UDC: 624. 012. 454: 539.3

ELASTIC ANALYSIS OF

REINFORCED CONCRETE SECTIONS

By: WIRATMAN WANGSADINATA C.E.

DAN TENAGA

C I PTA

LISTRIK

K A R Y A ~ DEPARTEMEN PEKERJAAN UMUM

DIREKTORAT JENDERAL

DIREKTORAT PENYELIDIKAN MASALAH BANGUNAN

Jalan Tamansari no: 84 * BAN DUNG * Tilpon: 81082/81083

UDC: 624. 012. 454: 539.3

ELASTIC ANALYSIS OF

REINFORCED CONCRETE SECTIONS

By: WIRATMAN WANGSADINATA C.E.

Dep. Pekerjaan Umum & Tenaga Ustrik PUSLITBANG

PERPUSTAKAAN

DEPARTEMEN PEKERJAAN UMUM

DIREKTORAT JENDERAL

DIREKTORAT PENYELIDIKAN

DAN TENAGA LISTRIK

C I PTA K A R Y A

MASALAH BANG UNA N

Jalan Tamansari no: 84 * BAN DUNG * Tilpon: 81082/81083

----------------------REGIONAL HOUSING CENTRE

Published by : Yayasan Lembaga Penyelidikan Masalah Bangunan- Bandung March197B

N. I. :

N.K.:

Dt:?ARTEMf.N PEKER.JAAN UMUM

PUSL!TBANG - ;~RPUST .L\KAAN

1/?tj,J jq &,zl_ 01~ , tj.r / ¢-1/#A/. 'e ...:._j

Dep. Pek .rb:lll Umu, & r n<~ga Listrik t

> 1 'SL1T'J\ , ....., ~ , J.:; , ~ N

.J~t/~//ry ~l ...... ......,__,...., - .-- - -

Synopsis.

ELASTIC ANALYSIS OF REINFORCED CONCRETE SECTIONS By:

Wiratman Wangsadinata, C.E.*)

3

The paper deals with the elastic analysis of reinforced concrete sections based on the new Indonesian Reinforced Concrete Code 1971. Although theory of elasticity is now gradually being abandoned as a basis for the design of sections and replaced by more rational theories based on ultimate strength concepts, nevertheless it will remain important as a tool for computing working stresses. Therefore, a more realistic approach in the determination of the value of the modular ratio 'n' should be assessed taking into consideration deformation properties of concrete due to actual loading cases in practice. Background of the stipulations concerning values of 'n' in the new Code is briefly discussed. Based on the derived equations charts are presented, with which sections subjected to pure bending can easily be analised. It is shown that the charts are also applicable to bending with normal force, provided that the tension steel remains in tension. A method is further presented on how to treat flanged sections as equivalent rectangular sections.

NOTATIONS.

A, A' At A2 b =

bm =

bo ca,cb d' D, Da, Db ea ~ = tg if> =

Bt,o = tg if>o h i M =

n = Ea/~ = N = t = T = y z ~= aa/nab 'Ym• 'Yma• 'Ymb =

l' 'Ypa• 'Ypb s

S =A'/ A €' b ~ = z/h =

flY

A.= bm/bo ~ = y/h, ~0 =

aa, a~, at,

aa, at, abk abu

area of steel; with accent (apostrophe) means compression steel. area of steel farthest to the application point of tensile normal force. area of steel nearest to the application point of tensile normal force. width of section; for flanged section width of equivalent rectangular section. effective flanged width of flanged section. stem width of flanged section. section coefficient; eq. (20), (21), (22), (23), (31), (32). distance of compression steel to compressed edge of section. resultant compressive force; subscripts a and b refer to steel and concrete. eccentricity of normal force with respect to the axis of tension steel. secant modulus of concrete. elasticity modulus of concrete; eq. (1). effective height of section. coefficient for bending with normal force; eq. (28). bending moment. modular ratio. normal force. flange thickness of flanged section. resultant tensile force. distance of neutral axis to compressed edge of section. internal lever arm. ratio of steel and concrete stress; eq. (11). material coefficient; subscripts a and b refer to steel and concrete. depreciation coefficient; subscripts a and b refer to steel and concrete. load coefficient. compression steel ratio. compression strain of concrete. internal lever arm coefficient; eq. (17), (41). distance of application point of resultant compressive force D to compressed edge of section. transformation coefficient for equivalent rectangular section of flanged section. neutral axis coefficient; subscript o refers to elastic balanced condition. working stress; subscripts a and b refer to steel and concrete; with accent (apostrophe) means compressive stress. allowable working stress; subscripts a and b refer to steel and concrete (compressed). characteristic compressive strength of 15 em concrete cubes at 28 days. flexural compressive strength of concrete.

*) Research member Building Research Institute Bandung; lecturer Dept. of Civil Engineering Institute of Technology Bandung; chairman Committee on the Indonesian Reinforced Concrete Code 1971.

4

¢o

slope of straight line determining the secant modulus of concrete; coefficient to determine 'Ymb·

w= ""'bh w = rKf ' 0

slope of straight line determining the elasticity modulus of concrete. tension steel coefficient; subscript o refers to elastic balanced condition.

1. INTRODUCTION.

Theory of elasticity is the historical starting point for the analysis of reinforced concrete sections, whereby the concrete is considered as a perfectly elastic material. Because of the assumption of straight line relationship between stress and strain, this theory is often called "straight line theory". From equality of strains in the concrete and steel during deformation a specific number appears in the calculation which is known as the modular ratio, which is the ratio of the elasticity modulus of steel and concrete and is denoted by 'n'. Therefore, the analysis of sections using this method is also known as the "n-method". Furthermore, as the stresses involved in the calculation are working stresses, the design of sections using this method is also often called "working stress design".

At present theory of elasticity as a basis for the design of reinforced concrete sections is gradually being abandoned to be replaced by more rational concepts based on ultimate strength. There are two reasons for this, i.e. (1) the n-method does not reflect the real safety against ultimate strength and (2) by using the n-method as generally practised in the past less economical sectional dimensions are obtained compared to that obtained using ultimate strength theories. However, theory of elasticity will remain important, not only as a branch of science but also as an important tool for computing working stresses in structural members which behave elastically under certain loading conditions. It is known that highly redundant structures will remain statically indeterminate at its ultimate stage, either because collapse is due to instability before all plastic hinges can developed [7] or because the structure is intentionally designed to develop plastic hinges in beams only rather than in columns [21]. Thus, even though a structure is analised using collapse criteria, the n-method will still be needed for computing stresses in structural members which remain elastic. Apart from that, for the computation of deformations consideration of elastic behaviour of sections may facilitate the analysis [20] , while for the computation of deflections and crack widths using formulas like given in the new Code, working stresses in the member must be calculated first.

From the above discussion it is clear that since the n-method will remain taking an important part in modern design of structures, the value of the modular ratio 'n', being the fundamental starting point, should be better assessed and more practical methods of application should be facilitated. Several stipulations concerning the elastic analysis of sections, which is more realistic than it was stipulated in the past, is given in the new Code [ 1] [2]. Several studies which lead to the above stipulations have been presented by the author in previous papers [ 4] [ 5] [ 6] (7] and this paper is a complement of the author's previous work, in which charts will now be presented for more practical application of the n-method. Although important principles given in the new Code are briefly reviewed here, the reader is recommended to examine the specific articles in said Code for more detailed informations.

2. HISTORICAL REVIEW.

The elastic analysis of reinforced concrete sections was introduced for the first time by Koenen in Germany in 1886. At that time he considered the elasticity modulus as being equal for steel and concrete so that the modular ratio became n = 1. It was in 1890 that Neumann pointed out that the elasticity modulus of concrete was much more smaller than that of steel and he proposed to use n = 14. Since then no general consensus has ever been reached concerning the value of the modular ratio. Only in the European continent uniformity in the use of a modular ratio of n = 15 has been agreed upon and only after an appeal of Emperger in 1936 [ 13] . Thus, based on a fixed modular ratio n = 15, several methods of analysis for practical purposes have been developed in Europe using tables and charts (14] [15] (16] (17] (18] (19].

In the American continent the modular ratio has always been considered variable depending on the concrete quality. The better the quality of the concrete the smaller the value of n. This is closer to the actual property of concrete. Several formulas have been proposed to compute the value of n and in the literature the following values of n can be found: n = 6, 7, 8, 9, 10, 12 and 15. Thus, based on a certain fixed value of n also in the American continent several methods of analysis for practical purposes have been developed using tables and charts [22] (23] (24] [25].

In so far the influence of the speed of loading was not being considered in the determination of the modular ratio, although it was known that concrete behaves completely different under long-time and short-time loading. The influence of the speed of loading was taken into account for the first time in Japan, where it was felt necessary to distinguish permanent loading due to dead load and live load from temporary loading due to any loading in combination with earthquake. For increasing quality of

5

concrete, the following values of n can be found in the literature: n = 30, 24, 21 for permanent loading and n = 20, 16, 14 for temporary loading [26]. Thus, also in Japan several methods of analysis for practical purposes have been developed using tables and charts based on a certain fixed value of n.

As regards Indonesia it has its own unique development. For a very long time it was under strong influence of Europe via the Netherlands. Since the first reinforced concrete code in 1921 up till the 1955 code, a fixed value of n = 15 was stipulated. Based on this several practical methods of analysis have been introduced from Europe and widely used in Indonesia [ 1 0] [ 11] . Around the year 1960 influence from the American continent started taking root, so that the use of variable low values of n became more and more common. Values of n stipulated in the ACI codes as mentioned above were recommended in official handbooks [12]. Then, in the following years also influence from Japan gradually found acceptence and by using relatively high variable valoes of n like mentioned above more economical structures had been produced in Indonesia. This trend was mainly caused by the increasing number of big projects constructed in Indonesia under technical cooperation with Japanese designers. Thus, non-uniformity arose in the use of the modular ratio in Indonesia. This disorderly situation came fortunately to an end by the issue of the new Indonesian Reinforced Concrete Code 1971. In this new code the modular ratio is defmitely stipulated as being variable depending on the concrete quality and the loading case, similarly stipulated like in Japanese ,codes, which is more realistic since Indonesia belongs to one of the important seismic countries in the world. Background of the determination of said modular ratio will be further discussed in 3.2.

Apart from the problem of determining the right value of n, it is felt necessary to find out more practical methods of analysis in which the variability of n is automatically taken into account. If each table or chart is valid only for a certain fixed value of n, then a great number of tables or charts will be required for practice. To overcome this difficulty in 1963 the author has developed a method of analysis using a table which is valid for any value of n [3] [ 5]. A modification of this table with more accurate figures calculated using an electronic computer was introduced by the author in 1970 [ 6] [7]. Finally those tables are further developed into charts presented in this paper, which are more practical than tables.

3. ASSUMPTIONS AND EXPLANATIONS.

3.1. Stress-strain relationship for concrete.

Concrete is an elasto-plastic material, it means that every deformation always consists of two components, the elastic and the plastic component, the latter also being known as creep. The elastic component will disappear as soon as the load ceases to act, while the plastic component is permanent and is a function of the speed of loading. If the loading is very fast the plastic component has not the chance to develop so that the concrete behaves perfectly elastic with a rectilinear stress-strain relationship. If the speed of loading is lowered down the plastic component, developed after every additional load application, becomes larger. The stress-strain relationship is no longer rectilinear but determined by a curve of which the curvature is approximately inversely proportional to the speed of loading. In Fig. 1 stress--strain relationships for concrete for several speeds of loading according to CEB [27] are shown.

As explainded above, the elasticity modulus Ebo = tg cf>o (see Fig. I) is a property of concrete which is related to fast or instantaneous loadings only. Its value can lij)parently be computed with the following formula:

Ebo = 19000 vfabk (1)

in which abk is the characteristic compressive strength of concrete determined with 15 em cube specimens. Here the elasticity modulus will be involved in strength analysis, that is why characteristic strength is used in the above formula. If for instance the elasticity modulus is required for rigidity computations then mean strength of concrete should be used in the above formula.

Figure 1 Stress-strain relationship for concrete.

6

From Fig. 1 it can be noticed that if the concrete is subjected to sustained pressure such that o'b/afm < 0.83, e.g. denoted by a, creep will develop continuously at a constant stress without causing failure untul the strain reaches its ultimate value, called the creep limit, which in Fig.1 is denoted by c. The most intensive creep occurs during the first 3 or 4 months of loading, while the creep limit will be reached after 3 or 4 years. The value of concrete strain at its creep limit is approximately three times its elastic strain. If for instance before the creep limit is reached an instantaneous loading is applied to the concrete then momentarily the concrete will behave perfectly elastic with a rectilinear stress-strain relationship, e.g. denoted by b-d in Fig.l. Such loading case actually happens in practice (see 3.2). If now sustained pressure occurs such that ab/ai,u ~ 0.83, Fig.1 clearly shows that the crushing strength of concrete will be reached sooner that its creep limit. Thus, due to continuous pressure a decrease in concrete strenght occurs. To account for this the compressive strength of concrete abu must be devided by a coefficient 'Ypb = 1/0.83 = 1.2 in the case of sustained loading. In the new Indonesian Reinforced Concrete Code 1971 this coefficient is called the depreciation coefficient for concrete. A similar decrease of concrete strength will also occur due to repeated loading. To account for that a suitable depreciation coefficient should be assessed.

I

iii

EJ..:: 1, ,;, = elasl/c,-ly ,.,,_,_

~ : i!J (' :: S~Cit~rl 11><t11/.

Figure 2 Elasticity modulus and secant modulus of concrete.

Because concrete is an elasto-plastic material, in the use of the elastic theory for the analysis of sections, the rectilinear relationship between stress and strain is no longer defined by the elasticity modulus of concrete Ebo = tg ¢0 , but by the secant modulus Eb = tg ¢, where ¢ is the slope of the straight line connecting the coordinate origin and the point corresponding to the stress-strain condition concerned (see Fig.2).

3.2. Stress-strain relationship for concrete for loading cases in practice and modular ratio.

The stress-strain relationship shown as curves in Fig. I are obtained using gradually increased loads with constant acceleration. Actual loadings in practice do not show such properties.

For the analysis of building structures, the Indonesian Loading Specification 1970 (NI 18), which is also refer to by the new Indonesian Reinforced Concrete Code 1971, specifies two categories ofloadings, that is permanent loading and temporary loading. Permanent loading is defined as loading caused by loads that act for a long period, such like dead load and live load, while temporary loading is defined as any

combination of loadings with short-time loading, such like caused by wind and earthquake. Based on the concrete property discussed in 3.1, it is possible to estimate approximately the stress-strain

relationship for concrete for actual loading cases in practice. Let consider Fig.3, which shows the progress

of stress and strain in the extreme compressed fiber of a reinforced concrete section. The straight line 0-4

represents the elastic deformation line for very fast loadings, so that the tangent of the angle 4-0-7 is the

elasticity modulus Ebo according to eq. (1). The line 0-S represents the creep limit, which approximately may be idealized by a straight line (compare with the creep limit shown in Fig. I). As the ultimate strain is

approximately three times the elastic strain, the tangent of the angle 5-0-7 is 1/3 Ebo· Hence, the secant modulus after the creep limit is reached may be considered as independent of the concrete stress with a valu~:

-t ~0 = 6333 vabk 0 0 0 0 0 0 0 0 0 0 0 0 (2)

/ /

/

6 /

/ /

6

ti..' t 2 ~~ t----+---.. -t-r---+-+.;--+-----,f'---+---+--+------1 'lu

(7.' II

Figure 3 Stress-strain relationship for concrete for loading cases in practice; (a) and (b) are

idealized curves for ultimate strength analysis.

7

Suppose now the dead load plus the live load (=the permanent loading) of the structure bring about a concrete stress of abl· In the progress of increase of these loads to their maximum value, in Fig.3 the stress-strain relationship is represented by the curved line 0-1. Thereafter creep develops at a constant stress abl represented by the horizontal line 1-2. Point 2 at the creep limit will almost be reached within 3 or 4 months but the full creep limit. will be reached not before 3 or 4 years. During the progress of creep a continuous decrease in the value of the secant modulus occurs until the minimum secant modulus is reacheq at the creep limit according to eq. (2). Suppose after the creep limit is reached due to permanent loading earthquake load appears, it means that per definition temporary loading occurs, then a series of increments or decrements of the loads occurs within a very short time (a few seconds) so that the concrete behaves perfectly elastic. If the earthquake load increases the previous loads, in the diagram shown in Fig.3 the stress-strain relationship will follow the straight line 2-3 parallel to line 0-4. The level of point 3 is intentionally limited in such a way that the concrete stress due to temporary loading will never exceed approximately two times the concrete stress due to permanent loading. In this way the secant modulus at the maximum concrete stress due to temporary loading is (see Fig.3) :

~2 or

~2 = 9500 v'Obk

_]_ 2 ~I

(3)

During the action of the earthquake a continuous increase in the value of the secant modulus occurs from Ebl to Eb2· If the earthquake happens before the creep limit is reached then the stress-strain relationship will follow for instance the straight line 2'-3'.

From the above discussion it is very clear that the stress-strain relationship for concrete fo~ actual loadings in practice is very complicated. That is the reason why we have to idealize the stress-strain relationship when it is required for analysis.

For the ultimate strength analysis of reinforced concrete sections the new Indonesian Reinforced Concrete Code 1971 specifies as a compromise an idealized stress-strain diagram represented by a second degree parabola up till the fiber where the strain reaches a value of 2 o/oo and continued by a horizontal line till the ultimate strain, which is specified as 3.5 o/oo. The maximum stress ordinate is defined as afm for temporary loading and 0.83 abu for permanent loading (see Fig.3). Ultimate strength analysis has been discussed elsewhere by the author [9] and will not be discussed again here.

In using the elastic theory for the analysis of reinforced concrete sections a problem arises in choosing the right value for the secant modulus. If we follow again the stress-strain progress due to permanent and temporary loading (Fig.3), then we can state the following. Due to the continuous decrease in the secant

8

modulus during the development of creep a continuous redistribution of stresses in the section occurs, whereby at a constant concrete stress the steel stress progressively increases until the maximum value is reached when the concrete reaches its creep limit. Hence, the secant modulus Ebl is for the analysis of sections due to permanent loading significant. If thereafter earthquake load happens, then due to the continuous increase in the secant modulus during the action of the earthquake a redistribution of stresses in the section again occurs, whereby both the concrete stress and the steel stress increase. It is clear now that for t}le analysis of sections due to temporary loading the secant modulus Eb2 is significant.

Now we can define the modular ratio. According to its defmition the modular ratio is:

n = . . . . . . . . . . . . . . . . (4)

so that by substituting Eb according to eq. (2) and (3) in the above equation we obtain the modular ratios for building structures stipulated in the new Indonesian Reinforced Concrete Code 1971

for permanent loading :

n = 2.1 x 106 = ___.1JQ_ •• • • . . • . • . • • • . • • • (5) 6333 Vabk v'Obk

for temporary loading :

n = 2.1 x 106 = _nQ____ . . . . . . . . . . . . . . . . (6) 95 00 v'Ubk {<1bk

For standard quality of concrete according _to the new Code values of its modular ratio are listed in Table I.

Table I

Concrete n

0 bk quality (kg/cm2 )

permanent temporary loading loading

B 1 100 33 22 K 125 125 30 20 K 175 175 24 16 K 225 225 21 14

It will be shown later that it is safer for the analysis to use too low estimated values of the modular ratios than too high estimated values. That is the reason why the values of the modular ratios listed in Table I are maximum values according to the new Code.

If a more accurate value of the modular ratio for permanent loading in combination with accidental loading is required, the mathod given by Borges [20] may be followed to assessed the related secant modulus.

3.3. Allowable working stress.

Although the design of reinforced concrete sections based on the elastic theory is not recommended, however it may still be used according to the new Code as an alternative. For that purpose allowable working stresses are stipulated.

Concerning material strength the new Code require the use of characteristic strength; these are strengths below which 5% of test results may allow to fall. This requirement applies for concrete as well as for steel. To account for the decrease in material strength in practice, to obtain the design strengths the characteristic strengths must be divided by the product of depreciation coefficient 'Y p and the material coefficient 'Y m· The value of the depreciation coefficient for conCEete 'Y pb has been derived in 3.1, where we have found a value of 1.2 for permanent loading and a value of 1.0 for temporary loading (no reduction in strength due to creep). The depreciation coefficient for steel is always taken as 1.0 for all kind of loadings. The material coefficient for concrete 'Y mb is specified as 1.4/ifJ, where if> = 1 for normal conditions. For excellent

9

conditions a value of ¢ > 1 may be considered, while for unfavourable conditions a value of¢< 1 must be used. The material coefficient for steel 'Yma is specified as 1.15 for all conditions.

In connection with the determination of the allowable working stresses, there is still one other safety coefficient to be considered, that is the load coefficient 'Ys· This coefficient must be considered to account for the effects of possible increase of the working load up till the ultimate load causing collapse of the structure. In statically determinate structures between the working load and the ultimate load there is a proportional relationship, but in statically indeterminate structures, due to the redistribution of moments and forces before collapse of the structure, the ultimate load is no longer proportional to the working load. However, for the determination of the allowable working stress it is always assumed that there exists a proportional relationship between the ultimate load and the working load. ln this way we can define the allowable working stress as follows:

allowable working stress characteristic strength 'Yp 'Ym 'Ys

According to the new Code, 'Y s = 1.5 .for permanent loading and 'Ys = 1.05 for temporary loading. For standard qualities of concrete and steel the allowable working stresses are given in the new Code. The allowable working stress is denoted by the stress symbol with a bar, e.g. Db and O'a·

4. DERIVATION OF EQUATIONS.

4.1. Rectangular section subjected to pure bending.

Consider a rectangular section, doubly reinforced and subjected to pure bending with notations as shown in Fig.4, in which the following assumptions have been used: (1) plane sections remain plane after bending and remain perpendicular to the member axis (hypothese of Bernoulli), (2) strains in the sections are assumed to be directly proportional to the distance from the neutral axis (Navier's law), (3) tensile stresses are assumed to be resisted by the reinforcement only.

In the following we will first define successively neutral axis coefficient, internal lever arm coefficient, compression steel ratio and tension steel coefficient :

~ L h

~ z h

(j A' A

w A bh

From proportionality of stesses we also get the stress coefficient:

(3

A . . . . .

T .!1= r 1z j_ _l __ _

I /

Figure 4

/ /

T

Rectangular section subjected to pure bending.

(7)

(8)

(9)

(10)

(11)

10

In our further discussions we will always neglect the reduction of concrete area by the area of the compression steel and assume that the modular ratio for the compression steel is the same as for the tension steel (=n).

Internal forces acting on a section are (see Fig.4) :

T A aa h...=_y_ n A ab (12) y

1\ ~by q; (13)

Da o A a· = Y- d' on A at, . (14) a y

Equilibrium of forces T = Db + Da gives, after substituting the above expressions of the internal forces and after derivation, the tension steel coefficient :

nw (15)

The distance of the resultant compressive force D = Db + Da from the compressed edge of the section will be denoted by 'T[Y ; it can be evaluated from the equilibrium requirement (Db + Da) 'T[Y = Db. (1/3) y + DW d', which after substituting internal forces according to eq. (12), (13) and (14) and further derivation, ytelds :

- 1 ~ 3 + onw.!t_(~-.!t_) 6 h h (16)

The internal lever arm is z = h- 77 y, which after substituting eq. (16) gives the internal lever arm coefficient :

-1 ~ 3 + onw_ct_(~-Jt_) 1 - ___,6 ____ -----..:;h,__ __ h:.:...__

le + onw(~- _ct_) 2 h

Equilibrium of moments M = T . z gives :

M A aa ~ h = n w b h ~ ~ h n

which yields the following expression :

_1_ n wz

M q; b

From equation (19) we find the expressions for the section coefficients

h

J1 nw~

(17)

(18)

n w (3 ~ (19)

(20)

(21)

=

=

h j_M_

t1hb

)n~,n By rearranging eq. (18) we get the area of the tension steel

A M

or the tensile stress in the tension steel

M

A t h

11

(22)

(23)

(24)

(24a)

The expression which gives the relationship between concrete and steel stresses are obtained from proportionality conditions :

a' a

4.2. Bending with normal force.

= ~- d'/h ~

~ .: d'/h 1 - ~ aa (25)

The analysis for bending with normal force can be solved by transfering the eccentric normal force so that it acts along the axis of the tension steel, where it may either decrease (Fig.S b) or increase (Fig.S c)

I,. 1

Figure 5 Rectangular section subjected to bending with normal force.

the force in the tension steel, depending whether the force is a compressive or a tensile force. If the eccentricity of the normal force with respect to the tension steel is ea, the bending moment produced by the transfer of force isM= N ea. An arbitrary section subjected to a bending moment M = N ea combind with a normal force N acting along the axis of the tension steel will have an area of tension steel A with a tensile stress a a which will satisfy the following equation:

A = (26)

12

which after derivation gives

i A (27)

with

= (28)

In the above expressions N must be given a negative sign when it is a tensile force and in that connection also ea must be given a negative sign when N is a tensile force. Hence, i > 1 for compressive norinal force and i < 1 for tensile normal force.

If now we compare eq. (27) and (24) then we will notice that the two equations are identical. The only difference lies in the fact that for pure bending N ea = M and i = 1. What is then the meaning of i = 1 ? If we examine eq. (28) we will notice that i = 1 if ea = oo and this is indeed true since for pure bending the eccentricity of the normal force N (=0) is ea = M/N = M/0 = oo and N ea changes to M.

An important conclusion we can draw from the above discussion is that bending with normal force can be treated just exactly like pure bending, provided that N ea be used instead of M and i A instead of A, notwithstanding the shape of the section. Parallel to this, for rectangular sections subjected to bending with normal force we may write the compression steel ratio, the tension steel area and the section coefficients successively as follows :

=

i A =

=

A' iA

wb h =

h

h

(29)

(30)

(31)

(32)

In case the section has symmetrical reinforcement, the value of 8 is such that A' = A. From eq. (28) and (29) it follows that this condition will be satisfied if :

= _I_ i

= (33)

As we have seen, the analysis for bending with normal force with the above method (the i-method), is only valid for tension steel that remains in tension.

If the normal force is a compression force with a small eccentricity, the whole section will be in compression, so that the i-method is no longer valid. Bending with normal force causing compression over the whole section will not be further discussed here:

If the normal force is a tensile force with small eccentricity, the whole section will be in tension, so that the i-method is also no longer valid. Since we always neglect the tensile strength of the concrete, in this case the tensile force must be resisted by the reinforcement only. If A1 and A2 are the area of the reinforcement farthest and nearest to the application point of the normal force, and the stress in both of the reinforcement is designed to be equal (= aa), then from equilibrium conditions we will fmd :

N ea

aa (h- d') (34) =

I3

(35)

In the above discussion about bending with normal compressive force buckling effects are not being considered. According to the new Indonesian Reinforced Concrete Code I97I these are taken into consideration by adding additional eccentricities to the original eccentricity of the normal force. This problem will not be further discussed here.

4.3. Flanged sectio~.

When flanged sections (T or L section) are being considered two cases can occur. When the neutral axis falls within the flange the problem of analysis is similar to that of analysing a rectangular section, where for the width b the effective width of the flange bm must be used. When the neutral axis falls below the bottom of the flange then we have a true flanged section. For practical analysis it is convenient to distinguish two types of sections for this case, i.e. narrow flanged sections with bm/b0 EO;; 5 and wide flanged sections with bm/bo > 5 (see Fig.6).

Narrow flanged section (bm/b0 EO;; 5).

The analysis of narrow flanged sections can be simplified by treating them as rectangular sections by transforming the compression zone into an equivalent rectangular zone, which produces an equal resultant compressive force [I 0] .

Based on the real compression zone the concrete compressive force is :

= I b h , [ ~ + ( bm _ I) _t 2 f- t/h -20 Ubt;; b h

0 ~

while based on the equivalent rectangular compression zone the concrete compressive force is :

= 4bhaj,~

~'-f --h,. -~)1 t 1 '(r---- b ~

1 T ~~~~ .Y~r/, .~----ez~D ..,l -+·-· 1 j / ~)H

Figure 6

~~ T ~ o.;~

n

Narrow flanged section treated as an equivalent rectangular section.

Equating eq. (36) and eq. (37) yields the width b of the equivalent rectangular compression zone :

b = A bm

with:

bm I) _t 2~-t/h ~ + ( 0::: -

A 0 h ~ = ~ bm ~

(36)

(37)

(38)

.• (39)

In Table I the transformation coefficients A are listed as a function of the shape of the compression zone, which is determined by the neutral axis coefficient ~ and· the ratios bm/b0 and t/h.

Although the equivalent rectangular and the real compression zone produce an equal resultant compressive force, the equivalent rectangular zone gives a slightly smaller internal lever arm. Therefote, the

14

analysis based on this equivalent rectangular compression zone is on the safe side. To start the analysis we have to estimate first ~ to obtain A from Table I for known values of bm/b0 and t/h. Treating the flanged section further as a rectangular section with width b the analysis must show the correctness of the estimated ~. otherwise the analysis must be rep:ated starting with a corrected value of~-

Wide flanged section (bm/b 0 > 5).

When the effective width of the flange is relatively large, the compression zone in the stem below the bottom of the flange will be relatively very small, so that it can be neglected. In this case it is on the safe side if we consider the resultant compressive force Db as acting in the mid-depth of the flange. By considering equilibrium conditions like for rectangular sections, then we will find the direct relationship between the bending moment and the reinforcement :

0.50 0.45 0.40 0.35

0.50 0.45 0.40 0.35

0.50 ~0.44 0.39

0.50 0.44

0.50

with

Table I Transformation coefficient for the equivalent rectangular

compression zone of flanged section.

t/h bm/bo

0.30 0.25 0.20 0.15 0.10 0.05 1.5 2.0 2.5 3.0

~ A

0.30 0.25 0.20 0.15 0.10 0.05 1.00 1.00 1.00 1.00

0.33 0.28 0.22 0.17 0.11 0.06 1.00 1.00 1.00 0.99

0.38 0.31 0.25 0.19 0.12 0.06 0.99 0.98 0.98 0.97

0.43 0.36 0.29 0.21 0.14 0.07 0.97 0.96 0.95 0.94

0.50 0.42 0.33 0.25 0.17 0.08 0.95 0.92 0.90 0.89

0.50 0.40 0.30 0.20 0.10 0.92 0.88 0.85 0.83

0.50 0.38 0.25 0.12 0.88 0.82 0.78 0.76

0.50 0.33 0.17 0.84 0.76 0.71 0.67

0.50 0.25 0.79 0.68 0.62 0.57

0.50 0.73 0.60 0.51 0.46

3.5 4.0 5.0

1.00 1.00 1.00

0.99 0.99 0.99

0.97 0.97 0.97

0.94 0.93 0.93

0.88 0.88 0.87

0.82 0.81 0.80

0.74 0.73 0.71

0.65 0.63 0.61

0.54 0.52 0.49

0.42 0.36 0.35

M A aa ~ h . . . . . . . . . . . . . . . . . (40)

~ ( _t_ ) 2 + 8 n w hd' 1 - 2 h (41)

t + 8 n w h

When the section is subjected to bending with normal force and the tension steel remains in tension, in eq. ( 40) Mmust be replaced by N ea and A by i A, where ea and i have the same meaning as defind in 4.2.

5. CHARTS.

Based on the derived equations for rectangular sections it is possible to arrange tables or charts which give the relationship between the coefficients t ~. n w, Ca and Cb. From the equations we can

15

see that ~. nw, Ca and Cb can be calculated for variable values of~. provided that 6 and d'/h are known. A computer program has been prepared by which the precented charts have been obtained using IBM 1130. In the computation of the charts a fixed value of d'/h = 0.10 has been used, which in general will be satisfied by beams in practice. Furthermore, the following compression steel ratios have been used 6 = 0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.25, 1.67 and 2.5.

Concerning the relationship between Ca and 100 nw which is shown in Chart II we can state the following. Following eq. (21), the relationship between Ca and 100 nw can be written in the following expression:

= J100 J 1 ~ 11 J 1 ~ · 100nw 100 nw

which is valid for values of ~ encountered in practice. From the above expression we can see that the relationship between Ca and 100 nw will be represented by lines which are practically straight lines if the coordinates are chartered following a logarithmic scale, like shown in Chart II.

From both charts it can be further noticed that a certain value of Ca or Cb gives a certain value of 100 nw for the known 6. This means that for the analysis it is safer to use too low estimated values of n than too high estimated values.

In the following the simple use of the charts will be reviewed:

Computation of stresses.

For pure bending compute 100 nw from eq. (IO); for bending with normal force estimate first~ to obtain i from eq. (28), then compute 100 nw from eq. (30). Compute 6 from eq. (9) for pure bending and from eq. (29) for bending with normal force. Next we can choose one of the two charts to proceed. However, it is better to use both of the charts to obtain better control of the results. With the known 100 nw and 6 from Chart I we get Cb, ~and~ and from Chart II we get Ca. ~and~ (as a check). For bending with normal force the obtained ~ must be in accordance with the estimated value, otherwise the analysis must be repeated started with a corrected value of ~- The working stresses can now be calculated from eq. (24a) and (25). Although stresses may also be calculated from eq. (24a) and (22), however this procedure is not recommended as it gives less accurate results. If a flanged section is being considered, assume first that ~ .;;;; t/h and treat the section as a rectangular section with width bm. Check from charts that ~is really .;;;; t/h; if it is so the analysis can be proceeded. If apparently the neutral axis falls below the bottom of the flange, estimate then~ to obtain A from Table I for the known t/h and bm/b0 . Treat the section as a rectangular section with width b = A bm. The analysis must show the correctness of the estimated~. otherwise the analysis must be repeated starting with a corrected value of ~-

Design.

As the new Indonesian Reinforced Concrete Code 1971 still allows the use of the elastic theory for the design of sections, both of the charts can serve that purpose ea:sily, with due consideration of the limitation given in 6.

Compute Ca or Cb from eq. (20) and (22). Estimate 6, taking into consideration ductility requirements and/or the value of the opposite moment. With the known Ca or Cb and 6 from the charts we get 100 nw, ~ and ~- For pure bending compute tension steel from eq. (I 0) or more accurate from eq. (24) after aa is being known. The compression steel follows from eq. (9). The working stress at the obtained steel areas can be calculated from eq. (24a) and (25) For bending with normal force compute first i from eq. (28) and then tension steel from eq. (30) and compression steel from eq. (29), while the working stress at the obtained steel areas can be calculated again from eq. (24a) and (25).

Coefficient Ca is chosen as a strarting point of the analysis if the steel stress is aimed to be fully utilized (aa = U'a), which can be accomplished for pure bending, provided that the amount of compression steel is sufficient; if it is not sufficient then the steel stress can no longer be fully utilized (aa < aa). Coefficient Cb is chosen as a starting point of the analysis if the concrete stress is aimed to be fully utilized (Ub' = CJ6), which may be the case for bending with normal compressive force.

For bending with normal force, where symmetrical reinforcement is required, ~ must be estimated first to compute 6 from eq. (33). The analysis must show the correctness of the estimated value, otherwise the analysis must be repeated starting with a corrected value of ~-

For bending with normal compressive force with small eccentricities, by fully utilizing the steel stress it is possible to get a value of ~ equal to ea/h, so that i = oo and A = 0. This means that the application point of the normal force N coincides with the application point of the resultant compressive force D. Another

Clr.arr I Cltaris for '"~ eta.~iic a.n.a!y~i.s e~F reinForced COI'Ic,.e-le ~ec-lion.s

z.o

So \. \. \. \\\

1\. cJI'-.\. \. \. \....,. \ .:r --+-----jf--H-+-+-+-++++-+-f-+-+-f--+-+---1 \. _\.\. \. \~ \""

t----t---+~~--l---"~d' '\.'\:~ ·- \ \ \ 1----+----+--"-".---~-"'~~ ~ \ \ \ '1----+---+--~~,, '\.~ ~ \ \ 1\

~..3' "\. ~ \ \ \ \

'\. '\~\ \\ \ '\ \.~\\ \ '

I-~ £?6+---+--+--+--+---1 ~S"

().-If)

4.'J6 -f.O

"-Jf-:t ay:z \1"1

D.'JO -1.~ '-~ 1;:: ~ 1/ L.d/1/ ,

1--r- v~--'VI/ V I) "f.D. '

;, \.\. .\

Ch = {ji i i '=- { " \~~\~ ,., {- t:- "''\\

fib'b ejJ 1\\ ~\ v Q v'- r- d'/h v.' _ r- "l.t rz '-'l\\

a- f n '/J - (- r a. ~'\

For pure 6end.it.~ : e .. = oo i i == 1' ' ~ F~r ben.da#a.! will, nor,_/ !Dree : 11~ N. e... ~ .,

~\

.o ~.0 .

... A~·

,~I\ 2.2 ~~--r-~~~~~\\-4~~

'\~\ \

t-+-HH-i-++++1f--+-+-+~++~ ~f---1--++ o = A' S:2.s A

.6,

-r-t~~Dj!!.H-t1 ......... ,//II

';ooi I;' 1/ II

111

v

1\.

\

.z.6 \

~:2. +--+--+-+--1

18

possibility is that ~ > ea/h, which gives a negative value of i. This means that the position of the application point of the normal force N is lower than the application point of the resultant compressive force D. This condition is impossible since it violates the equilibrium condition. Equilibrium can be restored only if the application point of the compressive force D comes lower, at least until it reaches the same level as the application point of the normal force N. The position of the application point of the compressive forceD can be lowered down by increasing the value of ~. For a constant ab the value of~ can be increase d only by decreasing the value of aa. This means that for small eccentricities of the normal compressive force both the concrete stress and the steel stress can not be fully utilized.

6. LIMITATIONS FOR THE DESIGN OF SECTIONS USING THE ELASTIC THEORY.

If the design of a section is based on the elastic theory, the new Indonesian Reinforced Concrete Code 1971 stipulates some limitations. In the following limitations for pure bending will be discussed.

First of all a minimum reinforcement is stipulated in relation to the prevention of brittle fracture. It is known that when the bending moment applied to a section reaches its cracking moment value, a sudden rise in the tension steel stress occurs due to the abrupt change in effectiveness of the section from the uncracked to the cracked condition. If the amount of tension steel is too small the steel stress will suddenly rise beyond its breaking strength producing the so called brittle fracture. To prevent this phenomena the new Code specifies a minimum area of tension reinforcement Amin = (12/ atu) b h, which gives a minimum tension steel coefficient of:

~in bh

(42)

In Table II values of UWin according to eq. (42) are listed for standard qualities of steel according to the new Code.

Table II

u aa'li 100. wmin (kg/cm 2 )

u 22 1910 0.628 u 24 2080 0.577 u 32 2780 0.432 u 39 3390 0.354 u 48 4170 0.287

Another limitation according to the new Code is the maximum reinforcement. This maximum reinfircement has been derived from limitations obtained from ultimate strength analysis of sections [2]. In terms of the n-method this limitations may be formulated as follows. For beam sections in general the area of tension steel minus the area of compression steel must not exceed 150% of the area of single reinforcement at elastic balanced condition, while for member sections of lateral force resisting structures the above percentage is I 00%.

At elastic balanced condition, where both the steel and the concrete stress reach the maximum allowable stress, the neutral axis coefficient is:

~0 + f3o

--~-=-...... . I+~

nab

(43)

so that the tension steel coefficient of a single reinforced section at elastic balanced condition is obtained by substituting eq. (43) in eq. (15) with 5 = 0, which gives:

19

. . . . . . . . . . . . . . (44)

Maximum reinforcement according to the new Code may now be expressed as follows:

- for beam sections in general:

0.75 . . . . . . . (45) a a

(1 + _a) --!- (1 - 8) nab nq;

- for member sections of lateral force resisting structures:

= 1.0 nw0

1 - 8 -------=0=.5"------- . . . . . . . (46)

a a (1 + _a)~(1-8)

nab nab

In Table III and Table N maximum tension steel coefficients according to eq. ( 45) and ( 46) are listed.

Table III

100 n wmax for beams in general

----- aa! n at, 8 --~ 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

0 52.1 37.5 28.4 22.3 18.0 14.9 12.5 10.6 9.2 0.2 65.1 46.9 34.3 27.9 22.5 18.6 15.6 13.3 11.5 0.4 86.8 62.5 47.4 37.2 30.0 24.8 20.8 17.8 15.3 0.6 130.2 93.8 71.0 55.8 45.1 37.2 31.2 26.6 23.0 0.8 260.4 187.5 142.0 111.6 90.1 74.4 62.5 53.2 46.0 1.0 00 00 00 00 00 00 00 00 00

Table IV

100 n wmax for lateral force resisting structures

~ 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

0 34.7 25.0 18.9 14.9 12.0 9.9 8.3 7.1 6.1 0.2 43.4 31.2 23.7 18.6 15.0 12.4 10.4 8.9 7.7 0.4 57.9 41.7 31.6 24.8 20.0 16.5 13.9 11.8 10.2 0.6 86.8 62.5 47.3 37.2 30.0 24.8 20.8 17.8 15.4 0.8 173.6 125.0 94.7 74.4 60.1 49.6 41.7 35.5 30.6 1.0 00 00 00 00 00 00 00 00 00

20

7. EXAMPLES.

Example 1.

Given

Find

Solution

Example 2.

Given

Find

Solution

Example 3.

Given

Find

Solution

Frame section with b = 30 em, h =55 em, A= 22.81 cm2 , A'= 14.18 cm2 , M = 21400 kgm for temporary loading, abk = 175 kg/cm2 •

Working stresses.

Obk = 175 kg/cm2 , temporary loading: n = 16.

100 nw = 100 n _A_ = 100 x 16 x 22.81 - = 22.2. b h 30x 55

~ = ..A:_ = 14·18 = 0.62 :::::: 0.6 A 22.81

With 100 nw = 22.2 and 8 = 0.6 from Chart 1: Cb = 1.80, ~ = 0.42, ~ = 0.874 and from Chart II: Ca = 2.28, ~ = 0.42 (o.k.), ~ = 0.874 (o.k.).

Working stresses:

a a M 2140000 1950 kg/cm2

A~h 22.81 x 0.874x55

q; t a a 0.42 X 1950 88 kg/cm2 • 1- ~ n 0.58 16

a' t - d'Lh a a 0.32

X 1950 1080 kg/cm2 . a 1 - ~ 0.58

Column section with b = 40 em, h= 35 em, A= 10.05 cm2 , A' = 8.04 cm2 , N ea = 4660 kgm (including buckling effects) and N = 3500 kg for permanent loading, a{,k = 175 kg/cm2 •

Working stresses.

a{,k = 175 kg/cm2 , permanent loading: n = 24.

ea 4660 = 1.33 m. Estimate ~ = 0.87. 3500 __ I __

1-~_h_ ea

1 - 0.87 X l_i 133

1.30

100 nw = 100 n i A = 100 X 24 X 1.30 X 10.05 b h 40 X 35

8 = _A'_ = 8·04 = 0.6. i A 1.30x 10.05

22.4

With 100 nw = 22.4 and 8 = 0.6 from Chart 1: Cb = 1.78, ~ = 0.42, ~ = 0.873 and from Chart II: Ca = 2.25, ~ = 0.42 (o.k.), ~ = 0.873 (o.k.). Hence, the obtained~ is in accordance with the estimated (o.k.).

Working stresses:

a a N ea 466000 1170 kg/cm2 • iA~h 1.30 X 10.05 X 0.873 X 35

a'b _L_~ J11L_ X ___l!lQ_ 35 kg/cm 2 • I - ~ n 0.58 24

a' = t - d'Lh a a = ____Q]_.L_ X 1170 645 kg/cm2 . a I - ~ 0.58

Beam section with b = 35 em, h = 65 em, A= 38.01 cm2 , A'= 20.11 cm 2 , N ea = 19400 kgm and N = -15300 kg (tension) for permanent loading, abk = 225 kg/cm2 •

Working stresses.

a'bk 225 kg/cm2 , permanent loading: n = 21.

ea 19400 -1.27 m. Estimate ~ = 0.88. -15300

1 - 0 88 X 0·65 . -1.27

0.69

Example 4.

100 nw = 100 n ~ ~ = 100 x 21 x

8 = _K_ = 20.11 = 0.8. i A 0.69 x 38.01

0.69 X 38.01 35 X 65

24.2.

21

With 100 nw = 24.2 and 8 = 0.8 from Chart 1: Cb = 1.68, ~ = 0.41, ~ = 0.878 and from Chart II: Ca = 2.16, ~ = 0.41 (o.k.), ~= 0.878 (o.k.). Hence, the obtained ~is in accordance with the estimated (o.k.).

Working stresses:

a a N ea 1940000 1290 kg/cm2 •

i A ~ h 0.69 X 38.01 X 0.878 X 65

a' ~ ~= 0.41 X

1290 43 kg/cm2 • b I - ~ n 0.59 21

a' I - d'Lh a a

0.31 X 1290 = 680 kg/cm2 • a I - ~ 0.59

Given Flanged section of beam with b0 = 40 em, bm = 160 em, h = 115 em, t = 12 em, A= 35.47 cm2 , A'= 13.90 cm2 , M = 68500 kgm for permanent loading, at,k = 225 kg/cm2 •

Find

Solution

Example S.

Given

Find

Solution

Working stresses.

06k = 225 kg/cm 2 , permanent loading: n = 21. t/h = 12/115 = 0.10. Assume neutral axis falls within the flange: b = bm = I60 em.

100 nw = 100 x 21 x 35.47 = 4.05 · 100x115

8 = _K_ = 13.90 = 0.4. A 35.47

With 100 nw = 4.05 and 8 = 0.4 from Chart 1: Cb = 2.80, ~ = 0.24 > t/h = 0.10. Hence, assumption was wrong.

Estimate now~= 0.25. With t/h = 0.10, bm/b0 = 160/40 = 4.0 and ~ = 0.25 from Table 1: A= 0.73, so that b =A bm = 0.73 x 160 = I17 em.

100 nw = 100 n _A_ = 100 x 21 x 35.47 = 5.54. bh 117xll5

With I 00 nw = 5.54 and 8 = 0.4 from Chart 1: Cb = 2.60, ~ = 0.27 (approx. in accordance with the estimated value, o.k.), ~ = 0.91 and from Chart II: Ca = 4.40, ~ = 0.27 (o.k.), ~= 0.91 (o.k.).

Working stresses:

a a M 6850000 = 1840 kg/cm2 .

A~h 35.47 X 0.91 X 115

a' ____L_ _5__ = 0.27 X 1840 32 kg/em: b 1 - ~ n 0.73 21

a' ~ - d'/h a = 0.17 X I840 430 kg/em: a I - ~ a 0.73

Beam section with b = 30 em, h =55 em, M = I5300 kgm for permanent loading, concrete quality K 225, ¢ = I and steel quality U 39.

Reinforcement using the n-method.

Permanent loading, K 225, ¢ = I : a\) = 75 kg/cm2 , n = 21. U 39 : aa = 2250 kg/em:

The steel stress is going to be fully utilized.

= h 55 - 2.52. J 21 X 15300 2250 X 0.30

22

Example 6.

Given

Find

Solution

Taking the value of the opposite moment and ductility requirements into consideration, it i~ estimated sufficient to use 6 = 0.2. With Ca = 2.52 and 6 = 0.2 from Chart II: ~ = 0.43, .

Working stresses:

Db = _L 0 a = 0.43 X 2250 = 81 kg/cml > Ob. 1- ~ n 0.57 21

This means that with 6 = 0.2 the steel stress can not be fully utilized.

Estimate now 6 = 0.6. With Ca = 2.52 and 6 = 0.6 from Chart II: ~ = 0.39, t = 0.878, 100 nw = 18.1.

Working stresses:

a.; = _t_ oa = 0.39 X 2250 =

1 - ~ n 0.61 21 69 kg/cm2 (o.k.).

a' = ~- d'/h oa = Jhf2..__ X 2250 = a 1 - ~ 0.61 1070 kg/cm2 (o.k.).

Reinforcement:

A M = 1530000 aa r h 2250 X 0.878 X 55

A' 6 A = 0.6 x 14.1 = 8.5 cm2 •

Check of limitation:

U 39, from Table II: 100 n wmin = 21 x 0.354 = 7.4

< 100 nw (o.k.).

6 = 0.6, 'Oa/n 'Ob = 2250/21 x 75 = 1.4. From Table III: 100 nwmax = 55.8 > 100

nw (o.k.).

Suppose now symmetrical reinforcement is required; with Ca = 2.52 and 6 = 1 from Chart II; ~ = 0.36, t = 0.885, 100 nw = 17.8.

Working stresses:

~ ~ 0.36 2250 60 kg/cm2 (o.k.). ut, = 1--=1 = X = n 0.64 21

a' = ~- d'/h a a = 0.26 X 2250 = 910 kg/cm2 (o.k.). a 1 - ~ 0.64

Symmetrical reinforcement:

A= A'= M = ----=1=53::..c0:..;:0:.;:::0=0 ____ = 13.9 em~ 2250 X 0.885 X 55

Limitation of reinforcement need not be considered.

Column section with b = 45 em, h = 40 em, N ea = 12200 kgm (including buckling effects) and N = 28300 kg (compression) for temporary loading, concrete quality K 125, q, = 1.08, steel quality U 32.

Symmetrical reinforcement using the n-method:

Temporary loading; K 125, q, = 1.08 : 0(, = 37 kg/cm2 , n = 20

U 32 : oa = 2650 kg/cm2 •

e = 16200 a 28300

= 0.43 m.

Assume the steel stress can be fully utilized.

= h 40 J 20 X 12200

2650 X 0.45

= 2.80

Estimate r = 0.87, ~ = 1 - r (h/ea) = 1 - 0.87 (40/43) = 0.2.

With Ca = 2.80 and ~ = 0.2 from Chart II: ~ = 0.40, t = 0.87 (same as estimated, o.lc).

Working stresses:

a: = _L_ (ja = 0.40 x 2650 = 90 kg/cm2 > lJ.' -o 1-~ n 0.60 20 -o

23

1his means that for symmetrical reinforcement the steel stress can not be fully utilized.

= h

j _Nea

<Ji,b

40

j 12200 37 X 0.45

= 1.48

Estimate r = 0.83, ~ = 1 - r (h/ea) = 1 - 0.83 (40/43) = 0.23 ~ 0.2.

With Cb = 1.48 and ~ = 0.2 from Chart 1: ~ = 0.59, t = 0.826 (approx. in accordance with the estimated value, o.k.).

Working stresses:

a a = 1 - ~ n U-j, = 0.49 X 20 X 37 ~ 0.59

= 514 kg/cm2 < oa (o.k.).

o'b ~ - d'/h not,= 0.49 X 20 X 37

~ 0.59 = 615 kg/cm2 < oa (o.k.).

Reinforcement:

= _1_ = _1_ = 5. ~ 0.2

i A = ~= 1220000 = 71.8 em~ aar h 514 X 0.826 X 40

A ___l_LL = 5

14.4 em~

A' = ~ i A = 0.2 X 71.8 = 14.4 cm2 (o.k.).

24

8. REFERENCES.

[1] "Indonesian Reinfor:ced Concrete Code 1971, NI 2" (in Indonesian). Ed. Building Research Institute, Bandung, Oct. 1971.

[ 2] ''Explanation and discussion of the Indonesian ~einforced Concrete Code 1971" (in Indonesian), Ed. Building Research Institute, Bandung, Sepl _1971.

(3] Wiratman Wangsadinata: "Bending for rectangular section in the cracked stadium" (in Indonesian), Reinforced Concrete II, Dept. of Civil Eng. ITB Pub!., 1963.

(4) Wiratman Wangsadinata: ''Basic principles for the newlndonesianconcrete code" (in Indonesian), Concrete Symposium, Bandung, Jan. 1970.

(5) Wiratman Wangsadinata: ''A general method for the analysis of reinforced concrete section subjected to bending based on the elastic theory and the influence of 'n' towards the design" (in Indonesian), Concrete Symposium, Bandung, Jan. 1970.

(6] Wiratman Wangsadinata: "A general method for the analysis of reinforced concrete·section subjected to bending with an arbitrary value of n" (in Indonesian), 2nd. Seminar on orderly construction, Bandung, April 1970.

[ 7) Wiratman Wangsadinata: "flexural analysis by means of the n-method" (in Indonesian), Building Research Institute Publ., Bandung, Ocl 1970.

[ 8] Wiratman Wangsadinata: "Structural safety in the design of reinforced concrete" (in Indonesian), Building Research Institute Pub!., Bandung, Oct. 1970.

[ 9] Wiratman Wangsadinata: "Ultimate strength analysis of reinforced concrete sections", to be published, 1972.

(1 0] Roosseno, R.: "Reinforced concrete"'(in Indonesian), P.T. Pembangunan, Djakarta, 1954. [ 11] Soemono, R: "Reinforced concrete nomograms" (in Indonesian), Djambatan, Djakarta, 1961. [ 12] Depl of Public Works: "Concrete handbook" (in Indonesian), Djakarta, 1966. (13] Emperger, F.: "Normalisatie van de factor n bij gewapend beton", Beton No. 12, Bijlage· De Ingenieur,

4 Dec. 1936.

(14] Morsch, E.: "Die Bemessungin Eisenbetonbau", 5. Aufl., Wittwer, Stuttgart, 1950. (15) Saliger, R: "Der Stahlbetonbau", 7. Aufl., Deuticke, Wien, 1949. (16] Pucher, A.: "Lehrbuch des Stahlbetonbaues", 2. Aufl., Springer, Wien, 1953. [17) LOser, B.: "Bemessungsverfahren", 15. Aufl., Ernst & S., Berlin, 1955. (18] Schrier, W.v.d.: "Gewapend beton", 9e dr., Argus, Amsterdam, 1953. [ 19] Plantema; Bisch: "Beton grafieken", L.J. Veen, Amsterdam, 1950. (20] Borges, J.F.: "Cracking and deformability of reinforced concrete beams", Int. Ass. for Bridge and

Structural Engineering, Pub!. Vol. 26, Zurich, 1966. [21] Cismigiu, A; Titaru, E; Velkov, M.: "Energetic interpretation of the structure beha.viour during the

earthquake of 26 July 1963 in Skopje and conclusions concerning the elasto-plastic design", Institute of Seismology, Earthquake Engineering and Town Planning, Univ. of Skopje, Pub!. No.5, Jan. 1967.

[22] Dunham, C.V.: "Theory and practice of reinforced concrete", McGraw Hill, 1953. (23] Peabody, D.: "The design of reinforced concrete structures", 2nd. ed., John Wiley, 1956. [24] Cernica, J.N.: "Fundamentals of reinforced concrete", Addison-Wesley, 1964. (25] ACI Committee 317: "Reinforced Concrete Design Handbook", 2nd ed., 1955. (26] AIJ Structural Standards Committee: "Architectural Institute of Japan Structural Standards", Tokyo,

1961.

(27] CEB: "Praktische richtlijnen voor de berekening en uitvoering .van gewapend-betonconstructies", . Betonvereniging, Okt. 1%7.

-=:~~-: :..·-- -

TGL. PINJAM l HAHUS KEMBALI TGL. KEMBALI

+ ~(-

KLAS

PENGARANG

JUDUL

No. STB.

Nama Peminjam

rY i ~~ ~? --tt ) f - (.,-(' '7\7 •

._ d () l

: Wiratmon Wangsa.dinata. CE*' .Elastic analysis of reinforced · concrete sections•

: 164/79•' Ala mat Peminjam

Tanggal Peminjaman

-·

Tanggal Kembali

M.ILIK. PER PUSTA~~AN iWSLITIA.G t\1