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Theory of Structures: With Worked Examples

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These notes on Structural Analysis are suitable for all Civil Engineering, Building Construction, Architecture, and Quantity Surveying students, in both Diploma and Degree levels of education, who wish to gain an understanding of the theory of structures. It will be especially helpful as a supplementary reading book and as standard teaching material. The worked examples included at the end of every chapter contain standard exam-type problems that will enable the learner to know how to tackle typical problems in structural analysis. The book also contains clear, well-labeled diagrams that will assist the learner in painting a mental picture in order to relate what he or she is studying to real-life examples. It is the hope, wish, and prayer of the author that this book will be instrumental in providing a sound foundation for the learner as he or she furthers their education.

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1.0 TYPES OF SUPPORTS AND LOADINGS

1.1 Types of Supports
Structural systems transfer their loading through a series of elements to the ground. This is made possible
by designing the joints of the elements at their intersections. Each connection is designed so that it can
transfer, or support, a specific type of load or loading condition.

To be able to analyze a structure, it is important to know the forces that can be resisted and transferred
at each support throughout the structure. Structural systems have either welded or bolted connections.
Precast reinforced concrete systems can be mechanically connected in many ways. Timber systems are
connected mainly by nails, glue or connectors.

The actual behavior of a support or connection can be quite complicated. Therefore, we idealize (imagine)
the supports/connections in order to be able to analyze the structure. The three types of common
connections that join a built structure to its foundation are:

(i) Roller Supports: see examples in Figure 4.1 below.




Figure 4.1: Practical examples of roller supports

, (ii) Pinned Supports: see examples in figure 4.2 below.




Figure 4.2: Practical examples of pinned supports

(iii) Fixed Supports: see examples in Figure 4.3 below.




Figure 4.3: Practical examples of fixed supports

, It is necessary to idealize the behavior of a support in order to facilitate its analysis, as earlier stated.
Therefore, friction and mass are often ignored in the consideration of the behavior of a connection or
support.

1.1.1 Roller Supports
Roller supports, as the name suggests, are free to rotate and move along the surface upon which the roller
rests. The surface may be horizontal, vertical or sloped at any angle. Therefore, the resulting reaction
force is always a single force that is perpendicular to and away from the surface. Roller supports are
normally located at one end of long bridges. This enables the bridge structure to expand and contract with
temperature changes, thus avoiding stresses and strains which could fracture the structure.

Practical forms of roller supports are rubber bearings, rockers or a set of gears. A roller support cannot
resist a horizontal force thus practical structures will have other types of supports in addition to roller
supports, to prevent a runaway situation.

We shall be illustrating roller supports and the resultant reactions as shown in Figure 4.4 below.




Figure 4.4: Reaction on a roller support

1.1.2 Pinned Supports
Pinned supports resist both vertical and horizontal forces but not a moment (a turning force). A pinned
support allows a structure to rotate but not move in any direction (horizontal or vertical). A majority of
connections are assumed or taken to be pinned connections even though they might resist a small amount
of moment in reality. It is also true that a pinned connection could allow rotation in only one direction,
providing resistance to rotation in any other direction.

The knee or a human being can be imagined as a connection which allows rotation in only one direction
and provides resistance to lateral movement.

A single pinned connection is usually not sufficient to make a structure stable. Another support must be
provided at some point to prevent rotation of the structure. The representation of a pinned support
includes both horizontal and vertical forces as shown in Figure 4.5
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