QUESTION SOLVING SESSION

STRUCTURAL ANALYSIS 2
Course Code: CET 302

Course Name: STRUCTURAL ANALYSIS 2

MODULE 1

1. Explain the concept of plastic hinge.

It is a section at which all the fibers have yielded and hence for any further load rotation takes place at the
section without resisting any additional moment. Plastic moment of a section is defined as the moment which
makes all the fibers at that section to yield and thereby form a plastic hinge. It is assumed that whenever a fully
plastic moment is attained at any cross section, a plastic hinge forms which can undergo rotation of any magnitude,
but the bending moment remains constant at the fully plastic value.
2. Define the terms shape factor and load factor
Load factor is the ratio of ultimate load to the working load that can be applied on the structure. Shape
factor is the ratio of plastic modulus (or plastic moment) to the elastic modulus (or yield moment) of a section.

3. List the assumptions in plastic theory

• Stress strain relationship is idealized to two straight lines, ie., strain hardening effect is neglected.
• Plane section before bending remains plane even after bending, ie., shear deformations are neglected.
• The relationship between compressive stress and compressive strain is the same as between tensile stress and
tensile strain.

4. Define a) Plastic Hinge b) Plastic moment c) plastic section modulus

a) Plastic Hinge:- It is a section at which all the fibers have yielded and hence for any further load rotation takes
place at the section without resisting any additional moment.
b) Plastic moment:- Plastic moment of a section is defined as the moment which makes all the fibers at that section
to yield and thereby form a plastic hinge.
c) Plastic section modulus:- The plastic section modulus is the sum of the areas of the cross section on each side of
the PNA (which may or may not be equal) multiplied by the distance from the local centroids of the two areas to
the PNA.

5. Define shape factor. Obtain the shape factor for solid circular section of diameter D.
Shape factor is the ratio of plastic modulus (or plastic moment) to the elastic modulus (or yield moment) of
a section.
6. Mention the advantages and disadvantages of approximate method of analysis
Advantages:
1. Simplicity: Approximate methods are usually easier to understand and implement than more complex analytical
methods. They involve fewer mathematical calculations and are more accessible to engineers and designers with
limited analytical skills.
2. Speed: Since approximate methods require less computation and analysis, they can significantly reduce the time
required to assess a structure's behavior. This can be particularly beneficial for quick preliminary evaluations or
during the early design stages.
3. Initial Design Estimation: Approximate methods can provide useful estimates for initial design purposes. Engineers
can use these estimates to assess the feasibility of various structural configurations before investing time and
resources in more detailed analyses.
4. Insight and Visualization: Approximate methods often provide more intuitive insights into structural behavior,
allowing engineers to gain a better understanding of how loads are distributed and how the structure deforms
under different conditions.
5. Practicality for Simple Structures: For relatively simple and common structural configurations, approximate
methods can yield reasonably accurate results, making them a practical choice for routine engineering tasks.
Disadvantages:
1. Accuracy: The most significant drawback of approximate methods is their reduced accuracy compared to more
rigorous analytical techniques. For complex or irregular structures, the approximations may lead to significant errors
in predicting the actual behavior.
2. Limited Applicability: Approximate methods are not suitable for all types of structures or loading conditions. They
are best suited for relatively simple and linear problems and may not capture the behavior of non-linear or highly
dynamic systems.
3. Conservative Results: In some cases, approximate methods tend to produce conservative results, overestimating
the required strength or size of structural elements. This conservatism can lead to inefficiencies in design and
potentially higher construction costs.
4. Lack of Detail: Approximate methods provide a broad overview of structural behavior but may not offer detailed
insights into critical factors, such as local stresses, strain distributions, or failure modes.
5. Sensitivity to Assumptions: Approximate methods heavily rely on assumptions to simplify the analysis. Small
changes in these assumptions can lead to significantly different results, making it crucial for engineers to be aware
of the method's limitations and potential inaccuracies.

7. Differentiate between structure and mechanism with suitable example

A structure is a stable arrangement of parts, often designed to support or enclose something. A mechanism
is focused on action and movement, while a structure is focused on stability and support. When a structure is
subjected to a system of loads, it is stable and hence functional until a sufficient number of plastic hinges have been
formed to render the structure unstable. As soon as structure became unstable it is considered to have been failed.
The segments of beam between the plastic hinges are able to move without an increase in load. This condition of
membrane is called a mechanism.

8. Write short note on the approximate method of analysis of multi storeyed building for
vertical load.
 In case of multi storeyed building degree of indeterminacy is very high. Hence to solve the frame, you
cannot use conventional methods. Thus for a quick solution design engineers use approximate method. The
statically indeterminate structures is simplified to statically determinate structure and the analysis is carried out
using the principle of statics. The validity of the result is based upon the assumption made in the analysis. This
method is used for determining the moment and shear force at any floor or roof level due to the vertical loads. Here
we assume that the moment transfer from one floor to another floor are negligible and hence analysis can be made
from the floor to floor.

9. Find the plastic moment capacity of the beam shown in the figure' Assume uniform section
throughout.
10. In a multi-storeyed building, the frame shown is spaced at 4m intervals. Dead load from the
slab is 3kN/m2 and the live load is 5kN/m2. Analyze the beam BC for mid span positively. Self-
weight of the beams may be ignored. Use 2 cycle moment distribution method for solving.
11. a) Define shape factor. Derive an expression for shape factor for circular cross section of
diameter D
b) Determine the collapse load for the fixed beam by Kinematic Method

a) Shape factor is the ratio of plastic modulus (or plastic moment) to the elastic modulus (or yield moment)
of a section.
13. a) Discuss the features of substitute frame
b) In a multi-storeyed building the frame shown is spaced at 3.5m intervals. Dead load and Live
load from the slab are 3 kN/m2 and 5 kN/m2respectively. Analyze the beam BC for maximum
mid span negative bending moment. Self weight of the beams of 4m span is 4kN/m and that of
6m span is 5 kN/m.

a) It is used for the analysis of complex structures i.e multi-storeyed buildings because analysis of multi-story
buildings is very complicated and long calculations are needed. Substitute frame method used for sudden analysis
and gives approximate values.
14. Determine the plastic moment carrying capacity Mp for the continuous beam (14) shown in
figure below. Take load factor: 1.5
15. A multi storeyed building consists of 4 storeys and 3 bay frames spaced at 3m c/c. LL on
floor slab is 3kN/m2. CL on floor slab is 3.5 kN/m2. The spacing of beam from left to right are
6m,4m, and 4m respectively. Storey height is 3.5m. I of beam is 1.5 times that of column. Self
weight of beam is 3.5 kN/m. Determine max. BM in the beam at junction of first span and
second span of an intermediate floor.
16. Determine the plastic moment capacity Mp of the continuous beam shown in Fig.
 MODULE 2
1. What are the assumptions involved in cantilever method of analysis?
The assumptions used in this method are that the points of contraflexure (or points of inflection of the
moment diagram) in both the vertical and horizontal members are located at the midpoint of the member, and that
the direct stresses in the columns are proportional to their distances from the centroidal axis of the frame. The
frame is analyzed in step-wise (iterative) fashion, and the results can then be described by force diagrams drawn up
at the end of the process. The method is quite versatile and can be used to analyze frames of any number
of storeys or floors.

2. Define flexibility influence coefficient and stiffness influence coefficient

Flexibility coefficient refers to the displacement at coordinate i due to unit force at coordinate j in a
structure. The flexibility influence coefficient aij is defined as the deflection at i due to a unit load at j. The Stiffness
Coefficient is a critical parameter in engineering that measures the resistance of a material or structure to
deformation under applied force. kij is the stiffness influence coefficient defined as the force developed at 'i' due to
unit force applied at 'j'.

3. Write the properties of Flexibility Matrix

• The flexibility matrix will always be a square matrix (nxn).
• Order of flexibility matrix will be equal to degree of static indeterminacy. (i.e.; no. of redundant)
• Order of matrix is the no. of co-ordinate chosen for solution of problem.
• Elements of flexibility matrix are displacements.
• Elements along the main diagonal will always be positive. Other elements can be zero or negative.

4. Explain the term equivalent joint loads

Equivalent load is defined as the load for linear analysis that generates the same response field as that for
nonlinear analysis. An equivalent point load is a single point force that will have the same effect on a body as the
original loading condition, which is usually a distributed force.

5. What are the assumptions made in the portal method of analysis for horizontal loads?
1. The points of inflection are located at the mid-height of each column above the first floor. If the base of the
column is fixed, the point of inflection is assumed at mid height of the ground floor columns as well; otherwise it is
assumed at the hinged column base.
2. Points of inflection occur at mid span of beams.
3. Total horizontal shear at any floor is distributed among the columns of that floor such that the exterior columns
carry half the force carried by the inner columns.

6. Find the force transformation matrix for the given element.

7. Differentiate between portal method and cantilever method of analysis of multi storeyed
building.
The portal and cantilever methods are two approximate methods for analyzing tall buildings for lateral
(wind or seismic) loads. The portal method is suitable for buildings likely to deform in shear mode. It is based on the
distribution of base shears proportional to influence areas. The cantilever method is suitable for slender buildings
likely to deform in flexure mode. It is based on the distribution of base moment as axial forces on columns
proportional to their distance from the centroidal axis of the building.

8. Derive force transformation matrix for the coordinates shown in Fig

9. Analyze the continuous beam shown in figure below by flexibility method and draw the
BMD.
10. Analyze the rigid frame loaded as shown in the figure using cantilever method. (14)
compute the BM in beams and columns and draw the BMD for beams and columns.
11. a) b) Module II List the assumptions in Cantilever method
b)Using portal method, analyze the frame shown below and draw the BMD

a) The assumptions used in this method are that the points of contraflexure (or points of inflection of the moment
diagram) in both the vertical and horizontal members are located at the midpoint of the member, and that the
direct stresses in the columns are proportional to their distances from the centroidal axis of the frame. The frame is
analyzed in step-wise (iterative) fashion, and the results can then be described by force diagrams drawn up at the
end of the process. The method is quite versatile and can be used to analyze frames of any number of storeys or
floors.
12. Analyze the beam using flexibility method and raw BMD and SFD
13. Analyze the continuous beam shown in figure below by flexibility method and draw the
BMD.
15. Analyze and determine the beam and column moments for the frame shown in figure
below by Portal method.
 MODULE 3
1. Write down the properties of stiffness matrix'
• Order of stiffness matrix corresponds to total dofs
• Singular stiffness matrix means structure is unconstrained and rigid body motion
• Each column of stiffness matrix is an equilibrium set of nodal force required to produce unit respective dof
• Symmetric stiffness matrix shows force is directly proportional to displacement
• Diagonal terms of the matrix are always positive i.e. force directed in say left direction cannot produce a
displacement in right direction.
• Diagonal terms will be zero or negative only if the structure is unstable.

2. Define equilibrium and compatibility

The equilibrium equations mean every small element of the material is in equilibrium , i.e. the forces
balance at every point. The compatibility equations mean the deformed material is "continuous" everywhere, i.e. it
doesn't have any internal holes, cracks, or overlapping regions.
3. Differentiate between force method and displacement method of analysis
In the force method of analysis, various forces are considered unknown parameters, while displacement
and slopes are considered unknown parameters in the displacement method of analysis. This forced method and
displacement method of analysis are further classified into other methods based on the analysis requirements. The
displacement method treats the displacement of structures in the nodes as unknowns. The force of each member is
then determined utilizing the equilibrium and stress–strain equations [4]. In the force method, the forces of some
members are chosen as unknowns.

4. Derive the stiffness matrix

5. Define Nodal load, element load, and equivalent joint load.
Nodal loads are forces, moments, or masses that act on nodes. The load vector associated with a finite
element is derived from the work function expressed in terms of nodal displacements. Equivalent load is defined
as the load for linear analysis that generates the same response field as that for nonlinear analysis.

6. Analyze the frame shown using stiffness method and draw the bending moment diagram
7. Analyze the beam shown in Fig using stiffness method and draw the bending moment
diagram.
8. Determine the end moments for the structure shown in figure below by stiffness method.
9. Find the forces for the truss shown in figure below by stiffness method.
10. Find the forces for the truss shown in figure below by stiffness method.
 MODULE 4
1. Explain direct stiffness method?
It is a matrix method that makes use of the members' stiffness relations for computing member forces and
displacements in structures. The direct stiffness method is the most common implementation of the finite element
method (FEM).

2. Explain the steps in direct stiffness method

The first step when using the direct stiffness method is to identify the individual elements which make up
the structure. Once the elements are identified, the structure is disconnected at the nodes, the points which
connect the different elements together.

3. Differentiate between local coordinates and global coordinates.

The global coordinate system defines the position and translation of a body in space. Local coordinate
systems define how limbs and body segments articulate about joints.
4. Analyze the beam shown Fig. by direct stiffness method.
5. Analyze the frame shown Fig. by direct stiffness method and draw BMD and SFD.
6. Analyze and draw bending moment diagram for the frame shown in figure using direct
stiffness method. Assume constant EI for all the members.
 MODULE 5
1. Explain the components of a basic dynamic system
The four basic components of a dynamic system are mass, energy dissipation (damper), resistance (spring),
and applied load. As the structure moves in response to an applied load, forces are induced that are a function of
both the applied load and the motion in the individual components.

2. Define logarithmic decrement.

It represents the rate at which the amplitude of a free under damped vibration decreases. It is defined as
the natural logarithm of the ratio of any 2 consecutive amplitudes. The displacement of an under damped
displacement is a sinusoidal oscillation with decaying amplitude.

3. Explain critical damping

For a system to be in critical damping, the term under the root is zero and the damping coefficient at this
stage is called the critical damping coefficient and is denoted by .
4. State and explain D' Alembert's principle
D’ Alembert’s principle is a method to convert a dynamic system into a equivalent system into and
equivalent static system by adding the inertia force taken in the reverse direction to the restoring force. It states that
the resultant force acting on a body along with the inertia force is zero, then the body is said to be in dynamic
equilibrium. If the mass centre of a body of mass ‘m’ is subjected to a resultant force F and if it acquires an
acceleration ‘a’ then,
F + (-ma) = 0

5. Compare transient and steady state response of a SDOF system subjected to harmonic load.
In a steady-state process, the response of the system, whether it is stress, temperature, or otherwise, does
not change over time. In a transient analysis, this response is time-dependent. Structural vibration displays a
transient response in displacement and acceleration when exposed to dynamic loads such as wind, traffic, or
earthquake. The steady-state response is the constant displacement and acceleration that the structure exhibits
when it is vibrating at its natural frequency.
6. Define the following terms: (i) free and forced vibration (ii) damped and undamped vibration
Free vibration is where there is no externally applied vibrating forcing. In free vibration, energy will remain
the same, and energy is not added or removed from the body.
Forced vibration is a type of vibration in which a force is repeatedly applied to a mechanical system.
Undamped oscillations occur when there is no friction or other force acting against the motion.
Damped oscillations occur when there is some frictional force, or damping, that acts to slow down the
motion.

7. Define the term 'degrees of freedom' as used in dynamic analysis

The degrees of freedom (DOF) are defined as the variables that must be specified to define the
process. Degrees of freedom (DOF) is the number of independent variables that define the possible positions or
motions of a mechanical system in space.
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