Analysis and Design of Mid-Rise Building_2023.docx

1
1 Analysis and Design of Mid-Rise Tall Building
Analysis and Design
of
Mid-Rise Building
A graduation project
Submitted to the department of civil engineering at
The University of Baghdad
Baghdad - Iraq
In partial fulfillment of the requirement for the degree of Bachelor of
Science in civil engineering
By
Muthena’
Supervised by
AL. ADNAN NAJEM LAZEM
July /2023

2
Analysis and Design of
Mid-Rise Building
ABSTRACT:

3
The objective of this study is to develop a better understanding for the basic principles of the
structural analysis and design of Multistory Building so they can be efficiently implemented into
modern computers.
Develop a structural stiffness matrix that takes into the building variable elements prosperities
(vertical, horizontal, and foundation bearing) into consideration.
In addition several important parameters have been incorporated in the analysis and design
process; Buckling and stability of building columns and shear walls, shear buckling, maximum
allowable deflection due to live load, maximum allowable flexural strength according to ACI-318
and IBC-2020 design limitations.
To evaluate the results of presented method were compared with result given in reference
number one. The agreement between both results was quit well.
PROJECT LAYOUT
The project is divided into five chapters as follows:
Chapter one: presents a general introduction to the subject .
Chapter two: presents the previous literatures published about this subject.
Chapter three: presents the theoretical bases for the Matrix analysis method and design.
Chapter four: presents a brief description of a computer program developed in this study.
Chapter five: discuses the results of this Analysis/Design method. And recommend future steps.
CONTENTS:
Title……………………………………………………………………………………2
Supervisor words……………………………………………………………….3

4
Committee words………………………………………………………………4
Thanks……………………………………………………………………………….5
Abstract…………………………………………………………………………….6
Project Layout………………………….……………………………………….6
Contents…….……..………………………………………………………………7
Chapter one; introduction…………..….…………………………………8
Chapter two; literature………………….………………………………..12
Chapter three; theory………………………………………………………17
Chapter four; computer program…………………………………….25
Chapter five; conclusions and recommendations…………….36
References…………………………………………………………………..….39
Appendix I…………………………………………………………………..……40

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Chapter one
Introduction
INTRODUCTION

6
In order to compete in the ever-growing competent market, it is very important for a structural
engineer to save time. As a sequel to this an attempt is made to analyze and design a multistoried
building by using a software package staad.pro. For analyzing a multi storied building one has to
consider all the possible loadings and see that the structure is safe against all possible loading
conditions. There are several methods for analysis of different frames approximate frame
method, cantilever method, portal method, and Matrix method. The present project deals with
the design & analysis of a multi storied residential building of G+12 consisting of office building.
The dead load &live loads are applied and the design for beams, columns, footing is obtained
STAAD.Pro with its new features surpassed its predecessors and compotators with its data sharing
capabilities with other major software like AutoCAD. Staad.pro is a very powerful tool which can
save much time and is very accurate in Designs.
The structural design of a building should ensure that the building can stand safely, operate
without excessive deformation or movement that could lead to fatigue of structural elements,
cracks or failure of fixtures, fittings or partitions, or failure. Inconvenience to occupants. It must
take into account the movements and forces due to temperature, creep, cracks, and imposed
loads. It must also verify that the design is nearly buildable within acceptable manufacturing
tolerances of the materials. It must allow the architecture to function and the building services to
adapt to the building functionally (ventilation, lighting, etc).
This project work is to analyze a 12-storeyed building for different load combinations using STAAD
Pro software, Staad foundation Advanced, RCDC software.

7
Chapter two
Literature
2.1 LINEAR ANALYSIS OF IN-PLANE STRUCTURES USING STIFFNESS MATRIX METHOD

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The theoretical foundation for matrix (stiffness) method of structural analysis was laid and
developed by many scientists:
James, C. Maxwell, [1864] who introduced the method of Consistent Deformations (flexibility
method).
Georg, A. Maney, [1915] who developed the Slope-Deflection method (stiffness method).
These classical methods are considered to be the precursors of the matrix (Flexibility and Stiffness)
method, respectively. In the pre-computer era, the main disadvantage of these earlier methods was
that they required direct solution of Simultaneous Equations (formidable task by hand calculations
in cases more than a few unknowns).
The invention of computers in the late-1940s revolutionized structural analysis. As computers could
solve large systems of Simultaneous Equations, the analysis methods yielding solutions in that form
were no longer at a disadvantage, but in fact were preferred, because Simultaneous Equations could
be expressed in matrix form and conveniently programmed for solution on computers.
Levy, S., [1947] is generally considered to have been the first to introduce the flexibility method,
by generalizing the classical method of consistent deformations.
Falkenheimer, H., Langefors, B., and Denke, P. H., [1950], many subsequent researches extended
the flexibility method and expressed in matrix form are:
Livesley, R. K., [1954], is generally considered to have been the first to introduce the stiffness
matrix in 1954, by generalizing the classical method of slop-deflections.
Argyris, J. H., and Kelsey, S., [1954], the two subsequent researches presented a formulation for
stiffness matrices based on Energy Principles.
Turner, M. T., Clough, R. W., and Martin, H. C., [1956], derived stiffness matrices for truss
members and frame members using the finite element approach, and introduced the now popular
Direct Stiffness Method for generating the structure stiffness matrix.
Livesley, R. K., [1956], presented the Nonlinear Formulation of the stiffness method for stability
analysis of frames.
Since the mid-1950s, the development of Stiffness Method has been continued at a tremendous
pace, with research efforts in the recent years directed mainly toward formulating procedures for
Dynamic and Nonlinear analysis of structures, and developing efficient Computational Techniques
(load incremental procedures and Modified Newton-Raphson for solving nonlinear Equations) for
analyzing large structures and large displacements. Among those researchers are: S. S. Archer, C.
Birnstiel, R. H. Gallagher, J. Padlog, J. S. przemieniecki, C. K. Wang, and E. L. Wilson and
many others.
LIVESLEY, R. K. [1964] described the application of the Newton- Raphson procedure to nonlinear
structures. His analysis is general and no equations are presented for framed structures. However,
he did illustrate the analysis of a guyed tower.

9
Chapter three
Theory

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3.1. ANALYSIS METHOD USING STIFFNESS MATRIX
Stiffness Matrix method is one of the most efficient means for solving a space elastic Structures
(framed building) type of problem based on following steps. It is easy to account for Boundary
Conditions, and self weight .
It is more versatile (multi-purposes) than the Finite Difference method, which requires a different
equation formulation for ends and the boundary conditions, and great difficulty is had if the Beam
elements are of different lengths.
Only the basic elements of the Stiffness Matrix Method will be introduce here, and the researcher
is referred to KassimAli (1999) (15) or Bowles (1974) if more background is required. This method
was interpolated to computer program which is given in appendix A. The program algorithm is
explained in details in chapter four and it conveniently coded for the user. Also the same program
was used to obtain the results of the numerical examples given in chapter four of this study.
3.1.1 GENERAL EQUATION AND THEIR SOLUTION
For the Beam Element, shown in Fig.(3.1), at any node (i) (junction of two or more members) on
the in-plane structure the equilibrium equation is:
𝑃𝑖 = 𝐵𝑖𝐹𝑖……………………Eq.(3.1)
Which states that the external node force P is equated to the internal member forces F using bridging
constants A. It should be is understand that (Pi, Fi) are used for either Forces (Shear) or Bending
Moments. This equation is shorthand notation for several values of Ai, Fi summed to equal the ith
nodal force.
For the full set of nodes on any in-plane structure and using matrix notation where P, F are Columns
Vectors and A is a Rectangular Matrix, this becomes:
{𝑃𝑖} = [𝐵𝑖]{𝐹𝑖}……………………Eq.(3.2)
Fig.(3.1) Beam Element, global and local forces-deformations designation.
An Equation relating internal-member deformation e at any node to the external nodal
displacements is:
{𝑒𝑖} = [𝐴𝑖]{𝑋𝑖}………………………..(a)

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Where both e and X may be rotations (in radians) or translations. From the Reciprocal Theorem in
structural mechanics it can be shown that the [A] matrix is exactly the transpose of the [B] matrix,
thus:
{𝑒𝑖} = [𝐵]𝑇
{𝑋𝑖}……………………..(b)
The internal-member forces {F} are related to the internal-member displacements {e} as:
{𝐹𝑖} = [𝑘]{𝑒𝑖}…………………………(c)
These three equations are the fundamental equations in the Stiffness Matrix Method of analysis:
Substituting (b) into (c),
{𝐹𝑖} = [𝑘]{𝑒𝑖} = [𝑘][𝐵]𝑇
𝑋…………………………(d)
Substituting (d) into (a),
{𝑃𝑖} = [𝐵]{𝐹𝑖} = [𝐵][𝑘][𝐵]𝑇
𝑋…………………………(e)
Note the order of terms used in developing Eqs. (d) and (e}. Now the only unknowns in this system
of equations are the X’s: so the BKBT is inverted to obtain
{𝑋𝑖} = ([𝐵][𝑘] [𝐵]𝑇
)−1
{𝑃𝑖}…………………………(f)
And with the X’s values we can back-substitute into Eq. (d) to obtain the internal-member forces
which are necessary for design. This method gives two important pieces of information: (1) design
data and (2) deformation data.
The BKBT
matrix above is often called Overall assembly Matrix, since it represents the system of
equations for each P or X nodal entry. It is convenient to build it from one finite element of the
structure at a time and use superposition to build the global BKBT
from the Local element EBKBT
.
This is easily accomplished, since every entry in both the Global and Local BKBT with a unique
set of subscripts is placed into that subscript location in the BKBT; i.e., for i = 2, j = 5 all (2, 5)
subscripts in EBKBT
are added into the (2, 5) coordinate location of the global BKBT.
3.1.2. DEVELOPING THE ELEMENT [B] MATRIX
Consider the in-plane structure, simple beam, shown in Fig.(3.2) coded with four values of P-X
(note that two of these P-X values will be common to the next element) and the forces on the
element Fig.(3.2). The forces on the element include two internal Bending Moments and the shear
effect of the Bending Moments. The sign convention used is consistent with the developed
computer program BEF.

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Fig.(3.2) In-plane structure divided into finite element, (b) Global coordinate system coding in (P-X) form,
(c) Local coordinate system coding in (F-e) form, (d) Summing of external and internal nodal forces.
Now at node (1), summing Moments (Fig.(3.2d))
𝑃1 = 𝐹1 + 0. 𝐹2
Similarly, summing forces and noting that the soil reaction (spring) forces are Global and will be
considered separately, we have:
𝑃2 =
𝐹1
𝐿
+
𝐹2
𝐿
𝑃3 = 0. 𝐹1 + 𝐹2
And 𝑃4 = −
𝐹1
𝐿
−
𝐹2
𝐿
Placing into conventional matrix form, the Element Transformation Matrix [EB] in local coordinate
is:
Force-Displacement relationships (P-X indexing)
P1 P2
(a) Local force-displacement relationships (F-e indexing)
FEM FEM

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EB =
F1 F2
P1 1 0
P2 1/L 1/L
P3 0 1
P4 -1/L -1/L
In same manner the EA matrix for element (2) would contain P3 to P6.
3.1.3. DEVELOPING THE [k] MATRIX
Referring to Fig.(3.3) and using conjugate-beam (Moment Area Method)principle, the end slopes
e1, and e2 are:
𝑒1 =
𝐹1𝐿
3𝐸𝐼
−
𝐹2𝐿
6𝐸𝐼
………………………(g)
𝑒2 = −
𝐹1𝐿
6𝐸𝐼
+
𝐹2𝐿
3𝐸𝐼
…………………….(h)
Fig.(3.3) conjugate-beam method Moments and rotations of beam element.
Solving Eqs.(g) and (h) for F, obtaining:
𝐹1 =
4𝐸𝐼
𝐿
𝑒1 +
2𝐸𝐼
𝐿
𝑒2
𝐹2 =
2𝐸𝐼
𝐿
𝑒1 +
4𝐸𝐼
𝐿
𝑒2
Placing into matrix form, the Element Stiffness Matrix [ES] in local coordinate is:
Ek =
e1 e2
F1
4𝐸𝐼
𝐿
2𝐸𝐼
𝐿
F2
2𝐸𝐼
𝐿
4𝐸𝐼
𝐿

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3.1.4. DEVELOPING THE ELEMENT [kBT
] AND [BKBT
] MATRICES
The EkBT
matrix is formed by multiplying the [Ek] and the transpose of the [EB] matrix (in the
computer program this is done in place by proper use of subscripting) AT
goes always with e and
X. The EBkBT
will be also obtained in a similar.
Multiplying [Ek] and [EBT
] matrices and rearrange them, yields:
EkBT
=
1 2 3 4
1
4𝐸𝐼
𝐿
6𝐸𝐼
𝐿2
2𝐸𝐼
𝐿
−
6𝐸𝐼
𝐿2
2
2𝐸𝐼
𝐿
6𝐸𝐼
𝐿2
4𝐸𝐼
𝐿
−
6𝐸𝐼
𝐿2
Multiplying [EB] and [EkBT
] matrices and rearrange them, yields:
EBkBT
=
X1 X2 X3 X4
P1
4𝐸𝐼
𝐿
6𝐸𝐼
𝐿2
2𝐸𝐼
𝐿
−
6𝐸𝐼
𝐿2
P2
6𝐸𝐼
𝐿2
12𝐸𝐼
𝐿3
6𝐸𝐼
𝐿2
−
12𝐸𝐼
𝐿3
P3
4𝐸𝐼
𝐿
6𝐸𝐼
𝐿2
4𝐸𝐼
𝐿2
−
6𝐸𝐼
𝐿2
P4 −
6𝐸𝐼
𝐿2
−
12𝐸𝐼
𝐿3
−
6𝐸𝐼
𝐿2
12𝐸𝐼
𝐿3
From Fig.(3.4), summing of the vertical forces on a node 1 will produce:
𝑃2 −
𝐹1 + 𝐹2
𝐿
= 0.0
Since (F1+F2)/L is already included in the Global BkBT
, we could rewrite above equation to:
𝑃2 = 𝐵𝑘𝐵2𝑋2
𝑇
𝑋2 = [𝐵𝑘𝐵2𝑋2
𝑇
]𝑋2
A check on the correct formation of the EBkBT
and the global BkBT
is that it is always symmetrical
and there cannot be a zero on the diagonal.

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3.1.5. DEVELOPING THE P MATRICES
The P matrix (a column vector) consists in zeroing the array and then inputting those node loads
that are nonzero. The usual design problem may involve several different loading cases or
conditions, as shown in Appendix II, so the array is of the form P(I, J) where (i) identifies the load
entry with respect to the node and P-X coding and (j) the load case.
It is necessary to know the sign convention of the (P-X) coding used in forming the [EA] matrix or
output may be in substantial error. Therefore; the sign convention will be as follow: the joint
translations are considered positive when they act in positive direction of Y-axis, and joint rotations
are considered positive when they rotate in counterclockwise direction.
For columns that are intermediate between two nodes, we may do one of two things:
1. Transfer the column loads to adjacent nodes prier to make problem sketch using superposition
concept.
2. Transfer the column loads to adjacent nodes as if the element has Fixed-Ends Actions so the
values include Fixed-End moments and shears (vertical loads).This procedure is strictly correct but
the massive amount of computations is seldom worth the small improvement in computational
precision.
3.1.6. BOUNDARY CONDITIONS
The particular advantage of the Stiffness Matrix method is to allow boundary conditions of known
displacement (translations or rotations). It is common in foundation analysis to have displacements
which are known to be zero (beam on rock, beam embedded in an anchor of some type, etc.). There
are two major cases of boundary conditions:
a. When the displacements are restrained (zero) in any particular node then the corresponding
rows and columns in the overall stiffness matrix will be eliminated (substitute by zeros).
b. When the (i) displacements are known (δ) in any particular node then the opposite position
in load vector [p] will have this known value (δ), and corresponding rows and columns in
the overall stiffness matrix will be eliminated (substitute by zeros) except the location of
(i,i) which will have unit value of (1.0).

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Chapter four
COMPUTER PROGRAM

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4.1 INTRODUCTION
This chapter presents a brief description of the computer program applied in this study which
governs the problem of analysis and design of space concrete structure (Multistory Building) using
Finite Element Method (FEM). The program was written using STAAD.Pro (version 22), it is
consist of two major parts; first the analysis of the space structure. last part is the checking of
Building elements according to the ACI-318 Design Manual using USD designing method.
4.2 DEVELOPMENTS OF COMPUTER PROGRAM
4.2.1. First part; Analysis of space plated structure using FEM Method.
At first, the computer program will develop the [EB] and [Ek] for each in-plane (beam) element
from input data describing the member geometry (coordinates) and cross-section properties
(modulus of elasticity, moment of inertia, area, angle of rotation…etc). Then, the program will
develop element stiffness matrix [EBkBT
] for each element in global coordinate system, throughout
series of matrix operations (inverse, multiplication, and addition). Later on, the program will
assemble the overall stiffness matrix [BkBT
] which is also represented by [K]. Finally, a direct
solution of the general stiffness equation {P}=[K]{d}, where {P} matrix containing the known
externally applied loads, will yield the global displacements {X} (translations and rotations). The
computer program then rebuilds the [EB] and [Ek] to obtain the [EkBT
] and computes the internal
element forces (axial, shear and moments) and node reactions.
The sign convention used in this program is as follow: the joint translations are considered positive
when they act in positive direction of Y-axis, and joint rotations are considered positive when they
rotate in counterclockwise direction.
It should be noticed that all above steps should be carried out with proper indices that identifies the
(P-X) coding so that the entries are correctly inserted into their right position of matrix.
Let the number of nodes NN and since DOF is three for each node. Each element stiffness element
[EBkBT
] has (6x6) size but the overall assembled stiffness matrix [BkBT
] or [K] has (NPxNP) size
because of the assembling process, where NP = NN * 3, therefore;
{𝑃𝑁𝑃} = [𝐵𝑘𝐵𝑁𝑃 × 𝑁𝑃
𝑇 ]{𝑋𝑁𝑃}
This indicates that the System of Equations is just sufficient, which yields a square coefficient
matrix [NPxNP], the only type which can be inverted. It also gives a quick estimate of computer
needs, as the matrix is always the size of (NP x NP) the number of {P}. With proper coding, as
shown in Fig.(3.4).
The global [BkBT
] is banded with all zeros except for a diagonal strip of nonzero entries that is
twelve values wide. These twelve nonzero entries, six are identical (the band is symmetrical). There
are matrix reduction routines to solve these type half-band width problems. As a consequence the
actual matrix required (with a band reduction method) is only (NP x 6) entries instead of (NP x
NP).
It may be convenient to store the [EkBT
] on a separate array when the [BkBT
] is being built and
recall it to compute the internal element forces of the {F} matrix.

18
4.4 COMPUTER PROGRAM APPLICATION:
Case study: Analysis and design a Multistory RC Building (including piled foundation), as shown
in fig.(4.1), with a total height of (36.0 m) and carrying combined Loadings consisted of a uniformly
distributed (WLL+WDL), Wind loading (two directions), and seismic loadings. The computed
results are listed below. Each load case was investigated for max stress and deformation in order to
demonstrate several factors effects.
Fig.(4.1) Building layout

19
Fig.(4.2-1) Building Plan Layout
Fig.(4.2-2) Building Elevation

20
4.2 STRUCTURAL MEMBERS SECTIONS:
The different parts used in the project are,
4.3 SUPPORT CONDITIONS:
Deep piles used as supports , with Winkler springs as follows in the project are;

21
4.4 LOAD COMBINATIONS:
The different combinations used in the project are,

22
Fig.(4.3) Building deflection diagram due to load combinations

23
Fig.(4.4) Building deflection diagram due to wind loadings
Table (4-1) Nodal displacement under critical load combinations (ACI-318)

24
Fig.(4.5) Bending Moment diagram

25
Fig.(4.6) Shear Force diagram

26
Fig.(4.7) Axial force diagram

27
Table(4.8-a) Beams Bending Stresses

28
Table(4.8-b) Beams Bending Stresses

29
Table(4.9-a) Columns Axial Stresses

30
Table(4.9-b) Columns Axial Stresses

31
Chapter five
CONCLUSIONS AND RECOMMENDATIONS

32
CONCLUSIONS
Depending on the results obtained from the present study, several conclusions may be established.
These may be summarized as follows:
Results indicate that Multistory Building structures (framed type) can be can be dealt with
successfully by the FEM Method.
Applied Program in this study (STAAD.Pro) is quite efficient and reliable for this type of analysis,
and the process of analyses can be carried out rapidly on electronic computer. Design criteria given
by ACI-318 (USD design) has been successfully implement inside presented program.
The aim of this project was bringing idea to plan, analysis and design of a multistoried, earthquake
resistant residential building. We were successful to fully complete the project in a successful and
efficient manner by considering all the relevant features given. The design is completely depend
on relevant American Standard Codes ACI-318. The analysis and design have been done with the
help of STAAD.Pro and RCDC software (Shop drawings are attached appendix).
All structural components of the building were checked and they proved to be safe in shear and
flexure.
RECOMMENDATIONS
Many important recommendations could be suggested, for the given analysis method for in-plane
structures, to include the following factors:
1- Soil structure interaction could be furtherly studied in details
2- Seismic analysis could be altered according to specific zone conditions
3- Shear walls locations could be changed to study the best resistant framed structure.
4- Steel diagrid could studied as additional lateral supporting system.

33
REFERENCES
1- Corrin, M.E., & Swensson, K.D. (1992). Eccentrically braced frames: Not just for seismic design.
Modern Steel Construction, 33-37.
2- Connor, J.J. (2003). Introduction to Structural Motion Control. New York: Prentice Hall.
3- Huxtable, A.L. (1984). The Tall Buildings Artistically Reconsidered: The Search for a Skyscraper
Style. New York: Pantheon Books.
4- Livesley, R. K., and Chandler D. B., "Stability Functions for Structural Frameworks." Manchester
University Press, Manchester, 1956.
5- Livesley, R.K., "The Application of an Electronic Digital Computer to Some Problem of Structural
Analysis." The Structural Engineer, Vol. 34, no.1, London, 1956, PP. 1-12.
6- Argyris, J.H., "Recent Advances in Matrix Methods of Structural Analysis." Pergamon Press,
London, 1964, PP. 115-145.
7- Livesley, R.K., "Matrix Methods of Structural Analysis." Pergamon Press, London, 1964. PP. 241-
252.
8- Khan, F.R. (1967). The John Hancock Center. Civil Engineering, 37(10), 38-42.
9- Khan, F.R. (1969). Recent structural systems in steel for high-rise buildings. In Proceedings of the
British Constructional Steelwork Association Conference on Steel in Architecture. London: British
Constructional Steelwork Association.
10- Khan, F.R. (1972). Influence of design criteria on selection of structural systems for tall buildings,
In Proceedings of the Canadian Structural Engineering Conference. Toronto: Canadian Steel
Industries Construction Council, 1-15.
11- Khan, F.R. (1973). Evolution of structural systems for high-rise buildings in steel and concrete. In
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Scientific and Technical Association.
12- Khan, F.R., & Sbarounis, J. (1964). Interaction of shear walls and frames in concrete structures
under lateral loads. Structural Journal of the American Society of Civil Engineers, 90(ST3), 285-
335.
13- Kowalczyk, R., Sinn, R., & Kilmister, M.B. (Eds) (1995). Structural Systems for Tall Buildings
(Council on Tall Buildings and Urban Habitat Monograph). New York: McGraw-Hill.
14- Moon, K. (2005). Dynamic Interrelationship between Technology and Architecture in Tall
Buildings. Unpublished PhD Dissertation, Massachusetts Institute of Technology.
15- Popov, E.P. (1982). Seismic framing systems for tall buildings. Engineering Journal/American
Institute of Steel Construction, 19(Third Quarter), 141-149.
16- Schueller, W. (1986). High-Rise Building Structure (2nd ed). Malabar, Florida: Krieger.

34
17- Taranath, B. (1998). Steel, Concrete, & Composite Design of Tall Buildings. New York: McGraw-
Hill.
18- Bowles, J. E., "Analytical and Computer Methods in Foundation Engineering." McGraw-Hill Book
Co., New York, 1974, pp. 190-210.
19- Bowles, J. E., "Foundation analysis and design" McGraw-Hill Book Co., New York, 1986, fourth
edition, pp. 380-230.
20- Bowles, J. E., "Mat Design." ACI Journal, Vol. 83, No.6, Nov.-Dec. 1986, pp. 1010- 1017.
21- Timoshenko, S.P. and Gere, J.M., "Theory of Elastic Stability." 2nd Edition, McGraw-Hill Book
Company, New York, 1961, pp. 1-17.
22- KassimAli, A., "Large Deformation Analysis of Elastic Plastic Frames," Journal of Structural
Engineering, ASCE, Vol. 109, No. 8, August, 1983, pp. 1869-1886.
23- LAZEM, A. Najem, "Large Displacement Elastic Stability of Elastic Framed Structures Resting On
Elastic Foundation" M.Sc. Thesis, University of Technology, Baghdad, 2003, pp. 42-123.

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