Non-Linear Analysis of Concrete Members

HERON contains contributions based mainly on research work performed in I.B.B.C. and STEVIN and related to strength of materials and structures and materials science.

Jointly edited by: STEVIN-LABORATORY of the Department of Civil Engineering of the Delft University of Technology, Delft, The Netherlands and

I.B.B.C. INSTITUTE TNO for Building Materials and Building Structures, Rijswijk (ZH), The Netherlands.

EDITORIAL STAFF:

F. K. Ligtenberg, editor in chief M. Dragosavic H. W. Reinhardt J. Strating A. C. W. M. Vrouwenvelder J. Witteveen Secretariat: L. van Zetten P.O. Box 49

Delft, The Netherlands

HER_N

Contents

vol. 22 1977 no.2

NON-LINEAR ANALYSIS OF CONCRETE MEMBERS

Ir. A. van den Beukel

Summary . . . . Acknowledgement Notation

1 Introduction 2 Critical load .

3 The uncracked cross-section 4 General method

Appendix A: Values of the parameters

'1 and f{J. • • . •

Appendix B: Determination of et and Mt Appendix C: Determination of a realistic Ee-value. . . . 3 3 4 5 5 7 12 16 18 20

NON-LINEAR ANALYSIS OF CONCRETE MEMBERS

Summary

This report deals with the calculation of the ultimate load of an axially compressed concrete member. It does not matter whether the member is straight or circular curved. Attention is mainly devoted to geometrical non-linearity (2nd order effect) and to physical non-linearity. The latter phenomenon is caused by the dependence of the bending stiffness on the state of load. The methods described offer a rather easy way of calculating the ultimate load. They are illustrated by two examples.

Acknowledgement

The analysis of concrete shell stability has become urgent following the recent growth in offshore activity. The amount of experimental data directly related to offshore construction was limited. Moreover there was a need for a theoretical analysis for the implosion pressure of concrete shells. The two methods of analysis described in this publication aim at satisfying that need.

Thanks are due to Ir. A. K. de Groot, Ir. W. J. Copier and Ir. F. B. J. Gijsbers, fellow-members of the Institute, for their constructive comments.

Notation

(additional symbols of the appendices are not given here) b e m n p A,A' E Ee EeD G I M Mo Mt N width of member

eccentricity of the normal load or deflection initial deflection

second order deflection

compressive strength of concrete yield stress of reinforcement steel length of member

number of half waves of the deflected shape relative normal force

Nlbt!:

uniform pressure

total thickness of member

cross-sectional area of reinforcement modulus of elasticity

modulus of elasticity of concrete

modulus of elasticity of concrete at the origin geometrical factor

moment of inertia bending moment

bending moment (lst order) bending moment (2nd order) normal force

actual normal force critical normal force maximum normal force ultimate normal force radius

bending stiffness safety margin steel strain

concrete compressive strain ultimate compressive strain half the aperture of an arch parameter, depending on () curvature

curvature (2nd order) Poisson's ratio steel stress

concrete compressive stress

concrete compressive stress in extreme fibre parameter, depending on

tlR, tlR

and

m

Non-linear analysis of concrete members

1 Introduction

In calculating bending moments and deflections of an axially compressed concrete member we can, in general, distinguish between two cases. In the first case the bending moments are supposed not to be influenced by the deflections. This case is mostly referred to as "1 st order". In the second case, referred to as "2nd order", the bending moments depend, apart from the axial load, on the magnitude of the deflections. This may be called "geometrical non-linear" too.

The considerations concerning 1st and 2nd order effects apply not only to simple columns, but also to circular curved beams or plates (rings and shells) with uniform radial load. In these cases there will occur a tangential force in the curved member which is quite analogous to the normal force in a straight member.

If the bending stiffness of the member is a constant, we may state that the 2nd order analysis is not a serious problem in most cases. But there could be difficulties, for example shells with varying edge conditions.

This report deals mainly with the complication that the bending stiffness of (reinforced) concrete is not a constant. The bending stiffness S is defined here as the bending moment M divided by the appropriate curvature u. A typical M - u relation is given in Fig. l.

In the following, a realistic approach to the 2nd order analysis is given. Two slightly different methods have been distinguished. The first is based on the assump-tion of an uncracked cross-secassump-tion (Chapter 3). In Chapter 4 a more general method is developed. Both methods make use of the so-called critical load which is first dealt with in Chapter 2.

fv1 - - N=·

o "

Fig. 1. Moment-curvature relation, typical.

2 Critical load

The critical load is defined as the load at which deflections, and therefore bending moments, increase indefinitely and when these deflections are initiated by an infinitely small lateral deflection of the member. For example the critical load Ncr of an axially compressed member with length I and constant stiffness S( = EI) is given by:

The critical load can be considered as a hypothetical one, for the initial deflection, which cannot be avoided in practice, causes an ultimate load Nu always less than Ncr. Nevertheless the magnitude of Ncr is of interest for it greatly influences the ultimate load Nu •

The critical load can be written as:

(1) where G is a geometrical factor with the dimension of one per square unit length (l/m2 ).

To obtain some idea of G, some specific cases are mentioned. See also Figs. 2 to 5.

Fig. 2. Critical load of a hinged column.

G= _ _ 3 _ _ {1_v 2)R2

Ncr = S. G (= tangential force)

Fig. 3. Critical load of an infinitely long cylinder (cross-section shown).

P Ncr

cr=7?

~ = function of 8

Fig. 4. Critical load of an infinitely long cylinder segment (cross-sections shown).

-r{~-I

• I ; R-.J , .

-!

i , I L

---1-m = number of half waves of half the circumference

- For a hinged column, length I, axially loaded:

- For an infinitely long cylinder with radius R under uniform radial pressure:

- For an infinitely long cylinder segment with radius R and aperture 28, under radial pressure:

where 1] is a function of the aperture.

- For a cylinder with length I, radius R, thickness t, uniformly loaded:

where <p is a function of

II

R,

tl

R and the number m of waves of the deflected shape. These expressions can be found in several well-known books on elastic stability, for example Timoshenko and Gere's "Theory of elastic stability".

*

A very comprehensive collection of Ncr-values is given in "Handbook of structural stability".

**

In the following treatment of the subject it is assumed that the geometrical factor

G is known. For the cases mentioned above the parameters 1] and <p are given in Appendix A.

3 The uncracked cross-section

Assuming an uncracked section and neglecting possible reinforcement, the bending stiffness depends only on the (variable) modulus of elasticity Ee of the concrete and on the constant moment of inertia [.

According to the conventional approach, we have:

(2)

*

Int. Stud. Edition, McGraw-Hill, Tokyo.

f~

2 nd. degree parabola

Fig. 6. Stress-strain diagram adopted for concrete.

For the stress-strain relation of concrete a 2nd degree parabola is adopted - Fig. 6. This relation can be written as:

where

,

'( ')

CTe = ~ 2 _ ~

f;

e~ e~

CT~ = compressive stress of concrete f~ = compressive strength of concrete

8~ = compressive strain of concrete 8~ = ultimate strain of concrete

The modulus of elasticity is defined as the tangent modulus:

At the origin of the stress-strain relation: E = 2f~

eo ,

8u

We can now write:

(3)

(4)

It is emphasized that expression (4) is mainly introduced to account for a decreasing Ee-modulus with increasing stress level. For a given stress level however, the

Ee-modulus is assumed to be a constant value across the section. In other words, there exists a linear stress distribution across the section.

Consider a rectangular section b· t. The section is loaded by a normal force N

and a 1st order moment M o = Noeo, where eo is the initial deflection. The 1st order compressive stress in the extreme fibre will be:

(j~

= N

+

M 0 = N

+

6. N

eo

= N (1

+

6.

eo)

bt W bt bt2 _{bt t} (5)

It is well-known that deflections and bending moments increase until certain limiting values are reached, belonging to a stable equilibrium (2nd order effect). These limiting values can be expressed as:

and

where t denotes the final value.

1

M, = M_{01 _N / N}

For a derivation of e, and Mt , see Appendix B.

(6) er

Note that these expressions are based on the assumption that all sections of the considered member have the same bending stiffness.

The extreme fibre stress will be:

(7) Using:

the bending stiffness is realistically taken into account.

But what concrete stress (j~ should be used in the expression of Ee?

A safe approximation is to use the maximum compressive stress that occurs anywhere in the member. Then we get the lowest, thus safe values of Ee and N er.

It is more likely that the mean stress in the section should be taken. We may assume that the overall behaviour depends more on some mean bending stiffness rather than on a local low value.

In Appendix C it is deduced that a realistic value of Ee can be based on the stress at the centroidal axis corresponding to a parabolic stress distribution across the section, see Fig. 7.

I u

r-

-l

l

":"-;/z

~ I L'" ~"'~ parabolic stress drstrlbutlon

~stress to be taken for a

t __

realistic Ec - value

-L __ _

I

I

The appropriate expression (Eq. (13) of Appendix C) depends on Nlbtj; and the relative eccentricity elt of the normal force with respect to the centroidal axis. That expression is not a very simple one. But for the purpose of calculating the ultimate load, in the appendix a very good approximation is derived, stating:

(8)

which is independent of the eccentricity!

On the assumption that the ultimate load Nu is reached when O"~t equals the concrete compressive strength

j;,

Nu can be taken from:

(9) where

This requires the solution of a 3rd degree equation in Nil' The solution for Nu can very conveniently be done graphically by calculating values of O"~t as a function of some values of N.

The method is illustrated by the following example. Given a rectangular cross-section with: b

=

1.00 m = 0.55 m Eeo = 29000 N/mm2 f~ = 40 N/mm2 G = 68.4.10-3 _11m2 eo = 0.03 m

Nae

=

6 MN (actuai load)

What is the safety margin y against failure? The 1 st order stress, according to Eq. (5) is:

, N ( 6 0.03) 2 4

O"e

=

1.00,0.55 1

+ .

0.55

= .

1· N

Modulus of elasticity, according to Eq. (8):

Ee

=

29000

J

i

(1 -

~)

Critical load :

MN

Second order stress, according to Eq. (7):

, N ( 0.03 1 )

(jct = rOO.0.55 1 +6· 0.55 ·1-NIN_cr

Some calculated values are listed below.

N a' _{c Ncr} (MN) (N/mm') (MN) 6 14.5 20.3 8 19.3 19.0 10 24.1 17.6 11 26.6 16.9 12 29.0 16.1 a' ct (N/mm') 16.0 22.8 31.9 38.8 50.0 (lOb) (lOc)

Plotting (j~t as a function of N, we obtain Fig. 8. The vertical line, indicated by N:"

is the asymptote of the calculated relation. Its value follows from the condition N =N:r • For this specific case, from Eq. (lOb):

45 f:' 40 --,- ~---35 30 'N' E ₂₅ ~ 6 20 --v b 15 -v b t 10 I I II I I I I 0 4 8 10 12 _1f* Nu Ncr - - - - N [MNJ Fig. 8. Determination of Nu .

The intersection of the (J~t-N relation with the concrete strength

f;

= 40 N/mm2 gives Nu = ILl MN.

The safety against failure, defined as the ratio NIII NaG' is:

'Y

=

1~2=

1.85

This method gives a clear view of what is happening when the load increases. It is evident that failure is always caused by reaching the concrete strength before the critical load causes instability, unless the initial deflection is zero. So it is not correct to define the safety as the ratio N:rl Nac.

Because of (J~t(minimum) = 2 . (J;t(mean) -

f:

= 0.4 N/mm2 _{(> 0), the condition of} an uncracked section is satisfied. But we can be sure that in the case of a greater initial deflection the assumption of a non-cracked section is not correct.

4 General method

The determination of the ultimate load in the case of an arbitrary cross-section (cracked, reinforced) needs a more general method. Such a method is described here.

The M-N-x relations, reflecting all material properties, are assumed to be known.

Furthermore, use is made of the Mu-Nu interaction diagram which can be deduced from the M-N-x relations. Typical diagrams are given in the Figs. 9a and 9b.

An M-N-x diagram gives, at a certain normal load N, the internal relation between

the bending moment and the curvature belonging to the considered section. The bending stiffness is taken as S = Mix.

It is also possible, as a consequence of the 2nd order effect, to speak of an external relation between the 2nd order moment Mt and the 2nd order curvature xt • This depends on the geometry of the member.

The external relation is derived from the expression (6): 1 Mt=Mo· -I-NINer ::;:

t

I ---+x

Fig. 9a. Typical M-N-x relations.

i

Nu1

o

With N = M

0/

eo and Nc/' = S· G follows: 1 M, = MO '_l

-M

_{o/e o}

GS

or By substituting S =

MJ;,c,

we get: (11) Equation (11) gives the external relation between Mt and x"

Both relations must be satisfied to obtain equilibrium. So the desired value of Mt is found by the intersection of the relation M = Mo(1

+

x/eo G) and the given M-x

relation.

A typical graph is presented in Fig. 10.

:;,:

t

I ~<"'" \ internal ' \ relation external relation -

..

"

Fig. I O. Graphical determination of Mt . Fig. 11. Graphical determination of Nmax •

Repeating this procedure for several values of the 1 st order moment M o( = N· eo),

we find a family of N - Mt relations. Plotting these relations in the given Mu - Nu interaction diagram we readily get the extreme value N max' being the load at which failure is initiated. See Fig. 11.

lt is of interest to note that Nmax , at the top of the N - Mt curve, corresponds with the situation where the line of external relation is the tangent line to the M - x curve. This situation reflects a state of unstable equilibrium. Only values smaller than Nmax cause stable equilibrium. Beyond Nmax equilibrium is not possible. Assuming that N is gradually increased, a slightly greater value of Nmax will cause a considerable increase of the deflection and bending moment until failure occurs at Nu equal to

Nmax •

The method is illustrated by means of the following example.

The same dimensions and characteristics of the previous example of Chapter 3 are given. In addition symmetrical reinforcement is considered. Data:

401---~ 38 ~ 30,5 _-30 ..3.. _"U

i

_2,8

Fig. 12. Stress-strain relations adopted for concrete and steel.

1,8 1,6 _ _ - - -... N=6 N=4 4 5 6 7 8 9 10 11 12 13 14 _ _ ,,[10-3 YmJ

Fig. 13. Calculated M-N-u diagram.

24 'Z' 12 Nmax 6 10 ~~ ~. 8

r

6 1.6 1,8

b = 1.00111 t = 0.55 m G =68.4'10 -311m2 eo = 0.03 m Nac = 6 MN Eco = 29000 Njmm 2

f;

= 40 N/mm2 A = A' = 0.002bt fy = 420 N/mm2

The M-N-x diagrams (Fig. 13) and the Mu-Nu interaction diagram (Fig. 14) are calculated by means of a computer. The stress-strain relations of concrete and steel which are used, are drawn in Fig. 12.

From Fig. 14 follows the failure load Nmax = 12 MN with the corresponding 2nd order moment Mt = 1.1 MN.m. The safety factor is:

It is not surprising that this safety is 8% greater than that found in Chapter 3: the reinforcement will cause a decrease in the concrete stresses and an increase in the bending stiffness.

APPENDIX A.I

The parameter '1 as a function of the aperture 2 (}

21, - --- ---- ---1 I I 20-- -

~

_! ---~o~-

--I 18 -16

-~

I 14 _I ---0 I 12 I 10

-11

8---

-l-t-6

--:-t--o

APPENDIX A.2

Graph of the parameter qJ

10-4 _{4 5 6 7 8 10-3 4 5 6 7 8 10-'} 4 5 6 7,1 10-' \0-' 1,5 1.5

,.5

lr

If- 10-' 8 ~I 1+ 8 7 7 6 ~.~

,

1-6 :~t 1-5 4 -qJ 4 3 - _·2 2

-1

15

Ii

i >"f~~ 1.5 '.5 I~ =i 10-' "AI'

WI

10-' 8 8 7 H- 7 6 6 s i T 4 V

El4

t,t

6' 4 3 -

_L

v

J:.

t

1-3

~

R H' +-+- /' ;)f : j - ~ '.5 H-1'lt

''''

~ .j Ft-j 1.5 bf I 4' 10-" 10-3 8 7 2.5 6 .~ .1- _I" 6 6' ~ 4 _! ~t 5 4 4 /T 4

++

~If-3 6 ~ 1-3

~

, j~ -.:.:L 1.5 3 1.5

~1h

,

, -~

:~

l

10-4 1- ~ J 15 10-4 8 ~.L 8 7 7 6

-F'

6 5 4 0=>

~H

4 H} H-3 i l fH ~

t .;.:

_Itt

J!-, _~ 1.5 t-- --j-+~I ~t , '.5

It

1+

rtI;tlitir~~~1 ~

t

10-5 H _, Ii It ]']'] 10-5 1.5 10-3 1.5 '.5 10- 4 2 3 4 5 6 7 8 3 4 5 6 7 B 10-' 3 4 5 6 7 8 10-' qJ 1 - , ' __

+

-'-a' (

m'-1

+

2m'-1~')

( 212y 12 m212 (m2-l) 1

T~

1 + -n2R2 n2R2 IX = tlR ₁₇

APPENDIX B

Determination of et and M t

Given a member with length I and deflection eox = eo sin

~x.

The load N will cause an additional deflection YX. The total deflection becomes ex

=

e ox

+

Y x' which is assumed to be a sine surve. So ex

=

et • sin re; and

(1) In general: M = _EI.d2yx x dx2 From Eq. (1): So: (2)

From the external load :

M _x

=

N· e _x

=

N· e . _t sin rex _I (3)

From Eqs. (2) and (3):

With:

APPENDIX C

Determination of a realistic Ee-value Assumptions

a. Parabolic (J'~-a~ diagram of concrete:

b. Linear strain distribution across the section. c. All strains remain positive (compression). d. Reinforcement is neglected.

See also Fig. C-l.

Fig. C-l.

Determination of Nand e as functions of the strains Strains: Stresses: (J" _x = _{J c ,} 1" •

~

(2 -

~)

_, au au Normal force: Distance a: With follows: t

S

(J'xbxdx a=~--­ N (1) (2) (3) (4)

Eccentricity: e = !t-a or:

(5)

With the Eqs. (3) and (5) the load N and the eccentricity e are known as functions of the strains. Because of M = N· e the bending moment M will be:

(6)

Capacity of the cross-section

This capacity will be reached when e~ = e~. Substituting e'l = e~ in Eq. (5) we find:

Solution of e~:

e~

= e'

{I -

J

12e_}

u 4e+t

Substitution of Eq. (7) into Eq. (3) gives:

N(= N) = btf"

-~-u Jc 4e 1

+-t

Bending stiffness and modulus of elasticity The curvature is:

So: e~ -e~ M U = = -t

S

M·t S = -e~ --e~

From Eq. (6) follows:

(7)

(8)

Because of the assumption of an uncracked section, we may state S =

EJ,

where

1= /2bt3.

With

Eq. (9) becomes:

(to)

where 8 = ·Hs~ +s~) = mean strain (at centroidal axis). From Eq. (2) can be derived:

1 -

~

=

J

1 -

O"~a

e~

f:

So:

E = E

/1 _

O"~a

e eo \

Ie

(10)

where O"~a = concrete compressive stress at the centroidal axis (parabolic stress distribution).

Using the notation:

where

8 I X = -_,

Su

we have to determine IX as a function of Nand e.

Putting S'l

+

s~ = 28 and S'l - s~ = xt, it follows B~ s~ =

e -

(-txt)2. Substitution into Eqs. (3) and (5) gives:

Using

f3

=

~

and 2s~ N n = -bt/~

we get:

e

3n-= f3(1-a)

t

Solution of these equations gives:

3n .:

f3=_t

1-a

The realistic modulus of elasticity is thus given by:

i;;c~

EmJ

l(t-n{l

+

Jl-12C~S

(i)l

L~____

____________ ,

_________________ . _______

.~_ .~l_J_J

Furthermore, there are two conditions to be satisfied: e~ ~ e~ and 0 ~ e~ ~ e'l' For the boundary conditions can easily be derived:

a

=

1-f3

=

l - l

J

12e if 2 4e+t and t-6e a = f3 = ~- if e~ = 0 t-4e

So, with

;c

= 1-a: co

~

_lJ

12e if

Eco - 2 4e+t

2e

t-4e if e~ = 0

The Eqs. (13), (16) and (17) are graphically presented in Fig. C-2.

(11) (12) (13) (14) (15) (16) (17)

Ec Eco

r

e q. (13) r -______ ~~~~n~=~o ____ _,---~ 1 - - - ' 0 " ' - 1 - - - i - - - _ _ _ _ -> r---__________ .1!.QZL-__ - / _ _ _ _ _ _ _ _ _ _ _ _ _ _ """ 1.0 0.9 0.8 0.7 r - - -_ _ _ -"'0.!!-4 _ _ -1 _ _ _ _ _ -0.5 ----~ 1 - - - -__ ----!.OU.3 _____ I ___________ - - - > 0.6 0.5 o O.OZ 0.04 0.06 0.08 010 0.12 014 0.16 0.18 0.20 0.22 024 0.26 QZ8 0.30 - -.... !-. t

Fig_ C-2_ Ec/Eco as a function of n(=N/btf~) and eft.

Modulus of elasticity at ultimate (B~

=

B~)

From the condition B~

=

B~ it has been found that I-IX

= p.

So, with Eq. (12) follows:

1-1X

=

J3nf

(18)

The modulus of elasticity at ultimate is now found by equating Eq_ (13) to Eq. (18). Thus:

From this expression:

hence:

e I-n

4n

(19)

In Eq. (19) the modulus of elasticity is expressed as a function of the relative normal force n(

=

N/btj;) and independent of the eccentricity. This expression is easier to use than Eq. (13) and gives the correct Ee-value when B~

=

B~. Eq. (19) is