Frame Analysis Design of Reinforced Concrete Two-Way Slabs A

7/24/2019 Frame Analysis Design of Reinforced Concrete Two-Way Slabs A

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Chapter : Structural, On-shore For additional information on this subject, contact

File Reference: CSE10803 C.C. Baldwin on 873-1567

Engineering EncyclopediaSaudi Aramco DeskTop Standards

Design Of Reinforced Concrete Two-Way

Slabs and Columns; Frame Analysis Techniques


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Engineering Encyclopedia Structural, On-shore

Design of Reinforced Concrete Two-way


Saudi Aramco DeskTop Standards

Contents Pages

INTRODUCTION TO DESIGN OF TWO-WAY SLABS ............................................................... 1

Direct Design Method......................................................................................................... 1

Equivalent Frame Method................................................................................................... 6

Yield Line Method.............................................................................................................. 8

General .................................................................................................................. 8

Yield Line Patterns................................................................................................ 8

Analysis Methods ............................................................................................................... 9

Slab Reinforcement ............................................................................................. 11

COLUMN DESIGN ....................................................................................................................... 12

Column Design Basis ....................................................................................................... 12

Determining Slenderness Effects......................................................................... 13

Using Axial Load-Moment Interaction Diagrams............................................................. 19

DETERMINING REINFORCING STEEL REQUIREMENTS ..................................................... 25

FRAME ANALYSIS TECHNIQUES ............................................................................................ 29

Beam Formulas ................................................................................................................. 30

ONE-STORY FRAMES................................................................................................................. 33

Computer Programs .......................................................................................................... 36

WORK AID 1: RECTANGULAR COLUMN: LOAD-MOMENT INTERACTION

DIAGRAMS .................................................................................................... 37

COLUMNS 7.4.3Load-moment strength interaction diagram for R4-60.75

columns ....................................................................................... 37

COLUMNS 7.4.4Load-moment strength interaction diagram for R4-60.90

columns ....................................................................................... 38

WORK AID 2: SPIRALLY REINFORCED COLUMN: LOAD-MOMENTINTERACTION DIAGRAMS......................................................................... 39

COLUMNS 7.23.1Load-moment strength interaction diagram for C5-

60.45 spirally reinforced columns ............................................. 39

COLUMNS 7.23.1Load-moment strength interaction diagram for C5-

60.60 spirally reinforced columns ............................................. 40


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Saudi Aramco DeskTop Standards

WORK AID 3: MINIMUM FACE DIMENSIONS OF RECTANGULAR TIED

COLUMNS ACCOMMODATING VARIOUS NUMBERS OF BARS

PER FACE....................................................................................................... 41

WORK AID 4: FRAME 39, SYMMETRICAL RECTANGULAR TWO-HINGED FRAME, CASE #1 .......................................................................... 42

WORK AID 5: FRAME 39, SYMMETRICAL RECTANGULAR TWO-

HINGED FRAME, CASE #2-4 ....................................................................... 43


HINGED FRAME, CASE #5-7 ....................................................................... 44


HINGED FRAME, SYMMETRICAL LOADING - CASE #8-

10 ..................................................................................................................... 45

WORK AID 8: FRAME 41, SYMMETRICAL RECTANGULAR FULLY-FIXED FRAME, CASE #1 .............................................................................. 46

WORK AID 9: FRAME 41, SYMMETRICAL RECTANGULAR FULLY-

FIXED FRAME, CASE #2-3........................................................................... 47


FIXED FRAME, CASE #4-5........................................................................... 48


FIXED FRAME, CASE #6-7 ....................................................................... 49


FIXED FRAME, CASE #8-9 ....................................................................... 50

WORK AID 13: APPENDIX LOAD TERMS; GENERAL NOTATIONS............................. 51

WORK AID 14: APPENDIX LOAD TERMS; NOTATIONS, CASE 1-3.............................. 52

WORK AID 15: APPENDIX LOAD TERMS; GENERAL NOTATIONS, CASE 4-6........... 53

WORK AID 16: APPENDIX LOAD TERMS; NOTATIONS, CASE 7-11............................ 54

GLOSSARY................................................................................................................................... 55


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Saudi Aramco DeskTop Standards 1

Introduction to Design of Two-Way Slabs

In CSE10802, different types of slabs were identified and the methodology for design of one-way slabspresented. For two-way slabs, however, the methods for determining the distribution of moments throughout

the slab are considerably more complex than for beams or one-way slabs. Consequently, the analysis and

design of two-way slab systems is usually performed using computer programs.

The following three methods for performing two-way slab analysis and design will be outlined in this module.

The first two methods are specified in the ACI 318 Code and are based on elastic analysis; the third is a

collapse method of analysis. An introduction to some of the concepts employed in the methods is presented

below. Details of all three methods, however, are considered beyond the scope of this course:

Direct Design Method

Equivalent Frame Method

Yield Line Method

Code-specified minimum thickness of slabs and deflection limitations will also be presented, as will some

details of slab reinforcement.

The designer must understand the concepts and provisions governing the analysis and design of two-way slabs.

This understanding is necessary to develop the necessary input data for computer programs and to be able to

hand-check the validity of the output. For a few limited cases, the Direct Design Method (DDM) per the Code

can be used to analyze and design the slab without a computer. Of equal importance, the designer can use thesteps and concepts contained in the DDM to evaluate computer-aided designs.

Direct Design Method

This semiempirical method consists of a set of rules for the proportioning of slab and beam sections to resist

flexural stresses.

Step 1 - Limitations Check

The first step checks the validity of certain assumptions stated in the Code. These critical limitations are stated

in Figure 1. If any of these assumptions are not true, the designer must resort to an alternative approach.


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Step 2 - Thickness/Shear Check

The second step selects a trial thickness and checks the perimeter or punching shearcapacity at the critical

section near each column. Refer to ACI Code for details. For two-way slabs, Section 9.5.3 of the Code

provides guidelines defining the minimum thickness for deflection and serviceability requirements; theguidelines are also summarized in

Figure 2. For slabs without drop panels, h 120 mm; for slabs with drop panels, h 100 mm per ACI 318 M.

For two-way slabs, a good first estimate is: h/l -- 1/20 to 1/30

The trial thickness can be based on guidelines in Figure 2 (serviceability requirements) or on experience.

Usually, a uniform slab thickness is selected and checked for shear. Note that an allowance should be made for

the slab-to-column moment transfer capacity, which is to be checked.

Step 3 - Detailed Design

Steps in the detailed design procedure are given in the ACI Code. Detailed design is beyond the scope of this

course.

Slabs with thicknesses less than the minimum given in Figure 2 may be used if it can be shown that deflections

will not exceed the limits specified in Figure 3. Deflections shall be computed taking into account the size and

shape of the panel, the conditions of support, and the nature of restraint at panel edges.


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DIRECT DESIGN METHOD LIMITATIONS

ACI 318-89, Section 13.6.1

Step 1 Determine whether slab geometry and loading satisfy conditions for use of

Direct Design Method.

A) Minimum of three continuous spans in each direction.

B) Panels shall be rectangular with ratio of longer to shorter span within a

panel not greater than 2.0.

C) Successive span lengths in each direction must not differ by more than

one-third of longer span.

D) No column offset more than 10% of span in the direction of offset.

E) All loads are due to gravity loading only and are uniformly distributed.

The services live loads are not greater than three times total dead load.

F) If the panel is supported by beams on all sides, the relative stiffness of the

beams in two perpendicular directions shall not be less than 0.2 nor greater

than 5.0.

G) No negative moment redistribution is permitted.

H) Variations to the above are permitted if:

- Conditions of equilibrium and compatibility are met.

- Design strength is at least equal to required strength.

- Serviceability conditions are met.

FIGURE 1


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TABLE 9.5 (c) MINIMUM THICKNESS OF SLABS WITHOUT INTERIOR BEAMS

Without drop panels With drop panels

Yield Stress

fy,

Exterior panels Interior

panels

Exterior panels Interior

panels

psi Without

edge beams

With edge

beams

Without

edge beams

With edge

beams

40,000 l n

33

l n

36

l n

36

l n

36

l n

40

l n

40

60,000 l n

30

l n

33

l n

33

l n

33

l n

36

l n

36

Min.

Thk.

9.5.3.2

5 in. (120 mm) 4 in. (100 mm)

Authorized reprint from ACI 318-89, Table 9-5(c), Page 99, with permission from the American Concrete Institute.

FIGURE 2


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TABLE 9.5(b) MAXIMUM PERMISSIBLE COMPUTED DEFLECTIONS

Type of member Deflection to be considered Deflection limitationFlat roofs not supporting or

attached to nonstructural

elements likely to be damaged

by large deflections.

Immediate deflection due to live

load L l

180

*

Floors not supporting or

attached to nonstructural

elements likely to be damaged

by large deflections.

Immediate deflection due to live

load L l

380

Roof or floor construction

supporting or attached to

nonstructural elements likely tobe damaged by large

deflections.

That part of the total deflection

occurring after attachment of

nonstructural elements (sum ofthe long-time deflection due to

all sustained loads and the

immediate deflection due to any

additional live load)

l

480

Roof or floor construction

supporting or attached to

nonstructural elements not

likely to be damaged by large

deflections.

l *

240

*

* Limit not intended to safeguard against ponding. Ponding should be checked by suitablecalculations of deflection, included added deflections due to ponded water,

and considering long-term effects of all sustained loads, camber, construction

tolerances, and availability of provisions for drainage.

** But not greater than tolerance provided for nonstructural elements. Limit may be

exceeded if camber is provided so that total deflection minus camber does not

exceed limit.

Authorized reprint from ACI 318-89, Table 9-5(b), Page 98, with permission from the American Concrete Institute.

FIGURE 3


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Equivalent Frame Method

The Equivalent Frame Method involves the representation of the three-dimensional slab system by a series of

two-dimensional frames which are then analyzed for loads acting in the plane of the frames. The negative and

positive moments so determined at the critical design sections of the frame are distributed to the slab sections in

accordance with rules in the ACI Code governing column strips, beams, and middle strips.

Application of the equivalent frame to a regular structure is illustrated in Figure 4. The three-dimensional

building is divided into a series of two-dimensional frame bents (equivalent frames) centered on column or

support centerlines with each frame extending the full height of the building. The width of each equivalent

frame is bounded by the centerlines of the adjacent panels. The complete analysis of a slab system for a

building consists of analyzing a series of equivalent (interior and exterior) frames spanning longitudinally and

transversely through the building.

The equivalent frame comprises three parts: (1) the horizontal slab strip, including any beams spanning in the

direction of the frame; (2) the columns or other vertical supporting members, extending above and below the

slab; and (3) the elements of the structure that provide moment transfer between the horizontal and vertical

members.

Details of the method are provided in Section 13.7 of the ACI Code.


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DEFINITION OF EQUIVALENT FRAME

l 1

l 1

l 1

l 2l 2l 2

l 2

l 22

SlabBeamStrip 2

Centerlineof Panel

Interior Equivalent Frame

ExteriorEquivalentFrame

Edge

CenterlineAdjacent Panel

Column Strip

One-Half Middle Strip

Authorized reprint from ACI 318-89, Figure 13.7.2.1, Page 231, with permission from the American Concrete Institute.

FIGURE 4


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Yield Line Method

General

In the conventional methods of slab design considered previously, the design moments and shears are

determined by simplified procedures which are based mainly on elastic concepts of slab behavior, with the final

proportioning of the slab to determine the reinforcement requirements carried out by means of the ultimate

strength method. An alternative approach to reinforced concrete two-way slab design is to consider the

inelastic distribution of moments and shears in the statically indeterminate structure just prior to collapse.

The proportion of flexural reinforcement which is used in the construction of a slab is typically much lower

than in beams and girders. This ensures a high degree of ductility, and since a slab is a highly indeterminate

structural system, an extensive redistribution of moments can thus occur prior to collapse. This very

satisfactory aspect of the overload behavior of slabs has led to the application of the basic concepts of simpleplastic theory to their analysis and design.

The collapse methods of slab analysis and design were originally developed in the Scandinavian countries, and

have obtained greater acceptance in Europe than in the

United States. One such method, the yield line theory, is an acceptable basis for slab design and is briefly

discussed below.

One of the important practical advantages in using plastic methods is that they can be applied to slabs of

irregular and complex shape. Therefore, they can be used to design slabs which cannot be designed by the

"elastic" methods. However, it must be emphasized that the collapse load methods are concerned solely with

the design objective of adequate strength; problems of serviceability, such as excessive deflection, needseparate and careful consideration. Although not included in the ACI Code, slab analysis by yield line theory

may be useful in providing the needed information for understanding the behavior of irregular or single-panel

slabs with various boundary conditions.

Yield Line Patterns

Yield line theory for two-way slabs requires a different treatment to that of limit (plastic) analysis of continuous

beams, because in this case the yield lines will generally not be parallel to each other, but will instead form a

yield line pattern. The entire slab area will be divided into several segments which can rotate along the yield

lines as rigid bodies at the condition of collapse or unstable equilibrium. Some yield line patterns for typicalsituations are shown in Figure 5.


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The slab of Figure 5(a) has nonparallel supports. At the collapse condition, this slab will break into two

segments; one segment will have an edge rotating about I and the other will have an edge rotating about II. The

positive moment yield line must then intersect lines I and II at their intersection, point 0. The exact position of

yield line III will depend on the reinforcement amount and direction, both in the positive and negative moment

regions.

For the case of Figure 5(b) where a rectangular panel is either simply supported or continuous over four linear

supports, the collapse mechanism consists of four slab segments. The exact locations of points aand bwill

depend on the moment strengths at the supports and the positive moment reinforcement in each direction.

The slab in Figure 5(c) is supported along two edges and, in addition, is supported by two isolated columns.

The rotational axes for the slab segments at collapse must occur along the supports (lines I and II), and

additional rotational axes must pass through the isolated columns. The critical position of the positive moment

yield lines, a, b, c, d and eis a function of the reinforcement amount and direction; in the meantime,

compatibility of deflection along the yield lines must be maintained during the rigid body rotations of the slab

segments.

For a concentrated load at a significant distance from a supported edge, the yield line pattern will be circular as

shown in Figure 5(d). The circular pattern will be a yield line of negative bending moment while the radial

yield lines are due to positive bending moment. For concentrated loads near a free edge, a fan or partial circular

pattern is typical.

Analysis Methods

There are two methods of yield line analysis of slabs: the virtual work method and the equilibrium method.

Based on the same fundamental assumptions, the two methods should give exactly the same results. In either

method, a yield line pattern must be first assumed so that a collapse mechanism is produced. For a collapse

mechanism, rigid body movements of the slab segments are possible by rotation along the yield lines while

maintaining deflection compatibility at the yield lines between slab segments. There may be more than one

possible yield line pattern, in which case solutions to all possible yield line patterns must be sought and the one

giving the smallest ultimate load would actually happen and thus should be used in design.

Most texts on advanced reinforced concrete design contain a discussion on the application of yield line

methods.


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TYPICAL YIELD LINE PATTERNS

III

O

(a)

Free Edge

Supported

Edges

Free Edge

III

e

a

b

c

d

II

IV

IIII

Supported Edges

Columns

(c)

(d)(b)

Negative

Moment

Y ield Line

Positive

Moment

Y ield Line

Four Supported Edges

a b

P

FIGURE 5


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Slab Reinforcement

When the shear design is satisfactory and the slab design moments and column moment transfer aredetermined, the designer selects the slab reinforcement. (Details are provided in the ACI Code).

The area of flexural reinforcement required to resist the factored moments in each strip is computed using the

flexure formulas/work aids for beams and one-way slabs. A uniform bar spacing is usually used within a given

strip. Note that reinforcement for any required code-specified additional column moment transfer must be

placed within a width not exceeding the column width plus one and one-half times the slab thickness on each

side of the column. Bar diameters and spacing are selected using the same considerations as for one-way slabs,

covering minimum reinforcement ratio for shrinkage and temperature, crack control, and minimum bar spacing.

Maximum bar spacing for two-way slabs shall not exceed two times the slab thickness. Bar development for

two-way slabs shall follow the provisions stated in Section 13.4 of the code (see Figure 13.4.8).


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Column Design

Column Design Basis

The design of columns and other structural members subjected to compressive loads is based on the same

assumptions covered in Module 2 in the discussion of ultimate strength design of beams. The procedures

described herein are based on the provisions in Section 10 of the Code. The design of compression members is

based on forces and moments determined from an elastic analysis. The designer needs to consider the influence

of axial loads and variable moment of inertia on member stiffness, the effect of deflections on moments and

forces, and the effects of the duration of loads (creep).

Column design notation is presented in Figure 6. Also, refer to Work Aid 1 in Module 2 for notation specific to

flexure and shear design.

For a short column without bending moments, the design axial load strength, Pn,is given by

For spiral reinforced:

Pn (max)= 0.85 0.85 fc' ( Ag- Ast) + fyAst

For tie reinforced:

Pn ( max )= 0.80 0.85 fc' ( Ag- Ast ) + fyAst

For members with spiral reinforcement conforming to ACI Code Section 10.9.3, is 0.75. For tie reinforcedmembers, is 0.7. See Section 9.3.2 of ACI Code.


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Determining Slenderness Effects

The Code defines a column as "short" based on a maximum value of klu/r as given below:

For columns braced against sidesway,

klu/r < 34 - 12 M1/M2

and for columns not braced against sidesway,

klu/r < 22

M1and M2are the smaller and larger factored end moments on the column, respectively. M1is positive if the

column is bent in single curvature and negative if the column is bent in double curvature. M2is always

positive.

The effective length factor, k, may be obtained from the nomograph shown on Figure 7. The terms Aand Bare the ratios of the sum of column stiffnesses, EI/lc, for each end joint divided by the sum of the beam

stiffnesses, EI/lb.

The unsupported length, lu, shall be taken as the clear distance between slabs, beams, or other members

providing lateral support. The radius of gyration, r, may be set equal to 0.3 times the overall dimension of the

rectangular member in the direction stability is considered, and equal to 0.25 times the diameter for circular

members. Note that if the moments at both ends of the column, M1and M2, are zero, the ratio M1/M2shall be

set equal to one.

If the value of klu/r exceeds that given above, column slenderness effects must be considered. Columns are

typically sized such that klu/r values are less than 50. Although permitted by the Code, experience indicates that

columns should not be sized such that klu/r values exceed 100.

The Code accounts for the effects of column slenderness by a moment magnification procedure provided in

Section 10.11. Columns are designed using the factored axial load and a magnified factored moment. Details

of this method are beyond the scope of this course.


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COLUMN DESIGN NOTATION

Ab

Ac

Ag

Ast

b

Cm

d

db

e

ex

ey

Ec

EI

=

=

=

=

=

=

=

=

=

=

=

=

=

Area of an individual bar.

Area of core of spirally reinforced columnmeasured to outside diameter of spiral.

Gross area of column cross section.

Total area of longitudinal reinforcement in

a column cross section.

Overall cross-section dimension of a

rectangular column.

A factor relating actual moment diagram to

an equivalent uniform moment diagram.

(For members braced against sidesway and

without transverse loads between supports,

Cm= 0.6 + 0.4 [Mlb/M2b] but not less

than 0.4. For all other cases, Cmshall be

taken as 1.0).

Depth to reinforcement.

Nominal diameter of bar.

Eccentricity of axial load at end of

member, measured from the centroid of the

tension reinforcement, calculated byconventional methods of frame analysis.

Eccentricity e along x-axis.

Eccentricity e along y-axis.

Modulus of elasticity of concrete, psi. (For

normal weight concrete. Ecmay be taken

as 57,000 fc'psi.)

Flexural stiffness term.

c

fs

fy

h

he

I

Ig

Ise

k

l

lc

ldb

ldc

lu

=

=

=

=

=

=

=

=

=

=

=

=

=

=

Specified compressive strength of

concrete.

Stress in reinforcement.

Specified yield strength of reinforcement.

Diameter of a round column or side of a

rectangular column.

Effective thickness of a column for

slenderness considerations.

Moment of inertia of section resisting

externally applied factored loads.

Moment of inertia of gross concrete

section about centroidal axis, neglecting

reinforcement.

Moment of inertia of reinforcement about

centroidal axis of member cross-section.

Effective length factor for compression

members.

Span length of beam or slab, as defined in

ACI 318-89, Section 8.7.

Height of column, center-to-center of

floors or roof.

Development length.

Development length for bars in

compression.

Unsupported length of member.

Authorized reprint from ACI, SP-17(91), Vol. 2, Notation, Page xi, with permission from the American

Concrete Institute.

FIGURE 6


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COLUMN DESIGN NOTATION (CONT'D)

Mc

Md

Mn

Mu

M1

M2

M2s

Pc

Pn

Pnx

Pny

Pno

=

=

=

=

=

=

=

=

=

=

=

=

Magnified factored moment to be used for

design of column.

Moment due to dead load.

Nominal moment strength at section.

Factored moment at section.

Value of smaller factored end moment on

compression member due to the loads that

result in no appreciable sidesway,

calculated by conventional elastic frame

analysis. Positive if member is bent in

single curvature, negative if bent in

double curvature.

Value of larger factored end moment on


result in no appreciable sidesway,

calculated by conventional elastic frame

analysis. Always positive.

Value of larger factored end moment on


result in appreciable sidesway, calculated

by conventional elastic frame analysis.Always positive.

Critical axial load.

Nominal axial load strength at given

eccentricity.

Nominal axial load strength for

eccentricity ey, along y-axis only,

x-axis being axis of bending.

Nominal axial load strength for

eccentricity ex along x-axis only.

y-axis being axis of bending.

Nominal axial load strength at zero

eccentricity, kips.

Pu

r

s

S

Sb

d

(gamma)

b(delta)

s

(xi)

g (rho)

(phi)

(psi)

=

=

=

=

=

=

=

=

=

=

=

=

=

Factored axial load at given eccentricity

Pn.

Radius of gyration of cross section of a

compression member.

Center-to-center spacing of bars.

Pitch of spiral, center-to-center of bar.

Clear spacing between bars.

Ratio of maximum factored dead load

moment to maximum factored total load

moment. Always positive.

Ratio of distance between centroid of

outer rows of bars and thickness of cross

section, in the direction of bending.

Moment magnification factor for columns

braced against sidesway.

Moment magnification factor for frames

not braced against sidesway, to reflect

lateral drift resulting from lateral and

gravity loads.

Dimensionless constant used in

computing Igand Ise.

Ast/Ag= ratio of total reinforcement area

to cross-sectional area of column.

Strength reduction factor as defined in

Section 9.3 of ACI 318-89.

Ratio of sum of stiffness _ (EI/l) of

compression members to _(EI/l) of

flexural members in a plane at one end of

a compression member.

FIGURE 6 (CONT'D)


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EFFECTIVE LENGTH FACTORS

A k B

A k B

50.0

10.0

5.0

3.0

2.0

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

1.0

0.9

0.8

0.7

0.6

0.5

50.0

10.0

5.0

3.0

2.0

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

50.0

10.0

5.0

3.0

2.0

1.0

9.0

8.0

7.0

0 1.0

20.0

3.0

2.0

1.5

100.0

30.0

20.0

4.0

6.0

10.0

5.0

4.0

50.0

10.0

5.0

3.0

2.0

1.0

9.0

8.0

7.0

0

100.0

20.0

4.0

6.0

(a) (b)

Braced Frames Unbraced Frames

= Ratio of (EI/lc) of compression members to (EI/lb) of flexural members in a plane at one end of acompression member.

k = Effective length factor.

For pinned support, = 10

For fixed support, = 1

FIGURE 7


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Example Problem 1

For a rectangular tied column with bars equally distributed along four faces, determine whether slenderness

effects can be ignored.

Given: Loading

Factored axial loadPu= 560 kips

Factored momentMu= 3920 kip-in.

Materials

Compressive strength of concrete f'c = 4 ksi

Yield strength of reinforcement fy= 60 ksi

Nominal maximum size of aggregate is 1 in.

Design conditions

Unsupported length of columns lu= 10 ft

Column is braced against sidesway

b = 1 6 I n .

h

h = 20 In.

e

P

u

= 5 6 0 k i p


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Example Problem 1 (Cont' d)

ACI 318-89

Reference Procedure Calculation

Handbook

Reference

Step 1--Determine column section size

Given: h= 20 in. b= 16 in.

10.11.4.1

10.11.5.1

10.11.3

10.11.4.1

10.11.2.1

10.11.4.1

Step 2--Check whether

slenderness ratio klu/ris

less than critical value. If

so, slenderness effects may

be neglected.

If not, slenderness effects

must be considered.

A) Compute M1/M2and

read critical value of klu/r

B) Compute klu/r and

compare with critical value;

determine whether slender-

ness effects must be

considered.

In this case, assume

conservatively that M1= M2=

3920 kip-in. Hence, upper limitof klu/r is 22.

For columns braced against

sidesway: k = 1.0

Given:

l u =10 ft =120 in.

r =0.3 h = 0.3 x 20 = 6.0 in .

kl u

r=

(1.0)(120 )

6 .0= 20

< 22

Slenderness effects may be neglected

Authorized reprint from ACI, SP-17(91), Vol. 2, Columns Example 1, Pages 1-3, with permission from the

American Concrete Institute.


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Using Axial Load-Moment Interaction Diagrams

As discussed previously, the strength of a member under axial load only, P n, and the strength of a member

under flexure only, Mn, are given by:

For tied columns

Pn = 0.80 0. 85 f c Ag Ast( )+ fyAst

For beams

Mn = fybd2

1 0.59fy

fc

For members subjected to both axial load and moment, the strength of the member depends on the interaction

between load and moment. As the applied moment increases, the axial load strength decreases. Due to the

nonlinear stress-strain behavior of concrete, the relationship between axial load strength and moment capacity is

nonlinear. Solutions to the tedious calculation for determining the axial load strength for a given moment

capacity are presented in the form of interaction diagrams or charts in the ACI SP-17 Design Handbook for

Columns. The charts cover both rectangular and circular columns, with varying 28-day strengths, reinforcement

arrangement, and yield strength. Note that these charts are applicable to the design of columns subjected to

moment about one axis or direction only.

A typical chart (from Work Aid 1) is shown on Figure 8. This chart is applicable to rectangular columns with

equal reinforcing in each face, fc' = 4 ksi, fy = 60 ksi, and = 0.75 referring to the ratio of the distance betweenreinforcing layers in opposite faces to the total thickness or dimension of the column.

Figure 8 relates values of the axial load strength Pn to the design moment strength (capacity)Mn, based onthe ratio e/h of load eccentricity to column thickness and ratio of the area of longitudinal reinforcing to the

column cross-section area, g. Generally the axial load and moment have been previously computed andtherefore are "known". The designer will assume a column shape and size, and reinforcing pattern, and will use

the charts to find the ratio .

Knowing the design moment and axial loads, the designer computes the abscissa and ordinate based on the

column size and enters the chart to obtain the reinforcing steel ratio required (g). Note that for a successfuldesign, the point on the chart corresponding to the combination of load and moment must fall below the solid

(or dashed) line corresponding to the strength curve for the steel ratio to be used.


23/58





LOAD MOMENT STRENGTH INTERACTION DIAGRAM

FOR R4-60.75 COLUMNS

Columns 7.4.3 Load-moment strength interaction diagram for R4-60.75 columns

References: ACI 318-89 Sections 9.3.2.2., 10.2, and 10.3: ACI Publication SP-7. pp 152-182

PnAg

eh

= MnA gh

,ksi

Authorized reprint from ACI, SP-17(90) Vol. 2, Columns 7.4.3, Page 82, with permission from the


FIGURE 8


24/58





Depending on the value of gobtained, the designer will revise his assumptions and reuse the chart until adesired value is found. The desired value of gis discussed in the following section.

A table of he/h values is provided on each chart. The value for hereflects the additional flexural stiffness

provided by the reinforcement and is used as an adjustment factor when computing slenderness ratios and

moment magnification factors in the event slenderness effects need to be considered.

Figure 9 illustrates some additional information regarding development and use of these charts. Example

Problem 2 indicates use of the charts to further design the column as defined in Example Problem 1.

For columns subjected to biaxial bending or moments about both column axes, such as corner columns in a

building, the axial load strength follows the form of the interaction surface. See the ACI Code and Design

Handbook for details.


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INTERACTION DIAGRAMS

P

o

Maximum Design Axial

Load Strength,

P

n m ax

Changes

Balanced Design

Design Moment Strength Index

Authorized reprint from ACI, SP-17(91) Vol. 2, Figure 1, page 206, with permission from the


FIGURE 9

1. The coordinates have been reduced by the appropriate values of the strength reduction factor inaccordance with Section 9.3.2.2 of ACI 318-89. Over most of the curve, is 0.75 for spirally reinforcedcolumns and 0.70 for tied columns. But increases toward 0.90 at low values of axial load, as providedin Section 9.3.2.2 of ACI 318-89, which accounts for the discontinuity at the low end of curves.

2. Curves of fs= 0 and fs = fyare given. The first of these two conditions represents the point at which the

stress in tension face steel changes from compression to tension as moment increases. The second

condition represents the point at which a longitudinal bar first reaches yield stress; that is, the point of

balanced load Pnb where simultaneously the concrete reaches a 0.3% strain and the steel reaches itstensile yield stress level. These values are useful in selecting splice lengths for bars in columns.

3. Dashed lines are used for curves for reinforcement ratios gof 0.05, 0.06, 0.07, and 0.08 to call attentionto the likelihood of difficulty at splices due to congestion of bars where ratios greater than 0.04 are used.

4. At the low end of the moment range, as moment decreases, design axial load strength Pnreaches amaximum and levels off at that value, in accordance with Section 10.3.5 of ACI 318-89.


26/58





Example Problem 2

This is the same column as presented in Example Problem 1.

For a rectangular tied column with bars equally distributed along four faces, select the

required area of reinforcement.

Given: Loading



Materials

Compressive strength of concretef'c= 4 ksi



Design conditions



b = 1 6 I n .

h

h = 20 In.

e

P

u

= 5 6 0


27/58






ACI 318-89


Handbook

Reference

Step 1--See Example Problem 1.

Step 2--See Example Problem 1.

9.3.2.2(b)

10.2.7

9.3.2.2(b)10.2

10.3

Step 3--Determine

reinforcement ratio using known values of

variables on appropriate

interaction diagram(s) and

compute required cross

section areaAst of

longitudinal reinforcement.

A) Compute PuAg

B) ComputeMu

Agh

C) Estimate h 5

h

D) Determine appropriate interaction diagram(s).

E) Read gforPu/Ag (

< Pn/Ag) andMu/Agh (< Mn/Agh)

F) Compute requiredAst from

Ast = gAg

Given:

Pu= 560 kips

Mu= 3920 kip-in

h= 20 in.

b= 16 in.

Ag=bx h= 20 x 16 = 320 in2

PuAg

= 560320 = 1.75 ksi

Mu

Agh=

3920

320 20= 0.61 ksi

20 5

20= 0.75

For a rectangular tied column with bars along

four faces, f'c= 4 ksi,

fy= 60 ksi, and an

estimated of 0.75, use R4-60.75.

ForPu/Ag= 1.75 from

Step 3A andMu/Agh=

0.61 from Step 3B:

g= 0.040

RequiredAst = 0.040 x 320

= 12.8 in.2

Work Aid 1

Authorized reprint from ACI, SP-17(91), Vol. 2, Columns Example 1, Pages 1-3, with permission from the



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Determining Reinforcing Steel Requirements

Requirements for reinforcing steel in columns are covered in sections 7.10, 10.9, and 12 of the Code. For

longitudinal reinforcing, the ratio gmust be within the range of 0.01 to 0.08. Based on typical columndimensions and loading and bar congestion, which occurs at high steel ratios, usually falls within the range of0.01 to 0.04. The Code does not specify a minimum or maximum diameter for longitudinal bars. Generally, #5

to #11 size bars are used. The minimum number of bars is 4 for columns with lateral ties and 6 for columns

with spirals. The maximum number of bars that can be placed in each face of the column depends on the

column dimension, cover, longitudinal bar diameter, tie or spiral bar diameter, a clear spacing between bars of

1.5db, but not less than 1.5 inches, and the type of column splicing.

Often, due to the height of the column and concrete placing/forming sequence, splices of longitudinal bars are

necessary. Column reinforcing bars may be spliced by welding, mechanical connectors, or lapping of bars.

Welding and mechanical connectors are not often used due to quality control and bar placement difficulties.

Note that the Code limits the size of bars that are lap spliced to #11 maximum. The length of the lap dependswhether the bars are in tension or compression and the percentage of bars to be spliced for bars in tension.

Minimum length of lap for tension lap splices shall be as required for Class A or B splices, but not less than 12

inches, where:

Class A splice. . . . . . . . . . . . . . . . . . . . . .1.0 ldb

Class B splice. . . . . . . . . . . . . . . . . . . . . .1.3 ldb

where ldbis the tensile development length for the specified yield strength fy,in accordance with Work Aids 6

through 8 of Module 2 adjusted to SAES-Q-001.

For bars in compression, use ldb, using Work Aid 9 of Module 2.

It is preferred to splice no more than 50% of the bars at one splice location. Splices should be spaced at least 24

inches apart. Each splice shall involve at least one bar in each face. The 1.5dbspacing requirement also applies

to the clear distance between sets of lapped bars. Work Aid 3 contains minimum face dimensions of

rectangular tied columns accommodating various numbers of bars per face.


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For lateral ties and spirals, the Code sets the minimum diameter at 3/8 inch. For spirals, the ratio of spiral

reinforcement, s(defined as the volume of spiral reinforcement to total volume of core (out-to-out of spirals)),shall not be less than:

s= 0.45AgAc

- 1fc'

fy

where Agis gross column area and Acis area of core measured to the outside diameter of the spirals.

TENSION LAP SPLICES

Maximum percent of Asspliced within required lap length

Asprovided*

Asrequired

50 100

Equal to or greater than 2 Class A Class B

Less than 2 Class B Class B

*Ratio of area of reinforcement provided to area of reinforcement required by analysis at splice location.

Authorized reprint from ACI 318-89, Table 12.15.2, Page 204, with permission from the American Concrete

Institute.

FIGURE 10

The spacing between ties shall not exceed 16 longitudinal bar diameters, 48 tie bar diameters, or the least

dimension of the column. As mentioned in CSE 108.01, every corner and alternate longitudinal bar shall be

laterally supported by a corner of a tie, with no more than a 135 included angle. The spacing between supported

longitudinal bars shall not exceed 6 inches. The clear spacing between spirals must fall within the range of 1 to 3

inches. Lap splices for spiral reinforcement shall be 48db, but not less than 12 inches.


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Example Problem 3

This is the same column as presented in Example Problem 1 and 2.

For a rectangular tied column with bars equally distributed along four faces, select the

reinforcement.

Given: Loading



Materials

Compressive strength of concrete f'c= 4 ksi



Design conditions



b = 1 6 I n .

h

h = 20 In.

e

P

u

= 5 6 0 k

ACI 318-89


Handbook

Reference

Step 1--See Example Problem 1

Step 2--See Example Problem 1

Step 3--See Example Problem 2 Required area of steel = 12.8 in2

10.9.2

Step 4--Select optimum

reinforcement

A) Assume trial bar quantities

4

(2/face)

8

(3/face)

12

(4/face)

16

(5/face)

10.9.1

7.10.5.1

B) Determine smallest bar size

to provide Ast list resulting Ast in.2

compute resulting g= Ast/Ag and check that 0.01


31/58






ACI 318-89


Handbook

Reference

3.3.3(c)

7.6.3

7.6.4

7.7.1

7.10.5.1

12.14.2.1

E) Check whether reinforcement can be accommodated along smaller

face with

--Bearing splices

--Normal lap splices

--Tangential lap splices

OK

Not per-

mitted

Not per-

mitted

OK

Not per-

mitted

Not per-

mitted

OK

NO

NO

NO

NO

NO

Work Aid 3

Work Aid 3

Work Aid 3

7.10.5.2

7.8

7.9

7.10

F)Determine tie spacing as least of 16 longitudinal bar diameters, in.

--48 tie bar diameters, in.

--Least dimension of column, in.

G)Consider special design details

H)Select most cost-efficient

reinforcement

36

24

16

Omitted

27

24

16

in this

20

18

16

example

(Probable)

first choice

Solution Use 12 #10 bars with bearing splices

and #3 ties spaced not more than 16 in.

apart. (Choice is based on minimum

steel requirement, use #3 ties instead of#4, and ease of handling #10 bars

instead of larger bars.)


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Frame Analysis Techniques

In this section of CSE 108.03, the participant will learn the use of beam formulas, as well as those for simple

one-story frames.

In previous sections in CSE 108.02 and CSE 108.03 covering the design of structural components (e.g., beams,

columns, slabs), the design moments, shears, and axial loads were given. As shown in Figure 11, the designer

must first determine the moments, shears, and axial loads from a structural analysis (Step 4), as well as compute

reactions and displacements. To conduct the analysis, the dimensions and sizes of the individual structure

members are assumed, based on experience and preliminary sizing guidelines.

STRUCTURAL DESIGN FLOWCHART

No

Obtain AllStructure

Requirements

SelectConstruction

Materials

DetermineDesign Loads

and LoadCombinations

StructuralAnalysis

StructuralMemberDesign

DoesDesign Match

AnalysisModel?

Develop

Drawings/Specifications

Review/MonitorConstruction

Owner / ClientOther Engineering Disciplines

Saudi Aramco Standards

Saudi Aramco Building Code

Uniform Buidling Code

SAES-Q-001ACI 318M

Saudi Aramco Standards

ACI Codes / Reports

ASTM / SASO Standards

FIGURE 11


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Using an assumed structure, the structural analysis is performed to convert the design loadings covered in CSE

108.01 (dead, live, wind, etc.) to moments and shears for each member in the structure.

Structural analysis is based on two principles:

1. Equilibrium of Forces.The structure as a whole must be in static equilibrium under the action of the

applied loads and reactions. Each member of the structure alone must also be in equilibrium with applied

loads and boundary forces. This means that the resultant of all forces and moments acting at a given

point in the structure or member must be zero.

2. Compatibility of Displacements. For any loading, the displacements of all the members of the structure

due to their respective stress-strain relationships must be consistent with respect to each other; that is, the

continuity of the structure must be satisfied. Continuity applies to deflections and rotations in the

member and structure.

For all but simple beams and one-story frames, hand calculation methods for structural analysis have been

replaced with computer analysis programs. For beams and one-story frames, published solutions may be

directly used. These solutions are also helpful in performing quick rough checks of computer-generated output.

Computer structural analysis is beyond the scope of this course.

Beam Formulas

Diagrams and formulas for moments, shears, and deflections covering a wide variety of loading patterns and

end restraint conditions are provided in standard texts, including the American Institute for Steel Construction

(AISC) Manual.

Diagrams for a uniformly distributed load for different end restraint conditions are shown in Figure 12, which

also shows the effect of end restraint on moments and deflections. Maximum positive moments in the beam

vary between 1/24 and 1/8 wl2. Maximum negative moments vary from 0 to 1/12 wl2. The maximum beam

deflection varies from 1/384 to 5/384 wl4/EI. Diagrams for a concentrated load at midspan are shown in

Figure 13. Similar variations in moment and deflection occur for the different end restraint conditions.

Note that when Figures 12 and 13 are compared by setting wl= P, moments for the concentrated load are 1.5 to

3.0 times higher than for the uniform load. As expected, deflections are higher for the concentrated load. These

diagrams can be used to check the upper and lower bounds to solutions for beams with more complicated

loadings or partial end restraint.


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BEAM DIAGRAMS AND FORMULAS

Beam Fixed At One End, Supported At Other - Uniformly Distributed Load

Beam Fixed At Both Ends Uniformly Distributed Load

R = V . . . . . . . . . . . . . . . . . . . . . . =w

2

. . . . . . . . . . . . =

. . . . . . . . . . =

Simple Beam Uniformly Distributed Load

w 2

12

w 2

24

w 4

384EI

3w8

. . . . . . . . . . . . . . . . . . . . . . =

M1 (at x = 38

) . . . . . . . . . . . . . =

R1 = V1

. . . . . . . . . =

. . . . . . . . . . . . . . . . . =

. . . . . . . . . . . . . . . . . . . . . =

w 4

185EI

9128

w 2

R2 = V2 max

Mmax

max (At x = 0.4215 )

R

V

V

Moment

Shear

2 2

M1

R

M maxM max

R = V . . . . . . . . . . . . . . . . . . . . . . . =

. . . . . . . . . . . . =

w

2

. . . . . . . . . . . . = 5w4

384EI

w 2

8M max (At Center)

max (At Center)

max (At Center)

M m ax (At Ends) . . . . . . . . . . . . =

M1(At Center)

Moment

38

4

R1

V1

V2

M1

MmaxMoment

w

x

2 2

R Rw

V

V

M max

Moment

w 2

8

R2

w

5w8

FIGURE 12


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BEAM DIAGRAMS AND FORMULAS

Simple Beam Concentrated Load at Center

R = V . . . . . . . . . . . . . . . . . . . . . . . . . =

. . . . . . . . =

. . . . . . . . . =

P2

P4

P

2 2

R

VShear

Moment

P 3

48EI

Beams Fixed at Both Ends ConcentratedLoad at Center

R = V . . . . . . . . . . . . . . . . . . . . . . =

. . . =

. . . . . . . . . . . . =

P2

P8

P

2 2

Shear

Moment

R

V

R

V

4

P3

192E

R 1 = V1

R 2 = V 2 max

. . . . . . . . . . . =

5P

16

11P16

3P16

5P

32

. . . . . . . . . . =

. . . . . . . . . . =

. . . . . . . . . . =

. . . . . . . . . . . . . . . . . . . =

. . . . . . . . . . . . . . . . . . . . . . =

P

2 2

Shear

Moment

P 3

48EI 5

7P 3

768EI

Beam Fixed at One End, Supported at Other Concentrated Load at Center

M max (At Point of Load)

max (At Point of Load)

M max (At Fixed End)

M 1 (At Point of Load)

x (At Point of Load)

R 1 R 2

V1

V 2

M 1

M max311

x

M max

M max

M max (At Center and Ends)

max (At Center)

Mmax

max (At x = 0.4472 )

FIGURE 13


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One-Story Frames

Published solutions to moments, shears, and axial forces for simple one-story frames subjected to certain

loading patterns are provided in the book entitled "Rigid Frame Formulas," by Kleinlogel, and are reproducedin part in Work Aids 4-16. Work Aids 4-12 cover the generalized solutions for different frames, with load terms

defined in Work Aids 13-16.

Figure 14 illustrates a fully fixed symmetrical rectangular frame. The solutions for this frame depend on

coefficients k, N1, and N2, and on load factors:

, ; r, 1; S, W;Mx

,

My

The coefficients are determined based on the ratio of the moment of inertia of the girder (beam), J2, to that of

the column, J1

. The load factors are determined using Work Aids 8-12 for the specific loading pattern. The

generalized solutions for two common load cases are shown in Figure 15.

For the special case of uniformly distributed load in Figure 15, note that the negative moment at the end of the

girder is given by the same expression as on the beam diagrams, except that it is modified by the factor 2/N 1.

Using the coefficients from Figure 14, based on a frame with height, h, equal to the span l, and similar moments

of inertia for the beam and column (J2/J1= 1), the value of 2/N1is 2/3. Therefore, the negative end moment for

the girder is 2/3 of that for a fully fixed restraint condition. The resulting positive moment in the span is equal

to (ql2/8 - 2*ql 2/(3*12) = 5ql2/72, or 5/3 the value for a fully-fixed beam, or 5/9 the value of a simply

supported beam

These frame solutions are directly appropriate for analyzing one-story building frames and pipe racks. Analysis

of more complicated frames can often be simplified and approximated using these published solutions.


37/58





FRAME 41

Fully fixed symmetrical rectangular frame

B C

DA

h

Frame Dimensions andNotations

B Cx x'

Coefficients:

y'

HA

MA

HD

VDVA

MD

J1 J1

J2

k =J2J1

. h N1 = k + 2 N2= 6k + 1

This Sketch Shows the PositiveDirection of the Reactions and theCoordinates Assigned to Any Point. ForSymmetrical Loading of the Frame Usey and y'. Positive Bending MomentsCause Tension at the Face Marked by aDashed Line.

y

Source: Kleinlogel "Rigid Frame Formulas", Frederick Ungar Publishing Co. New York, 1980, Frame 41, Page

147.

FIGURE 14


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GENERALIZED SOLUTION FOR TWO COMMON LOAD CASES

-M

J1

J1

J2

P

h

A D

2B C

y2

-M y1

2M

x

-HA

-MA-V

A-V

D

MD

HD

MAMD

= + Ph2. 3k + 1

N2M BM C

= Ph2

. 3kN 2

;

H D= HA=P2

; VD= V A=2 M B ;

Case 41/8: Horizontal Concentrated Load at the Girder

Case 41/3: Girder Loaded by Any Type of Vert ical Load , Acting Symmetricall y.

B C

S

A D

J1 J1

+

B C

-My

HA HD

MA MDVA

VD

y

+

2

MA= M D= 3N 1

;

My= MA HA

Special Case: Uniformly Distributed Load S = q l

HA= HD=3M A

hVA= VD= S

2;

MB= MC= - 2MA

MA= MD= +q 2

12 N1V A= VD=

q

2

M x= M xo+ M B

max M x=q 2

8+ M B

y

-M

J2

+

+ +

+-

Source: Kleinlogel, "Rigid Frame Formulas", Frame 41, Pages 148 and 151.

FIGURE 15


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Computer Programs

During the past ten years, numerous computer programs have been developed for public use. These programs

are used for traditional structural analysis and design of structural components. The analysis and design often

use different programs.

The designer must develop the input data covering materials, structure layout, framing, and preliminary

member sizes and determine the applicable design loads and load combinations. The designer must be aware of

any limitations of the program. The use of consistent units is essential. The accuracy of the program output

depends directly on the quality of the input prepared by the designer and whether the assumptions/limitations

of the program have been checked and satisfied. After the analysis is performed and validated, the analysis

output (moments, shears, forces, reactions, and deflections) is used as input to the component design program.

The same checks regarding consistent units, program limitations, and assumptions must be made. Computer

structural analysis and design is beyond the scope of this course.


40/58





Work Aid 1: Rectangular Column: Load-Moment Interaction Diagrams

COLUMNS 7.4.3Load-moment strength interaction diagram for R4-60.75 columns


Authorized reprint from ACI, SP-17(90) Vol. 2, Columns 7.4.3 , Page 82 with permission from the American

Concrete Institute.


41/58





COLUMNS 7.4.4Load-moment strength interaction diagram for R4-60.90 columns


Authorized reprint from ACI, SP-17(90) Vol. 2, Columns 7.4.4, Page 82 with permission from the American

Concrete Institute.


42/58





Work Aid 2: Spirally Reinforced Column: Load-Moment Interaction

Diagrams

COLUMNS 7.23.1Load-moment strength interaction diagram for C5-60.45 spirallyreinforced columns

References: ACI 318-89 Sections 9.3.2.2. 10.2. and 10.3; ACI Publication SP-7, pp. 152-182

Authorized reprint from ACI, SP-17(90) Vol. 2, Columns 7.23.1 , Page 119 with permission from the American

Concrete Institute.


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COLUMNS 7.23.1Load-moment strength interaction diagram for C5-60.60 spirally

reinforced columns

References: ACI 318-89 Sections 9.3.2.2. 10.2. and 10.3; ACI Publication SP-7, pp. 152-182

Authorized reprint from ACI, SP-17(90) Vol. 2, Columns 7.23.2, Page 119 with permission from the American

Concrete Institute.


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Work Aid 3: Minimum Face Dimensions of Rectangular Tied Columns

Accommodating Various Numbers of Bars per Face

Note: This Work Aid needs to be adjusted for specific Saudi Aramco cover requirements per SAES-Q-001.

Authorized reprint from ACI, SP-17 (90) Vol. 2, Reinforcement 2, Page 165 with permission from the



45/58





Work Aid 4: Frame 39, Symmetrical Rectangular Two-Hinged Frame, Case

#1

Source: Kleinlogel, "Rigid Frame Formulas," Page 138, Frederick Ungar Publishing Company, New York,

1980


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#2-4


1980


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#5-7

Source: Kleinlogel, "Rigid Frame Formulas," Page 140 ,Frederick Ungar Publishing Company, New York,

1980


48/58





Work Aid 7: Frame 39, Symmetrical Rectangular Two-Hinged Frame,

Symmetrical Loading - Case #8-10


1980


49/58


50/58





Work Aid 9: Frame 41, Symmetrical Rectangular Fully-Fixed Frame, Case

#2-3


1980


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#4-5


1980


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#6-7


1980


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#8-9


1980


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Work Aid 13: Appendix Load Terms; General Notations


1980


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Work Aid 14: Appendix Load Terms; Notations, Case 1-3


1980


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Work Aid 15: Appendix Load Terms; General Notations, Case 4-6


1980


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Work Aid 16: Appendix Load Terms; Notations, Case 7-11


1980


58/58




GLOSSARY

biaxial bending Bending moments about two perpendicular axes.

column strip Area of slab adjacent to column line used to analyze and design two-way

slabs.

compatibility of

displacements

Defected shape of frame or member. Must be smooth (no kinks) unless

pinned joints are used.

drop panels Areas of thickened slab surrounding columns for enhancing shear transfer.

equilibrium of forces All forces (internal and external) must equate to zero.

inflection point Point along axis of member where moments change sign.

lap spliced Describes how load transfer occurs from one rebar to another, bars are

overlapped.

middle strip Strip outside of column strip used for analysis and design of two-way slabs.

one-way slabs Reinforcement and loads carried in one principal direction.

orthogonal direction Perpendicular directions (two).

perimeter or punchingshear

Shear failure mode where column or load breaks through slab on four sidesor circle.

sidesway Lateral sway or deflection of frame due to lateral or unsymmetric gravity

loads.

spiral reinforced Lateral reinforcing in columns, made by a continuous vertical spiral.

stiffness matrix

approach

Method used to analyze frames and structures using matrix operations using

member stiffness.

tie reinforced Lateral reinforcing in columns by one or more discrete enclosure bars.

two-way slab Reinforcement and loads carried in both orthogonal directions.

Frame Analysis Design of Reinforced Concrete Two-Way Slabs A

Documents

Transcript of Frame Analysis Design of Reinforced Concrete Two-Way Slabs A