The purpose of this example is to investigate the effect of different kinematic (geometry) assumptions on theresponse of a shallow truss with one free dof. We limit our discussion to linear material response.
8
1
1U1P
EA=25000
The above truss has only one non-zero free dof, as shown. The relation between the applied force and thecorresponding displacement is, in general, nonlinear. Given the applied force P it is not easy to establishthe corresponding displacement without a good knowledge of solution strategies for nonlinear equation(s). For now, we limit ourselves to establishing the nonlinear relation between global dof displacement and corresponding force by specifying a range of displacementvalues and calculating the corresponding force that the structure can resist.
We drop the subscript from P and U in the following discussion. The following data are given:
Geometry
ΔX 8:= ΔY 1:= L ΔX2 ΔY2+:= original position L 8.062=
Material properties E 50:= A 500:= EA E A⋅:= EA 25000=
The compability relation between truss element deformation v and global dof displacement(s) U isnonilnear. Since there is no danger of confusion we drop the bar over the local element displacementvector in the local coordinate system and denote these by u. We use the Green-Lagrange straindefinition in the following which when applied to the problem at hand reads
εGLΔuxL
12
ΔuxL
⎛⎜⎝
⎞⎟⎠
2
⋅+12
ΔuyL
⎛⎜⎝
⎞⎟⎠
2
⋅+= (1)
For the truss element deformation we write: vGL εGL L⋅=
Noting the truss element orientation (direction cosines ΔX/L and ΔY/L) and the fact that the lower end ofthe truss element does not displace we express the relative axial and transverse deformation in the localcoordinate system in terms of the single global dof displacement
Δux UΔYL
⋅= and Δuy UΔXL
⋅=
Substituting in equation (1) we obtain vGL U( )1LU⋅
ΔYL
⋅1
2L2U
ΔXL
⋅⎛⎜⎝
⎞⎟⎠
2⋅+
1
2L2U
ΔYL
⋅⎛⎜⎝
⎞⎟⎠
2⋅+
⎡⎢⎢⎣
⎤⎥⎥⎦L⋅:= (2)
It is worth noting that ΔX2 ΔY2+ L2=
thus the expression in (2) for the Green-Lagrange deformation simplifies to vGL U( )UL
ΔYL
⋅U2
2L2+
⎛⎜⎜⎝
⎞⎟⎟⎠L⋅:=
Clearly, the relation between truss element deformation and global dof displacement is nonlinear.
For small values of U/L we can neglect the quadratic term and obtain a linear relation between trusselement deformation and global dof displacement(s).
vl U( )ΔYLU⋅:=
Let us also set up the rotated engineering strain. We define the new length of the element as function of U
Ln U( ) ΔX2 ΔY U+( )2+:=
and the rotated engineering deformation vRE U( ) Ln U( ) L−:=
we also look at the common approximation of the rotated engineering strain for many structural engineeringapplications, namely
vap U( )1LU⋅
ΔYL
⋅1
2L2U
ΔXL
⋅⎛⎜⎝
⎞⎟⎠
2⋅+
⎡⎢⎢⎣
⎤⎥⎥⎦L⋅:= which is arrived at by power series expansion of the square
root and truncation of higher than quadratic order terms
Page 2
To plot the relation between v and U specify a range of displacement values for the free global dof
U 0.5− L⋅ 0.498− L⋅, 0.5 L⋅..:=
We note that the linear strain and the nonlinear strains deviate significantly only for large values ofU/L. The corresponding deformation is very large at these values (v approaches 10% of initiallength!!) We also conclude that the approximation of the rotated engineering strain is excellent overthe selected range of displacement values and that there is a difference between the two nonlinearstrain (deformation) definitions
0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.40.1
0.08
0.06
0.04
0.02
0
0.02
0.04
0.06
0.08
0.1
vl U( )
L
vGL U( )
L
vRE U( )
L
vap U( )
L
UL
In order to establish a relation between global dof displacement U and corresponding resisting force P weneed to relate the basic force to the element deformation by the element force-deformation relation andthen the basic force to the global force by equilibrium. The former is straightforward for a linear material. Weuse the same expression for the different strain measures. The resulting constitutive laws are not the same!
qEALv⋅= and thus qGL U( )
EALvGL U( )⋅:= qRE U( )
EALvRE U( )⋅:= ql U( )
EALvl U( )⋅:=
Page 3
Now, we need to relate the basic force q to the applied force P by equilibrium. We write the equilibriumin the deformed configuration for the nonlinear strain measures and the undeformed configuration forthe linear strain measure by using PVD. To this end we need the variation of strains (see notes). We obtain:
for equilibrium in deformed configuration PΔY U+Ln
q⋅= for rotated engineering strain
PΔY U+L
q⋅= for Green-Lagrange strain
for equilibrium in undeformed configuration PΔYLq⋅= by setting U=0 in any one of above eqs
With these we obtain three different force-displacement relations for the truss
We now have for Green-Lagrange strain: PGL U( )ΔY U+L
EALvGL U( )⋅⎛⎜
⎝⎞⎟⎠
⋅:=
for rotated engineering strain PRE U( )ΔY U+Ln U( )
EALvRE U( )⋅⎛⎜
⎝⎞⎟⎠
⋅:=
for linear strain Pl U( )ΔYL
EALvl U( )⋅⎛⎜
⎝⎞⎟⎠
⋅:=
The applied force-global dof displacement relation becomes
0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2100
50
0
50
100
Pl U( )
PGL U( )
PRE U( )
UΔX
Page 4
Behavior of shallow truss in compression (downward load)
U 0.3− L⋅ 0.298− L⋅, 0..:=
PGL U( )ΔY U+L
EALvGL U( )⋅⎛⎜
⎝⎞⎟⎠
⋅:=
PRE U( )ΔY U+Ln U( )
EALvRE U( )⋅⎛⎜
⎝⎞⎟⎠
⋅:=
0 0.05 0.1 0.15 0.2 0.25 0.310
5
0
5
10
15
20
PGL U( )−
PRE U( )−
UL
−
We conclude that for the shallow truss, both nonlinear strain measures yield practically identical results!
