a study of the differential deflections occurring in …

A STUDY OF THE DIFFERENTIAL DEFLECTIONS OCCURRING IN FULL-SCALE RESIDENTIAL SLAB-ON-GROUND FOUNDATIONS

by

CHAUR-SONG SHIH, B.S.

A THESIS

IN

CIVIL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE IN

CIVIL ENGINEERING

Approved

r

Dean of the Graduate School

August, 1985

1'^

ACKNOWLEDGEMENTS

I would like to express my deepest gratitude to Dr. W.

K. Wray for his helpful guidance and encouragement through

out the progress of this research and during my graduate

studies. I also wish to express appreciation to Dr. J. E.

Minor and C. V. G. Vallabhan who have given me valuable sug

gestions and recommendations for this thesis during this

period of study. I should also like to thank the Department

of Civil Engineering for supporting me during the graduate

studies in Texas Tech University. I am deeply grateful to

my parents for their support and encouragement. Finally, I

am very much thankful to Miss Wanfang Fu and especially Miss

Menwen Wang, who had offered their free services to type the

manuscript.

11

ABSTRACT

A model is described which reproduces measured deforma

tions in a slab-on-ground foundation for each of two resi

dential structures. The study then investigates the effect

on differential deflection produced by different combina

tions of stiffening beams to the foundation model. The mod

el uses the finite element method to analyze bending mo

ments, shear forces and deflections occurring in the slab

foundation. Analyses of results show that the relative dif

ferential deflections and deflection ratios are reduced as

the spacing of stiffening beams is reduced. Additionally,

increasing the number of stiffening beams results in gradu

ally increasing magnitudes of bending moment and shear

force, but decreasing magnitudes of the shear stresses. The

results of this study show that interior stiffening beams

can reduce differential deflections and maximum deflection

ratios that may occur in slab-on-ground foundations under

service loads and that interior stiffening beams produce

only a small increase in construction cost.

Ill

CONTENTS

ACKNOWLEDGEMENTS i i

ABSTRACT i i i

CHAPTER

I. INTRODUCTION 1

II. DISCUSSION OF PROBLEM . 3

What Is The Problem 3 Factors Affecting Slab-On-Ground Soil-Struc

ture Interaction 4 Objectives Of This Investigation 10

III. MODEL FOR STIFFENED SLAB ON GRADE 11

Addressing The Problem 11 Describe The Model 13 Accuracy of The Specific Method 14 Analytical Procedures 17 Analytical Parameters in This Model 34

IV. RESULTS OF ANALYSIS 45

Comparison of Results 46

V. CONCLUSIONS AND RECOMMENDATIONS 55

Conclusions 55

Recommendation 56

LIST OF REFERENCES 57

APPENDIX

A. INPUT DATA FOR CASE I AND CASE II 59

IV

B. MAXIMUM DIFFERENTIAL DEFLECTION FOR CASE I

AND CASE II 69

C. CALCULATED MOMENT CAPACITY 81

D. CALCULATION OF ESTIMATED COST 82

LIST OF TABLES

3-1 Slab Classifications and Thickness 35

3-2 Typical Range of Values for Poisson's

Ratio of Soil 38

3-3 Typical Range of Values for 'Es' 40

3-4 Calculation of Perimeter Loading

(per unit length) 42

3-5 Structural Parameters Used in This Model 44

4-1 Comparison of Differential Deflections and

Deflection Ratios 50

4-2 Comparison of Maximum Shear Stress 51

4-3 Comparison of Maximum Bending Moments and Calculated Moment Capacity 52

4-4 Comparison of Cost Increase with Increasing the Number of Stiffening Beams (per ft^) 54

VI

LIST OF FIGURES

3-1 Deflections due to Loading 15

3-2 Stresses due to Loading 16

3-3 A Sketch of A Typical Roof-Wall-Floor Section 18

3-4 Plan View of Case I with Finite Element Grids 19

3-5 Plan View of Case II with Finite Element Grids 20

3-6 Contour Lines of Differential Deflections of Case I (actual measurement) 22

3-7 Contour Lines of Differential Deflections of Case II (actual measurement) 23

3-8 Contour Lines of Differential Deflections of Case I 26

3-9 Contour Lines of Differential Deflections of Case II 27

3-10 Contour Lines of Differential Deflections with 3-Stiffening Beams for Case I 28

3-11 Contour Lines of Differential Deflections with 3-Stiffening Beams for Case II 29





4-1 Comparison of Beam Spacings with Maximum Differential Deflection and with Percentage Decrease in Maximum Differential Deflection 47

4-2 Comparison of Beam Spacings with Maximum Deflection Ratio and with Percentage

Vll

Decrease in Deflection Ratio 48

Vll 1

CHAPTER I

INTRODUCTION

Apartments, residences, and light commercial buildings

have used concrete slab-on-ground foundations as a main

floor surface since World War II (16). The slab-on-ground

foundation has also been used for industrial and warehouse

structures for many years.

Slab-on-ground foundations in apartments, factories,

residences, or warehouses are expected to give satisfactory

service for many years without damage from heavy loads,

traffic, abrasive wear, and chemical attack. Many of these

slab-on-ground foundations were developed as a result of in

vention and trial and error rather than through design or

analysis.

Many slab-on-ground foundations have uniformly thick

sections, except for thickened edges required for frost

depth, (i.e., the depth to which freezing temperatures can

cause volume expansion and contraction of water in the soil)

or structural shear consideration. Although many of these

slabs perform successfully, significant numbers of slabs

have been considered to be failures. According to Wray

(16), poor material properties are not usually the primary

factor which contributes to cause poor slab performance.

Poor slab performance is frequently caused by poor design,

analysis and construction techniques, including inadequate

structural sections, inadequate bending strength, and, espe

cially, inadequate or improper site preparation. In con

struction, oftentimes insufficient attention is given to

site conditions or site preparation before concrete is

placed. According to Jones and Holtz (8), the structural

damage caused by poor slab performance annually exceeds that

caused by floods, tornadoes, hurricanes, and earthquakes,

and produces a great deal of monetary losses every year.

Differential deflections which occur in slab-on-grade foun

dations in residential and light commercial buildings are a

very serious problem.

CHAPTER II

DISCUSSION OF PROBLEM

What Is The Problem

Whenever a building load is placed on soil, some set

tlement results and, unless the building is perfectly rigid,

the settlement is almost invariably unequal even though the

applied load is uniformly distributed. A flexible super

structure normally adjusts itself to a dish-like deflection

of the soil surface.

It is common practice to place gas, sewer, and water

service lines in trenches in the soil beneath the slab be

fore the concrete is placed. Excessive settlement may occur

and a gap may develop between the soil and slab due to neg

ligence in compacting the backfill in these trenches. Ex

cessive deflections can result in structural damage to a

building frame, sticking doors and windows, cracks in tile

and dry wall or plaster, and excessive wear or equipment

failure from misalignment due to excessive foundation de

flections. Obviously, serious stresses may result from un

equal deflections which tends to occur. These differential

deflections are of concern in this study.

Factors Affecting Slab-On-Ground Soil-Structure Interaction

There are many factors influencing the interaction that

occurs between a structural slab-on-ground foundation and

the soil supporting it. These factors include soil, struc

ture, and climate properties. They are:

1. type and amount of clay mineral;

2. site preparation;

3. stiffening beams;

4. climate;

5. vegetation before and after construction; and

6. bearing capacity.

These important factors are presented and briefly dis

cussed as follows.

1. Type and amount of clay mineral: The type and amount

of clay controls the magnitude of the change in vol

ume of soils as water content changes. In other

words, soil structure and mineralogical composition

and the intereffect with water, have influence on the

properties and behavior which are considered impor

tant for design and construction of slab-on-ground

foundations. For civil engineering purposes, clay

minerals are generally divided into three primary

groups, i.e., kaolinite, illite, and montmorillonite

(10). Typical properties of these clays are

summarized as follows:

a. Kaolinite: Very stable, little tendency for

volume change when exposed to water. A soil with a

low percentage of clay sized particles comprised of

kaolinite will be expected to experience only a rela

tively small amount of shrink-swell.

b. Illite: Generally, illite is more plastic

than kaolinite. It does not expand when exposed to

water unless a deficiency in potassium exists.

c. Montmorillonite: Because of the weak bond

between layers and the high negative surface charges,

the clay readily adsorbs water between layers. This

mineral has a great tendency for large volume change

because of this property. That is, a soil consisting

predominantly of montmorillonite will experience a

great amount of shrink-swell when subjected to chang

es in moisture content. As Mitchell (11) pointed

out: "Montmorillonites and vermiculites undergo

greater volume changes on wetting and drying than

kaolinites and illites. Experience clearly indicates

this to be the case."

2. Site preparation: The supporting subgrade should

provide adequate support for finished

slabs-on-ground. Slabs-on-ground must have a

subgrade with uniform bearing to avoid creating dif

ferential deflections (hence, critical stresses)

which will crack the slab.

Specifications usually require a minimum "in-

place density" for the subgrade soils of at least 95

percent of the laboratory maximum. If the soil has

been disturbed during previous construction opera

tions, it must be compacted uniformly. This includes

filling and compaction of holes, utility trenches,

and irregularities using uniform fill material that

is free of vegetable matter, lumps, large stones, or

frozen soil. Soil may be partially replaced in areas

that have expansive or shrinking soils if they will

be subjected to changes in moisture during the life

of the floor, or if the soil has high moisture reten

tion. Subgrades which will support residential slabs

must be prepared with compacting or tamping equip

ment. Improper site preparation may result in gaps

between the slab and subgrade, i.e., not full contact

conditions between the soil and slab in some

locations.

3. Stiffening beams: Stiffening beams are constructed

monolithically with the uniformly thick slab to

increase the moment of inertia of the structural

section. Usually, the slabs are stiffened with grade

beams around the perimeter. These perimeter beams

are not constructed for stiffening purposes, but be

cause the building codes require them for frost con

siderations. For a constant depth slab, a thicker

slab will create more stiffness to resist the bending

stress. Without a significant increase in the amount

of concrete, the stiffness of the slab can also be

increased by adding stiffening beams in the interior

of the slab in both directions. The slab-on-ground

foundation should be designed to permit deformation

with as soil movements occur, but the differential

deflections occurring in the slab should be limited

to some allowable value as specified for the type of

construction it is supporting. In general, the more

flexible structures can tolerate greater deformation

without cracking than can rigid structures. For ex

ample, wood-frame construction can withstand more de

formation than can a concrete block structure. The

stiffness of a stiffened slab is dependent on the

beam depth, beam width, and beam spacing. Especially

the beam depth is the major factor influencing the

stiffness.

8

4. Climate: Soil volume change is influenced by the

soil moisture content change. As the moisture con

tent decreases, the ground surface shrinks and moves

downward. As moisture content increases, the ground

surface swells and moves upward. If a slab is con

structed at the end of a wet period, the slab should

be expected to experience some loss of support around

the perimeter when the wet soil begins to dry out and

shrink. Conversely, if the slab is constructed at

the end of a dry period, the slab should be expected

to experience some uplift around the perimeter if the

soil at the edge of the slab gains in moisture con

tent.

5. Vegetation before and after construction: The type,

amount, and location of vegetation that exists before

or after construction can cause localized desicca

tion. Certain types of vegetation growing on the slab

site prior to construction may have desiccated the

site to some degree. Construction over this desic

cated soil can result in some heave if moisture is

introduced. Vegetation planted or permitted to grow

close to the slab after construction can produce a

loss of foundation support when the water is removed

from the soil by the vegetation. On the other hand.

9

if the vegetation is watered excessively, it can

result in swelling of the soil at the edge of the

slab.

6. Bearing capacity: The bearing capacity of soils

plays a important part in the design of buildings and

structures. The safe bearing capacity of a soil is

usually defined as the load which may be applied to

that soil without causing detrimental settlements of

the building or other structure. The safe bearing

capacity of a given soil for structures of different

character might be and in fact should be different,

depending on the allowable settlement for each class

of structures. The soil must be capable of carrying

loads from any engineered structure placed upon it

without a shear failure and with the resulting set

tlements being tolerable for that structure. It is

necessary to investigate both base shear resistance

and settlements for any structure.

Objectives Of This Investigation

1. To study the degree or extent to which interior

stiffening beams affect the settlement deflection

10

performance of a slab-on-ground under service loading

conditions by comparing the effect of different com

binations of interior stiffening beams on the modeled

slab performance of two actual foundation.

2. To investigate how closely interior stiffening beams

must be spaced to provide a significant influence on

the deflection and deflection ratios. There is like

ly to be a "point of diminishing return" with re

spect to beam spacing.

3. To study the economics of constructing interior

stiffening beams with different spacings.

CHAPTER III

MODEL FOR STIFFENED SLAB ON GRADE

Addressing The Problem

Two residential structures experiencing residential

distress were thoroughly investigated to determine the caus

es and significance of the distress (6,7). The data ob

tained during these investigations were made available for

this study on the conditions that the exact locations of the

residences not be identified. Subsurface and laboratory in

vestigations of the soils were not conducted in the report;

however, from the USDA Soil Conservation Service County Soil

survey report (14,15) the surface soils at both locations

were identified as Pullman Clay loam, 0-1 percent slope.

Based on information in the reports, which was obtained from

field and laboratory studies, this soil class typically

shows the following soil property characteristics:

Depth 0"-7" 7"-54"

Classification

Unified CL CL, CH

AASHTO A-6, A-7-6 A-7-6

Percent Passing Sieve No.

#4 100 100

#10 100 100

11

12

#40 95-100 95-100

#200 70-90 85-98

Liquid Limit

Plasticity Index

30-50

15-30

41-55

22-35

Although this soil informations indicates the soil on

which the existing exterior grade beams were bearing has a

potential for some volume change as the soil moisture con

tent changes, measurements and observations made during the

damage inspections indicate very little, if any, of the ob

served structural distress could be attributed to expansive

soil movement. Thus, it was concluded that the observed

structural distress and differential movement of the slab

foundation was due principally to settlement beneath the in

terior of the structure (6,7), and not expansive soil move

ment. Therefore, those factors that affect the performance

of a slab-on-ground foundation as discussed in Chapter II

which are expansive soil related do not enter into this

problem.

Describe The Model

As a result of the conclusions reached regarding the

type of foundation distortion being observed in the two

13

structures, it was concluded that this distortion could be

reasonably modeled using an appropriate plate-on-elastic-

foundation computer code. A code originally developed by Y.

H, Huang was found to be well-suited for the proposed inves

tigation of a settling slab foundation.

The computer program originally developed by Huang (5)

and subsequently modified by Wray (16) was used as the mod

el. This program, named SLAB2, used the finite element

method to analyze a plate supported on an elastic continuum.

(We assume that the foundation resists the loads transmitted

by the slab in a linearly elastic manner; that is, the pres

sure developed at any point between the slab and the founda

tion is proportional to the deflection of the slab at that

point.) It is based on the classical theory of thin plates

that assumes planes before bending remain planes after bend

ing. The program can determine the bending moments, deflec

tions, stresses, and shear forces in the slab due to the

loading transferred to the slab plate from perimeter walls,

interior walls, or both. Modeling the problem of a

slab-on-ground, the plate is divided into rectangular finite

elements of various sizes. The model can be used to analyze

either a slab of constant thickness, or a stiffened slab.

14

Accuracy of The Specific Method

Huang showed the model to accurately predict stresses

and deflections by comparing model-predicted results to

field experimental measurements obtained from the AASHO Road

Test (5). These results are shown in Figs. 3-1 and 3-2

which compare the experimentally measured and computer-pre

dicted deflections and stresses due to specific loadings.

If part of the slab near the edge is not in contact with the

subgrade, i.e., it is in partial contact with the subgrade,

a closer relationship between the theoretical solution and

the experimental measurement is obtained than by assuming

the slab and the subgrade to be in full contact. The meas

ured stresses generally fall between the model results based

on full contact and those based on partial contact, indicat

ing that the true contact condition may lie between the two.

The comparison shows this computer program can produce rea

sonably accurate results of stresses and deflections.

Therefore, this computer code is reasonable and suitable for

modeling either a slab-on-ground foundation of uniform

thickness or a slab with stiffening beams. The computer

model can incorporate perimeter loads (apartment,

residential or commercial buildings), interior line loads

(load-bearing walls), or isolated interior point loads

(columns), as well as soil and concrete properties.

15

Free Edge h LonglCudlnal Joint

o

I" 20

•'"''^ (

o / /

/ /

/

f^^

20 In

"—-.,».

— F\ill contact Partial contact

o Ebcperlmental

l-""''*'''^ •

7 In slab 7,000 lb load

I

8 il 10

1 15

20

/ -

/ 20 In

1^


en •o

5 0

5

10

15

20

»

y^ ( / /

/ /

/ ^20 IT

1

•

0.1

0.2 . g

0.3 d

tO.4 §

0.5

o

_ _ _ _ ^ _ _ _ _ ^ _ _ _ _ ^ . ^ ^ _ _ ^ . ^ I ___^___ _ _ ^ _ _ _ ^ _ _ _ _

_ ^ ^ _ ^ ^ _ - ^ ^ ^ ^ - ^ ^ ^ ^ ^ ^ ^ _ ^ _ - ^ ^ P V ^ ^ B ^ B

0

<"•' i

0.2 g*

0.3 ^

0.4 a

0.5


Fig. 3-1. Deflections due to Loading.

16

Free Edge Longitudinal Joint

200 (NJ

- -

o

X „

V

Theoretical Experimental

- F\J11 Contact o Transverse

- Partial contact x Longltiidlnal

e

. V at

0 I 3 1 ^ w

3

4

(0

(O

7 In slab, 9,000 lb load

200|

600"

X

f̂ -

SI = u 0) OS > in

1 ^ ^

1-' i IS h


—«—a -fc

" ^

»

V V

K

,

• X

- — 0 —

,„„—-—

5 %

CM

B

e -u ^ s

a g 1 5 i^

CM

B

1 u m J •»! as

A I ^ 4 J CO


Fig. 3-2. stresses due to Loading.

17

Analytical Procedures

six tasks were employed in evaluating and modeling this

slab settlement problem. They are discussed step by step as

follows.

1. Use field study data: Two actual wood-frame-struc

ture buildings were studied (6,7). A typical sketch

of their roof-wall-floor construction is shown in

Fig. 3-3. A single line drawing of the floor plan of

each structure is shown in Figs. 3-4 and 3-5, respec

tively, with the finite element grids used for each

study indicated. Each of the residential buildings

had exterior perimeter grade beams having overall di

mensions of 12 inches wide (assumed), 18 inches deep

(measured), and the thickness of the floor slab was

measured to be a nominal 4 inches.

2. Actual and model dimensions of the two actual cases:

In order to simplify the problem, it was assumed each

of the slabs studied is square. The dimensions as

sumed were 48 feet x 48 feet for Case I and 50 feet x

50 feet for Case II can be seen from Figs. 3-4 and

3-5. The actual plans of each foundation were not

quite square but were slightly irregular with the

foundation for Case I having a width of 48 feet but

Ceiling joist 2"x6" (5)16" o.c.n

Roof sheathing 2"x8" @24" o.c. (or shingle lath)

18

Ccnposition roofing with f e l t

V Plywood

2"x4" Sole plate

4" Slab

TWO 2"x4" plates

Brick veneer

Concrete Footing

F i g . 3 - 3 . A Sketch of A T y p i c a l Roof -Wal l -F loor S e c t i o n .

1 1 1 1

2 1

3 1

4 1

5 1

48 feet

6 7 8 1 t 1

9 1

10 1

11 1

12 1

13 1

19

4J (U 0)

5

L-

K-

J-

I-

H-

G-

F_

E-

D-

C-

B-

A-l

10 15 FEET

Fig. 3-4. Plan View of Case I with Finite Element Grids.

I L 4 ±

5 J.

50 feet

6 L

7 8

J L

10 TI. 20

20Ft.

Fig. 3-5. Plan View of Case II with Finite Element Grids.

21

with two length dimensions of 48 feet and 52 feet

respectively; Case II had a width of 50 feet but had

two length dimensions measuring 50 feet and 56 feet

respectively. Thus, the simplifying assumption of a

square foundation was not unreasonable.

3. Calculate the relative differential deflections and

draw the contour lines from field survey values: The

relative differential elevations of the slabs from

the field survey values were computed. These values

were measured in the field using conventional civil

engineering surveying equipment and procedures. The

contour lines of differential deflections at various

points on the slabs are shown in Fig. 3-6 and Fig.

3-7 for each structure, respectively. The lowest

relative elevation was chosen to be elevation 0.00.

4. Reproduce the measured relative differential eleva

tions using modeling technique A: Using the computer

code described above an attempt to reproduce the

measured relative differential elevations was made.

Initially, the problem was modeled by assuming the

relative differential elevations obtained from

the field survey constituted gaps that vere

existed between an initially horizontal

22

DINING

GARAGE

BATH ''̂ ^̂ ASTER

0 10 15FEEr

Fiq. 3-6. Contour Lines of Differential Deflections of Case I (actual measurement).

23

^ry^\ MASTER

CL0^\\2( "

I^Z^kap.

GARAGE

0 10

>>y>^ »>>•*

20Ft

Fig. 3-7. Contour Lines of Differential Deflections of Case II (actual measurement).

24

plane (represented by the slab surface) and a new

deformed soil subgrade beneath the slab. An analysis

of this problem was performed and the ensuing soil-

structure interaction produced a distorted final con

dition of the slab surface (which was initially hori

zontal) similar to that measured in the field. Then

adjustments were made by trial and error to obtain

the distortion shape closer to that found in the

field. However, it was found that it was impossible

to create an acceptable final result that closely re

sembled the actual slab. Thus, it was concluded that

this method of modeling the problem was invalid, so

it was abandoned.

5. Reproduce the measured relative differential eleva

tions using modeling technique B: Because of negli

gence in compacting the backfill in the utility line

trenches excavated in the subgrade beneath the slab

at the time of construction, frequently excessive

settlement occurs in these trenches and the result is

a gap between the slab and the subgrade soil in the

vicinity of the trench. Thus, there may not be any

contact at all between the slab and subgrade at those

locations. By trial and error, using different

magnitudes of gaps located along the presumed lines

25

of the utility trenches and trying different values

of modulus of elasticity of soil, the model was grad

ually shaped so that the model distortion closely ap

proximated that of the actual slab. The deflection

results of the final model is shown in Fig. 3-8 and

Fig. 3-9 for Case I and Case II, respectively.

6. Add stiffening beams to the model and study their ef

fect on the predicted deflection: After the final

model which resembled the measured field conditions

was obtained, one interior stiffening beam in each

direction was added at mid-dimension to the founda

tion model for each case. These beams extended the

full width and length and were continuous across the

foundation. All other conditions used to produce the

distorted model shape (loading location and magni

tude, dimensions, and support conditions) were held

constant. The respective contours of resulting rela

tive differential deflections are shown in Fig. 3-10

and Fig. 3-11. Then models with two and, finally,

three interior beams in each direction (i.e., with

smaller equal spacings) were analyzed and relative

deflections calculated and plotted. Their results

are plotted in Figs. 3-12, 3-13, 3-14, and 3-15.

26

10 15FEEr

Fig, 3-8. Contour Lines of Differential Deflections of Case I.

Note: (1). All values are in inches. (2). Assuming the lowest point is equal to zero. (3). The bigger value is the higher elevation.

27

cz^zz

0 10

^^^^^^^\

20Ft.

F i q . 3 - 9 . Contour Lines or Case II.

f Differential Deflections of

vT̂ vo. M ) All values are in inches. ^°^ 2) Assuming the lowest point is equal to zero.

(3): TheTigger value is the higher elevation.

28

0 5 10 15FEEI

Fig. 3-10. Contour Lines of Differential Deflections with 3-Stiffening Beams for Case I.

29

20Ft.

Fiq. 3-11. Contour Lines of Differential Deflections with 3-Stiffening Beams for Case II.

30

Fig. 3-12. Contour Lines of Differential Deflections with 4-Stiffening Beams for Case I.

31

ESZZZ

Vr^,Vr'r'/A|

0 10 20Ft,

Fig. 3-13. Contour Lines of Differential Deflections with 4-Stiffening Beams for Case II.

32

100 GARAGE

\ N

0 10 15FEET

Fiq. 3 -14 . Contour Lines of D i f f e r e n t i a l De f l ec t i ons with 5 - S t i f f e n i n g Beams for Case I .

33

C2ZZZZZ3QB1

' > > ! • ' - ' - "

0 10 20Ft,

Fiq 3-15. Contour Lines of Differential Deflections with 5-Stiffening Beams for Case II.

34

The input data used in the computer codes for these

models are listed in Appendix A. All the computer results

of differential deflections for each node in both cases are

listed in Appendix B.

Analytical Parameters in This Model

There are many important variables involved in develop

ing this model; some of these factors are discussed below:

1. Slab length and width: Both the slab length and

width are usually unchangeable parameters which are

decided either by the property limitation, e.g., the

adjoining property is a public sidewalk or alley, or

by the owner for functional purposes.

2. Slab thickness: The required thickness may vary ac

cording to the intended use of the floor. From prac

tical considerations, the minimum thickness for resi

dential slabs should be a nominal 4 inches due to

construction limitations and building code specifica

tions (1). Generally, the controlling factor in de

termining the thickness of a slab-on-ground is the

heaviest concentrated load it will carry. The

various uses of slabs and their recommended thickness

are shown as Table 3-1.

35

Table 3-1

Slab Classifications and Thickness (1)

Class

1

2

3

4

5

Use

Residential or tile covered

Offices, churches, schools,

hospitals, ornamental

residential

Drives, garage floors, and

sidewalks for residences

Light industrial commercial

Single-course industrial.

integral topping

Thickness(in)

4

4

5

5

6

3. Moment of inertia: The bending stiffness of a non-

stiffened slab is a function of the moment of inertia

of the structural section. It is calculated by the

familiar equation:

1='-̂ - b h^ 1 2

(3-1)

where b = width of slab

h = thickness of slab

36

A thicker slab will produce a greater stiffness

or resistance to bending than a thinner depth. For

stiffening beams, the moment of inertia of the com

posite beam-slab section is dependent on the beam

spacing and beam width, especially beam depth.

4. Beam width, depth, and spacing:

a. Beam width—A range from 8 inches to 12 inch

es is commonly used. If the width is less than 8

inches, an excavation problem will happen because of

excavation equipment limitation, particularly if it

is a very deep excavation. Unless it is necessary to

have a wider beam to resist shear forces and to ob

tain higher bearing capacity, the beam width is typi

cally no more than 12 inches. Thus, a width ranging

between 8 inches and 12 inches is usually used in

practice.

b. Beam depth—It is not an economical approach

to increase the slab thickness and to make the foun

dation so strong and rigid that there is not any dif

ferential movement at all. One method to reduce the

magnitude of differential deflection without using an

excessive amount of concrete is to add a beam to the

underside of the thin uniformly thick slab. These

beams are called "grade beams" or "stiffening beams."

38

b. Soil--Typical poisson's ratio values for

several soils are given in Table 3-2.

Table 3-2

Typical Range of Values for Poisson's Ratio of Soil(3)

Type of Soil

Saturated clay

Unsaturated clay

Sandy clay

Silt

Sand (dense)

Coarse

(void ratio=0.4

Fine grained

(void ratio=0.4

Loess

-0.7)

-0.7)

Poisson

0.4 -

0.1-

0.2 -

0.3 -

0.2 -

0.15

0.25

0.1 -

's Ratio

0.5

0.3

0.3

0.35

0.4

0.3

Poisson's ratio for soil is not as well defined

as it is for concrete. Values ranging between 0.40

and 0.50 are usually reported for weak, stiff or

semisolid saturated clay (9). For swelling clays,

the value of 0.4 is a reasonable ratio (16).

39

6. Modulus of elasticity (Young's Modulus):

where Aa = stress variation

Ae = strain variation

a. Concrete—The value of Ec is a function ofi

the compressive strength of the concrete. The ACI

Building Code uses the empirical formula (2):

EC = 57,0007fc' •• (3-4)

where fc' is the 28-day compressive strength of con-

6 Crete. In this study, a value of 3 x 10 psi is

adopted.

b. Soil—Typical Es values for several soilsi

are given in Table 3-3. The Es value is not a con

stant but has a nonlinear relationship between ap

plied stress and resulting strain. It also increases

with depth within each stratum. For small strains it

is usually assumed to be a constant. It is obvious

that there is wide variation in the 'Es' values.

Table 3-3

Typical Range of Values for 'Es' (3)

40

Type of Soil

Clay

very soft

soft

medium

hard

sandy

Glacial till

loose

dense

very dense

Loess

Sand

silty

loose

dense

Sand and gravel

loose

dense

Silt

Es (psi)

347 -1736

674 -3472

2083 -6944

6944 -13889

3472 -34722

1389 -22222

20833 -104167

69444 -208333

2083 -8333

1042 -3125

1389 -3472

6944 -11806

6944 -20833

13889-27778

278-27778

41

Beginning with assumed stiffer values and pro

gressively reducing the value to create more distor

tion in the slab model, a value of 50 psi of 'Es' was

ultimately used for both cases in this study. Com

paring this value with typical values given in Table

3-3, 50 psi would seem to be an unreasonable value

for modeling the problem. However, the intent was to

produce a distortion in the slab model equal to what

was observed in the field. Thus, the very low value

of Es used was not really a variable in the study.

In fact, it probably would have been difficult to de

termine a field value of Es, particularly if the sub-

grade soil (a fill) was as poorly compacted as it ap

parently was.

7. Load: Each exterior wall may support different loads

from the roof, but in the model the contact area of

the walls was adjusted slightly to make the loading

pressure on the slab equal. The interior walls sup

port only the weight of the ceiling joists and the

sheet rock ceiling in addition to their own weight.

Thus, the perimeter of the slab experiences the

greatest portion of the loading that the

superstructure transmits to the slab. Perimeter wall

loads for a typical wood-frame structure, including a

42

brick veneer exterior wall, were calculated to be

approximately 400 pounds per foot. The calculation

of perimeter load is listed in Table 3-4.

Table 3-4

Calculation of Perimeter Loading (per unit length)

Item

Wall

Stud (2 in.x 4 in.)

Ceiling joist (2 in.x 8 in.)

Sheet rock (wall and ceiling, 1/2 in.)

Plate (3 layers, 2 in.x 4 in.)

Brick veneer

Celotex sheathing

Roof

Roof sheathing (shingle lath)

Plywood (1/2 in.)

Felt (20 lb)

Total

Weight (lb)

24

48

40

27

120

8

73

36

24

400

For a residence or a light commercial building, in

addition to the perimeter wall loads transmitted to

43

the slab, there is some additional interior loading,

such as plumbing and mechanical systems, appliances,

non-load bearing interior partitions, and furnish

ings, applied to the slab. Because of the difficulty

of calculating the magnitude and location of these

loadings, a uniformly distributed loading of 40 psf

is applied over the entire slab. This value was cho

sen in accordance with the American National Standard

Building Code Requirements for Minimum Design Loads

in Buildings and Other Structures . The weight of

the slab is also needed to be taken into account. It

is found by calculating the volume of concrete in the

slab and multiplying it by the unit weight of the

concrete being used, usually 145 pcf.

There are various parameters involved in this

section discussed previously. The following limits

or boundaries appear to represent the extreme condi

tions encountered in these two models. They are

shown in Table 3-5.

Table 3-5

Structural Parameters Used in This Model

44

Beam depth

Beam width

Beam spacings

Perimeter loads

Slab dimensions

Case I

18 inches

12 inches

48 ft, 24 ft

16 ft, 12 ft

400 lb/ft

48 ft X 48 ft

Case II

18 inches

12 inches

50 ft, 25 ft

16.7 ft, 12.5 ft

410 lb/ft

50 ft X 50 ft

CHAPTER IV

RESULTS OF ANALYSIS

The principal purpose of the analysis was to study the

effect of interior stiffening beams on imiting differential

deflections occurring in slabs-on-ground experiencing in

terior settlement. Two actual slab distortions were modeled

in this investigation. Applying the computer program which

used the finite element method reproduced the measured dif

ferential deflections quite closely. Then, the stiffening

beams were added to the model slab to control the deflec

tions. These results were compared to the results from the

model with no interior stiffening beams. The results of

maximum shear stress and bending moments with different

spacing of stiffening beams were also studied and compared

with permissible values of shear stress, which is the margin

of causing shear failure, and calculated bending capacity.

The results of this study are presented in four parts.

45

46

Comparison of Results

1. Differential deflection: The maximum differential

deflection, 'A', was calculated after the deflections

for each node were determined by computer models.

The maximum differential deflection was found to oc

cur between the center and the perimeter of the slab.

The maximum differential deflections for each case

and the percentage of decrease in maximum differen

tial deflections as a function of the number of

stiffening beams from the computer model's analysis

are plotted in Fig. 4-1. The maximum differential

deflection was found to decrease as the number of

stiffening beams increased. It also shows that the

smaller is the beam spacing, the greater is the per

centage decrease in maximum differential deflections,

which range from 17.6% to 34.2% in Case I, and from

11.5% to 24.7% in Case II. The deflection ratio and

percentage decrease in deflection ratio are plotted

in Fig. 4-2. In Case I, one additional interior beam A'

resulted in the deflection ratio, --- , being reduced 1

to the permissible deflection ratio, ̂ QQ ' ̂ ^i^h is

the ratio for slabs-on-ground of wood-frame structure

to limit damage to superstructure (4). In Case II,

although the ratios are greater than the permissible

ratio.

47

2.60-,

2.40>

1-1

2.20-

& 2.00-

iS 1.80-

1.60

1.40 •

1.20

0

Case I I

Case I

•r-l

1-1

40 3 4J

- 30 Maxinum Differen

t i a l Deflection

7o Decrease in

MaxiniLiTi Differen

t i a l Deflection

20

• 10

0 10 20 30 40 50

Beam Spacings (Ft)

Fig. 4-1. Comparison of Beam Spacings with Maximum Differential Deflection and with Percentage Decrease in Maximum Differential Deflection.

48

o •H 4J

•U

o I-l

1 350

1 300

1 250

200

1 150

1 100

0

Deflection Ratio, A/L

7o Decrese in Deflection

Ratio

- 50

- 40

,10

se I

Case II se II

o

5

30 U

20 5 V)

- 10

2 ^

r?

0

20 30 40 50

Beam Spacings(Ft)

Fig. 4-2. Comparison of Beam Spacings with Maxi Deflection Ratio and with Percentage Decrease in Deflection Ratio.

mum

49

the results show that interior beams can serve to A

reduce the ratios. There is much more percentage LI

decrease in deflection ratio as the number of interi

or stiffening beams increased. In Case I, it even

decreased 51.7% in deflection ratio by adding 3 in

terior stiffening beams. The comparison of maximum

differential deflections and deflection ratios are

shown in Table 4-1.

2. Shear stress: The computer results, in both cases,

show that the absolute value of the shear stress de

creased gradually as the number of stiffening beams

increased. From the ACI code (2), the permissible

shear stress

Vc=1.5v^'=75 psi (4-1)

In this analysis, a conservative fc'=2500 psi was

assumed. The maximum shear stress is always much

less than the permissible stress in both cases.

Therefore, shear failure is not the cause of the poor

performance of the studied slab-on-ground founda

tions. Table 4-2 indicates the results.

50

Table 4-1

Comparison of Differential Deflections and Deflection Ratios

Case I

No. of beams

Beam spacing (feet)

Maximum differential

deflection (inches)

Percentage decrease

in maximum deflection (%! A

Deflection ratio L

Percentage decrease in

deflection ratio (%)

2 3 4 5

48 .0 24 .0 16 .0 12 .0

1.93 1 .59 1.40 1.27

0 .0 17 .6 27 .5 34 .2

._!__ _J-__ _i__ _A_ "l99 242 274 302

0 .0 21 .6 37 .7 51 .7

Case II

No. of beams

Beam spacing (feet)

Maximum differential

deflection (inches)

Percentage decrease

in maximum deflection (%)

Deflection ratio -^-

Percentage decrease in

deflection ratio (%)

2 3 4 5

50.0 25.0 16.7 12.5

2.51 2.22 2.03 1.89

0.0 11.5 19.1 24.7

A _A_ _i__ _!__ 143" 162 177 190

0.0 13.3 23.8 32.9

51

Table 4-2

Comparison of Maximum Shear Stress

No.- of

beams

Case I

2

3

4

5

Case II

2

3

4

5

Beam

spacing

48.0

24.0

16.0

12.0

50.0

25.0

16.7

12.5

Max. shear stress

from analysis

(psi)

4.7

3.5

3.6

3.3

23.2

17.5

14.2

12.0

Max. permissible

shear stress

(psi)

75

75

75

75

75

75

75

75

3. Bending moment: The slab with more stiffening beams

will have a greater stiffness. In both cases inves

tigated, as the number of stiffening beams increased,

the bending moment also increased. The comparison of

the computer values for bending moment and the

calculated moment capacity is listed in Table 4-3.

52

Table 4-3

Comparison of Maximum Bending Moments and Calculated Moment Capacity

No. of

beams

Case I

2

3

4

5

Case II

2

3

4

5

Beam

spacing

(feet)

48.0

24.0

16.0

12.0

50.0

25.0

16.7

12.5

Maximum

bending moment

from analysis

(kips-ft/ft)

Mx My

5.885 5.183

6.879 5.923

7.482 6.595

8.030 7.079

8.121 6.804

9.834 8.194

11.019 9.150

11.959 9.828

Calculated

moment capacity

(kips-ft/ft)

Mcap

1.133

1.622

2.095

2.554

1.093

1.567

2.021

2.463

In each case investigated. The computer results are

greater than the calculated maximum moment

capacities, which were calculated from the familiar

flexure formula:

53

ft y Mcap=—Y" (4-2) where f^=allowable tensile stress of concrete, psi

The allowable tensile stress of 6Jfc' is used to de

termine the maximum moment capacity by the ACI Code

318-83 (2). All the calculations of moment capacity

for both cases are listed in Appendix C. It is ap

parent that the tensile strength of the concrete can

not resist the tensile strength produced by the slab

deformation. This is the reason why the slab

cracked.

4. Construction cost:Three primary construction costs

were considered:

a. the quantity of concrete and its costs;

b. the excavation cost; and

c. the steel reinforcement bar, assuming 2 ea #5

bar were used in the bottom of each stiffening

beam. Total cost was estimated and divided by

the area of the slab. Table 4-4 shows the cost in

crease for each additional interior beam in each

Case. A calculation of the estimated cost is listed

in Appendix D. The $0.12 to $0.37 increase in cost

per square foot is only a small fraction of the

current (1985) average cost of approximately $40 per

54

square foot for new construction (excluding cost of

the building lot). It indicates the amount of the

cost increases only slightly as the interior stiffen

ing beams are added.

Table 4-4

Comparison of Cost Increase with Increasing the Number of Stiffening Beams (per ft^)

Number of

Interior stiffening beams

Case I

1

2

3

Case II

1

2

3

Cost increase

$0.13

$0.25

$0.37

$0.12

$0.24

$0.36

CHAPTER V

CONCLUSIONS AND RECOMMENDATIONS

Conclusions

Using the SLAB2 computer code, which proved to be able

to reasonably model the the two distorted residential slab-

on-ground foundations, the following conclusions were

reached in the study.

1. Maximum differential deflections are reduced as the

spacing of equal-depth stiffening beams decreases.

It obviously indicates that interior stiffening beams

do limit the deflection in slab-on-ground founda

tions.

2. Magnitudes of service maximum bending moments in

crease as the spacing of the stiffening beams is re

duced in both cases. Increasing the number of stiff

ening beams increases the stiffness of a given

section which results in greater service moments un

der a given service loading, but increasing the num

ber of stiffening beams also increases the moment ca

pacity of the slab.

3. This study shows the shear stresses in slabs are

reduced when the spacing of the stiffening beams are

55

56

decreased. The small values of shear stress indicate

the slab is thick enough to resist the shear stress.

The structural damage or poor performance of the

slabs in this study were not caused by shear failure.

4. From an economical point of view, although it is not

required to construct interior beams on the underside

of the slab in the building codes or construction

specifications, using stiffening beams is an effec

tive measure to prevent the slab from experiencing

large differential deflections. This study shows

that adding interior stiffening beams increases the

initial cost of construction a very small amount.

Recommendat ion

It is recommended that interior stiffening beams be

used in the design and construction of slab-on-ground foun

dations in order to control differential deflections which

might occur due to settlement of the support subgrade be

neath the slab interior. As this study showed, it is prac

tical and economical to use interior stiffening beams when

constructing residential and light commercial buildings.

Thus, if site preparation and subgrade compaction cannot be

controlled during construction, use of interior stiffening

beams is especially recommended.

LIST OF REFERENCES

1. ACI Committee 302, "Recommended Practice for Concrete Floor and Slab Construction," Title No 65-42, ACI Journal, August 1968, p. 581.

2. American Concrete Institute, "Building Code Requirements for Reinforced Concrete," ACI Standard 318-83, American Concrete Institute, Detroit, Mich., 1983.

3. Bowles, J. E., Foundation Analysis and Design, 3rd edition, McGraw-Hill Book Co., New York, N.Y., 1968, pp. 86-90.

Building Research Advisory Board, "National Research Council Criteria for Selection and Residential Slabs-on Ground," U.S. National Academy of Sciences Publication 1571, Wash., D.C., 1968.

5. Huang. Y. H., "Finite Element Analysis cf Slabs on Elastic Solids," Transportation Engineering Journal, ASCE, Vol. 100, No. TE2, May 1974, pp. 403-410.

6. Insurance Company of North America, report of damage inspection, (4602 Scotswood Drive), August 30, 1983.

7. Insurance Company of North America, report of damage inspection, (5708 Hillside Avenue), February 18, 1984.

8. Jones, D. E., and Holtz, W. G., "Expansive Soils-The Hidden Disaster," Civil Engineering--ASCE Vol. 43, No. 8, New York, N.Y., August 1973, pp. 49-51.

57

:8

Leonards, G. A., Editor, Foundation Engineering, McGraw-hill Book Co., New York, N.Y., 1968, pp. 86-90.

10. McCarthy, D. F., Essential of Soil Mechanics and Foundations, Reston Publishing Company Inc., Reston Virginia, 1982, pp. 47-52.

11. Mitchell, J. K., Fundamentals of Soil Behavior, John Wiley & Sons Inc., New York, N.Y., 1976, pp 169-185.

12. Pierce, David M., "A Numerical Method of Analyzing Prestressed Concrete Members Containing Unbonded Tendons," Dissertation presented to the' University of Texas at Austin, Texas, 1968.

13. Singer, F. L. , Strength of Materials, 2nd Edition, Harper and Row, New York, N. Y., 1962, p. 314 and p. 545.

14. United States Department of Agriculture Soil Conservation Service in cooperation with Texas Agricultural Station. "Soil Survey of Potter County," February 1980, p. 129.

15. United States Department of Agriculture Soil Conservation Service in cooperation with Texas Agricultural Station. "Soil Survey of Randall County," June 1970, p. 22, sheet No. 4 and 11.

16. Wray, W. K., "Development of Design Procedure for Residential and Light Commercial Slabs-On-Ground Constructed over Expansive Soils," Dissertation presented to the Texas A & M University, College Station, Texas, December 1978.

APPENDIX A

INPUT DATA FOR CASE I AND CASE II

1. Notation for Input Data

ASPACE

BEAMLL

BEAMLW

BEAMSL

BEAMSW

BSPACE

DEL

DELF

ICLF

ISOTRY

Center-to-center spacing of longitudinal

stiffening beams, in inches.

Long dimension of longitudinal beam cross-

section, in feet.

Long dimension of transverse beam cross-

section, in feet.

Short dimension of longitudinal cross-

section, in feet.

Short dimension of transverse beam cross-

section, in feet.

Center-to-center spacing of transverse

stiffening beams, in inches.

Tolerance to control convergence, general

ly use 0.001 for coarse control.

Tolerance to control convergence, general

ly use 0.0001 for fine control.

Maximum number of iterations allowed, gene

rally use 30 for fine control.

Switch to determine if stiffness of con

stant thickness slab or of stiffened slab

is to be used in problem solution—> assign

59

60

MOIX

MOIY

NB

NCYCLE

NGAP

NLOAD

NOBEAM

NPRINT

NPROB

NSLAB

NSYM

0 if constant thickness; assign 1 if

stiffened slab.

Moment of inertia of stiffened slab section

in longitudinal direction, in inches.

Moment of inertia of stiffened slab section

in transverse direction, in inches.

Half band width, equal to or greater than

(NY + 2)*3.

Maximum number of cycles for checking sub-

grade contact, generally use 10.

Total number of nodes at which a gap exists

between slab and subgrade. Assign 0 if no

gap exists or the gap is very large.

Number of elements on which load is applied

use 0 if there is no load.

Number of interior stiffening beams.

Number of nodes at which stresses are to be

printed.

Number of problems to be solved.

Number slabs.

Condition of symmetry. Assign 1 when no

symmetry exists, 2 when symmetric with re

spect to Y axis, 3 when symmetric with re

spect to X axis, 4 when symmetric with re

spect to both X and Y axes, and 5 for four

NWT

61

slabs symmetrically loaded.

Method employed. Assign 0 when weight is

not considered, assign 1 when weight is

considered for slab of non-constant cross-

section, and assign -1 when slab is of con

stant, rectangular cross-section.

NX Number of nodes in the X direction.

NY Number of nodes in Y direction.

PR Poisson's ratio of the concrete.

PRS Poisson's ratio of the soil.

Q Loading on slab, in psi.

QSLAB Weight of slab expressed as a uniformly

Distributed load, in psi.

T Thickness of the constant depth slab, in in.

XDA(l) Lower and upper limits of loaded area in X.

& XDA(2) direction. Use -1 to +1 if the load

covers whole width of element.

XXL Slab length, in feet.

XXS Slab width, in feet.

YDA(l) Lower and upper limits of loaded area in Y.

& YDA(2) direction. Use -1 to +1 if the load covers

whole width of element.

YM Young's modulus of the concrete, in psi.

YMS Young's modulus of the soil, in psi.

62

2. Input Data for Case I

a) 2-Stiffening beams

NPR0B=4

N0BEAM=2

XXL=48, XXS=48

BEAMLW=14.0, BEAMSW=12.0, BEAMLL=14.0, BEAMSL=12.0

ASPACE=576.0, BSPACE=576.0, MOIX=MOIY=32310

NSLAB=1, PR=0.15, T=4.0, YM=3X10 , YMS=50, PRS=0.4

NSYM=1, NB=45, NX=13, NY=13, NCYCLE=10, NPRINT=169

VALUES OF X ARE: 0.0 4.0 8.0 12.0 16.0 20.0 24.0

28.0 32.0 36.0 40.0 44.0 48.0

VALUES OF Y ARE: 0.0 4.0 8.0 12.0 16.0 20.0 24.0

28.0 32.0 36.0 40.0 44.0 48.0

NGAP=73, NLOAD=66, NICL=10, NCK=5, NWT=1, Q=5.65,

DEL=0.001, DELF=0.0001, RFJ=0.5, ICLF=30

NODES HAVE GAPS: 2-D, 2-E, 2-F, 2-G, 2-H, 2-1, 2-J,

2-K, 2-L, 3-D, 3-E, 3-F, 3-G, 3-H, 3-1, 3-J,

3-K, 3-L, 4-C, 4-D, 4-E, 4-F, 4-G, 4-H, 4-1,

4-J, 4-K, 4-L, 4-M, 5-E, 5-F, 5-G, 5-1, 5-J,

5-K, 5-L, 6-F, 6-1, 6-J, 6-K, 6-L, 6-M, 7-F,

7-G, 7-H, 7-1, 7-J, 7-K, 7-L, 8-A, 8-B, 8-C,

8-G, 8-H, 8-1, 8-J, 8-K, 8-L, 9-B, 9-C, 9-H,

9-1, 9-J, 9-K, 9-L,10-C,10-D,12-D,12-E,12-F,

12-G,13-E,13-F

ALL GAP INPUT VALUES WERE 0.5 INCHES EXCEPT 3-F WHICH

63

WAS 1.00 INCH, AND 13-E AND 13-F WHICH WERE 0.25

INCHES EACH.

QSLAB=0.61

LOADED ELEMENTS : 1 , 2, 3, 4, 5, 6, 7, 8,

9, 10, 11, 12, 17, 24, 29, 30,

31, 32, 33, 34, 35, 36, 41, 45,

48, 49, 53, 57, 60, 62, 65, 69,

72, 74, 77, 81, 84, 86, 89, 93,

96, 97, 98,101,105,108,109,113,

117,119,121,125,129,132,133,134,

135,136,137,138,139,140,141,142,

143,144.

XDA(1)=-1.00, XDA(2)=-0.75, XYA(1)=0.00, XYA(2)=0.00

FOR ELEMENTS 1 TO 11.

XDA(1)=-1.00, XDA(2)=-0.75, XYA(1)=0.75, XYA(2)=1.00

FOR ELEMENT 12.

XDA(1)=0.00, XDA(2)=0.00, XYA(1)=-1.00, XYA(2)=0.00

FOR ELEMENTS 17, 41, 53, 65, 77, 89, 101, 113, 125, 137

XDA(1)=0.00, XDA(2)=0.00, XYA(1)=0.75, XYA(2)=1.00

FOR ELEMENTS 24, 48, 60, 72, 84, 96, 108, 119, 132.

XDA(1)=0.90, XDA(2)=1.00, XYA(l)=-0.10, XYA(2)=0.00

FOR ELEMENT 29.

XDA(1)=0.90, XDA(2)=1.00, XYA(1)=0.00, XYA(2)=0.00

FOR ELEMENTS 30 TO 35-

XDA(1)=0.90, XDA(2)=1.00, XYA(1)=0.75, XYA(2)=1.00

64 FOR ELEMENT 36.

XDA(1)=0.00, XDA(2)=0.00, XYA(1)=0.90, XYA(2)=1.00

FOR ELEMENTS 45, 57, 69, 81, 93, 105, 117, 129.

XDA(1)=0.75, XDA(2)=1.00, XYA(1)=0.00, XYA(2)=0.00

FOR ELEMENTS 49, 134, 135, 136, 138, 139, 140, 142, 143

XDA(1)=0.00, XDA(2)=0.00, XYA(l)=-0.35, XYA(2)=-0.10

FOR ELEMENTS 62, 74, 86, 98.

XDA(1)=0.00, XDA(2)=0.00, XYA(1)=-0.10, XYA(2)=0.15

FOR ELEMENT 97.

XDA(1)=0.00, XDA(2)=0.00, XYA(1)=-1.00, XYA(2)=-0.75

FOR ELEMENTS 109, 121.

XDA(1)=0.75, XDA(2)=1.00, XYA(1)=-1.00, XYA(2)=-0.75

FOR ELEMENT 133.

XDA(1)=0.75, XDA(2)=1.00, XYA(1)=0.90, XYA(2)=1.00

FOR ELEMENT 141.

XDA(1)=0.75, XDA(2)=1.00, XYA(1)=0.75, XYA(2)=1.00

FOR ELEMENT 144.

b) 3-Stiffening Beams

All input data are the same as in the 2-stiffening beams

model except N0BEAM=3, ASPACE=BSPACE=288,

MOIX=MOIY=44800.

c). 4-Stiffening Beams



MOIX=MOIY=51690.

65

d). 5-Stiffening Beams



MOIX=MOIY=66650.

3. Input Data for Case II

a). 2-stiffening beams

NPR0B=4

N0BEAM=2

XXL=50, XXS=50

BEAMLW=14.0, BEAMSW=12.0, BEAMLL=14.0, BEAMSL=12.0

ASPACE=600.0, BSPACE=600.0, MOIX=MOIY=32560

NSLAB=1, PR=0.15, T=4.0, YM=3X10 , YMS=50, PRS=0.4

NSYM=1, NB=39, NX=11, NY=11, NCYCLE=10, NPRINT=121

VALUES OF X ARE: 0.0 5.0 10.0 15.0 20.0 25.0 30.0

35.0 40.0 45.0 50.0

VALUES OF Y ARE: 0.0 5.0 10.0 15.0 20.0 25.0 30.0

35.0 40.0 45.0 50.0

NGAP=54, NLOAD=56, NICL=10, NCK=5, NWT=1, Q=5.70,

DEL=0.001, DELF=0.0001, RFJ=0.5, ICLF=30

NODES HAVE GAPS: 1-F, 1-G, 1-H, 2-D, 2-E, 2-F, 2-G,

2-H, 2-1, 2-J, 3-D, 3-E, 3-F, 3-G, 3-H, 3-1,

3-J, 3-K, 4-D, 4-1, 4-J, 4-K, 5-D, 5-1, 5-J,

5-K, 6-D, 6-1, 6-J, 6-K, 7-E, 7-1, 7-J, 7-K,

8-1, 8-J, 8-K, 9-E, 9-F, 9-H, 9-1, 9-J,10-E,

66

10-F,10-G,10-H,10-I,10-J,11-C,11-D,11-E,11-F,

11-G,11-H.

ALL GAP INPUT VALUES WERE 0.5 INCHES EXCEPT 2-F, 4-K,

5-J, 5-K, 6-K, 7-K, 8-J, 8-K, 10-E, 10-F, 10-H, lO-I,

AND 11-E WHICH WERE 1.0 INCH EACH, AND 5-1, AND 11-F

WHICH WERE 0.75 INCHES EACH, AND 4-J, AND 6-J WHICH WERE

1.25 INCHES EACH, AND 9-H, AND 9-1 WHICH WERE 1.50

INCHES EACH.

QSLAB=0.61

LOADED ELEMENTS: 1, 2, 3, 4, 5, 6, 7, 8,

9, 10, 14, 20, 24, 25, 26, 27,

28, 29, 30, 34, 40, 42, 44, 48,

50, 52, 54, 58, 59, 60, 61, 62,

64, 68, 70, 71, 74, 75, 76, 77,

78, 80, 81, 84, 88, 90, 91, 92,

93, 94, 95, 96, 97, 98, 99,100.

XDA(1)=-1.00, XDA(2)=-0.80, XYA(1)=0.00, XYA(2)=0.00


XDA(1)=-1.00, XDA(2)=-0.80, XYA(1)=0.80, XYA(2)=1.00

FOR ELEMENT 10.

XDA(1)=0.00, XDA(2)=0.00, XYA(1)=0.50, XYA(2)=0.60

FOR ELEMENTS 14, 24, 34, 44, 54, 64, 74, 84.

XDA(1)=0.00, XDA(2)=0.00, XYA(1)=0.80, XYA(2)=1.00

FOR ELEMENTS 20, 30, 40, 50, 70, 80, 90.

XDA(l)=-0.20, XDA(2)=-0.10, XYA(1)=0.00, XYA(2)=0.00

67


XDA(1)=0.00, XDA(2)=0.00, XYA(l)=-0.30, XYA(2)=-0.10


XDA(1)=-1.00, XDA(2)=1.00, XYA(1)=0.00, XYA(2)=1.00

FOR ELEMENT 48.

XDA(1)=-1.00, XDA(2)=-0.90, XYA(l)=-0.30, XYA(2)=-0.20

FOR ELEMENT 58.

XDA(1)=-1.00, XDA(2)=-0.90, XYA(1)=0.00, XYA(2)=0.00

FOR ELEMENT 59.

XDA(1)=-1.00, XDA(2)=-0.90, XYA(1)=0.80, XYA(2)=1.00

FOR ELEMENT 60.

XDA(1)=0.00, XDA(2)=0.20, XYA(1)=-1.00, XYA(2)=-0.80

FOR ELEMENT 61.

XDA(1)=0.00, XDA(2)=0.20, XYA(l)=-0.30, XYA(2)=-0.10

FOR ELEMENT 62.

XDA(1)=-0.10, XDA(2)=0.00, XYA(1)=0.00, XYA(2)=0.00

FOR ELEMENTS 75, 76, 77.

XDA(1)=0.00, XDA(1)=0.00, XYA(l)=-0.30, XYA(2)=-0.20


XDA(1)=0.00, XDA(2)=0.00, XYA(1)=-1.00, XYA(2)=-0.80

FOR ELEMENT 81.

XDA(1)=0.80, XDA(2)=1.00, XYA(1)=-1.00, XYA(2)=-0.80

FOR ELEMENT 91.

XDA(1)=0.80, XDA(2)=1.00, XYA(1)=0.00, XYA(2)=0.00

FOR ELEMENTS 92, 93, 95, 96, 97, 99.

68

XDA(1)=0.80, XDA(2)=1.00, XYA(1)=-0.30, XYA(2)=-0.20

FOR ELEMENT 98.

XDA(1)=0.80, XDA(2)=1.00, XYA(1)=0.80, XYA(2)=1.00

FOR ELEMENT 100.

b). 3-Stiffening Beams



MOIX=MOIY=45250.

c). 4-Stiffening Beams



MOIX=MOIY=56700.

d). 5-Stiffening Beams



MOIX=MOIY=67320.

APPENDIX B

MAXIMUM DIFFERENTIAL DEFLECTION FOR CASE I AND CASE II

1. Case I

Coordinates Differential deflection

2-Beams 3-Beams 4-Beams 5-Beams

1-A

1-B

1-C

1-D

1-E

1-F

1-G

1-H

l-I

1-J

1-K

1-L

1-M

2-A

2-B

2-C

2-D

2-E

69

1 .93

1 .65

1 .39

1 .17

1 .01

0 . 9 0

0 . 8 5

0 . 8 7

0 . 8 5

1 .08

1 .27

1 .50

1 .75

1 .68

1 . 4 1

1 .15

1 .43

1 .26

1 .59

1 .36

1 .14

0 . 9 5

0 . 8 1

0 . 7 2

0 . 6 7

0 . 6 8

0 . 7 5

0 . 8 6

1 .01

1 .20

1 . 4 1

1 .38

1 .15

0 . 9 4

1 .26

1 . 1 1

1 .40

1 .19

1 .00

0 . 8 3

0 . 7 0

0 . 6 2

0 . 5 8

0 . 5 8

0 . 6 3

0 . 7 3

0 . 8 6

1.02

1 .20

1 .21

1 .01

0 . 8 2

1 .16

1 .04

1.27

1.08

0 . 9 0

0 . 7 5

0 . 6 3

0 . 5 5

0 . 5 1

0 . 5 2

0 . 5 6

0 . 6 5

0 . 7 7

0 . 9 1

1 .08

1 .09

0 . 9 1

0 . 7 5

1.10

0 . 9 8

70 2 - F

2-G

2-H

2 - 1

2 - J

2-K

2-L

2-M

3-A

3-B

3-C

3-D

3-E

3 -F

3-G

3-H

3 - 1

3 - J

3-K

3-L

3-M

4-A

4-B

4-C

4-D

1 . 1 5

1 .10

1 .12

1 .20

1 .33

1 . 5 1

1 .73

1 .48

1 .45

1 .18

0 . 9 3

1 . 2 1

1 .04

1 .43

0 . 8 7

1 .45

1 .18

0 . 9 3

1 .28

1 .50

1 .23

1 .26

0 . 9 9

1 .25

1 .03

1 .02

0 . 9 8

0 . 9 9

1 .05

1 .16

1 .30

1 .48

1 .18

1 .18

0 . 9 6

0 . 7 6

1 .08

0 . 9 4

1 .34

0 . 8 0

0 . 8 1

0 . 8 7

0 . 9 7

1 . 1 1

1 .28

0 . 9 7

1 . 0 1

0 . 8 0

1 .10

0 . 9 3

0 . 9 5

0 . 9 1

0 . 9 2

0 . 9 7

1 .06

1 .18

1 .33

1 .00

1 .03

0 . 8 4

0 . 6 6

1 .01

0 . 8 8

1.30

0 . 7 6

0 . 7 7

0 . 8 1

0 . 9 0

1 .02

1 .16

0 . 8 1

0 . 8 8

0 . 7 0

1.02

0 . 8 8

0 . 9 1

0 . 8 7

0 . 8 7

0 . 9 2

1 .00

1 .11

1.24

0 . 8 9

0 . 9 3

0 . 7 6

0 . 6 0

0 . 9 6

0 . 8 5

1 .27

0 . 7 4

0 . 7 4

0 . 7 8

0 . 8 6

0 . 9 6

1 .09

0 . 7 3

0 . 7 9

0 . 6 3

0 . 9 7

0 . 8 4

71 4 - E

4 - F

4-G

4-H

4 - 1

4 - J

4-K

4 -L

4-M

5-A

5-B

5-C

5-D

5-E

5 -F

5-G

5-H

5 - 1

5 - J

5-K

5-L

5-M

6-A

6-B

6-C

0 . 8 6

0 . 7 5

0 . 7 0

0 . 7 1

0 . 7 9

0 . 9 1

1 .09

1 .30

1 .53

1 . 2 1

0 . 8 5

0 . 6 1

0 . 5 0

0 . 7 3

0 . 6 2

0 . 5 7

0 . 0 8

0 . 6 5

0 . 7 8

0 . 9 5

1 .16

0 . 8 9

1 .00

0 . 7 5

0 . 5 2

0 . 7 9

0 . 7 0

0 . 6 6

0 . 6 7

0 . 7 2

0 . 8 2

0 . 9 6

1 .02

1 .30

0 . 8 8

0 . 6 8

0 . 4 9

0 . 3 2

0 . 6 9

0 . 6 0

0 . 5 6

0 . 0 6

0 . 6 2

0 . 7 1

0 . 8 4

1 .00

0 . 6 8

0 . 7 9

0 . 6 0

0 . 4 2

0 . 7 6

0 . 6 8

0 . 6 4

0 . 6 4

0 . 6 9

0 . 7 7

0 . 8 8

1 .02

1 .16

0 . 7 6

0 . 5 8

0 . 4 2

0 . 2 8

0 . 6 6

0 . 5 9

0 . 5 5

0 . 0 5

0 . 5 9

0 . 6 7

0 . 7 8

0 . 9 1

0 . 5 6

0 . 6 7

0 . 5 1

0 . 3 5

0 . 7 3

0 . 6 6

0 . 6 3

0 . 6 3

0 . 6 7

0 . 7 4

0 . 8 4

0 . 9 6

1.09

0 . 6 8

0 . 5 2

0 . 3 8

0 . 2 5

0 . 6 5

0 . 5 8

0 . 5 5

0 . 0 5

0 . 5 9

0 . 6 5

0 . 7 5

0 . 8 6

0 . 4 9

0 . 6 6

0 . 4 5

0 . 3 2

72 6-D

6-E

6 - F

6-G

6-H

6 - 1

6 - J

6-K

6-L

6-M

7-A

7-B

7-C

7-D

7-E

7 - F

7-G

7-H

7 - 1

7 - J

7-K

7-L

7-M

8-A

8-B

0 . 3 2

0 . 1 6

0 . 5 5

0 . 0 0

0 . 0 1

0 . 5 8

0 . 7 0

0 . 8 7

1 .09

1 .32

0 . 9 5

0 . 7 1

0 . 4 9

0 . 3 0

0 . 1 4

0 . 5 4

0 . 4 9

0 . 4 9

0 . 5 6

0 . 7 9

0 . 8 6

1 .08

0 . 8 1

1 .46

1 .23

0 . 4 6

0 . 1 3

0 . 5 4

0 . 0 0

0 . 0 1

0 . 5 6

0 . 6 5

0 . 7 6

0 . 9 4

1 .12

0 . 7 5

0 . 5 6

0 . 3 9

0 . 2 4

0 . 1 2

0 . 5 3

0 , 4 9

0 . 4 9

0 . 5 5

0 . 6 4

0 . 7 7

0 . 9 4

0 . 6 1

1 .26

1 .07

0 . 2 2

0 . 1 1

0 . 5 3

0 . 0 0

0 . 0 0

0 . 5 4

0 . 6 2

0 . 7 2

0 . 8 6

1 .00

0 . 6 3

0 . 4 7

0 . 3 2

0 . 1 9

0 . 0 9

0 . 5 2

0 . 4 8

0 . 4 9

0 . 5 3

0 . 6 1

0 . 7 1

0 . 8 5

0 . 5 0

1 .13

0 . 9 8

0 . 2 0

0 . 1 0

0 . 5 3

0 . 0 0

0 . 0 0

0 . 5 4

0 . 6 0

0 . 7 0

0 . 8 1

0 . 9 4

0 . 5 6

0 . 4 2

0 . 2 9

0 . 1 8

0 . 0 9

0 . 5 2

0 . 4 8

0 . 4 9

0 . 5 3

0 . 6 0

0 . 6 9

0 . 8 1

0 . 4 4

1 .06

0 . 9 2

73 8-C

8-D

8-E

8 -F

8-G

8-H

8 - 1

8 - J

8-K

8-L

8-M

9-A

9-B

9-C

9-D

9-E

9 -F

9-G

9-H

9 - 1

9 - J

9-K

9-L

9-M

10-A

1 . 0 1

0 . 3 3

0 . 1 8

0 . 0 8

0 . 5 3

0 . 5 4

0 . 6 1

0 . 7 4

0 . 9 2

1 .13

0 . 8 7

1 .04

1 . 3 1

1 .09

0 . 4 1

0 . 2 7

0 . 1 8

0 . 1 4

0 . 6 5

0 . 7 3

0 . 8 6

1 .04

1 .25

0 . 9 9

1 .18

0 . 8 9

0 . 2 6

0 . 1 5

0 . 0 7

0 . 5 3

0 . 5 3

0 . 5 9

0 . 6 5

0 . 8 2

0 . 9 8

0 . 6 7

0 . 8 2

1 .14

0 . 9 7

0 . 3 3

0 . 2 2

0 . 1 5

0 . 1 1

0 . 6 2

0 . 6 8

0 . 7 8

0 . 9 2

1 .09

0 . 7 7

0 . 9 4

0 . 8 3

0 . 2 1

0 . 1 1

0 . 0 4

0 . 5 1

0 . 5 2

0 . 5 6

0 . 6 4

0 . 7 5

0 . 8 9

0 . 5 4

0 . 6 8

1 .03

0 . 8 8

0 . 2 6

0 . 1 7

0 . 1 1

0 . 0 8

0 . 5 9

0 . 6 3

0 . 7 2

0 . 8 4

0 . 9 8

0 . 6 3

0 . 7 8

0 . 8 0

0 . 1 9

0 . 1 0

0 . 0 4

0 . 5 1

0 . 5 2

0 . 5 6

0 . 6 3

0 . 7 2

0 . 8 4

0 . 4 8

0 . 6 1

0 . 9 7

0 . 8 4

0 . 2 4

0 . 1 5

0 . 1 0

0 . 0 7

0 . 5 8

0 . 6 2

0 . 7 0

0 . 8 0

0 . 9 2

0 . 5 6

0 . 6 9

74

10-B

10 -C

10-D

1 0 - E

1 0 - F

10-G

10-H

l O - I

1 0 - J

10-K

1 0 - L

10-M

11-A

11-B

11-C

11-D

11 -E

1 1 - F

11-G

11-H

l l - I

1 1 - J

11-K

1 1 - L

11-M

0 . 9 5

1 .23

1 . 0 5

0 . 4 1

0 . 3 2

0 . 2 9

0 . 3 1

0 . 3 9

0 . 5 3

0 . 7 1

0 . 9 3

1 .16

1 .37

1 .13

0 . 9 1

0 . 7 3

0 . 5 9

0 . 5 1

0 . 4 8

0 . 5 2

0 . 6 0

0 . 7 4

0 . 9 3

1 .14

1 .38

0 . 7 6

1 .08

0 . 9 4

0 . 3 3

0 . 2 6

0 . 2 4

0 . 2 5

0 . 3 2

0 . 4 2

0 . 5 7

0 . 7 4

0 . 9 2

1.10

0 . 9 1

0 . 7 4

0 . 5 9

0 . 4 8

0 . 4 1

0 . 3 9

0 . 4 2

0 . 4 9

0 . 6 0

0 . 7 5

0 . 9 2

1 .11

0 . 6 2

0 . 9 8

0 . 8 5

0 . 2 6

0 . 2 0

0 . 1 7

0 . 1 9

0 . 2 4

0 . 3 4

0 . 4 6

0 . 6 0

0 . 7 6

0 . 9 1

0 . 7 5

0 . 6 0

0 . 4 7

0 . 3 7

0 . 3 2

0 . 3 0

0 . 3 2

0 . 3 8

0 . 4 8

0 . 6 1

0 . 7 6

0 . 9 2

0 . 5 5

0 . 9 2

0 . 8 1

0 . 2 3

0 . 1 8

0 . 1 6

0 . 1 7

0 . 2 2

0 . 3 0

0 . 4 1

0 . 5 4

0 . 6 8

0 . 8 1

0 . 6 7

0 . 5 3

0 . 4 2

0 . 3 3

0 . 2 8

0 . 2 6

0 . 2 8

0 . 3 4

0 . 4 3

0 . 5 5

0 . 6 8

0 . 8 3

75

12-A

12-B

12-C

12-D

12-E

12-F

12-G

12-H

12-1

1 2 - J

12-K

12-L

12-M

13-A

13-B

13-C

13-D

13-E

13-F

13-G

13-H

13-1

1 3 - J

13-K

13-L

1 .59

1 . 3 5

1 .13

1 .44

1 .29

1 .22

1 .19

0 . 7 5

0 . 8 4

0 . 9 9

1 .17

1 .39

1 .63

1 .84

1 .59

1 .36

1 .17

1 .28

1 .20

0 . 9 4

0 . 9 9

1 .10

1 .25

1 .44

1 .66

1 .29

1 .09

0 . 9 1

1 .26

1 .15

1 .08

1 .07

0 . 6 0

0 . 6 8

0 . 8 0

0 . 9 5

1 .13

1 .33

1 .50

1 .29

1 .10

0 . 9 5

1 .08

1 .02

0 . 7 6

0 . 8 0

0 . 8 9

1 . 0 1

1 .17

1 .36

1 .08

0 . 9 0

0 . 7 4

1 .11

1 .01

0 . 9 5

0 . 9 4

0 . 4 7

0 . 5 4

0 . 6 5

0 . 7 8

0 . 9 4

1 .11

1 .26

1.07

0 . 9 0

0 . 7 6

0 . 9 0

0 . 8 5

0 . 5 9

0 . 6 3

0 . 7 1

0 . 8 2

0 . 9 7

1.14

0 . 9 6

0 . 8 0

0 . 6 6

1.04

0 . 9 5

0 . 9 0

0 . 8 9

0 . 4 2

0 . 4 8

0 . 5 8

0 . 7 0

0 . 8 5

1 .00

1 .12

0 . 9 5

0 . 8 0

0 . 6 7

0 . 8 3

0 . 7 8

0 . 5 2

0 . 5 6

0 . 6 3

0 . 7 3

0 . 8 7

1 .02

76

13-M 1 .90 1 .54 1 .32 1 .19

2 . C a s e I I

1-A

1-B

1-C

1-D

1-E

1-F

1-G

1-H

l - I

1-J

1-K

2-A

2-B

2-C

2-D

2-E

2 - F

2-G

2-H

2 - 1

2 - J

2-K

2 . 5 1

2 . 1 1

1 .75

1 .45

1 .22

1 .58

1 .57

1 .69

1 .43

1 .75

2 . 1 0

2 . 2 0

1 . 8 1

1 .45

1 .64

1 .40

1 .77

1 .24

1 .33

1 .51

1 .76

1 .56

2 . 2 2

1 .86

1 .53

1 .25

1 .04

1 .42

1 .40

1 . 5 1

1 .15

1 .46

1 .76

1.94

1 .60

1 .28

1 . 5 1

1 .29

1 .66

1 .13

1 .19

1 .33

1 .54

1 .28

2 . 0 3

1 .70

1 .39

1 .14

0 . 9 4

1 .32

1 .29

1 .37

1 .04

1 .26

1 .53

1 .78

1.47

1.17

1 .42

1 .22

1.60

1 .06

1 .11

1.22

1 .39

1 .09

1.89

1.58

1.30

1 .05

0 . 8 7

1 .25

1.22

1 .28

0 . 9 2

1 .12

1 .35

1 .66

1.37

1.10

1 .36

1 .18

1.56

1.02

1 .05

1.14

1 .28

0 . 9 6

77 3-A

3-B

3-C

3-D

3-E

3 - F

3-G

3-H

3 - 1

3 - J

3-K

4-A

4-B

4-C

4-D

4 -E

4 - F

4-G

4-H

4 - 1

4 - J

4-K

5-A

5-B

5-C

1.92

1 .54

1 .18

1 .37

1 .14

1 .00

0 . 9 5

1 .00

1 .12

1 .32

1 .55

1 . 7 1

1 .33

0 . 9 7

1 .17

0 . 4 4

0 . 3 0

0 . 2 3

0 . 2 3

0 . 8 1

1 . 7 1

1 .66

1 .57

1 .19

0 . 8 4

1 .70

1 .37

1 .05

1 .28

1 .07

0 . 9 4

0 . 8 9

0 . 9 2

1 .01

1 .16

1.34

1 .51

1 .18

0 . 8 7

1 .11

0 . 4 1

0 . 2 8

0 . 2 1

0 . 2 0

0 . 7 5

1 .60

1 .49

1 .39

1 .06

0 . 7 6

1 .56

1 .26

0 . 9 8

1 .23

1 .04

0 . 9 1

0 . 8 6

0 . 8 7

0 . 9 4

1 .05

1 .20

1 .39

1.10

0 . 8 2

1 .08

0 . 4 0

0 . 2 7

0 . 1 9

0 . 1 8

0 . 7 1

1 .53

1 .39

1.28

0 . 9 9

0 . 7 2

1 .46

1 .18

0 . 9 2

1 .19

1 .01

0 . 8 9

0 . 8 3

0 . 8 3

0 . 8 8

0 . 9 8

1 .09

1 .29

1 .03

0 . 7 7

1 .06

0 . 3 8

0 . 2 6

0 . 1 9

0 . 1 6

0 . 6 2

1 .48

1 .31

1 .18

0 . 9 2

0 . 6 8

78 5-D

5-E

5 -F

5-G

5-H

5 - 1

5 - J

5-K

6-A

6-B

6-C

6-D

6-E

6 - F

6-G

6-H

6 - 1

6 - J

6-K

7-A

7-B

7-C

7-D

7-E

7 - F

1.04

0 . 3 2

0 . 1 7

0 . 0 8

0 . 0 6

0 . 8 6

1 .34

1 .42

1 .51

1 .13

0 . 7 8

0 . 9 9

0 . 2 7

0 . 1 1

0 . 0 3

0 . 0 0

0 . 5 5

1 .43

1 .35

1 .54

1 .16

0 . 8 1

0 . 5 2

0 . 7 9

0 . 1 4

1 .00

0 . 3 0

0 . 1 7

0 . 0 8

0 . 0 5

0 . 8 3

1 .16

1 .28

1 .33

1 .01

0 . 7 1

0 . 9 6

0 . 2 6

0 . 1 1

0 . 0 3

0 . 0 0

0 . 5 2

1 .35

1 .22

1 .35

1 .03

0 . 7 3

0 . 4 8

0 . 7 7

0 . 1 4

0 . 9 8

0 . 3 0

0 . 1 7

0 . 0 8

0 . 0 5

0 . 8 1

1 .11

1.20

1.22

0 . 9 4

0 . 6 7

0 . 9 4

0 . 2 6

0 . 1 3

0 . 0 4

0 . 0 0

0 . 5 1

1 .31

1.14

1.24

0 . 9 5

0 . 6 9

0 . 4 6

0 . 7 7

0 . 1 4

0 . 9 7

0 . 3 0

0 . 1 8

0 . 0 9

0 . 0 5

0 . 7 9

1 .07

1 .13

1 .13

0 . 8 8

0 . 6 4

0 . 9 3

0 . 2 6

0 . 1 4

0 . 0 5

0 . 0 0

0 . 4 9

1 .27

1 .08

1 .15

0 . 8 9

0 . 6 5

0 . 4 4

0 . 7 7

0 . 1 4

79 7-G

7-H

7 - 1

7 - J

7-K

8-A

8-B

8-C

8-D

8-E

8 -F

8-G

8-H

8 -1

8 - J

8-K

9-A

9-B

9-C

9-D

9-E

9 -F

9-G

9-H

9 - 1

0 . 0 6

0 . 0 6

0 . 6 3

0 . 7 7

1 .45

1 .65

1 .27

0 . 9 2

0 . 6 2

0 . 3 9

0 . 2 4

0 . 1 8

0 . 2 1

0 . 8 1

1 .50

1 .72

1 .84

1 .45

1 .08

0 . 7 7

1 .04

0 . 9 0

0 . 3 7

1 .93

2 . 0 9

0 . 0 6

0 . 0 4

0 . 5 8

0 . 6 8

1 .30

1 .45

1 .12

0 . 8 1

0 . 5 5

0 . 3 5

0 . 2 2

0 . 1 5

0 . 1 6

0 . 7 4

1 .37

1.54

1 .61

1 .27

0 . 9 5

0 . 6 8

0 . 9 7

0 . 8 4

0 . 3 0

1.84

1 .97

0 . 0 6

0 . 0 4

0 . 5 6

0 . 6 2

1.22

1.32

1 .03

0 . 7 5

0 . 5 2

0 . 3 3

0 . 2 1

0 . 1 4

0 . 1 4

0 . 6 9

1.30

1.43

1.46

1 .15

0 . 8 7

0 . 6 2

0 . 9 3

0 . 8 1

0 . 2 7

1.80

1.90

0 . 0 6

0 . 0 3

0 . 5 4

0 . 5 9

1 .15

1.22

0 . 9 5

0 . 7 1

0 . 4 9

0 . 3 2

0 . 2 0

0 . 1 2

0 . 1 2

0 . 6 6

1 .24

1 .35

1 .35

1 .07

0 . 8 1

0 . 5 8

0 . 9 0

0 . 7 9

0 . 2 5

1 .76

1.84

80

9 - J

9-K

10-A

10-B

10-C

10-D

10 -E

1 0 - F

10-G

10-H

l O - I

1 0 - J

10-K

11-A

11-B

11-C

11-D

11-E

1 1 - F

11-G

11-H

l l - I

1 1 - J

11-K

1 .34

1 .14

2 . 1 0

1 .68

1 .29

0 . 9 7

1 .73

1 . 6 1

1 . 1 1

1 .73

1 .95

1 .77

1 .64

2 . 3 9

1 .94

2 . 0 2

1 .68

1 .95

1 .60

1 .64

1 .57

1 .37

1 .74

2 . 1 6

1 .17

0 . 9 1

1 .83

1 .46

1 .12

0 . 8 3

1 .62

1 .51

1.00

1 .59

1.77

1.54

1.34

2 . 0 9

1 .68

1 .81

1.50

1 .79

1.44

1 .47

1.37

1 .13

1.44

1 .80

1 .06

0 . 7 6

1 .66

1.32

1 .01

0 . 7 5

1 .55

1 .45

0 . 9 3

1 .51

1 .66

1 .39

1 .15

1.89

1.52

1.67

1 .39

1.70

1 .35

1.37

1 .25

0 . 9 7

1 .25

1.56

0 . 9 8

0 . 6 4

1 .53

1 .22

0 . 9 4

0 . 6 9

1 .51

1 .41

0 . 8 9

1 .45

1 .58

1 .28

1 .00

1 .73

1 .39

1 .58

1 .32

1 .63

1 .29

1 .30

1 .16

0 . 8 6

1 .10

1 .39

APPENDIX C

CALCULATED MOMENT CAPACITY

Mcap .Au. where ft =6Jfc"'=300 psi for fc'=2500 psi

No. of beams

Case I

2

3

4

5

Case II

2

3

4

5

I

(in )

32,312

44,800

56,189

66,653

32,562

45,246

56,701

67,320

y

(in)

14.855

14.385

13.968

13.595

14.895

14.438

14.031

13.667

Mcap

(kips-ft)(kips-ft/ft)

54.379 1.113

77.859 1.622

100.567 2.095

122.569 2.554

54.653 1.093

78.345 1.567

101.028 2.021

123.143 2.463

81

APPENDIX D

CALCULATION OF ESTIMATED COST

No. of

1

Case I

Concrete

Quantity(yd-^) 3.93

Cost $220

Steel Reinforcement Bar

Quantity(lb) 200

Cost $50

Excavation Cost $20

Total Cost $290

Cost Increase $0.13

Case II

Concrete

Quantity(yd ) 4.10

Cost $230

Steel Reinforcement Bar

Quantity(lb) 208

Cost $52

Excavation Cost $21

Total Cost $303

Cost Increase $0.12

interior stiffening

2

7.78

$436

399

$100

$39

$575

$0.25

8.12

$455

416

$104

$41

$600

$0.24

beams

3

11.54

$646

599

$150

$58

$854

$0.37

12.06

$675

624

$156

$60

$891

$0.36

82

All the calculations are based on(local estimate):

1. $56.00 per cubic yard for concrete,

2. $0.25 per pound for steel reinforcement bar, and

3. $5.00 per cubic yard for excavation cost.

a study of the differential deflections occurring in …

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