Transactions on the Built Environment vol 20, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509
On the treatment of non-scalar quantities in the
response spectrum method
J. Plesek
K/4ME7Ifd, f/zzrzj&a 227, 7 JO 00 f ra/za J, CzecA
Abstract
In the response spectrum method, the complete time histories of modal responses are not known.
Therefore, we must adopt some approximate technique, for instance the SRSS rule, in order to
estimate the probable maximum reached by a quantity over the time interval of interest. It is generally
believed that any response, e.g. the component of a vector or a tensor, may be treated this way. The
opposite is true, however, and we show that the pseudotensors composed of the unsigned real
numbers violate the frame invariance principle. As a consequence of this, the use of approximate
summation methods that provide but the response amplitudes should be restricted to manipulation
with scalars. The implications for engineering analysis are examined in details. In conclusion, the
SRSS summation of invariant effective stresses leads to approximate but objective results. Similarly.
the magnitudes of reactions can be soundly estimated. In contrast, the direction of a force established
via the SRSS rule depends solely on the choice of a coordinate system, making the results physically
meaningless. These important observations are demonstrated -by two examples involving an
illustrative FEM analysis.
1 Introduction
The response spectrum method (RSM) has been widely used in engineering
practice for several decades. Meanwhile, the attention of analysts has shifted
from analytically based approximate approaches toward straight numerical
methods such as the FEM. This turnabout has given rise to some difficulties
with the interpretation of RSM results. One of those is a loss of physical
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Earthquake Resistant Engineering Structures
inspection over the summation process of reaction forces.
A key principle of RSM lies in that the classical sum of the response
contributions of individual modes to some quantity is replaced with the SRSS
sum (the square root of the sum of the squares). This summation rule,
however, erases not only signs but also directional information carried on by
vectors and tensors. As a result, only scalar quantities summed up in this way
have any physical meaning.
Calculation of stresses usually presents no problem since we are mostly
interested in effective stresses, which are scalars. The same cannot be said
about reaction forces, where both the magnitude and direction play important
roles. For simple problems, the force direction can be guessed by inspection
as in semi-analytical solutions. Unfortunately, when a FEM model with a large
number of variables is employed, one has little chance to determine reactions,
except for their magnitudes.
In this paper we want to prove the following facts: i) The SRSS rule
conserves scalars such as tensor norms (magnitudes of reactions and, to some
extent, the effective stresses) but ii) violates the frame invariance principle for
tensorial quantities (directions are lost). It is shown that the pseudovectors
obtained via the SRSS sum may assume arbitrary directions.
2 Combination of modal responses
In the modal superposition method we combine the contributions of modes
n = l,2,...,N to some quantity c at the time t as
c(t)
= yc^gn(t),
(l)
where -1 < g»(r) < 1 are the time functions and c^ are the maximum values
(amplitudes) reached by the modal responses over the time interval of interest.
This equation applies only when we know the response functions g.(f) for all
the modal components. This is not the case with RSM and we estimate the
probable maximum of c(f) with the Goodman-Rosenblueth-Newmark rule*
Sr\ax
=
=
(2)
known as the SRSS sum.
Now consider a Cartesian orthogonal tensor T with the standard transformation rule between two coordinate systems %, and xC
Transactions on the Built Environment vol 20, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509
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215
with A being the orthonormal matrix /W%=6y and 5, the Kronecker symbol.
Suppose that the tensor T consists of the sum of modal contributions as in (1)
We may define a pseudotensor f such that
The disadvantage of such a definition is that the components % of f do not
obey the transformation law (3). Writing the equation (4) at the primed system
jtj' we have
(6)
and the inverse transformation back to jCj gives
(7)
'
•'- ijk.., ~ ^ip^jq^ki-'
Had the right-hand side of (6) been a linear function of the coefficients of A,
those coefficients would have canceled out in the eqn.(7). This is not true,
however, thus
f
* ?"•£
.
(8)
As a consequence, the results of the SRSS summation viewed from a fixed
configuration *i will vary according to the choice of the coordinate system %/
that has been used for calculations. This leads us to the conclusion that there
is not much sense in defining the quantity f.
On the other hand, the approximate equation (2) remains objective if c(t)
represents a scalar. If, for instance, c™ denotes some invariant of T™, i.e.
c*"> = I[T™], we may use the SRSS sum (2) in order to estimate the magnitude
of/mas
^ax^=
£IMT<*>]
.
(^
Transactions on the Built Environment vol 20, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509
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Earthquake Resistant Engineering Structures
It should be pointed out that, generally,
with only one exception, which is the Euclidean norm of a tensor
(n) IIII - -f
For this particular case the equality in (10) holds, i.e
( N
u
3 Finite element method
The equations of motion for an N-degree of freedom structure have the form*
Mfl + Cli + JTu = -Mu^U
,
(13)
where Af, C and JST are mass, damping and stiffness matrices, respectively; if
is the relative displacement vector; ii> is the static displacement vector when the
base of the structure displaces by unity in the direction of the earthquake; and
#, is the ground acceleration. The system has ^-orthogonal modal vectors *.
satisfying
where <•>„ are the angular frequencies. In addition we define the modal
participation factors
in ~~ V n ** **jb
and denote 5. the spectral displacement obtained from the acceleration response
spectrum SA(&>) as
$n = A^(^n) •
n
(16)
Then the maximum relative displacement vector for each mode can be
computed using
Transactions on the Built Environment vol 20, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509
Earthquake Resistant Engineering Structures
217
and we may employ the SRSS rule (2) in order to estimate an amplitude of any
quantity c(t) derived from u. In the context of FEM we are especially
interested in stresses
where D is the constitutive matrix and B is the strain-displacement matrix c =
Bu, and reaction forces
(19)
3.1 Calculating stresses
Combination of effective stresses a*, which are invariants of the stress tensor,
does not pose serious problems. We can make use of the eqn.(18) and
compute,first,the effective stresses oi§ for each mode and, second, sum up
those contributions to obtain o* as in (9). Generally, the effective stress cannot
be computed from the pseudotensor a.
A typical situation arises when the effective stress is a function of the
deviatoric stress s only. In this frequent case we can write analogously to (1)
](t),
(20)
from which it follows
In order to obtain an upper bound we may take the norm of both sides of (21)
arriving at
N
_ / o_\ _ \""^ _ (n)
O0(t) < 2^ Oe
*
n=1
/g g\
(**)
where a, is the von Mises stress
S,,S,, .
(23)
In the inequality (22) we combined the maxima of modal effective stresses of >,
which was a safe but pessimistic estimate. Instead, we might want to employ
Transactions on the Built Environment vol 20, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509
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Earthquake Resistant Engineering Structures
the SRSS rule
It should be noted that Von Mises' stress is defined as the Euclidean norm of
the deviator rather than the stress tensor. Therefore, the equality (12) does not
hold for <Jt(a) and neither (22) nor (24) has any relation to the pseudotensor a.
Similar considerations apply to the maximum shear stress criterion, whose
effective function 7, is bounded by
(25)
3.2 Calculating forces
Employing (19) for each modal displacement #**, we obtain the force vectors
K* pertinent to modes /i = l,2,...,#. Let us denote as Rf\ Rf\ R™ the
components of the reaction force at a constrained node. We customarily use the
SRSS rule to estimate the probable maxima at the node as
(26)
According to (8), the relative orientation of the pseudovector R = {/?, ,^ ,R^ Y
with respect to the finite element model will depend on the alignments of
coordinate axes with this model.
Yet some useful information can be retrieved from the components of R.
Summation of the three squared equations (26) yields
(27)
in accordance with the equality (12). Apparently, the magnitude \\R\\ of the
force R can be estimated either way, while its decomposition into directional
components has no physical meaning.
Transactions on the Built Environment vol 20, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509
Earthquake Resistant Engineering Structures
219
4 Examples
The derived features of the SRSS rule are now verified with two examples. In
the first one we sum up 2D vectors to prove that the SRSS rule produces
arbitrary results as regards the direction of the computed pseudovector. In the
second example an illustrative FEM analysis is carried out and the influence of
the coordinate axes rotation on the results is studied.
4.1 Analytical example
Consider two vectors a and ft containing an angle <p. Let us specify some fixed
coordinate system jc,y, which in plane orientation is defined by the angle a see Fig.l.
Figure 1: The SRSS sum of two vectors.
The pseudovector C may now be computed as the SRSS sum of vectors a and
b as
c* = a* + tj = a^cos^a + jb^cos* (a + cp)
Cy = ay + by = a * s irfa. + £ * s in* ( a + <p )
g2 = gj + -2 = ^2 + 6= .
Clearly, the length \\c\\ of c is independent of the choice of a coordinate
system (the choice of a). Denote 0 the angle between the vector b and the
pseudovector c. Inserting the directional cosines c%, C, into the above equations
we have
Transactions on the Built Environment vol 20, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509
220
Earthquake Resistant Engineering Structures
(a + <p)
(a + <p) .
These two equations are identical. Solving for a from either one, we find
(2q>) -
tan(2o) =
cos
The last expression indicates that given an arbitrarily chosen 0, we may
determine a such that the response vector £ points in the desired direction ft.
4.2 Numerical example
A clamped beam made of steel is subjected to horizontal ground motion with
the acceleration a». Dimensions and material properties are given in Fig.2. The
object was discretized by four 12 degrees of freedom beam elements using the
finite element code PMD\
aH
Figure 2: Dimensions: L = 1 m, b - 1 cm, h — 2 cm. Material properties:
E = 2x10" Pa, v = 0.3, p = 7800 kg/m\ Total mass = 1.56 kg.
Transactions on the Built Environment vol 20, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509
Earthquake Resistant Engineering Structures 221
First, we computed the natural frequencies and modal participation factors
(15) for both the horizontal axes y and z. The results together with the
prescribed accelerations a,=a,=4H for each mode are listed in Tab.I. The
energy carried on by thefirstten modes approaches the total kinetic energy
(total mass) of the structure subject to horizontal motion. With the aid of (17),
(19) and the prescribed acceleration spectrum & in (16) we were also able to
calculate the reactions *, = 1.05 N, £ = 2.39 N, |*| = 2.61 N.
Table I: Dynamic characteristics.
N«
f[Hz]
a* [m/Sj]
Y^[kg]
Yl [kg]
1
8.18
2.5
0
0.9654
2
16.36
1.0
0.9564
0
3
51.32
0.7
0
0.2936
4
102.64
0.7
0.2936
0
5
144.65
0.7
0
0.1003
6
285.36
0.7
0
0.0469
7
289.30
0.7
0.1003
0
8
530.75
0.7
0
0.0241
9
570.72
0.7
0.0469
0
10
706.73
0.7
0
0
89%
91%
E
1.56kg =
100%
Next, the entire analysis was repeated, this time in the coordinate system
yy rotated by the angle a - see Fig.2. We obtained the components R\, R\
related to the primed system y' ,z'. Since we were interested in the force action
relative to beam's crossection, we transformed those components into the
principal axes y,z as pictured in Fig.2. The reactions computed for varying
angle a are plotted in Fig.3. The magnitude ||it|| should be recognized as the
only objective quantity.
5 Conclusions
We have discussed the implications of the approximate summation rules that
'erase signs' in the response spectrum method. It has been shown, both on a
theoretical basis and by means of examples, that there is not much sense in
computing amplitudes of tensorial components because of the loss of their
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222
Earthquake Resistant Engineering Structures
Ry
0
Rz
46
90
IRI
136
180
rotation of y' axis [degrees]
Figure 3: Force components vs. rotation angle.
frame invariance property. The invariance principle remains, however,
conserved for scalars.
Acknowledgements
This work was supported by the Grant Agency of the Czech Republic under
grant number 101/96/0910 and by VAMET Ltd.
References
1. Goodman, L.E., Rosenblueth, E. & Newmark, N.M. A seismic design of
elastic structures founded on firm ground, in ASCE/53, pp. 349-1 to 34927, Proceedings of the ASCE Conf., 1953.
2. Gupta, A.K. Response Spectrum Method In Seismic Analysis and Design
of Structures, Blackwell Scientific Publications, Cambridge, 1990.
3. PMD version 486.3, Reference guide, VAMET, Praha, 1994.