Castigliano's method, named after
Carlo Alberto Castigliano
Carlo Alberto Castigliano (9 November 1847, in Asti – 25 October 1884, in Milan) was an Italian mathematician and physicist known for Castigliano's method for determining displacements in a linear-elastic system based on the partial deriva ...
, is a method for determining the displacements of a
linear-elastic system based on the
partial derivatives of the
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
. He is known for his two theorems. The basic concept may be easy to understand by recalling that a change in energy is equal to the causing force times the resulting displacement. Therefore, the causing force is equal to the change in energy divided by the resulting displacement. Alternatively, the resulting displacement is equal to the change in energy divided by the causing force. Partial derivatives are needed to relate causing forces and resulting displacements to the change in energy.
* Castigliano's first theorem – for forces in an elastic structure
Castigliano's method for calculating forces is an application of his first theorem, which states:
:''If the strain energy of an elastic structure can be expressed as a function of generalised displacement q
i then the partial derivative of the strain energy with respect to generalised displacement gives the generalised force Q
i.''
In equation form,
:
where U is the strain energy.
If the force-displacement curve is nonlinear then the complementary strain energy needs to be used instead of strain energy.
[ History of Strength of Materials, Stephen P. Timoshenko, 1993, Dover Publications, New York]
* Castigliano's second theorem – for displacements in a linearly elastic structure.
Castigliano's method for calculating displacements is an application of his second theorem, which states:
:''If the strain energy of a linearly elastic structure can be expressed as a function of generalised force Q
i then the partial derivative of the strain energy with respect to generalised force gives the generalised displacement q
i in the direction of Q
i.''
As above this can also be expressed as:
:
Examples
For a thin, straight cantilever beam with a load P at the end, the displacement
at the end can be found by Castigliano's second theorem :
:
:
where
is
Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied le ...
,
is the
second moment of area
The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The ...
of the cross-section, and
is the expression for the internal moment at a point at distance
from the end. The integral evaluates to:
:
The result is the standard formula given for cantilever beams under end loads.
External links
Carlo Alberto CastiglianoCastigliano's method: some examples
References
{{Structural engineering topics
Structural analysis