ECV 4421: THEORY OF STRUCTURES IV

COURSE OUTLINE

 Energy principles: strain energy theorem, least work method, application of energy
principles to structural analysis - Castigliano’s theorem.
 Analysis of statically determinate and indeterminate structures: Conjugate-Beam
Method, Moment Area Method
 Elastic deformation of statically indeterminate structures

1.0 Analysis of Statically Indeterminate Structures

There are three factors that cause a structure to yield displacements. They are:

 Loadings
 Temperature changes and material expansion
 Support settlement

Under the action of the above three factors, the members of a structure will be deformed, which
is named the deformation of the structure. When a structure develops deformation, the
translation of a point, the translation or rotation of a section of a member of the structure are
referred to as displacement corresponding to the point and the section of the structure,
respectively.

There are two kinds of displacements for a structure. One is linear displacement or translation,
which indicates the distance of movement of a point of the structure along a line. Another is
angular displacement or rotation, which indicate the sectional rotation of a member of the
structure.

1.1. Principle of superposition

In the analysis of statically indeterminate structures, the knowledge of the displacements of a
structure is necessary. Knowledge of displacements is also required in the design of members.
Several methods are available for the calculation of displacements of structures. However, if
displacements at only a few locations in structures are required then energy-based methods are
most suitable. If displacements are required to solve statically indeterminate structures, then
only the relative values of EA, EI and GJ are required.

The principle of superposition may be stated as the deflection at a given point in a structure
produced by several loads acting simultaneously on the structure and can be found by
superposing deflections at the same point produced by loads acting individually.

Consider a cantilever beam of length L and having constant flexural rigidity EI subjected to
two externally applied forces P1 and P2 as shown in Fig. 1.1. From moment-area theorem we
can evaluate deflection below C, which states that the tangential deviation of point c from the

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tangent at point A is equal to the first moment of the area of the diagram between A and C
about C. Hence, the deflection u below C due to loads P1 and P2 acting simultaneously is

𝑢 = 𝐴 𝑥̅ + 𝐴 𝑥̅ + 𝐴 𝑥̅ … … 1.1

Figure 1.1: Cantilever beam with two concentrated loads

Where, u is the tangential deviation of point C with respect to a tangent at A. 𝑥 , 𝑥 , and 𝑥 are
the distances from point C to the centroids of respective areas

𝑥̅ = , 𝑥̅ = + , 𝑥̅ = +

( )
𝐴 = ,𝐴 = ,𝐴 =

Hence

𝑃 𝐿 2𝐿 𝑃 𝐿 𝐿 𝐿 (𝑃 𝐿 + 𝑃 𝐿)𝐿 2 𝐿 𝐿
𝑢= + + + + … … 1.2
8𝐸𝐼 3 2 4𝐸𝐼 2 4 8𝐸𝐼 32 2
Simplifying equation 1.2

𝑃𝐿 5𝑃 𝐿
𝑢= + … … 1.3
3𝐸𝐼 48𝐸𝐼
Consider the forces being applied separately and evaluate deflection at C in each of the case

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Figure 1.2: Deflection computation

𝑃𝐿
𝑢 = … … 1.4
3𝐸𝐼
Where u22 is deflection at C (2) when load P1 is applied at C (2) itself. And

1 𝑃 𝐿 𝐿 𝐿 2𝐿 5𝑃 𝐿
𝑢 = + = … … 1.5
2 2𝐸𝐼 2 2 3 2 48𝐸𝐼
The total deflection at C when both the loads are applied simultaneously is obtained by adding
u22 and u21.

𝑃𝐿 5𝑃 𝐿
𝑢=𝑢 +𝑢 = + … … 1.6
3𝐸𝐼 48𝐸𝐼
Hence it is seen from equations (1.3) and (1.6) that when the structure behaves linearly, the
total deflection caused by forces P1, P2, ……, Pn at any point in the structure is the sum of
deflection caused by forces acting independently on the structure at the same point. This is
known as the Principle of Superposition.

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Strain energy
When an external load acts on a structure, the structure undergoes deformation and hence, work
is done. To resist these external forces, the internal forces develop gradually from zero to their
final value and the work is done. This internal work is stored as energy in the structure and it
helps the structure to spring back to the original shape and size, whenever the external loads
are removed, provided the material of the structure is still within the elastic limit. This internal
work which is stored as energy is due to the straining of the material and hence, is called strain
energy.

When equilibrium is reached the work done by the external forces must equal the strain energy
stored. This concept of energy balance is utilized in structural analysis to develop a number of
methods to find the deflections of structures.

Consider an elastic spring which is slowly pulled and deflects by a small amount u1. When the
load is removed from the spring, it goes back to the original position. When the spring is pulled
by a force, it does some work and this can be calculated once the load-displacement relationship
is known. The spring is a mathematical idealization of a rod being pulled by a force P axially.
It is assumed here that the force is applied gradually so that it slowly increases from zero to a
maximum value P. Such a load is called static loading, as there are no inertial effects due to
motion. Now, work done by the external force may be calculated as,

1 1
𝑊 = 𝑃 𝑢 = (𝑓𝑜𝑟𝑐𝑒 ∗ 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡) … … 1.7
2 2
Hence the elastic strain energy stored in a member of length s (it may be curved or straight)
due to axial force, bending moment, shear force and torsion is summarized as

Due to axial force

𝑃
𝑈 = 𝑑𝑠
2𝐴𝐼

Due to bending

𝑀
𝑈 = 𝑑𝑠
2𝐸𝐼

Due to shear

𝑉
𝑈 = 𝑑𝑠
2𝐴𝐺

Due to torsion

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 𝑇
𝑈 = 𝑑𝑠
2𝐺𝐽

J – polar moment of inertia

G – modulus of rigidity
T – applied torque

1.2 The Partial-Derivative Method – Castigliano’s Theorem

Castigliano published two important theorems in structural analysis (1879). The first theorem
helps in determining deflection and the second in determining redundant reaction component.

Castigliano’s First Theorem

For linearly elastic structure, where external forces only cause deformations, the
complementary energy is equal to the strain energy. For such structures, the Castigliano’s first
theorem may be stated as the first partial derivative of the strain energy of the structure with
respect to any particular force gives the displacement of the point of application of that force
in the direction of its line of action.

Figure 3.3: Castigliano’s first theorem

Let P1, P2, …. Pn be the forces acting at x1, x2, …xn from the left end on a simply supported
beam of span L. Let u1, u2, …un be the displacements at the loading points P1, P2 …. Pn

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respectively. Invoking the principle of superposition, the work done by the external forces is
given by

1 1 1
𝑊 = 𝑃 𝑢 + 𝑃 𝑢 + ⋯ + 𝑃 𝑢 … … 1.8
2 2 2
Work done by the external forces is stored in the structure as strain energy in a conservative
system. Hence, the strain energy of the structure is,

1 1 1
𝑈 = 𝑃 𝑢 + 𝑃 𝑢 + ⋯ + 𝑃 𝑢 … … 1.9
2 2 2
Displacement u1 below point P1 is due to the action of P1, P2…Pn acting at distances x1, x2 ….
xn respectively from left support. Hence, u1 may be expressed as,

𝑢 = 𝑎 𝑃 + 𝑎 𝑃 + ⋯ … … + 𝑎 𝑃 … … 1.10

In general,

𝑢 = 𝑎 𝑃 + 𝑎 𝑃 + ⋯ … … + 𝑎 𝑃 𝑤ℎ𝑒𝑟𝑒 𝑖 = 1,2, . . 𝑛 … … 1.11

Where, aij is the flexibility coefficient at i due to unit force applied at j. Substituting the values
of u1, u2…un in equation 1.9 from equation 1.11

1 1
𝑈 = 𝑃 [𝑎 𝑃 + 𝑎 𝑃 + ⋯ ] + 𝑃 [𝑎 𝑃 + 𝑎 𝑃 + ⋯ ]
2 2
1
+ 𝑃 [𝑎 𝑃 + 𝑎 𝑃 +. ]. . .1.12
2
From Maxwell-Betti’s reciprocal theorem aij = aji. Hence, equation 1.12 may be simplified as

1
𝑈= [𝑎 𝑃 + 𝑎 𝑃 + ⋯ 𝑎 𝑃 ] + [𝑎 𝑃 𝑃 + 𝑎 𝑃 𝑃 + ⋯ 𝑎 𝑃 𝑃 ] … … 1.13
2
Differentiating the strain energy with any force P1 gives,

𝜕𝑈
= 𝑎 𝑃 + 𝑎 𝑃 + ⋯ + 𝑎 𝑃 … … 1.14
𝜕𝑃

In general,

𝜕𝑢
= 𝑢 … … 1.15
𝜕𝑃

Hence, for determinate structure within linear elastic range the partial derivative of the total
strain energy with respect to any external load is equal to the displacement of the point of
application of load in the direction of the applied load, provided the supports are unyielding
and temperature is maintained.

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Differentiation under the integral sign
The use of Castigliano’s theorem for determining beam deflections may lead to lengthy
integrations, especially when more than two loads act on the beam. The reason being the
finding of the strain energy requires the integration of the square of the bending moment. After
the integrations are completed and the strain energy has been determined, we differentiate the
strain energy to obtain the deflections. However, we can bypass the step of finding the strain
energy by differentiating before integrating. This procedure does not eliminate the integrations,
but it does make them simpler.

𝜕 𝑀 𝑀 𝜕𝑀
𝛿 = 𝑑𝑥 = 𝑑𝑥 … 1.15
𝜕𝑃 2𝐸𝐼 𝐸𝐼 𝜕𝑝

We refer to this equation as the modified Castigliano’s theorem.

Mathematically

𝜕𝑈
= 𝑃 𝑤ℎ𝑒𝑟𝑒 𝑈 𝑖𝑠 𝑠𝑡𝑟𝑖𝑎𝑛 𝑒𝑛𝑒𝑟𝑔𝑦 𝑠𝑡𝑜𝑟𝑒𝑑 𝑖𝑛 𝑏𝑒𝑛𝑑𝑖𝑛𝑔
𝜕∆
𝜕𝑈
= 𝑀 𝐻𝑒𝑟𝑒 ∆ 𝑖𝑠 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝑤𝑖𝑡ℎ 𝑙𝑜𝑎𝑑𝑠 𝑎𝑛𝑑 𝜃 𝑤𝑖𝑡ℎ 𝑚𝑜𝑚𝑒𝑛𝑡
𝜕𝜃

Castigliano’s Second Theorem

In any elastic structure having n independent displacements u1, u2…un corresponding to
external forces P1, P2, …Pn along their lines of action, if strain energy is expressed in terms of
displacements, then n equilibrium equations may be written as follows. The Castigliano’s
second theorem may be stated as, the partial derivative of the total internal energy in a beam,
with respect to the load applied at any point, is equal to the deflection at that point.

𝜕𝑈
= 𝑃 , 𝑗 = 1,2, … 𝑛
𝜕𝑢

Note that differentiating the strain energy with respect to any displacement u1 gives the applied
force P1 at that point.

Mathematically

𝜕𝑈
=∆
𝜕𝑃
𝜕𝑈
=𝜃
𝜕𝑀

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Example 3.1

Find the displacement and slope at the tip of a cantilever beam loaded as shown using the
Castigliano’s theorem. Assume the flexural rigidity of the beam EI to be constant for the beam.

Solution

Moment at any section at a distance x away from the free end is given by

𝑀 = −𝑃𝑥

Strain energy stored in the beam due to bending is

𝑀
𝑈= 𝑑𝑥
2𝐸𝐼

Substituting the expression for bending moment M in the equation

(𝑃𝑥) 𝑃 𝐿
𝑈= 𝑑𝑥 =
2𝐸𝐼 6𝐸𝐼

According to Castigliano’s theorem, the first partial derivative of strain energy with respect to
external force P gives the deflection uA at A in the direction of applied force. Thus,

𝜕𝑈 𝑃𝐿
=𝑢 =
𝜕𝑃 3𝐸𝐼
To find the slope at the free end, we need to differentiate strain energy with respect to externally
applied moment M at A. As there is no moment at A, apply a fictitious moment Mo at A. Moment
at any section at a distance x away from the free end is given by

𝑀 = −𝑃𝑥 − 𝑀

Strain energy stored in the beam may be calculated as,

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 (𝑃𝑥 + 𝑀 ) 𝑃 𝐿 𝑀 𝑃𝐿 𝑀 𝐿
𝑈= 𝑑𝑥 = + +
2𝐸𝐼 6𝐸𝐼 2𝐸𝐼 2𝐸𝐼

Taking partial derivative of strain energy with respect to Mo, we get slope at A

𝜕𝑈 𝑃𝐿 𝑀 𝐿
=𝜃 = +
𝜕𝑀 2𝐸𝐼 𝐸𝐼

Since there is actually no moment applied at A, hence substitute Mo = 0, we get the slope at A

𝑃𝐿
𝜃 =
2𝐸𝐼