1

Structural Analysis CE2100

Deflections Using Energy Methods

2

Work done by an external force:Calculate the work done :

1) By gravity on a point mass(m) free falling a distance D under gravity (g)2) By gravity on a point mass(m) pulled vertically up by a distance D.3) By gravity on a car (m) travelling a distance (L) on an uphill slope(q).4) By gravity on a car (m) travelling a distance (L) on a downhill slope(q).

Work done by an External force:Calculate the work done :

1) By force ‘P’ on a massless spring extending (DlP) under load ‘P.’

2) By force ‘P’ on a massless spring extending (DlQ) under additional load ‘Q’, while ‘P’ was already acting.

3) By moment ‘M1’ on a rotational spring rotating (Dq1) under moment ‘M1’.

4) By moment ‘M1’ on a rotational spring rotating (Dq2) under moment ‘M2’, while ‘M1’ was already acting.

5) By a Moment ‘M’ on a beam deforming by (D), rotating by (q).

3

D

q

Work done by an Internal forces (Strain Energy):

Strain Energy (U) is the potential energy that is stored when solid elements undergo deformations.

Axial load ‘P’ is applied gradually to the bar:

Energy (dU) stored in length dx = e.dx. P/2

e= P/AE; therefore, dU = (P2/2AE)(dx)

U= P2L/2AE

Bending Moment M(x) is acting on a beam due to a set of gradually applied loads:

dq = (M/EI)dx

dU= M (d )q /2 = M2(dx)/2EI

U = 4

PD

LA, Edx

P

x

dx

dq

MM

Principle of Work and Energy:Internal Work (Strain Energy) = External WorkU = W =

=

=

U=W

D=

5

D

q

w

-PL

x

L

Principle of Work and Energy:Internal Work (Strain Energy) = External WorkU = W =

=

For more than one point load, we need more than one unknown variables to calculate W. But we have only one equation (W=U). This method is not suitable

6

D

q

M=-wL2/2

x

w

Principle of Virtual Work (Johann Bernoulli, 1717):

• Calculate the forces in each element due to the applied real loads.

• Calculate internal deformations in each element due to the applied real loads (dL).

• Apply a unit virtual load at the location (and direction) where we want to calculate the deformations due to the real loads.

• Calculate internal forces in each element due to the virtual load (u).

• External work by the virtual load while real loads are applied gradually = 1 . D

• Internal work by the virtual load while real loads are applied gradually = ∑u.dL

• 1 . = D ∑u.dL

• Therefore, = D ∑u.dL

Similarly, we can apply unit virtual moment to calculate

• q=∑uq.dL7

Not to be confused with Daniel Bernoulli, though they were contemporaries

Exam

ple

, Tr

uss

Ple

ase n

ote

th

at

there

is a

corr

ecti

on

8