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Chapter 12
ENERGETICAL METHOD FOR DISPLACEMENTS
CALCULATION
12.1. INTRODUCTION. HYPOTHESIS
From Theoretical Mechanics we know that, for a system of material points, the
variation of the kinetic energy, written with respect to a fix system of axis, is equal to
the sum of the external and internal work :
ie dLdLdE +=
This theorem of the kinetic energy and work may be written in a finite form:
ie L L E E +=− 0 (12.1)
where:
0: is the initial kinetic energy: is the kinetic energy of the system at a gi!en moment
"e and "i: is the work #rod$ced by e%ternal, res#ecti!ely internal forces,
between the initial and final config$ration of the system
To st$dy the abo!e theory, the following ass$m#tions are made:
& the material is s$b'ected ma%im$m $ntil the elasticity limit, so it has a
#erfect elastic beha!io$r the ook*s law is !alid
& the e%terior forces are statically loaded (increasing their intensity slowly,
from 0 to the final !al$e)
& the internal frictions and the friction in bearings are neglected
& the work lost by tem#erat$re !ariation is neglected+onsidering an elastic body, in e$ilibri$m $nder the action of a system of forces
statically a##lied, the kinetic energy and 0 will be n$ll, so e$ation (12.1) is
red$ced to:
0=+= ietotal L L L (12.2)
12.2. THE POTENTIAL ENERGY OF STRAIN
-n Theoretical Mechanics the rigid body is considered to be made from material
#oints, the distance between them remaining in!ariable. Therefore, internal work related to all #oints of the body is n$ll ("i 0).
For a deformable body the work is differently e!al$ated. "et*s consider a thin bar,
tensioned by e%terior forces F a##lied at both ends (Fig.12.1).
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Fig.12.1
+a$sed by the bar elongation, two material #oints sit$ated to a distance ds one from
another, are mo!ing away with a $antity Δ(ds), (Fig.12.1a). The attraction forces
between these two #oints will #rod$ce the interior work:( )ds dLi ∆⋅−=
with the min$s sign, beca$se the force F and the dis#lacement Δ(ds) ha!e o##osite
senses.
/ow, consider the same bar as a contin$$m and by the method of section an elementds is isolated. t each ends of this element the a%ial forces F m$st be introd$ced
(Fig.12.1b). eing sectional forces, they re#resent e%terior forces for the element ds.
The work #rod$ced by these sectional forces is an e%terior work:( )ds dLe ∆⋅+=
being #ositi!e beca$se F and Δ(ds) ha!e the same sense, b$t ha!ing contrary sign with
"i calc$lated abo!e:
ei dLdL −= , or:
ei L L −= (12.)
This work, #rod$ced by the internal stresses in a deformed body, a##ears as a
conse$ence of the deformations increasing, and it is called potential energy of strain, being noted with ! d .
From (12.) and (12.2) we ha!e:
d e ! L = (12.3)
4elation (12.3) is called "lapeyron#s first theorem, being en$nciated: if an elastic
body is in rest, the work of the exterior forces is equal to the potential energy of strain
accumulated by body$ 5ith other words: d$ring the loading of an elementary body the
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e%terior forces will #rod$ce a work e$al to the !ariation of the #otential energy of
these forces.
From (12.3), in what follows the #otential energy will be re#laced by the e%terior
work (called also work of strain "d).
12.3. THE WORK OF STRAIN
From Theoretical Mechanics the work " is defined as the force m$lti#lied by the
#ro'ection of the dis#lacement along the force direction (Fig.12.2a). s L ⋅= (12.6)
Fig.12.2
4e#resenting the de#endence between F and the s#ace co!ered by it, we obtain a
straight line #arallel to s a%is, the work being the area between this re#resentati!e
c$r!e F(s) and abscissa s (Fig.12.2b).
"et*s consider now a deformed bar of length l s$b'ected to centric tension (Fig.12.a).
Fig.12.
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The tensile force F #rod$ce the dis#lacement (elongation) 7l increasing in !al$e, in
the same time with the force F. The work won*t be written any more as in e$ation
(12.6): F87l, beca$se the !al$es of both factors !ary in time. The #roblem can be
written $nder a differential form. 9$ring the bar deformation the force F increase in
!al$e with a differential !al$e dF, #rod$cing an increasing d 7l of the dis#lacement.
The work can be written:( ) l d l d d dLe ∆⋅≅∆⋅+=
neglecting the small infinite of second degree ( )l d d ∆⋅ . From integration:l d Le ∆⋅= ∫ (12.a)
$t, between F and 7l a #ro#ortionality relation (ook*s ty#e) e%ist:
l l
E%
E%
l l ∆⋅=⇒
⋅=∆ (12.b)
-n (12.b) ;l is a constant for homogeny a%ial solicitation. 4e#lacing in (12.a):
l l
E%
l
l
E%l
l
E%l d l
l
E% Le ∆⋅=
∆⋅⋅=
∆⋅=∆⋅∆⋅= ∫ 2
1
22
2
4es$lting from the bar deformation, "e is called work of deformation and we shallnote it in what follows with "d:
l Ld ∆⋅=2
1 (for a%ial solicitation) (12.<)
The factor 2
1 take acco$nt the static a##lication of force, the linear increasing of the
force from 0 to the final !al$e (fig.12.b).
The work of deformation "d is acc$m$lated in the !ol$me of the bar as #otential
energy =d (#aragra#h 12.3).
12.4. THE POTENTIAL ENERGY OF STRAIN
The strain energy =d is $niform distrib$ted in the element !ol$me, so it sho$ld be
calc$lated first the strain energy acc$m$lated in a differential element of !ol$me
(fig.12.3a).
a. b. c.
Fig.12.3
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Fig.12.6
+onsidering an a%ial solicitation the
differential element from fig$re 12.3a
reached the deformed #osition from
fig$re 12.3b. ss$ming that the
material has elasto&#lastic beha!io$r,
with a non&linear characteristic c$r!eshown in fig$re 12.6, we ha!e to write
the strain energy for a s$##lementary
increasing of load (fig.12.3c), where
the linear strain&stresses (>&?) !ariation
may be considered.
The arc & may be re#laced in fig$re
12.6 by the secant &, so the medi$m
!al$es of the $nit stresses may be $sed
on this inter!al.
The second differential of the strain
energy1
( )2
d xd! d% dxσ = ∆ , will be:
( )( )dxd
d%d d%! d x x x
d ∆⋅++
=2
2 σ σ σ
(12.@)
4e#lacing ( ) ( ) dxd dxd dxd x x ⋅==∆ ε ε relation (12.@) becomes:
dxd d%d d%! d x x xd ⋅⋅
+= ε σ σ
2
12
neglecting the small infinites:( ) d& d dxd%d ! d x x x xd ⋅⋅=⋅⋅= ε σ ε σ 2
-ntegrating twice: once with res#ect to the s#ecific elongation ?% and then with res#ect
to the element !ol$me A, we find:
x x&
d d d& !
x
ε σ
ε
⋅= ∫ ∫ (12.B)
-n (12.B) the second integral re#resents the elastic strain energy stored into a $nit
!ol$me (A1) and it is called the specific strain energy (or the strain energy density):
x xd d u
x
ε σ ε
⋅= ∫ (12.10)
For linear&elastic deformations, ook*s law ε σ ⋅= E is !alid, so:
x x x x
x x x xd E
E d E d u
x x
ε σ σ ε
ε ε ε σ ε ε
2
1
22
22
====⋅= ∫ ∫ (12.11)
and the strain energy is:
∫ ⋅=& d d d& u! (12.12)
+onsidering the sim$ltaneo$s action of the linear s#ecific deformations with res#ect
to %, y and C a%is, relation (12.11) become, for linear&elastic deformations:
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)(2
1 ' ' y y x xd
u ε σ ε σ ε σ ++= (12.1)
-n a similar mode, for ang$lar s#ecific deformations D %y, D%C and DyC relation (12.1) is
written:
)(2
1 y' y' x' x' xy xyd u γ τ γ τ γ τ ++= (12.13)
for linear&elastic deformations, where ook*s law γ τ ⋅= ( is also !alid.Finally, the strain energy is written adding (12.1) and (12.13), and with (12.12):
1( )
2d x x y y ' ' xy xy x' x' y' y'
&
! d& σ ε σ ε σ ε τ γ τ γ τ γ = + + + + +∫ (12.16)
or: d& ( E
! y' x' xy ' y x
&
d )E(1
)(1
F2
1 222222τ τ τ σ σ σ +++++= ∫ (12.1)
s the differential !ol$me dA can be written:d' dydxdxd%d& ⋅⋅=⋅=
the #otential strain energy from relation (12.1) can be written generically:
∫∫∫ += dxdyd' ( E
! d )(21
22
τ σ (12.1<)
or: d%dx( E
! d )(2
1 22
∫∫ += τ σ
(12.1@)
12.4.1 The potent!" #t$!n ene$%& o' #t$!%ht (!$#) 'o$ !*!" !+ton
For an a%ially tensioned or com#ressed bar %
) x =σ and all other $nit stresses >C >y
G HyC 0, from (12.1@):
∫ ∫ ∫∫ =⋅⋅ = %
l d d%dx
E% ) d%dx
E % ) !
0 2
22
211
21
-f the a%ial force / and the cross sections are constants along the bar, the abo!e
integral may be #erformed, obtaining:2
0
1
2
l
d
) ! dx
E%= ∫
2
2
1
2
l %
E%= × × × (12.1Ba)
E%
l ) ! d
2
2
= (12.1Bb)
or l ) ! d ∆⋅=2
1(12.1Bc)
/ote that the strain energy =d from e$ation (12.1Bc) is e$al to the work of deformation "d from (12.<), knowing that the a%ial force / F constant.
12.4.2 The potent!" #t$!n ene$%& o' #t$!%ht (!$#) 'o$ (en,n%
For a bent bar, /a!ier*s form$la for bending with res#ect to y a%is is: ' *
+
y
y
x =σ , and
relation (12.1@) become:
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∫ ∫ ∫∫ =⋅⋅
⋅=
%
l
y
y
y
y
d d% ' dx E*
+ d%dx
E *
' + ! 2
0 2
22
2
11
2
1
as: ∫ = %
y * d% ' 2 , re#lacing abo!e:
dx
E*
+ !
l
y
y
d
∫ =
0
2
2
1(12.20)
-n the #artic$lar case, of constant cross section and bending moment My , relation
(12.20) is:
y
y
d E*
l + !
2
2
= (12.20a)
or, for bending with res#ect to C a%is:
'
' d
E*
l + !
2
2
= (12.20b)
12.4.3 The potent!" #t$!n ene$%& o' #t$!%ht (!$#) 'o$ #he!$n%
I$ra!ski*s form$la for shearing along C a%is is: y
y '
b*
, - =τ and the strain energy from
(12.1@) is:
∫ ∫ ∫∫ ⋅
=⋅⋅
⋅
⋅=
% y
yl
'
y
y '
d d% * b
, %dx
(%
- d%dx
( * b
, - !
22
2
0
22
2
11
2
1
The second integral is a nondimensional coefficient noted with k , called shape factor.
d%b
, * %d%
* b, %k
%
y
y % y
y
2
222
2
∫ ∫
=⋅= (12.21)
This coefficient takes acco$nt the non$niform distrib$tion of shear stresses $#on the
section height.
5ith (12.21) the strain energy will be:
dx(%
- k !
l '
d ∫ =0
2
2
1(12.22a)
dx %(
- !
l '
d ∫ =0
2
2
1(12.22b)
where:k
% % = is called the cross section transformed area
%am#le: For a cross section (fig.12.) ha!ing a rectang$lar sha#e, k is established as
follows:
12
bh
* y = bh % = d' bd% ⋅=
−=
−+
−= 2
2
32232 '
hb ' h ' '
hb, y
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Fig. 12.
4e#lacing in (12.21):2
2 222
2
2
12 11, 2
3 3
h
h
bh hk ' bd'
b h −
×= − = ÷
∫
For rolled #rofiles - or =, the #rofile
web take o!er the entire shear force. Thesha#e factor can be e%#ressed in this
case by the relation:
web %
%k = , being k 2,0G.2,3
12.4.4 The potent!" #t$!n ene$%& o' #t$!%ht (!$#) 'o$ to$#on
"imiting the e%#lanations to circ$lar or t$b$lar cylindrical bars, ha!ing the torsion
moment of inertia -t e$al to the #olar moment of inertia - # : -t - # constant on the
entire cross section, the strain energy is, with r *
+
p
t ⋅=τ :
∫ ∫ ∫∫ =⋅⋅
⋅=
%
l
p
t
p
t d d%r dx
(*
+ d%dx
(r
*
+ ! 2
0 2
22
2
11
2
1
as: ∫ = %
p * d%r 2 , re#lacing abo!e:
dx(* + !
l
p
t d ∫ = 0
2
21 (12.2)
-f the tor$e Mt and the cross section are constant along the bar a%is, e$ation (12.2)
is:
p
t d
(*
l + !
2
2
=
12.-. ENERGY PRINCIPLES AND THEORES
The energetically a##roach of the strength of materials #roblems re#resents a !eryefficient alternati!e with res#ect to the statically a##roach.
-n Theoretical Mechanics which deals only with nondeformable bodies, the principle
of virtual work is $sed $nder two distinct forms:
& -he principle of virtual displacement , also called principle of virtual work ,
when forces a##lied to the str$ct$re are real e%ternal forces in e$ilibri$m and the
!irt$al work is #rod$ced im#osing to the str$ct$re !irt$al dis#lacements
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& -he principle of virtual forces, also called principle of complementary virtual
work , when the dis#lacements of the str$ct$re are real dis#lacements and the !irt$al
work is carried o$t by the !ariation of !irt$al forces a##lied to the str$ct$re.
12.-.1 The p$n+p"e o' /$t0!" ,#p"!+eent#
+onsider a deformed bar in e$ilibri$m (#osition (1), in fig.12.<), an infinite close
#osition arbitrary chosen (#osition (2), in fig.12.<), is im#arted, com#atible with the
geometrical conditions im#osed by s$##orts. The new #osition of the bar is fictitio$s,
!irt$al. The dis#lacements, between the #osition in e$ilibri$m and the infinite close
#osition, are called virtual displacements.
Fig. 12.<
The dis#lacement from the initial #osition
(0) to the e$ilibri$m #osition (1) is d and
the !irt$al dis#lacement, between (1) and
(2), is /d .
The system of forces from the statice$ilibri$m #osition (1) is com#osed
from the e%terior forces which deform the
bar and the interior forces #rod$ced by
the bar deformations.
-m#arting to the bar a com#atible !irt$al dis#lacement (#osition (2) from fig$re 12.<),
the #rinci#le of !irt$al work is:0=+ ie L L δ δ (12.23)
with: J"e: the !irt$al work #erformed by the e%terior forces
J"i: the !irt$al work #erformed by the interior forces$ation (12.23) re#resent the energetically condition of e$ilibri$m the system of
forces.
/oting with /d 0 the !irt$al dis#lacement with res#ect to the force K direction, the
e%terior !irt$al work is:
∑ ∑ ⋅=⋅=⋅= Ld 0 d 0 L d 0 d 0 e δ δ δ δ (12.26)
with: ∑ ⋅= 0 d 0 L : the work of final intensity of e%terior forces (missing the factor L
of the static mode of action), the !irt$al dis#lacement /d 0 being a !ariation of the real
dis#lacement d 0 .
Jd: the !irt$al !ariation of the deformation elements (s#ecific linear and ang$lar deformations, dis#lacements of the #oints of a##lication of the e%terior forces)
s we e%#lained in #aragra#hs (12.1) and (12.2), the work of interior forces, from
relation (12.) is:
ei L L −=
and the e%terior work Le is the strain energy =d , from relation (12.3):
d e ! L =
From these two relations, the interior work Li is:
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d i ! L −= (12.2)
From (12.12) with (12.1) and (12.13), written $nder a shorter form:
( ) i
& &
d d Ld& d& u! −=+=⋅= ∫ ∫ τγ σε 2
1(12.2<)
4et$rning to the initial #roblem referring to the #rinci#le of !irt$al work, the interior
!irt$al work can be written from (12.2<) (neglecting the factor L of the static
character):( ) d d
&
i ! d& L ⋅−=⋅+⋅−= ∫ δ δγ τ δε σ δ (12.2@)
4e#lacing (12.26) and (12.2@) in relation (12.23):0=⋅−⋅ d d d ! L δ δ or ( ) 0=− d d ! Lδ (12.2B)
where: " is the work of final intensity:
∑ ⋅= 0 d 0 L (12.0)
5e note with: L! d −=Π (12.1)
: re#resent the potential or the total potential energy (the ca#acity of thedeformed body to cede energy)
4elation (12.2B) becomes:0=Π⋅d δ (12.2)
$ation (12.2) re#resents the principle of a stationary value of the total potential
energy, which can be form$lated as: 12f all compatible displacements satisfying
given boundary conditions, those which correspond to the body equilibrium make the
total energy 3 assumes a stationary value4. -t m$st be remarked that the linearity of
the stress&strain relationshi# has not been in!oked abo!e, so that this #rinci#le is !alid
for nonlinear, as well as linear stress&strain law, as long as the body remain elastic.
The e%ternal forces m$st be conser!ati!e forces (in which energy is conser!ed).
12.-.2 The p$n+p"e o' the /$t0!" 'o$+e#
-n the #rinci#le of the !irt$al dis#lacement there were no restrictions concerning the
magnit$de of the body deformations. -n this #rinci#le we shall consider that the real
deformation of the bar is !ery small (the hy#othesis of the small deformations), this
deformation being considered a virtual displacement$
Formally in the #rinci#le of the !irt$al dis#lacement we re#lace /d with d and /d 0
with d 0 . This deformation is #rod$ced by a system of forces (e%terior and interior)arbitrary chosen, which f$lfill the condition of e$ilibri$m im#osed by the #rinci#le
of !irt$al work. These forces are generically noted with /0 being called virtual forces$
The #rinci#le of !irt$al work from (12.23):0=+ ie L L δ δ
is now written:( ) 0=⋅+⋅−⋅ ∫ ∑
&
0 d& d 0 γ τ ε σ (12.a)
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where in the e%#ressions of J"e from (12.26) and J"i from (12.2@) the !irt$al
dis#lacement JdK, J? and JD were re#laced by real !al$es dK, ? and D.
+onsidering now, in the im#osed condition of f$lfilling the statically e$ilibri$m, that
the forces and the $nit stresses !ary arbitrarily to KNJK, >NJ> and HNJH, the #rinci#le
of !irt$al dis#lacement is:0E)()F()(
=+++−⋅+ ∫ ∑ & 0
d& d 0 0 γ δτ τ ε δσ σ δ (12.b)
O$btracting (12.a) from (12.b) we obtain:( ) 0=⋅+⋅−⋅ ∫ ∑
&
0 d& d 0 γ δτ ε δσ δ (12.3)
4elation (12.3) re#resent the mathematical e%#ression of the principle of virtual
forces, which is in!erse to the #rinci#le of !irt$al forces, beca$se now the deformation
is real and the e%terior forces and the $nit stresses are !irt$al.
Oimilarly to e$ations (12.2B) and (12.2) this #rinci#le may be written:( ) 0=− L! d f δ (12.6a)
or: 0=Π⋅ f δ (12.6b)
whereJf : is the !irt$al !ariation of the force elements (e%terior forces and $nit
stresses)
12.-.3 C!#t%"!no# theo$e#
These theorems are res$lts of the principle of a stationary value of the total potential
energy, being #$blished by +astigliano in 1@<B.
Castiglian!s "irst there# is e%#ressed by the relation:
i
d i
0
! d
∂
∂=
(12.)and it may be stated as follows: 1or a linear structure, the partial derivative of the
strain energy with respect to any load 0 i is equal to the corresponding displacements
d i , provided that the strain energy is expressed as a function of loads4
This theorem is !ery $sef$l for the calc$l$s of dis#lacements in a linear str$ct$re,
where the stresses or the internal actions may be established statically or with other
methods.
Fig.12.@
"et*s a##ly this theorem, to calc$late the
ma%im$m deflection w of the sim#le
s$##orted beam loaded by a concentrateforce K sit$ated at middle s#an, like in
fig$re 12.@. -n the calc$l$s of the strain
energy we take acco$nt only the effect of
the bending moment My. From relation
(12.20) the strain energy:
dx E*
+ !
l
y
y
d ∫ =0
2
2
1
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The deflection w $sing relation (12.) (neglecting the factor L of the static
character), will be:
0
1l y
y
y
+ w + dx
E* 0
∂=
∂∫
$t as the moment in a section x is x 0
+ y
2
= (fig.12.@), the #artial deri!ati!e from the
abo!e relation is:
22
x x
0
0 0
+ y =
∂∂
=∂
∂
Taking acco$nt the load symmetry, the deflection w will be:
; 2 ; 2
2
0 0
12
2 2 2 3@
l l
y y y
0 x 0 0l w x dx x dx
E* E* E* = × = =∫ ∫
Castiglian!s se$n% there# is e%#ressed by the relation:
i
d i
d
! 0
∂∂
= (12.<)
and it states that: P-he partial derivative of the strain energy with respect to any
displacement d i is equal to the corresponding force 0 i , provided that the strain energy
is expressed as a function of displacements4
12.. OHRA5WELLS ENERGETICALLY FORULA FOR
DISPLACEENTS CALCULATION
For linear&elastic members s$b'ected to com#o$nd actions, the strain energy ! d is the
s$m of =d relati!e to each interior action (e$ations (12.1Ba), (12.20), (12.22a),
(12.2)):
∑ ∫ ∫ ∫ ∫
+++= dx
(*
+ dx
%(
- dx
E*
+ dx
E%
) !
p
t
d
2222
2
1(12.@)
ased on the "astigliano#s first theorem:
∑ ∫ ∫ ∫ ∫
∂∂
+∂∂
+∂∂
+∂∂
=∂
∂= dx
0
+
(*
+ dx
0
-
%(
- dx
0
+
E*
+ dx
0
)
E%
)
0
! d
i
t
p
t
iiii
d i
2
1(12.B.a)
Qn the other hand, in the case of a linear elastic str$ct$re, the internal actions may be
e%#ressed as homogeno$s linear f$nction of the e%ternal loads:......11 +++= ii 0 n 0 n ) , ......11 +++= ii 0 t 0 t - ,......11 +++= ii 0 m 0 m + , ......11 +++= itit t 0 m 0 m + (12.Bb)
where ni , t i , mi , mti re#resent the influence coefficients and are the internal stresses, in
any cross section, #rod$ced by a generaliCed !irt$al $nit force ( )1=i
0
From (12.Ba) and (12.Bb) we find +ohr5+axwell#s formula (neglecting the factor
L), which was first deri!ed by I.+.Ma%well in 1@3 and a##lied in design #ractice in
1@<3 by Q.Mohr, being called also the unit5load method for linearly elastic
structures.
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∑ ∫ ∫ ∫ ∫
⋅+
⋅+
⋅+
⋅= dx
(*
m + dx
%(
t - dx
E*
m + dx
E%
n ) d
p
tit iiii (12.30)
4elation (12.30) #ermits the com#$ting of the dis#lacement d i (which can be an a%ial
dis#lacement Δ, a deflection v or w, or a rotation 6) in a section i on the direction of
the force 0 i$
To calc$late a dis#lacement $sing the energetical form$la (12.30) we ha!e to followthe ste#s:
a. 9etermine the diagrams of internal forces and moments /, T, M and Mt in
str$ct$re, d$e to e%ternal real loads
b. Klace a !irt$al $nit load ( )1=i
0 in the section i where the dis#lacement d i is to
be fo$nd. -f d i re#resent an a%ial dis#lacement Δi or a deflection vi or wi , we
#lace an $nit !irt$al force in section i . -f d i re#resent a rotation 6i , we #lace an
$nit !irt$al moment in section i .
c. 9etermine the diagrams ni , mi , t i , mti d$e to this $nit load 1=i
0
d. -ntrod$ce the terms in e$ation (12.30), integrate each term and finally
s$mmariCe them
The integrals which inter!ene in +ohr5+axwell#s formula #resent the feat$re that the
infl$ence coefficients ni , mi , t i , mti $nder the integral, ha!e a linear !ariation. This is
a conse$ence of the fact that they res$lt from the a##lication on the straight bar of a
unit virtual concentrated force or moment$ For this reason these integrals can be
easily sol!ed $sing the Aereshciaghin*s integration r$le, #resented bellow.
ll integrals ha!ing identical str$ct$re, we shall refer to the first term from e$ation
(12.30), relati!e to bending:
dxm
*
+
E
dx
E*
m + 7 ⋅=⋅=
∫ ∫
1(12.31a)
For bars with constant cross section (-const.) and the fle%$ral rigidity -const.:
∫ ⋅⋅=⋅ dxm + 7 E* (12.31b)
5e gra#hically re#resent the diagram MM(%) in an orthogonal system of a%is x2+
(Fig.12.Ba). =nder this diagram another system of a%is x2m is taken (Fig.12.Bb),
re#resenting the linear f$nction m , which makes an angle R with the abscissa 2x$ The
origin of both system of a%is x2+ and x2m is #oint 2 where the linear f$nction m
intersect the x a%is (Fig.12.Bb). -n M diagram a differential element of area d8 can be
written: dx + d ⋅=Ω . -n the same x section in m diagram the corres#onding !al$e is:α tg xm ⋅= . 4e#lacing these in (12.31b) the integral is:
∫ ∫ ∫ Ω⋅=Ω⋅⋅=⋅=⋅ d xtg d tg xdx + m 7 E* α α )( (12.32a)
The last integral is a static moment:
∫ ⋅Ω=Ω⋅ ( xd x (12.32b)
where: S: is the area of + diagram
%: is the abscissa of this + diagram centroid
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Fig.12.B
4e#lacing in (12.32a):η α ⋅Ω=⋅Ω⋅=⋅ ( xtg 7 E* (12.32c)
Finally, the integral I is:
E* dx
E*
m + 7
η ⋅Ω=
⋅= ∫ (12.3)
enerally, from (12.3) we concl$de that in order to calc$late a dis#lacement, first we
ha!e to calc$late the area S of the real internal forces or moments diagrams (/, T, M,
or Mt) and than these areas ha!e to be m$lti#lied by the ordinate U of the !irt$al
internal forces or moments diagrams (n, t, m or mt), corres#onding to the centroid of
S diagram. -f the real internal forces or moments diagrams (/, T, M, or M t) ha!e alsolinear !ariation, the role of the two diagrams may be in!erted.