Djee B.

Tolentino
BSCE – 3B
CE 138
Assignment #1
1. Historical Background
Since the dawn of history, structural engineering has been an essential part of
human endeavor. However, it was not until about the middle of the seventeenth century
that engineers began applying the knowledge of mechanics (mathematics and science)
in designing structures. Earlier engineering structures were designed by trial and error
and by using rules of thumb based on past experience. The fact that some of the
magnificent structures from earlier eras, such as Egyptian pyramids (about 3000 b.c.),
Greek temples (500–200 b.c.), Roman coliseums and aqueducts (200 b.c.–a.d. 200),
and Gothic cathedrals (a.d. 1000–1500), still stand today is a testimonial to the
ingenuity of their builders.
Galileo Galilei (1564–1642) is generally considered to be the originator of the
theory of structures. In his book entitled Two New Sciences, which was published in
1638, Galileo analyzed the failure of some simple structures, including cantilever
beams. Although Galileo’s predictions of strengths of beams were only approximate, his
work laid the foundation for future developments in the theory of structures and ushered
in a new era of structural engineering, in which the analytical principles of mechanics
and strength of materials would have a major influence on the design of structures.
Following Galileo’s pioneering work, the knowledge of structural mechanics
advanced at a rapid pace in the second half of the seventeenth century and into the
eighteenth century. Among the notable investigators of that period were Robert Hooke
(1635–1703), who developed the law of linear relationships between the force and
deformation of materials (Hooke’s law); Sir Isaac Newton (1642–1727), who formulated
the laws of motion and developed calculus; John Bernoulli (1667–1748), who
formulated the principle of virtual work; Leonhard Euler (1707–1783), who developed
the theory of buckling of columns; and C. A. de Coulomb (1736–1806), who presented
the analysis of bending of elastic beams.
In 1826 L. M. Navier (1785–1836) published a treatise on elastic behavior of
structures, which is considered to be the first textbook on the modern theory of strength
of materials. The development of structural mechanics continued at a tremendous pace
throughout the rest of the nineteenth century and into the first half of the twentieth, when
most of the classical methods for the analysis of structures described in this text were
developed. The important contributors of this period included B. P. Clapeyron (1799–
1864), who formulated the three-moment equation for the analysis of continuous
beams; J. C. Maxwell (1831–1879), who presented the method of consistent
deformations and the law of reciprocal deflections; Otto Mohr (1835–1918), who
developed the conjugate-beam method for calculation of deflections and Mohr’s circles
of stress and strain; Alberto Castigliano (1847–1884), who formulated the theorem of
least work; C. E. Greene (1842–1903), who developed the moment-area method; H.
Muller-Breslau (1851–1925), who presented a principle for constructing influence lines;
G. A. Maney (1888–1947), who developed the slope-deflection method, which is
considered to be the precursor of the matrix stiffness method; and Hardy Cross (1885–
1959), who developed the moment-distribution method in 1924. The moment-
distribution method provided engineers with a simple iterative procedure for analyzing
highly statically indeterminate structures. This method, which was the most widely used
by structural engineers during the period from about 1930 to 1970, contributed
significantly to their understanding of the behavior of statically indeterminate frames.
Many structures designed during that period, such as high-rise buildings, would not
have been possible without the availability of the moment-distribution method.
The availability of computers in the 1950s revolutionized structural analysis.
Because the computer could solve large systems of simultaneous equations, analyses
that took days and sometimes weeks in the pre-computer era could now be performed
in seconds. The development of the current computer-oriented methods of structural
analysis can be attributed to, among others, J. H. Argyris, R. W. Clough, S. Kelsey, R.
K. Livesley, H. C.Martin, M. T. Turner, E. L.Wilson, and O. C. Zienkiewicz.
From ancient civilizations to the digital age, engineers and researchers have
continuously pushed the boundaries of knowledge and technology to create safer, more
efficient, and more resilient structures. As we look to the future, the legacy of past
achievements serves as a foundation for further progress and discovery in the field of
advanced structural analysis.
2. Classical, Matrix, and Finite-Element Methods of Structural Analysis
Structural analysis is a crucial aspect of engineering, especially in civil and
mechanical engineering fields, where the behavior of structures under various loads
needs to be understood and predicted. Classical, Matrix, and Finite-Element Methods
are three approaches used for structural analysis, each with its own advantages and
limitations.
 Classical Method
The classical method, also known as the "hand calculation" method, relies on
simplified assumptions and mathematical equations to analyze structures. It's commonly
used for simple and small-scale structures where assumptions of linear behavior and
uniform materials are valid. Engineers typically use equations derived from equilibrium
principles, such as statics and mechanics of materials, to calculate forces, stresses, and
displacements in a structure. It is simple and straightforward calculations and provides
insight into the behavior of structures. But it has limited applicability to complex and
non-linear structures and prone to errors due to oversimplified assumptions.
 Matrix Method
The matrix method, also known as the stiffness method or the displacement
method, represents the structure as an interconnected system of nodes and elements.
Suitable for analyzing structures with moderate complexity and non-linear behavior.
Involves assembling a system of linear equations based on the stiffness of individual
elements and their connections, which are then solved using matrix algebra techniques.
It is capable of handling moderate complexity and non-linear behavior. Allows for more
accurate modeling of structural behavior compared to classical methods. On the other
hand, it becomes computationally intensive for large-scale structures and requires
expertise in matrix algebra and programming.
 Finite Element Method
FEM is a numerical technique that divides a complex structure into smaller,
simpler elements to analyze its behavior under various conditions. Widely used for
analyzing structures with complex geometries, material properties, and loading
conditions. Involves discretizing the structure into finite elements, applying boundary
conditions and loads, and then solving the resulting system of equations to obtain
displacements, stresses, and other relevant information. It is highly versatile and
capable of modeling complex structures with accuracy and can handle non-linear
behavior and material properties. This method requires significant computational
resources, especially for large-scale simulations and expertise in numerical methods
and finite element analysis software.
In summary, while the classical method is suitable for simple structures, the
matrix method offers improved accuracy for moderately complex systems, and the finite
element method is the go-to choice for analyzing highly complex structures with non-
linear behavior. Each method has its place in structural analysis, and the choice
depends on the specific requirements of the project and available resources.
3. Flexibility and Stiffness Methods
The Flexibility Method and the Stiffness Method are two fundamental approaches
in structural analysis, each offering unique perspectives on how to solve for the internal
forces and displacements within a structure.
 Flexibility Method
The Flexibility Method, also known as the Displacement or Force Method,
represents a powerful tool particularly suited for analyzing statically indeterminate
structures. At its core, this method focuses on determining the displacements at various
points within the structure in response to applied loads. The procedure commences with
the formulation of a flexibility matrix, which delineates the relationship between applied
loads and resulting displacements. Through the application of boundary conditions,
constraints or supports are incorporated into the analysis, reducing the number of
unknown displacements. Subsequently, the system of equations representing the
structure's equilibrium is solved to ascertain the unknown displacements. Once the
displacements are determined, the method facilitates the calculation of member forces
and reactions, providing comprehensive insights into the structural behavior.
The advantages of the Flexibility Method lie in its efficacy for indeterminate
structures and its ability to offer valuable insights into structural behavior by focusing on
displacements. Moreover, its flexibility in handling complex boundary conditions
enhances its applicability to real-world engineering scenarios. However, the method's
reliance on the inversion of the stiffness matrix to obtain the flexibility matrix may entail
computational intensity, particularly for structures with numerous degrees of freedom.
 Stiffness Method:
The Stiffness Method, also referred to as the Displacement or Matrix Method,
represents a systematic approach to structural analysis that is applicable to both
statically determinate and indeterminate structures. Unlike the Flexibility Method, which
prioritizes displacements, the Stiffness Method focuses on finding the internal forces
and displacements within a structure by considering its stiffness properties. The
procedure begins with the formulation of a stiffness matrix, which relates the forces and
displacements at various points in the structure. Similar to the Flexibility Method,
boundary conditions are applied to account for constraints or supports. The subsequent
solution of the system of equations representing the structure's equilibrium yields the
unknown displacements, facilitating the calculation of member forces and reactions.
The Stiffness Method offers advantages in its versatility, as it can handle both
determinate and indeterminate structures, and its systematic approach to solving
complex structural problems. Additionally, its capability to handle a wide range of
boundary conditions enhances its applicability to diverse engineering contexts.
However, the method may necessitate significant computational resources, particularly
for large structures, and the formulation and manipulation of the stiffness matrix can be
complex, requiring expertise in linear algebra.
In conclusion, the Flexibility and Stiffness Methods represent indispensable tools
in structural analysis, offering complementary approaches to solving complex
engineering problems. While the Flexibility Method excels in analyzing statically
indeterminate structures and provides valuable insights into structural behavior through
its focus on displacements, the Stiffness Method offers a systematic approach
applicable to both determinate and indeterminate structures. The choice between the
two methods depends on factors such as the nature of the structure being analyzed, the
desired level of detail, and the available computational resources. Nonetheless, both
methods stand as testaments to the ingenuity and rigor inherent in the field of structural
engineering, driving advancements and innovations in the built environment.
4. Classification of Framed Structures
Framed structures are ubiquitous in modern construction, providing strength,
stability, and versatility in a wide range of architectural applications. These structures,
characterized by their interconnected framework of beams and columns, offer immense
flexibility in design and construction, catering to diverse spatial and load-bearing
requirements. To comprehend the myriad forms and functions of framed structures, it is
imperative to delve into their classification, which delineates their various types based
on structural configurations, material compositions, and load-carrying mechanisms.
 Classification Based on Structural Configuration:
Framed structures can be classified into several categories based on their
structural configuration, which pertains to the arrangement and interconnection of
structural elements. One common classification is based on the arrangement of beams
and columns, yielding the following types:
o Single-Storey Frames: These structures consist of a single level or storey and
typically comprise horizontal beams supported by vertical columns. Single-storey
frames are commonly employed in residential, commercial, and industrial
buildings, offering simplicity in design and construction.
o Multi-Storey Frames: Multi-storey framed structures encompass multiple levels
or storeys, often interconnected by vertical columns and horizontal beams. These
structures are prevalent in high-rise buildings, skyscrapers, and urban
developments, where space optimization and vertical expansion are paramount.
o Portal Frames: Portal frames are characterized by their distinctive configuration,
featuring rigid joints between horizontal beams and vertical columns. This design
imparts stability and resistance to lateral loads, making portal frames ideal for
industrial buildings, warehouses, and agricultural structures.

 Classification Based on Material Composition:

Framed structures can also be classified based on the materials used in their
construction, which significantly influence their structural properties, aesthetics, and
durability. Common classifications based on material composition include:
o Steel Framed Structures: Steel framed structures utilize steel beams and
columns as primary structural elements, offering high strength-to-weight ratios,
versatility in design, and rapid construction. Steel framed structures are prevalent
in commercial buildings, bridges, and infrastructure projects due to their durability
and cost-effectiveness.
 o Reinforced Concrete Framed Structures: Reinforced concrete framed
structures employ concrete as the primary structural material, reinforced with
steel bars or mesh to enhance tensile strength. These structures offer excellent
durability, fire resistance, and thermal mass, making them suitable for residential,
commercial, and infrastructure applications.
o Timber Framed Structures: Timber framed structures utilize timber beams and
columns as primary structural elements, offering sustainability, thermal insulation,
and aesthetic appeal. Timber framed structures are commonly employed in
residential buildings, cabins, and low-rise structures, where environmental
considerations and architectural aesthetics are paramount.

 Classification Based on Load-Carrying Mechanism:

Another criterion for classifying framed structures is based on their load-carrying
mechanism, which encompasses how loads are transferred and distributed throughout
the structure. Common classifications based on load-carrying mechanisms include:
o Gravity Load-Resisting Frames: Gravity load-resisting frames primarily resist
vertical loads such as dead loads and live loads, relying on the inherent strength
and stiffness of beams and columns to support the structure's weight. These
frames are prevalent in most building types, where gravity loads are the
predominant form of loading.
o Lateral Load-Resisting Frames: Lateral load-resisting frames are designed to
withstand horizontal loads such as wind loads, seismic forces, and lateral earth
pressure. These frames employ various mechanisms, including bracing systems,
shear walls, and moment-resisting frames, to resist lateral forces and maintain
structural stability.
o Combined Load-Resisting Frames: Combined load-resisting frames integrate
elements of both gravity load-resisting and lateral load-resisting frames to
withstand vertical and horizontal loads simultaneously. These frames offer
enhanced structural performance and resilience, making them suitable for high-
rise buildings, bridges, and infrastructure projects subjected to diverse loading
conditions.
Furthermore, the classification of framed structures provides a comprehensive
framework for understanding their diverse forms, functions, and applications in the built
environment. By categorizing framed structures based on structural configuration,
material composition, and load-carrying mechanism, engineers, architects, and
designers can effectively analyze, design, and construct structures that meet the
demands of modern construction while embodying principles of strength, stability, and
sustainability. As framed structures continue to evolve and innovate in response to
changing societal needs and technological advancements, a nuanced understanding of
their classification remains essential for advancing the art and science of structural
engineering and architecture.
5. Analytical Models
An analytical model is a simplified representation, or an ideal, of a real structure
for the purpose of analysis. The objective of the model is to simplify the analysis of a
complicated structure. The analytical model represents, as accurately as practically
possible, the behavioral characteristics of the structure of interest to the analyst, while
discarding much of the detail about the members, connections, and so on, that is
expected to have little effect on the desired characteristics. Establishment of the
analytical model is one of the most important steps of the analysis process; it requires
experience and knowledge of design practices in addition to a thorough understanding
 of the behavior of structures. One must remember that the structural response predicted
from the analysis of the model is valid only to the extent that the model represents the
actual structure.
Development of the analytical model generally involves consideration of the
following factors.
 Plane Versus Space Structure
If all the members of a structure as well as the applied loads lie in a single plane,
the structure is called a plane structure. The analysis of plane, or two-dimensional,
structures is considerably simpler than the analysis of space, or three-dimensional,
structures. Fortunately, many actual three-dimensional structures can be subdivided
into plane structures for analysis.
Although a great majority of actual three-dimensional structural systems can be
subdivided into plane structures for the purpose of analysis, some structures, such as
latticed domes, aerospace structures, and transmission towers, cannot, due to their
shape, arrangement of members, or applied loading, be subdivided into planar
components. Such structures, called space structures, are analyzed as three-
dimensional bodies subjected to three-dimensional force systems.
 Line Diagram
The analytical model of the two- or three-dimensional body selected for analysis
is represented by a line diagram. On this diagram, each member of the structure is
represented by a line coinciding with its centroidal axis. The dimensions of the members
and the size of the connections are not shown on the diagram. Note that two lines are
sometimes used in this text to represent members on the line diagrams. This is done,
when necessary, for clarity of presentation; in such cases, the distance between the
lines does not represent the member depth.
 Connections
Two types of connections are commonly used to join members of structures: rigid
connections and flexible, or hinged, connections. (A third type of connection, termed a
semirigid connection, although recognized by structural steel design codes, is not
commonly used in practice and, therefore, is not considered in this text.)
A rigid connection or joint prevents relative translations and rotations of the
member ends connected to it; that is, all member ends connected to a rigid joint have
the same translation and rotation. In other words, the original angles between the
members intersecting at a rigid joint are maintained after the structure has deformed
under the action of loads. Such joints are, therefore, capable of transmitting forces as
well as moments between the connected members. Rigid joints are usually represented
by points at the intersections of members on the line diagram of the structure.
A hinged connection or joint prevents only relative translations of member ends
connected to it; that is, all member ends connected to a hinged joint have the same
translation but may have different rotations. Such joints are thus capable of transmitting
forces but not moments between the connected members. Hinged joints are usually
depicted by small circles at the intersections of members on the line diagram of the
structure.
The perfectly rigid connections and the perfectly flexible frictionless hinges used
in the analysis are merely idealizations of the actual connections, which are seldom
perfectly rigid or perfectly flexible. However, actual bolted or welded connections are
purposely designed to behave like the idealized cases.
 Supports
Supports for plane structures are commonly idealized as either fixed supports,
which do not allow any movement; hinged supports, which can prevent translation but
permit rotation; or roller, or link, supports, which can prevent translation in only one
direction.
Furthermore, an analytical model is a simplified representation of a real structure
for the purpose of analysis. Development of the model generally involves determination
of whether or not the structure can be treated as a plane structure, construction of the
line diagram of the structure, and idealization of connections and supports.
6. Fundamental Relationships for Structural Analysis
Structural analysis serves as the cornerstone of engineering, providing essential
insights into the behavior of various structures under different loading conditions. At the
heart of structural analysis lie fundamental relationships that govern the equilibrium,
compatibility, and material behavior of structural elements. These relationships, rooted
in principles of mechanics and mathematics, form the basis for analyzing the stability,
strength, and performance of structures across diverse engineering disciplines.
 Equilibrium Equations:
Equilibrium equations, derived from Newton's laws of motion, establish the
balance of forces and moments acting on a structure in static equilibrium. These
equations, commonly known as the equations of equilibrium, govern the distribution of
internal forces and reactions within a structure and serve as the foundation for structural
analysis. The three primary equilibrium equations are:
o Summation of Forces in the x-direction: ΣFx = 0
o Summation of Forces in the y-direction: ΣFy = 0
o Summation of Moments about a point: ΣM = 0

These equations ensure that the structure remains stationary and does not
undergo translational or rotational motion under the applied loads. By solving the
equilibrium equations, engineers can determine the internal forces, reactions, and
support conditions necessary to maintain equilibrium within the structure.
 Compatibility Equations:
Compatibility equations govern the deformation or displacement compatibility of
interconnected structural elements within a system. These equations ensure that the
displacements of adjoining elements are compatible with each other, thereby
maintaining the overall integrity and stability of the structure. Compatibility equations
arise from the principle of virtual work, which states that the virtual work done by
external forces must be equal to the virtual work done by internal forces.
Mathematically, compatibility equations can be expressed as:
o Δu = 0
o Δv = 0
o Δw = 0

Where Δu, Δv, and Δw represent the displacements in the x, y, and z directions,
respectively. By satisfying compatibility equations, engineers can ensure that the
structure deforms in a manner consistent with its geometry and material properties,
thereby preventing excessive stresses and deformations that could compromise its
structural integrity.

 Material Behavior Equations:

Material behavior equations describe the relationship between applied loads and
resulting stresses and strains within a structural material. These equations are derived
from constitutive relationships, which characterize the mechanical properties of
materials under various loading conditions. For linearly elastic materials, such as steel
and concrete within their elastic limits, material behavior equations can be expressed
as:
o Stress-Strain Relationship: σ = Eε

Where σ represents stress, ε represents strain, and E represents the modulus of

elasticity or Young's modulus of the material. This relationship quantifies the linear
relationship between stress and strain and enables engineers to predict the deformation
and response of structural materials under different loading scenarios.
The fundamental relationships for structural analysis encompass equilibrium
equations, compatibility equations, and material behavior equations, each playing a
crucial role in analyzing and designing structural systems. These relationships provide a
rigorous framework for understanding the behavior of structures under various loading
conditions and enable engineers to ensure the safety, stability, and performance of
structural elements. By applying these fundamental principles, engineers can design
innovative and resilient structures that meet the demands of modern construction while
adhering to principles of safety, sustainability, and efficiency.
7. Linear versus Nonlinear Analysis
Linear and nonlinear analysis represent two distinct approaches in structural
engineering, each offering unique insights into the response of structures to external
forces.
Linear Analysis:
Linear analysis assumes that the structural response is directly proportional to
the applied loads, following principles of linear elasticity. This approach is based on the
assumption that material behavior is linear, meaning that stress is directly proportional
to strain within the elastic range. Linear analysis is commonly used for analyzing
structures with small deformations and linear material properties. Key characteristics of
linear analysis include:
o Linear Material Behavior: Linear analysis assumes that materials exhibit linear
stress-strain behavior within their elastic limits. This simplifies the analysis
process and enables engineers to predict the structural response with reasonable
accuracy.
o Superposition Principle: Linear analysis employs the superposition principle,
which states that the response of a structure to multiple loads can be obtained by
summing the individual responses to each load. This principle facilitates the
analysis of complex loading scenarios and structural configurations.
Despite its simplicity and computational efficiency, linear analysis has limitations,
particularly when applied to structures with large deformations, nonlinear material
behavior, or significant geometric nonlinearity. In such cases, linear analysis may yield
inaccurate results and underestimate the structural response.
 Nonlinear Analysis:
Nonlinear analysis considers the nonlinear behavior of materials and structures,
accounting for factors such as large deformations, material yielding, and geometric
nonlinearity. This approach provides a more comprehensive understanding of structural
behavior under realistic loading conditions. Key characteristics of nonlinear analysis
include:
o Nonlinear Material Behavior: Nonlinear analysis considers the nonlinear stress-
strain behavior of materials, including plastic deformation, yielding, and strain
hardening. By incorporating material nonlinearity, nonlinear analysis can
accurately predict the structural response under extreme loading conditions.
o Geometric Nonlinearity: Nonlinear analysis accounts for geometric nonlinearity,
which arises from large deformations and changes in the structure's geometry.
This includes effects such as large displacements, buckling, and instability, which
may significantly impact the structural response.
o Iterative Solution Methods: Nonlinear analysis typically requires iterative
solution methods to solve the nonlinear equations governing the structural
behavior. These methods, such as Newton-Raphson iteration or incremental-
iterative techniques, iteratively update the solution until convergence is achieved.
Applications and Considerations:
Linear analysis is commonly used for preliminary design, linear elastic analysis,
and assessing structures with small deformations and linear material behavior. It is well-
suited for simple structures and provides quick and cost-effective solutions.
Nonlinear analysis, on the other hand, is employed for analyzing complex
structures, considering material nonlinearity, large deformations, and geometric
instability. It is essential for assessing structures subjected to extreme loading
conditions, such as seismic events, blast loads, or progressive collapse scenarios.
Furthermore, linear and nonlinear analysis represent two essential approaches in
structural engineering, each offering distinct advantages and limitations. While linear
analysis provides simplicity and computational efficiency, nonlinear analysis offers a
more comprehensive understanding of structural behavior under realistic loading
conditions. By leveraging both approaches judiciously, engineers can design safe,
resilient, and efficient structures that meet the demands of modern construction while
ensuring structural integrity and performance.