Basic Structural Analysis.......................................................................................................... 2
Plane Truss.......................................................................................................................................2
Truss types............................................................................................................................................................2
Pratt truss .............................................................................................................................................................3
Bow string roof truss.............................................................................................................................................3
Difference between truss and frame ................................................................................................3
Perfect truss.....................................................................................................................................4
Imperfect truss......................................................................................................................................................4
Deficient Truss ......................................................................................................................................................4
Analysis of Plane trusses.......................................................................................................... 5
Method of joints ..............................................................................................................................5
Method of Sections..........................................................................................................................6
Zero force members.........................................................................................................................8
Reasons for Zero-force members in a truss system..............................................................................................8
Beams ..................................................................................................................................... 8
Types of beams ................................................................................................................................9
Statically Determinate Beams...............................................................................................................................9
Statically Indeterminate Beams........................................................................................................9
Shear force and bending moment in beams ................................................................................... 10
Shear force and bending moment diagrams ......................................................................................................11
Relationship between load, shear and bending moment................................................................ 12
Basic Structural Analysis
Plane Truss
A plane truss is one where all the members and joints lie within a 2-dimensional plane, while a
space truss has members and joints extending into 3 dimensions.
Truss types
There are two basic types of truss:
• The pitched truss, or common truss, is characterized by its triangular shape. It is most
often used for roof construction. Some common trusses are named according to their web
configuration. The chord size and web configuration are determined by span, load and
spacing.
• The parallel chord truss, or flat truss, gets its name from its parallel top and bottom
chords. It is often used for floor construction.
A combination of the two is a truncated truss, used in hip roof construction. A metal plate-
connected wood truss is a roof or floor truss whose wood members are connected with metal
connector plates.
Pratt truss
The Pratt truss was patented in 1844 by two Boston railway engineers; Caleb Pratt and his son
Thomas Willis Pratt. The design uses vertical beams for compression and horizontal beams to
respond to tension. What is remarkable about this style is that it remained popular even as wood
gave way to iron, and even still as iron gave way to steel.
The Southern Pacific Railroad bridge in Tempe, Arizona is a 393 meter (1291 foot) long truss
bridge built in 1912. The structure is composed of nine Pratt truss spans of varying lengths. The
bridge is still in use today.
Bow string roof truss
Named for its distinctive shape, thousands of bow strings were used during World War II for
aircraft hangars and other military buildings.
Difference between truss and frame
Truss structure is designed to support huge loads in comparison to its weight. In a truss, the
joints are of pin type and consists of 2 force members (tensile and compression - axial forces).
Here the the members are free to rotate about the pin.
A frame is a structure in which at least one of its individual member is a multi-force member.
The members of frames are connected rigidly at joints by means of welding and bolting. The
joints can transfer moments in addition to the axial loads.
In a plane frame all the moments are perpendicular to a single plane
Frames can be of 2 types - rigid noncollapsible and nonrigid collapsible frames.
Perfect truss
A truss is said to be perfect when the number of members in the truss is just sufficient to prevent
distortion of its shape when loaded with an external load.
A perfect truss has to satisfy the following equation:
where m is the number of members.
Imperfect truss
An imperfect truss is that which does not satisfy the equation, m = 2j – 3. Or in other words, a
truss in which the number of members is more or less than 2j – 3. The imperfect truss may be
further classified into the following two types:
1. deficient truss
2. redundant truss.
Deficient Truss
A deficient truss is an imperfect truss, in which the number of members is less than 2j – 3.
Analysis of Plane trusses
Because the forces in each of its two main girders are essentially planar, a truss is usually
modeled as a two-dimensional plane frame. If there are significant out-of-plane forces, the
structure must be modeled as a three-dimensional space. The analysis of trusses often assumes
that loads are applied to joints only and not at intermediate points along the members. The
weight of the members is often insignificant compared to the applied loads and so is often
omitted. If required, half of the weight of each member may be applied to its two end joints.
Provided the members are long and slender, the moments transmitted through the joints are
negligible and they can be treated as "hinges" or 'pin-joints'. Every member of the truss is then in
pure compression or pure tension – shear, bending moment, and other more complex stresses are
all practically zero. This makes trusses easier to analyze. This also makes trusses physically
stronger than other ways of arranging material – because nearly every material can hold a much
larger load in tension and compression than in shear, bending, torsion, or other kinds of force.
Structural analysis of trusses of any type can readily be carried out using a matrix method such
as the direct stiffness method, the flexibility method or the finite element method.
Method of joints
This method uses the force balance in the x and y directions at each of the joints in the truss
structure.
At A,
At D,
At C,
Although we have found the forces in each of the truss elements, it is a good practice to verify
the results by completing the remaining force balances.
At B,
Method of Sections
This method can be used when the truss element forces of only a few members wants to be
known. This method is used by introducing a single straight line cutting through the member
whose force wants to be calculated. However this method has a limit in that the cutting line can
pass through a maximum of only 3 members of the truss structure. This restriction is because this
method uses the force balances in the x and y direction and the moment balance, which gives us
a maximum of 3 equations to find a maximum of 3 unknown truss element forces through which
this cut is made. Let us try to find the forces FAB, FBD and FCD in the above example
Method 1: Ignore the right side
Method 2: Ignore the left side
The truss elements forces in the remaining members can be found by using the above method
with a section passing through the remaining members.
Zero force members
In the field of engineering mechanics, a zero force member refers to a member (a single truss
segment) in a truss which, given a specific load, is at rest: neither in tension, nor in compression.
In a truss a zero force member is often found at pins (any connections within the truss) where no
external load is applied and three or fewer truss members meet. Recognizing basic zero force
members can be accomplished by analyzing the forces acting on an individual pin in a physical
system.
NOTE: If the pin has an external force or moment applied to it, then all of the members attached
to that pin are not zero force members UNLESS the external force acts in a manner that fulfills
one of the rules below:
• If only two members meet in an unloaded joint, both are zero-force members.
• If three members meet in an unloaded joint of which two are in a direct line with one
another, then the third member is a zero-force member.
• If two members meet in a loaded joint and the line of action of the load coincides with
one of the members, the other member is a zero-force member.
Reasons for Zero-force members in a truss system
• These members contribute to the stability of the structure, by providing buckling
prevention for long slender members under compressive forces
• These members can carry loads in the event that variations are introduced in the normal
external loading configuration
Beams
A beam is a bar subject to forces or couples that lie in a plane containing the longitudinal section
of the bar. According to determinacy, a beam may be determinate or indeterminate.
Types of beams
Statically Determinate Beams
Statically determinate beams are those beams in which the reactions of the supports may be
determined by the use of the equations of static equilibrium. The beams shown below are
examples of statically determinate beams.
Statically Indeterminate Beams
If the number of reactions exerted upon a beam exceeds the number of equations in static
equilibrium, the beam is said to be statically indeterminate. In order to solve the reactions of the
beam, the static equations must be supplemented by equations based upon the elastic
deformations of the beam.
The degree of indeterminacy is taken as the difference between the umber of reactions to the
number of equations in static equilibrium that can be applied. In the case of the propped beam
shown, there are three reactions R1, R2, and M and only two equations (ΣM = 0 and ΣFv = 0) can
be applied, thus the beam is indeterminate to the first degree (3 - 2 = 1).
Shear force and bending moment in beams
The deformation of a beam is usually expressed in terms of its deflection from its original
unloaded position. The deflection is measured from the original neutral surface of the beam to
the neutral surface of the deformed beam. The configuration assumed by the deformed neutral
surface is known as the elastic curve of the beam.
Methods of Determining Beam Deflections
Numerous methods are available for the determination of beam deflections. These methods
include:
1. Double-integration method
2. Area-moment method
3. Strain-energy method (Castigliano's Theorem)
4. Conjugate-beam method
5. Method of superposition
Shear force and bending moment diagrams
Consider a simple beam shown of length L that carries a uniform load of w (N/m) throughout its
length and is held in equilibrium by reactions R1 and R2. Assume that the beam is cut at point C
a distance of x from he left support and the portion of the beam to the right of C be removed. The
portion removed must then be replaced by vertical shearing force V together with a couple M to
hold the left portion of the bar in equilibrium under the action of R1 and wx.
The couple M is called the resisting moment or moment and the force V is called the resisting
shear or shear. The sign of V and M are taken to be positive if they have the senses indicated
above.
Relationship between load, shear and bending moment
Since this method can easily become unnecessarily complicated with relatively simple problems,
it can be quite helpful to understand different relations between the loading, shear, and moment
diagram. The first of these is the relationship between a distributed load on the loading diagram
and the shear diagram. Since a distributed load varies the shear load according to its magnitude it
can be derived that the slope of the shear diagram is equal to the magnitude of the distributed
load. The relationship between distributed load and shear force magnitude is:
Some direct results of this is that a shear diagram will have a point change in magnitude if a
point load is applied to a member, and a linearly varying shear magnitude as a result of a
constant distributed load. Similarly it can be shown that the slope of the moment diagram at a
given point is equal to the magnitude of the shear diagram at that distance. The relationship
between distributed shear force and bending moment is
A direct result of this is that at every point the shear diagram crosses zero the moment diagram
will have a local maximum or minimum. Also if the shear diagram is zero over a length of the
member, the moment diagram will have a constant value over that length. By calculus it can be
shown that a point load will lead to a linearly varying moment diagram, and a constant
distributed load will lead to a quadratic moment diagram.
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