Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

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ENERgy METHOdS
procedure for Analysis
The following procedure is recommended when Castigliano’s second theorem is applied to
calculate deflections for beams:
1. Load P Expressed as a Variable: If an external load acts on the beam at the point
where deflections are to be calculated and in the direction of the desired deflection,
then designate that load as the variable P. As a result, subsequent calculations will be
performed in terms of the variable P rather than in terms of the actual numeric value
known for this particular external load. Otherwise, apply a fictitious load (a dummy
load) in the direction of the desired deflection at the particular point. Designate this
dummy load as P.
2. Beam Internal moment M in Terms of P: Establish appropriate x coordinates for
regions of the beam where there is no discontinuity of force, distributed load, or concentrated
moment. Develop expressions for the internal moment M in terms of the actual
external loads and the variable load P. The expression M for the internal moment of a
particular segment of the beam will likely include both a numeric value and a function
in terms of P. Use the standard convention for bending-moment signs illustrated in
Figures 7.6 and 7.7 for the M equations.
3. partial Derivatives: Differentiate the expressions for the internal moment with
respect to P in order to compute ∂M/ ∂P.
4. Substitute Numeric Value for P: Substitute the actual numeric value for load P
into the expressions for M and ∂M/ ∂P. If a dummy load has been used for P, its numeric
value is zero.
5. Integration: Perform the integration indicated in Equation (17.40) to calculate the
desired deflection. A positive answer indicates that the deflection acts in the same direction
as P, and a negative answer in the opposite direction.
The following procedure is recommended when Castigliano’s second theorem is
applied to calculate slopes for beams:
1. concentrated moment M′ Expressed as a Variable: If an external concentrated
moment acts on the beam at the point where slopes are to be calculated and in the
direction of the desired beam rotation, then designate that concentrated moment as the
variable M′. As a result, subsequent calculations will be performed in terms of the variable
M′ rather than in terms of the actual numeric value known for this particular external
concentrated moment. Otherwise, apply a fictitious concentrated moment (a dummy
moment) in the direction of the desired slope at the particular point. Designate this
dummy moment as M′.
2. Beam Internal moment M in Terms of P: Establish appropriate x coordinates for
regions of the beam where there is no discontinuity of force, distributed load, or concentrated
moment. Determine expressions for the internal moment M in terms of the actual
external loads and the variable moment M′. Be aware that the expression M for the internal
moment of a particular segment of the beam may include both a numeric value
and a function in terms of M′. Use the standard convention for bending moment signs
illustrated in Figures 7.6 and 7.7 for the M equations.
3. partial Derivatives: Differentiate the expressions for the internal moment with
respect to M′ in order to compute ∂M/∂M′.
4. Substitute Numeric Value for M′: Substitute the actual numeric value for moment
M′ into the expressions for M and ∂M/∂M′. If a dummy moment has been used for
M′, its numeric value is zero.