiined

procedure

^was

slightly

^{modified in the above mentioned}

program in order to

speed

up the convergence (see

Fig.

^6).

Generally,

^a

non-linear relation exists between the load factor and the

displacements

even if the

rigidities

^are

kept

^{constant. The reason is that the}

geometri¬

cal stiffness matrix

[K2] (Eq.

^2.1)

depends

^{on the axial forces N which, in} turn,

depend

^{on the load factor X. However, if the axial forces intro¬}

duced into the

geometrical

stiffness matrix are assumed

independent

^{of X,}

a linear relation between X and the

displacements

^{results. In this case}

it is suffieient to consider two

loading

^{cases, for instance X}

equal

^{to 1}

and X

equal

^{to 2.} The load factor

corresponding

^{to the}

speeified displace-

X

k

2-

' ^{\ elastic}

1. iteration

last iteration

^/ i.

/ >¦

^elastic

analysis, EA,El,N

^are

^assumed

/ /v

^constant

^{in each iteration}

"/' / X=2

real load-displacement

^curve

XP

-+-¦

controlled displacement

^w

Fig. ^6: Displacement controlled determination of a point ^{on the}

real load-displacement ^curve

ment w is found

by

^linear

interpolation

^{as indicated in}

Fig.

^{6. For this}

new load factor new

rigidities

^{and axial forces to be introduced in}

[K2]

are determined. The

procedure

^is

repeated

^{until the calculated}

rigidities

and axial forces agree with the assumed ones.

Under sustained loads the

long-time

^{load factor X, is}

given

^(see

Fig.

^7).

The

displacement

^{w is increased in}

steps

^{as before}

(Fig.

⁷

corresponds

^to

one value of w. The ^starting6 time is t ). For each

step

^{of w, the time is}

o

increased in

steps.

^{For each}

step

^{in time, the}

corresponding

^{load factor X}

is determined as before

considering

^the creep of concrete. When the cal¬

culated load factor X is

equal

^{to the}

given

^{load factor X ., the time cor¬}

responding

^{to the chosen}

displacement

^{has been found. If the calculated X}

at the ^starting° time t is less than X ,, creep

instability

^{failure has taken}

o d

place.

- 23

w

is constant

Xh-p

H

XdP

I-,

time

*0 U f2

Fig. ^7: Displacement controlled procedure under sustained loads

3.2 Assumed material

properties

The assumed short-time stress-strain relations* for steel and concrete un¬

der instantaneous

loading

^{to failure for a}

previously

^unloaded

specimen

^are

shown with solid lines in

Fig.

^{8. Tensile stresses o are taken}

^positive.

The maximum stress in the short-time stress-strain relation is the

strength

on the

material,

which will be denoted f. Material failure is related to the strain. A failure strain for concrete of -0.0035 is assumed. The stress- strain relations under instantaneous

unloading,

^and instantaneous

reloading

after ^a time of creep ^are indicated with dashed lines in

Fig.

^8.

Creep

^{under variable stress is calculated}

by dividing

^{the stress}

history

into time intervals, ^and

assuming

^{a constant stress within each interval}

as indicated in

Fig.

^{9. Two methods, outiined in ref.}

[24],

^are considered.

For both methods it is assumed that creep under

varying

^{stress can be ob¬}

tained from creep ^curves for constant stresses. Such curves, for two stress levels o and 0 , are indicated

by

^{solid lines in}

Fig.

^{9. The two methods}

ci c2

differ in the manner data from the creep ^curves for constant stress is

) The mathematical formulation of the assumed stress-strain-time rela¬

tions is

given

ⁱⁿ

Appendix

^A2.

arctan

E,

steel stress-strain relation

1 ^?

€i

0.010

concrete stress-strain relation

(Tr.

(neg)

rshort-time: ac

⁼

fc[2(ö|fe)

⁺

(äöoz)2] ^for 0>€t> ^-0.002

instantaneous loading

nstantaneous unloading

&C3^arctan ^Ec

*

creep under constant stress

0.002 -0.0035

€t(neg)

Fig. ^8: Stress-strain relations

stress history

^rate

^{of creep method}

<TC

k

°c2-]

ac\-

strain hardening ^method

t,

-1« ^-

**

-^t

Fig. ^9: Creep under variable stress

25 ^-

transformed to the

varying

^{stress case.}

The first method, the "rate of

creep"

method, is based on the

assumption

that the creep increase

during

^{the time interval At =}

t2

^-

ti (Fig.

⁹⁾

under the stress a is

equal

^{to the} creep increase between

ti

^and

t2

ⁱⁿ c2

the constant-stress _creep curve for the stress a .

Thus,

the creep increase c2

is

independent

^{of the}

previous

^stress

history.

The second method called the "strain

hardening" ^method,

^{is also based on}

the increase of strain in the time interval At taken from the constant- stress creep ^curve. However, instead of

starting

^{at the time}

tj,

^{the strain}

increase is calculated from a fictitious time t_

corresponding

^{to the time}

that would have been necessary to

develop

^{the total}

prior

creep ^e under

c c

the ^new stress level 0 ^. This is indicated

by

^{the horizontal} translation c2

in

Fig.

^{9. Hence, the increase of strain in the strain}

hardening

^{method de¬}

pends

^{on the}

previous

^stress

history.

A

comparision

^between recorded and

predicted

^strains

by

^{the two methods is}

shown in

Fig.

^{10 (taken from Ref.}

^[24]).

^Under

monotonically increasing

stress the rate of creep method

underestimates,the

strain

hardening

^method

overestimates the increase of creep strain. Thus, ^{the two methods seem to}

give

^{a lower and} upper bound for the increase of creep.

-2000

^-1600-1

|-12001

u

-800

"

-400 0

-500

-400

o

-300

I -200

o

B -100

strain hardening-v

observed

,VCUV"

^H

—j

rate

of _creep

0 8 40 80 120

age in days

Fig. ^10: Comparison of recorded strains (from Ref.[25]) ^{with pre}

dictions by ^rate of creep ^{and strain} hardening ^methods.

The figure ^{is taken from Ref.} [24]

- 27 ^-

4. DESIGN METHOD

4.1

Introductory

^remarks

The purpose of a

design

^{method is to determine cross sections and reinforce¬}

ments for

given design

^{loads. It is}

important

^{to note that the}

design

^method

outiined in the

following

^{can be}

applied

^to any limit state. For

instance,

if

displacements

^under

working

^loads govern the

design,

^the

design

^method

can be used as well. In this case the limited

displacements

^{introduce addi¬}

tional restraints.

Basis of the

design

^{methods are the elastic}

analysis,

^{the cross sectional}

analysis

^{and the force and strain}

equality requirements.

^{The essential of}

the methods concerns the cross sectional

analysis.

^{In an interative} proce¬

dure the cross section and reinforcement are

changed

^to

satisfy

^{the force}

and strain

equality requirements.

The

design

^{methods consider}

only

^the

design

^{loads and not the}

loading path

up to this load level. The stress in any

part

^{of a structure}

depends

^{on the}

prior

stress-strain-time

history

^{as outiined in Section 3.2, and it is}

generally

necessary to follow this

history step by step.

^The

only

^{case where}

this

step by step procedure

^is

superfluous,

^{is short-time}

loading

^{where the}

stress increases

monotonically

^{in all}

parts

^{of the structure. The}

general design

^{method will be limited to short-time}

loading,

^{and the short-time} stress-strain relations

given

ⁱⁿ

Fig.

^{8 will be assumed. For the}

simpli¬

fied

design method,

^an

approximate

way to account for sustained loads will be outiined.

4.2 General

design

^method

The

general procedure

^{is outiined in}

Fig.

^{11 in the form of a} flow-chart.

Given is a set of

design

^loads.

Rigidities

^{of the elements are assumed.}

Elastic forces and strain distributions are determined in a second order elastic

analysis.

^{The inelastic forces in the cross sectional}

analysis

^are calculated for the strain distributions found in the elastic

analysis.

^Thus

strain

equality

^{is satisfied. The cross section and} reinforcement for each section are

iteratively changed

^{until force}

equality

^{is satisfied. Then,}

according

^{to Section} 2.1, the Situation

corresponds

^{to a}

point

^{on the real}

load-displacement

^{curve of the frame.}

Two

major

difficulties are encountered in the outiined

design

^method.

geometry ^of frame, ^material proper¬

ties, design ^loads

El, ^EA

[k]= [><<]

⁺

[k2]

{w}.[k]-'.{p}

M N

*m (1/r)

> ^f ({•})

Mi 1

,

y ^{f (€m, 1/r,}

^cross

^section, Nj J reinforcement)

-cross

section

and

reinforcement

are

chosen in such

a

way that force equality is satisfied

T

finished

given

assumed

2

nd

order elastic analysis

cross

sectional analysis

Fig. ^{11: Flow-chart for} general design ^method

29 ^-

The first one concerne the choice of

rigidities.

^{This choice affects the}

moment

distribution,

the slenderness effect and

consequently

^{the strain}

distribution. This

aspect

^{will be considered further in connection with the}

design

^criterion

^ia

^{Section 4.5.}

The choice of cross section and reinforcement to

satisfy

^{the force}

equality requirement represents

^another

problem.

^{This is} illustrated with reference to a

simplified

^{section shown in}

Fig.

^12.

£

vA

o-2 ⁼

f (e2)

cr1=f(€<)

=»

N

M

Fig. ^{12 :} Determination ^{of cross} section

The elastic forces M and N, and the strain distribution

expressed by

^the

middle strain e and the curvature Vr have been determined in a second or- m

der elastic

analysis. Equality

^{between inelastic and elastic forces re¬}

quires

Ol ⁺ 02 ^• A N

These two

equations together

^{with the real} stress-strain relation for the material

give

^{the unknowns, i.e. the area A and the}

depth

^{h. In the case}

of an elastic material these can be obtained

explieitly

^as

A ⁼ N/(2*E«e )

m

h ⁼ /- 2-n

/(E-A'Vr))'

In the case of non-linear materials such as reinforced concrete, the un¬

knowns are determined

iteratively.