December 2011 

NASA/TP–2011-217315 

Dynamic Shape Reconstruction 

of Three-Dimensional Frame Structures 

Using the Inverse Finite Element Method 

Marco Gherlone, Priscilla Cerracchio, Massimiliano Mattone, and Marco Di Sciuva 

Department of Aeronautics and Space Engineering

Politecnico di Torino, Torino, Italy 

Alexander Tessler 

Langley Research Center, Hampton, Virginia 
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December 2011 

NASA/TP–2011-217315 

 

Dynamic Shape Reconstruction 

of Three-Dimensional Frame Structures 

Using the Inverse Finite Element Method 

Marco Gherlone, Priscilla Cerracchio, Massimiliano Mattone, and Marco Di Sciuva

Department of Aeronautics and Space Engineering

Politecnico di Torino, Torino, Italy 

Alexander Tessler 

Langley Research Center, Hampton, Virginia
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 

The use of trademarks or names of manufacturers in this report is for accurate reporting and does not 

constitute an official endorsement, either expressed or implied, of such products or manufacturers by the 

National Aeronautics and Space Administration. 
1

Abstract

A robust and efficient computational method for reconstructing the 

three-dimensional displacement field of truss, beam, and frame 

structures, using measured surface-strain data, is presented. Known as 

“shape sensing”, this inverse problem has important implications for 

real-time actuation and control of smart structures, and for monitoring 

of structural integrity. The present formulation, based on the inverse 

Finite Element Method (iFEM), uses a least-squares variational 

principle involving strain measures of Timoshenko theory for stretching, 

torsion, bending, and transverse shear. Two inverse-frame finite 

elements are derived using interdependent interpolations whose interior 

degrees-of-freedom are condensed out at the element level. In addition, 

relationships between the order of kinematic-element interpolations and 

the number of required strain gauges are established. As an example 

problem, a thin-walled, circular cross-section cantilevered beam 

subjected to harmonic excitations in the presence of structural damping 

is modeled using iFEM; where, to simulate strain-gauge values and to 

provide reference displacements, a high-fidelity MSC/NASTRAN shell 

finite element model is used. Examples of low and high-frequency 

dynamic motion are analyzed and the solution accuracy examined with 

respect to various levels of discretization and the number of strain 

gauges.

 Nomenclature

 x y z , , 

Cartesian coordinate system

, , x r

cylindrical coordinate system

 xxx 1 2 3 , , 



local “strain-gauge” coordinate system

E, G, ,  Young’s modulus, shear modulus, Poisson ratio, and density

L, A length and cross-sectional area of frame member

y

I , 

z

I

area moments of inertia with respect to the 

y - and 

z -axis

P

I

polar moment of inertia

ux

, 

y u , 

z

u

displacements along 

x -, y -, and 

z -axis 
2

u , v , w

displacements along along x -, y -, and 

z -axis

x, y, z rotations about

x -, y -, and 

z -axis

, , , , ,

T

x y z

   u v w      u

vector of the kinematic variables

h u

vector of the approximated kinematic variables

x

 , 

xz

 , 

xy



non vanishing components of the strain tensor for the Timoshenko 

beam theory in the (x,y,z) coordinate system

x

 , 



 , 

x



components of the strain tensor for the Timoshenko beam theory 

in the 

, x

plane

*

2



strain measured by the strain-gauge

 x

,  , 

x



components of the stress tensor for the Timoshenko beam theory 

in the 

, x

plane

( 1,...,6) ie i 

section strains

 1 2 3 4 5 6 , , , , , 

T

e  e e e e e e

vector of the section strains



e

experimentally evaluated section strains

N, Qy, Qz, Mx, My, Mz section forces and moments

Ax, Gy, Gz, Jx, Dy, Dz axial rigidity, shear rigidities, torsional rigidity and bending 

rigidities of the beam

2

y

k , 

2

z

k

shear correction factors

qx, qy, qz distributed loads along the x-, y-, and z-direction



least-squares functional

e u

vector of the nodal degrees of freedom of the inverse finite 

element

N

matrix of the shape functions relating the kinematic variables to 

the nodal degrees of freedom

B

matrix of the shape functions relating the section strains to the 

nodal degrees of freedom

0

, ( 1,...,6) w w k k k

weighting coefficients

,

e e L A , 

e

y

I , 

e

z

I , 

e

P

I

length, cross-sectional area and moments of inertia of frame 

inverse element

n , i

x

number and axial coordinate of the locations where the section 

strains are evaluated 
3

,

e e k f

matrix and vector of the single inverse element governing equation

Κ F ,

matrix and vector of the whole structure governing equation

 

(1) Li 

linear Lagrange polynomials

 

(2) Lj



quadratic Lagrange polynomials

   

(3) (3)

, k N N j 

special cubic polynomials

 

(4) Lk



quartic Lagrange polynomials



orientation of a strain-gauge with respect to the beam axis

Rext

external radius of the circular cross-sectio

R, t radius and thickness of the thin-walled circular cross section

Fz(t) time varying applied force

Fz0 amplitude of the applied force

f0 frequency of the applied force
4

1 Introduction

Real-time reconstruction of structural deformations using measured strain data, is a key technology for 

actuation and control of smart structures, as well as for Structural Health Monitoring (SHM) [1]. Known 

as “shape sensing”, this inverse problem is commonly formulated with the assumption that multiple strain 

sensors at various structural locations provide real-time strain measurements. Most inverse algorithms use 

some type of Tikhonov’s regularization, which is manifested by constraint (regularity) terms that ensure a 

certain degree of solution smoothness (refer to [2-5] and references therein). 

Most of the shape sensing efforts focused exclusively on beam-bending problems. Davis et al. [6] used 

optimized trial functions and weights to reconstruct a simple static-beam response from discrete strain 

measurements. To model more complex deformations, the approach requires a large number of trial 

functions and strain sensors. Kang et al. [7] used vibration mode shapes to reconstruct a beam response 

due to dynamic excitation. In their approach, modal coordinates are computed using strain-displacement 

relationships and measured surface strains; the method requires the same number of mode shapes and 

strain sensors. Kim et al. [8] and Ko et al. [9] used classical beam equations to integrate discretely 

measured strains to determine the deflection of a beam. By regression of experimental strain data and by 

accounting for the applied loading, Kim et al. [8] obtained a continuous curvature function, leading to the 

evaluation of beam deflection. Ko et al. [9] developed a load-independent method by approximating the 

beam curvature using piece-wise continuous polynomials; the authors demonstrated the validity of a onedimensional scheme by evaluating the deflection and cross-section twist of an aircraft wing.

To enable shape-sensing analyses of plates undergoing bending deformations, Bogert et al. [10] 

examined a modal transformation method that allows the development of suitable strain-displacement 

transformations. The approach makes use of a large number of natural vibration modes. When applied to 

high-fidelity finite element models, however, the method requires a computationally intensive eigenvalue 

analysis and a detailed description of the elastic and inertial material properties. Jones et al. [11] 

employed a least-squares formulation for shape sensing of a cantilever plate, where the axial strain was 

fitted with a cubic polynomial. The strain field was then integrated with the use of approximate boundary 

conditions at the clamped end to obtain plate deflections according to classical bending assumptions. 

Shkarayev et al. [12,13] used a two-step solution procedure: the first step involves the structural analysis 

of a plate/shell finite element model, and the second, a least-squares algorithm. The methodology 

reconstructs the applied loading first, which then leads to the displacements. In a series of four papers, 

Mainçon and co-authors [14-17] developed a finite element formulation that seeks the solution for the 

displacements and loads simultaneously, requiring a priori knowledge of a subset of applied loading and 

the material properties. The solution procedure minimizes a cost function consisting of unknown loads 

and differences between the measured and estimated quantities (displacements or strains); the cost 

function is regularized by way of equilibrium constraints. The number of unknowns is three times the 

number of the degrees-of-freedom in the finite element discretization. Importantly, the accuracy of the 

solution strongly depends on the choice of suitable weights; these are computed from a complex 

procedure involving the probability distributions of the unknown loads and measured data. In [16,17], 

sensitivity analyses were carried out for truss structures, investigating variations in the input data as well 

as the modeling errors. Nishio et al. [18] employed a weighted-least-squares formulation to reconstruct, 

on the basis of measured strain data, the deflection of a composite cantilevered plate. The weighting 

coefficients in the least-square terms were adjusted in order to account for the inherent errors in the 

measured strain data. The weights were computed for a given data-acquisition apparatus, load case, and 

test article, with the consequent difficulties in generalizing the procedure.

Many of the aforementioned inverse methods either lack generality with respect to structural topology 

and boundary conditions, or require sufficiently accurate loading and/or elastic-inertial material 

information – the kind of data that are either unavailable or difficult to obtain outside the laboratory 

environment; for these reasons, such approaches are generally unsuited for use in onboard SHM 
5

algorithms. A well-suited algorithm for SHM should be: (1) general enough to accommodate complex 

structural topologies and boundary conditions (e.g., built-up aircraft structures), (2) robust, stable, and 

accurate under a wide range of loadings, material systems, inertial/damping characteristics, and inherent 

errors in the strain measurements, and (3) sufficiently fast for real-time applications.

An algorithm that appears to fulfill the aforementioned requirements was recently developed by

Tessler and Spangler [1,19]. The methodology, labeled the inverse Finite Element Method (iFEM), 

employs a weighted-least-square variational principle which is discretized by C0-continuous finite 

elements that accommodate arbitrarily positioned and oriented strain-sensor data. The iFEM framework, 

providing accurate and stable solutions of the displacement and strain fields for a discretized structural 

domain, is amenable to any type of structural modeling including frame (truss and beam), plate, shell, and 

solid idealizations. Because only strain-displacement relations are used in the formulation, both static and 

dynamic response can be reconstructed without any a priori knowledge of material, inertial, loading, or 

damping structural properties. To model arbitrary plate and shell structures, Tessler [20] developed, using 

first-order shear-deformation theory, a three-node inverse shell element. Numerically generated [20] and 

experimentally measured-strain data [21,22] were used to assess robustness and accuracy of the 

formulation. 

The present paper consolidates the authors’ recent efforts [23-24] and presents the development and 

assessment of simple and efficient inverse-frame finite elements. The methodology permits effective and 

computationally efficient shape-sensing analyses to be performed on truss, beam, and frame structures 

instrumented with strain gauges. The kinematic assumptions are those of Timoshenko shear-deformation 

theory [25]; they incorporate stretching, torsion, bending, and transverse shear deformation modes in 

three dimensions. The formulation uses a least-squares variational principle that is specialized from [19] 

for three-dimensional frame analysis. The variational framework, in conjunction with suitable finite 

element discretizations involving inverse finite elements, yields a system of linear algebraic equations; the 

equations are efficiently solved for the unknown displacement degrees-of-freedom (dof’s), thus providing 

the deformed structural-shape predictions. 

In the remainder of the paper, the kinematic assumptions for a three-dimensional frame are discussed, 

followed by the description of the least-squares variational principle suitable for three-dimensional 

deformations of frame structures. This is followed by a discussion of two C0-continuous, inverse-frame 

elements that use the well-established interdependent interpolations that resolve the shear locking effect. 

Finally, to examine the predictive capabilities of the inverse elements for a given set of distributed strain 

gauges, shape-sensing studies are carried out for a cantilevered beam undergoing harmonic excitations in 

the presence of structural damping.

2 Governing equations

Consider a straight frame member of constant cross-section referred to the three-dimensional 

Cartesian coordinates (x,y,z) as depicted in Figure 1; the coordinate origin, O, is located at the crosssection’s center of mass, which is also coincident with the shear center. The longitudinal, elastic x-axis is 

normal to the cross-sectional plane (y, z), where y and z are the cross-section’s principal inertial axes. The 

frame member has length L and its cross-section has area A, area moments of inertia with respect to the 

y - and 

z -axis 

y

I

and 

z

I

, respectively, and polar moment of inertia 

P y z I I I  

(Figure 1). The frame 

member is made of an isotropic homogeneous material, represented by the elastic constants: 

E

(Young’s 

modulus), 

G

(shear modulus), and v (Poisson ratio). 

The three Cartesian components of the displacement vector that are consistent with the kinematic 

assumptions of Timoshenko theory [25] for three-dimensional deformations are given by 
6

       

     

     

, ,

, ,

, ,

x y z

y x

z x

u x y z u x z x y x

u x y z v x z x

u x y z w x y x

 





  

 

 

 (1)

where 

x u , 

y u

, and 

z

u

are the displacements along the 

x , y

, and 

z

axes, respectively, with 

u , v

, and 

w

denoting the displacements at 

y z   0

; 

 x

,  y, and 

 z

are the rotations about the three coordinate 

axes; positive orientations for the displacements and rotations are depicted in Figure 1. These kinematic 

assumptions neglect the effect of axial warping due to torsion, i.e., each cross-section remains flat and 

rigid with respect to thickness-stretch deformations along the y and z axes. The six kinematic variables

can be grouped in vector form as

 , , , , , 

T

x y z u  u v w   

(2)

Figure 1: Beam geometry and kinematic variables.

Adhering to the small-strain hypothesis, the non-vanishing strain components have the form

1 2 3

4 6

5 6

( , , ) ( ) ( ) ( )

( , ) ( ) ( )

( , ) ( ) ( )

x

xz

xy

x y z e x z e x y e x

x y e x y e x

x z e x z e x







  

 

 

(3)

where
7

( ) , , , , ,  1 2 3 4 5 6

T

e u  e e e e e e

(4)

denote the section strains of the theory, given by

1 , 4 ,

2 , 5 ,

3 , 6 ,

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

x x y

y x x z

z x x x

e x u x e x w x x

e x x e x v x x

e x x e x x



 

 

  

  

  

(5)

The section forces (N, Qy, and Qz) and moments (Mx, My, and Mz) are related to the section strains, 

i

e

, by 

way of the constitutive equations (refer to Figure 2)

1 6

5 2

4 3

N M

Q M

Q M

x x x

y y y y

z z z z

A e J e

G e D e

G e D e

 

 

 

(6)

where 

A EA

x



is the axial rigidity; 

2 G k GA y y



and 

2 G k GA z z



are the shear rigidities, with 

2

y

k

and 

2

z

k

denoting the shear correction factors; 

x P J GI 

is the torsional rigidity and 

D EI

y y



and 

D EI

z z



denote the bending rigidities.

Figure 2: Beam section forces and moments.

The equilibrium equations that correspond to the distributed loads 

( )

x q x , ( )

y q x

, and 

( )

z q x

along the x, 

y and z directions, are
8

dN dM 0

d d

dQ dM

Q

d d

dQ dM Q

d d

x

x

y y

y z

z z

z y

q

x x

q

x x

q

x x

  

  

  

(7)

To reconstruct the deformed shape of a frame-member for which certain in-situ strain measurements 

are known, a functional 

( ) u

that matches in a least-squares sense the complete set of analytic section 

strains,

eu( ) , to the in-situ section strains, 

ε

e

, is minimized with respect to the kinematic variables, 

u

; 

( ) u

functional can be written as

   

2

    u e u e

(8)

This functional may be used in a finite element framework by introducing a discretization in which the 

element kinematic field is interpolated by C

0

-continuous shape functions,

  ( ) h e u u N u x x 

(9)

where 

N( ) x

denotes the shape functions and 

e u

nodal dof’s. Thus, the total least-squares functional is a 

sum of the individual element contributions, 

( ) e h  u

, i.e., 

1

N

e

e

   

, with N denoting the total number 

of elements. Accounting for the axial stretching, bending, twisting, and transverse shearing, the element 

functional is given by the dot product of the weighting coefficient vector, 

        

0 0 0 0 0 0

1 2 3 4 5 6 , , , , ,

e e e e e e w   w w w I A w I A w w w I A k y z P

, and the least-squares 

component vector, 

 

e Φ  k

,

( ) e h    u w Φ

(10)

where 

0

( 1,...,6) w k k

denote dimensionless weighting coefficients; 

e A , 

e

y

I , 

e

z

I

, and 

e

P

I

are, 

respectively, the cross-section area, moments of inertia with respect to the 

y - and 

z -axis, and polar 

moment of inertia of the element, and 
9

 

2

1

( 1,...,6)

e n

e i

k k i k

i

L

e x e k

n





       

(11)

denote the least-squares components of the element functional, where 

e

L

denotes the element length; 

n

and 

i

x

(

0

e

i   x L

) are, respectively, the number and the axial coordinate of the locations where the 

section strains are evaluated, and the superscript i is used to denote the section strains that are computed 

from the strain-sensor values (experimental values) at the location 

i

x

. The 

0

wk

coefficients may be 

assigned different values to enforce a stronger or weaker correlation between the measured section-strain 

components and their analytic counterparts, i.e., a larger value of 

0

wk

enforces a stronger correlation, 

whereas a smaller value enforces a weaker correlation.

Substituting Eq. (9) into Eq. (3) gives the section strains in terms of the nodal dof’s as

( 1,...,6) e

k k e k   B u

(12)

A vector form of Eq. (12) that incorporates all six section strains is given by

   

e

e u B u  x

(13)

where the matrix 

B x

contains the derivatives of the shape functions 

N( ) x

(

B x

is defined in 

Appendix B for the case of shape functions presented in Sec. 3.) Substituting Eq. (13) into Eq. (11) and 

then Eq. (10) results in the following quadratic form

   

1

2

T T

e e e e e e e     u k u u f c

(14)

where 

e

c

is a constant while 

e

k

and 

e

f

are defined as follows

6 6

1 1

,

k k k k

k k

w w

 

    e e e e k k f f

(15)

with
10

     

1 1

, ( 1,...,6)

e e n n

T T i

k k i k i k k i k

i i

L L x x x e k

n n



 

             e e k B B f B

(16)

Note that 

e

k

resembles an element stiffness matrix of the direct finite element method and 

e

f

resembles 

the load vector; 

e

k

is a function of the measurement locations, 

i

x

, whereas 

e

f

possesses the measured 

strain values. Minimization of functional 

e 

(see Eq. (14)) with respect to 

e

u

leads to the inverse 

element matrix equation 

e e e k u f 

(17)

The assembly of the finite element contributions, while accounting for the appropriate coordinate 

transformations and by specifying problem-dependent displacement boundary conditions, results in a non 

singular system of algebraic equations of the form

ΚU F

(18)

The solution of these equations for the unknown dof’s is efficient: the K matrix is inverted only once, 

since it is independent of the values of the measured strains. The F vector, however, is dependent on the 

measured strain values that change during deformation. Thus, at any strain-measurement update during 

deformation, the matrix-vector multiplication provides the solution for the unknown nodal displacement 

dof’s, U = K

-1 F, where K-1

remains unchanged for a given distribution of strain sensors.

The remaining part of the element formulation involves the selection of suitable shape functions, 

symbolically defined by Eq. (9), and the computation of the experimental section strains, 

i

k

e



, appearing 

in Eqs. (12),(13). In Sec. 3, the shape functions for two alternative inverse-frame elements, each having 

two nodes and twelve dof’s, are derived. In Sec. 4, a procedure for computing 

i

k

e



is described; it relates 

the number of strain gauges to the interpolation order of the shape functions.

3 Element shape functions

In this section, inverse frame elements of 0th- and 1st-order are formulated. The elements use C

0

-

continuous interdependent interpolations that enable excellent predictions even for very slender frame 

members, without incurring any form of excessive stiffening due to shear locking [26]. The 0th-order 

shape functions are guided by Timoshenko equilibrium equations, Eqs. (7), that correspond to the forces 

and moments applied exclusively at the end nodes, resulting in constant distributions of the transverseshear section strains. The 1st

-order shape functions accommodate Eqs. (7) for uniformly distributed 

transverse loads, giving rise to linear distributions of the transverse-shear section strains.

A frame element is referred to a local axial coordinate 

0, ex L    

, where 

e L

denotes the element 

length. Furthermore, a non-dimensional coordinate 

(2 / 1) 1,1  

e

     x L

is used to define the 

element shape functions (Figure 3). The initial nodal configurations are defined by the two end nodes, 1 
11

(at 

 1

) and 2 (at 

 1

) and one or three interior nodes. Thus, the initial configuration for the 0th-

order element has the interior node, r (at the midspan, 

  0

); whereas the interior nodes of the 1st-order 

element are q (at 

  1 2

), r (at 

  0

), and s (at 

  1 2

) .

Figure 3: Inverse finite element geometry and nodal topology.

The initial nodal configurations of the 0th- and 1st-order elements are readily reduced to two nodes and 

twelve dof’s by condensing out the interior dof’s at the element level in a manner analogous to static 

condensation. The resulting elements have three-displacement and three-rotation dof’s at each end node 

(Figure 4); thus, the dof’s vector of the elements is

 1 1 1 1 1 1 2 2 2 2 2 2 , , , , , , , , , , , 

T

e

x y z x y z u  u v w u v w      

(19)

Figure 4: Two-node inverse finite element.

The process of condensing out the interior dof’s results in the reduced element equations, 

e e e k u f R R R

 , 

where 

e

k R

is a function of the partitioned parts of the original 

e

k

matrix, and 

e

uR

contains the end-node 

dof’s. Since the unreduced 

e

k

matrix is independent of the strain values, so is the 

e

k R

matrix. This 

implies that even for the elements with the condensed-out interior dof’s, the corresponding system matrix, 

Κ

, is strain-value independent (refer to Eq. (18).)

3.1 0

th

-order element 

The formulation of the 0th-order element is guided by Eqs. (7) for the loading case of end-node forces 

and moments. For this case, the axial force, twisting moment, and shear forces are constant along the 

element; whereas the bending moments are linear. Eqs. (7) in terms of the section strains (after Eqs. (6) 

have been introduced) indicate that the section strains ei (i=1,4-6) are constant, and ei (i=2,3) are linear. 

From Eqs. (5), it is deduced that 

u

and 

 x

are linear, 

 y

and 

 z

are parabolic, and 

v

and 

w

are cubic. 
12

The inter-relationship of the polynomial order of the deflection variables, 

v

and 

w

, and bending 

rotations, 

 y

and 

 z

, can also be inferred from the definitions of the transverse shear section strains [24]

e w e v 4 , 5 ,      x y x z   ,  

(20)

It is understood that in 

4

e

both 

w

,x

and 

 y

should be represented by the same polynomial order and, 

similarly, in 

5

e

both 

,x

v

and 

 z

should also have matching polynomial orders. The above interpolations 

give rise to quadratic interpolations for 

4

e

and 

5

e

; they permit a consistent reduction of interior dof’s for 

v

and 

w

by requiring a constant variation of these section strains across the element span

4 5 e const e const  

., .

(21)

The complete set of interpolations for this element is thus given by 

       

       

           

(1) (1)

1,2 1,2

(2) (2)

1, ,2 1, ,2

(1) (3) (1) (3)

1,2 1, ,2 1,2 1, ,2

,

,

,

i i x i xi

i i

y j yj z j zj

j r j r

i i j zj i i j yj

i j r i j r

u L u L

L L

v L v N w L w N

     

       

       

 

 

   

 

 

   

 

 

   

(22)

where 

 

(1) Li

 i 1,2

and 

 

(2) Lj

  j r 1, ,2

are, respectively, linear and quadratic Lagrange 

polynomials, and 

 

(3) Nj

  j r 1, ,2

are special-form cubic polynomials (refer to Appendix A.) Static 

condensation can be used to condense out the two interior dof’s (

 yr

and 

 zr

, refer to Eqs. (22)), thus, 

achieving a two-node element with twelve dof’s (Figure 4, Eq. (19)).

3.2 1

st

-order element

Consider a frame element loaded by uniformly distributed transverse loads, 

( )

y q x

and 

( )

z q x

. From 

Eqs. (7), after the substitution of Eqs. (6), it is readily deduced that 

4

e

and 

5

e

need to be linear and 

2

e

and 

3

e

parabolic. The 

u

and 

 x

variables remain linear as in Eq. (22). Moreover, 

v

and 

w

are quartic 

whereas 

 y

and 

 z

are cubic. Following the constraint strategy for 

4

e

and 

5

e

, these section strains are 

set to be linear for this element. The resulting interpolation polynomials are given by
13

       

       

           

(1) (1)

1,2 1,2

(4) (4)

1, , , ,2 1, , , ,2

(3) (3) (1) (1)

1,2 1, , , ,2 1,2 1, , , ,2

,

,

,

i i x i xi

i i

k k k k

k q r s k q r s

k k y i yi k z i zi k

i k q r s i k q r s

u L u L

v L v w L w

L N w L N v

     

   

         

 

 

   

 

 

   

 

 

   

(23)

where 

 

(3)

Nk  k q r s 1, , , ,2

are cubic polynomials that satisfy the conditions

4 5 e linear e linear  

,

(24)

For the detailed expressions of the 

 

(3)

Nk 

polynomials, refer to Appendix A. The interior dof’s (

q

v , 

r

v , 

s

v , wq

, wr, and 

ws

, refer to Eqs. (23)) are condensed out at the element level, leading again to a 

twelve dof’s inverse element as in Eq. (19).

4 Input data from surface strain measurements 

A key step in the iFEM formulation is to evaluate the section strains due to experimentally measured 

surface strains. In this section, the relationships between the measured surface strains and the six section 

strains, ei, are established. Also discussed are the strain gauge positions along the frame axis and their 

angular orientations that enable the complete description of the experimental section strains. For 

illustration, the present analysis is restricted to frame members with circular cross-sections; the adopted 

cylindrical coordinate system 

, , x r

is shown in Figure 5. 

Figure 5: Orthogonal and cylindrical coordinate systems.

4.1 Section strains derived from linear strain gauge measurements

Taking the usual assumption of negligible 

 y

and 

 z

, then 

 x

and 

x



are the only non-zero stress 

components acting on the external surface 

ext r R 

(Figure 6(a)). The corresponding strain state, 

represented in Figure 6(b), is
14

, ,

x x

x x x x E E G



 

  

          

(25)

(a) Stress state. (b) Strain state.

Figure 6: Stress and strain states on the frame external surface (r=Rext) in the cylindrical 

coordinate system.

Consider a linear strain gauge placed on the external surface at 

i

x x 

, at a particular 



and with an 

angle 



with respect to the beam axis (Figure 7); 

 xxx 1 2 3 , , 



is a local Cartesian coordinate system 

having its origin on the frame external surface, the 

2

x



-axis along the strain gauge measurement axis (

*

2



), 

the 

1

x



-axis on the frame surface and the 

3

x



-axis normal to the frame surface and coincident with r-axis.

Figure 7: Location and coordinate system of a linear strain gauge placed on the frame 

external surface.

Using appropriate strain-tensor transformations from the 

, , x r

to 

 xxx 1 2 3 , , 



coordinates [27], the 

relationship between the measured strain 

*

2



and the strain tensor components in Eqs. (25) becomes

* 2 2

2

cos sin cos sin 

x x  

          

(26)

or, using the second of Eqs. (25),
15

 

* 2 2

2

cos sin cos sin x x          

(27)

Substituting 

ext r R 

in Eqs. (3), yields

1 2 3

4 5 6

sin cos

cos sin

x ext ext

x ext

e e R e R



e e e R

  

  

  

  

(28)

Substituting Eq. (28) into Eq. (27) results in the relation between the measured strain 

*

2



and the six 

section strains at 

i

x x 

    

  

  

 

 

 

* 2 2

2 1

2 2

2

2 2

3

4

5

6

, , i i

i ext

i ext

i

i

i ext

x e x c s

e x c s s R

e x c s c R

e x c s c

e x c s s

e x c s R

 

  

  

  

  

 

   





 

 

 







(29)

where 

c cos    , s sin    , c cos   

, and 

s sin    .

4.2 Strain gauge distributions

The iFEM formulation minimizes, in the least-squares sense, Eq. (8), where 

ε

e

are the section strains 

computed from the measured strains. Thus, an important question arises what constitutes the minimum 

number of strain measurements.

For the 0th-order element, 

1

e , 

4

e , 

5

e

and 

6

e

are constant, whereas 

2

e

and 

3

e

are linear with respect to the 

axial coordinate, x; these section strains necessitate eight strain measurements. Similarly, for the 1st-order 

element, 

1

e

and 

6

e

are constant, 

4

e

and 

5

e

are linear, and 

2

e

and 

3

e

are quadratic, thus requiring twelve 

strain measurements. 

A further reduction of strain measurements is possible if one invokes Eqs. (7). Substituting Eqs. (6) 

into Eqs. (7) results in 

2, 4 3, 5 , D e G e D e G e y x z z x y

 

(30)

For the 0th-order element, Eqs. (30) give rise to two constraints equations, thus reducing the minimum 

number of strain measurements to six; whereas, for the 1st-order element, Eqs. (30) give rise to four 

constraint equations, thus reducing the minimum number of strain measurements to eight. It is worth 

noting that this procedure should be viewed as a convenient means of reducing the required number of 

strain gauges by solving for 

4

e

and 

5

e

analytically rather than measuring these quantities experimentally.

Since the strain gauges can be placed anywhere along the beam surface, the distributions considered 

in this study are summarized in Table 1 (also refer to Figures 8 and 9). To refer to a specific combination 

of the element type and strain gauge configuration, a compact notation, #-#E, is used; where the first 
16

position, #, refers to the element order (0 or 1), the second position, #, indicates the number of strain 

gauges per element (6 or 8), and the letter “E” indicates that Eqs. (30) have been used in the formulation. 

The strain gauges are placed at different positions x = (L

e

/3, L

e

/2, 2L

e

/3) along the element. The strain 

gauge angular orientations (

 ,

) are also allowed to be different; for example, (

 ,

) = (-2/3, /4) 

indicates that the strain gauge is placed at the circumferential angle =-2/3 and is oriented with an angle 

=/4 with respect to the frame x-axis (Figure 7).

Table 1: Strain gauge distributions x, (

 ,

) corresponding to the 0th– and 1st–order elements.

Elementstrain gauge 

notation

Orientation (

 ,

) of 

strain gauges at 

x=L

e

/3

Orientation (

 ,

) of 

strain gauges at x=L

e

/2

Orientation (

 ,

) of 

strain gauges at 

x=2L

e

/3

0-6E -

(-2/3,0), (-2/3,/4),

(0,0), (0,/4),

(2/3,0), (2/3,/4)

-

1-8E (-2/3,/4)

(-2/3,0), (-2/3,/4),

(0,0), (0,/4),

(2/3,0), (2/3,/4)

(2/3,/4)

Figure 8: 0-6E strain gauge distribution.

Figure 9: 1-8E strain gauge distribution.
17

5 Numerical results 

A simple cantilevered beam subjected to dynamic loading is analyzed to assess the accuracy of the 

inverse finite element method. The beam is made of an aluminum alloy (E=73,000 MPa, 



=0.3, and 

=2557 Kg/m3) and has a thin-walled circular cross-section with the average radius 

R

=39 mm, wall 

thickness s=2 mm, and length L=800 mm. The beam is fully clamped on one end and subjected at the 

other end to a harmonic vertical force Fz (t) (where t denotes time) acting in the z-direction at frequency f0, 

i.e., 

F t F f t

z z

( ) sin 2 

0 0   

(31)

where 

F

z0

is the force amplitude (

F

z0

=103 N.) To simulate the experimental-strain measurements and to 

assess the accuracy of the inverse method, high-fidelity direct FE analyses are performed using QUAD4 

shell elements in MSC/NASTRAN. The model consists of 114 elements along the cross-sectional 

circumference and 360 elements along the beam axis, for a total of 41,040 elements and 41,156 nodes. 

The tip force is applied at the cross-sectional center at a node which is connected to all other nodes within 

the cross-section by means of multi-point constraints (or MPC’s) [28].

The dynamic response of the beam is calculated using a modal transient analysis in MSC/NASTRAN 

and keeping the modes up to 5,000 Hz, with the inclusion of viscous damping of magnitude 5% with 

respect to the critical value at each frequency. In the frequency range from 0 to 5,000 Hz, 51 modes are 

present: these include the first lowest flexural beam modes, 1F-5F, appearing twice due to the crosssection symmetry, and the first membrane mode (1M). Table 2 summarizes the order of the global modes, 

their type, and corresponding frequency value f; the first three flexural mode shapes are shown in Figures 

10-12. The other modes in this frequency range are associated with shell modes describing cross-sectional 

distortion and are neither shown nor tabulated.

Table 2: Global modes of the cantilevered beam in the frequency range of 0-5,000 Hz. (F-type modes are 

flexural; M-type modes are membrane.)

Mode order 1

st and 2nd

modes

3

rd and 4th

modes 12th mode 13th and 14th modes

30th and 

31st modes

40th and 

41st modes

Mode type 1F 2F 1M 3F 4F 5F

f

frequency 

[Hz]

126.8 729.5 1,670 1,835 3,187 4,671
18

Figure 10: 1st flexural mode (1F, 

f

= 126.8 Hz).

Figure 11: 2nd flexural mode (2F, 

f

=729.5 Hz).

Figure 12: 3rd flexural mode (3F, 

f

=1,835 Hz).

To investigate the accuracy of the iFEM modeling for dynamic applications in both low- and highfrequency regimes, three different values of the applied-force frequency f0 have been considered, namely: 

f0=60 Hz (about half of the fundamental frequency), f0=450 Hz (halfway between the 1F and 2F modes), 

and f0=1,400 Hz (halfway between the 2F and 3F modes). Figures 13-15 compare the tip-deflection time 

histories, wmax(t), calculated by means of the high-fidelity FEM shell model using MSC/NASTRAN and 

the corresponding iFEM frame-element models. The tip deflection of the NASTRAN model corresponds 

to the cross-sectional center, and is computed at a node which is connected to all other nodes within the 

cross-section by means of MPC’s. The present iFEM models used the strain-gauge distributions in Table 

1 and the uniform weight coefficients 

0

1 ( 1,...,6) w k k 

in Eq. (10); the strain values were taken from 

the nodes (at the specific locations in Table 1) of the NASTRAN model. It is noted that slightly more 

accurate strain values reside at the element Gauss points. However, considering the high fidelity of the 

reference FEM model, the “measured” strains taken at the nodes are quite satisfactory. 

For the low-frequency loading of f0=60 Hz, a single 0th-order inverse element gives accurate results, 

with a maximum error in the tip deflection of 2.3% (Figure 13). At this excitation frequency, when 

z

y

x

z

y

x

z

y

x
19

t  0.1

, viscous damping has reduced the structural vibrations to a steady state response, proceeding at a 

constant amplitude and the same frequency as the forcing function. When the excitation frequency of the 

forcing function is increased, the response has a longer transient region, which is manifested by 

interactions between the natural modes of vibration and that due to the applied dynamic loading. To 

model the transient response at higher frequencies, finer discretizations are required. Thus, for f0 = 450 

Hz, a two-element, 1st-order model yields a 1.1% error in the maximum deflection (Figure 14). At the 

f0=1,400 Hz frequency, a three-element iFEM discretization using the 1st-order element results in the 

maximum deflection error of 2.0% (Figure 15). These results clearly demonstrate that the methodology is 

highly efficient, requiring only few inverse elements and strain gauge measurements, and is applicable not 

only for the steady state portion of the response but also for the transient regime at high frequencies.

Figure 13: Tip deflection wmax of the beam loaded by a transverse concentrated force Fz at 

f0=60 Hz.
20

Figure 14: Tip deflection wmax of the beam loaded by a transverse concentrated force Fz at 

f0=450 Hz.

Figure 15: Tip deflection wmax of the beam loaded by a transverse concentrated force Fz at 

f0=1,400 Hz.
21

6 Conclusions

In search of a suitable computational method for use in Structural Health Monitoring (SHM) systems, 

an inverse Finite Element Method (iFEM) has been formulated to perform the displacementreconstruction analysis (shape-sensing) of three-dimensional frame structures undergoing static and 

dynamic deformations. The methodology uses a least-squares variational principle, which is discretized 

by C0-continuous displacement-based inverse frame elements. Linear strain-displacement relations and 

their components, known as section strains, are based on the Timoshenko (first-order) shear deformation 

theory that includes the deformations due to stretching, torsion, bending, and transverse shear. The 

variational statement enforces experimentally measured strains to be least-square compatible with those 

interpolated within the inverse frame elements. The implementation of this least-square compatibility is 

accomplished using the individual section strains. 

Two inverse frame elements, each having two nodes and twelve dof’s, have been developed. The 0th-

order element has a constant shear-section strain along the element length, whereas the 1st-order element 

has a linear shear section strain. The element shape functions are based on interdependent interpolations 

that ensure locking-free bending of slender frame members. The element interpolation order is linked to 

the definition of the number and orientation of the uniaxial strain gauges that are necessary for the 

analysis. Two simple and effective strain gauge distributions have been selected and used in the numerical 

examples.

The present shape-sensing capability has been demonstrated on a thin-walled, circular cross-section 

cantilevered beam subjected to harmonic excitations in the presence of structural damping. To provide the 

simulated strain-gauge measurements, as well as the reference displacements, a high-fidelity shell finite 

element model was developed using the MSC/NASTRAN commercial code. Low- and high-frequency 

dynamic beam motions were analyzed and time history of the tip deflection examined, comparing several 

iFEM discretizations and strain-gauge schemes. The iFEM shape-sensing analysis, which is based only on 

the strain-displacement relations and the measured strain data (without any reliance on the material, 

inertial, or damping properties of the structure), has been shown to be highly effective and efficient in 

predicting the dynamic structural response of a damped beam. Accurate predictions of both the steadystate and transient response required only few elements and strain-gauge measurements, where the higherfrequency excitations necessitated somewhat higher fidelity of the iFEM models.

Although beyond the scope of the present effort, additional studies need to be performed, including: 

(a) shape-sensing analysis of spatial frame structures using the strains measured in a laboratory, and (b) 

studies of the strain-gauge distributions that provide optimal (or nearly optimal) solutions.

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Navigation, and Control Conference and Exhibit, San Francisco, California, 2005.
23

 [23] P. Cerracchio, M. Gherlone, M. Mattone, M. Di Sciuva, A. Tessler, Shape sensing of threedimensional frame structures using the inverse finite element method, 5th European Workshop on 

Structural Health Monitoring, Sorrento, Italy, 2010.

 [24] M. Gherlone, P. Cerracchio, M. Mattone, M. Di Sciuva, A. Tessler, Dynamic shape reconstruction 

of three-dimensional frame structures using the inverse finite element method, 3rd ECCOMAS 

Thematic Conference on Computational Methods in Structural Dynamics and Earthquake 

Engineering, Corfù, Greece, 2011.

 [25] S.P. Timoshenko, On the correction for shear of differential equations for transverse vibrations of 

prismatic bars, Philosophical Magazine, 41, 744–746, 1921.

 [26] A. Tessler, S.B. Dong, On a hierarchy of conforming Timoshenko beam elements, Computers & 

Structures, 14, 335–344, 1981.

 [27] A.I. Lurie, Theory of Elasticity, Springer-Verlag Berlin Heidelberg, New York, 2005.

 [28] MSC/MD-NASTRAN, Reference Guide, Version 2006.0, MSC Software Corporation, Santa Ana, 

CA.

Appendix A

The 1st, 2nd, and 4th-degree Lagrange shape functions are given as

 1

st degree

   

(1) (1)

1 2

1

, 1 , 1

2

  L L       

   

(A1)

 2

nd degree

     

(2) (2) (2) 2

1 2

1

, , 1 ,2 1 , 1

2

LLL

r

            

   

(A2)

 4

th degree

     

       

(4) (4) 2

1 2

(4) (4) (4) 2 2

1

, 4 1 1 , 1

6

1

, , 1 4 2 1 ,3 1 4 ,4 2 1

3

q r s

L L

LLL

   

     

           

            

(A3)

where 

2 / 1 1,1  

e

     x L

is a non-dimensional axial coordinate; 

0, ex L    

and 

e L

denotes the 

element length. The subscripts 1 and 2 represent the end nodes, whereas q, r, and s denote the uniformly 

spaced interior nodes.

The 3rd–degree shape functions, 

 

(3) Nj



, of the 0th-order element have the form
24

     

(3) (3) (3) 2

1 2 , , 1 2 3 , 4 , 2 3

24

e

r

L

  NNN               

(A4)

whereas the cubic 

 

(3)

Nk 

shape functions of the 1st-order element are 

         

(3) (3) (3) (3) (3) 2

1 2

4

, , , , 1 4 3 , 2 8 3 ,24 , 2 8 3 , 4 3

3

q r s NNNNN

e

L

                       

(A5)

Appendix B

The matrix 

B

relating the section strains to the element dof’s may be written as follows

 

1 11 12

2 21 2 22

3 31 3 32

4 41 4 42

5 51 5 52

6 61 62

c

c

c

c



   

   

   

   

     

   

   

   

   

B B 0 B

B B B B

B B B B

B

B B B B

B B B B

B B 0 B

(B1)

For a 0th-order element, 

0

is a 1x2 null matrix and the other sub-matrices are defined below

(1)

1 ,

(2)

2 ,

(2)

3 ,

(2)

(1) (3)

4 , ,

(2)

(1) (3)

5 , ,

(1)

6 ,

2

00000

2

0 0 0 0 0

2

00000

2

0 0 0 0

2

2

0 0 0 0

2

2

0 0 0 0 0

i i e

i i e

i i e

e

i

i i i e

e

i

i i i e

i i e

L

L

L

L

L

L

L L L N

L

L L L N

L

L

L







 

 





     



      



    

  



    

      

    



   

          

     

B

B

B

B

B

B

( 1, 2) i 











(B2)
25

(2)

2 ,

(2)

3 ,

(2)

(3)

4 ,

(2)

(3)

5 ,

2

0

2

0

2

0

2

2

0

2

c r e

c r e

e

r

c r e

e

r

c r e

L

L

L

L

L L N

L

L L N

L









    

     

   

         

   

          

B

B

B

B

(B3)

For a 1st-order element, 

0

is a 1x6 null matrix and the other sub-matrices are defined below

(1)

1 ,

(3) (2)

, 2 ,

(3) (2)

, 3 ,

(3) (2)

(4)

4 ,

(3) (2)

(4)

5 ,

6

2

00000

2

0 0 0 0

2

0 0 0 0

2

0 0 0 0

2 2

2

0 0 0 0

2 2

2

000

i i e

i

i i e

i

i i e

e e

i

i

i i e

e e

i

i

i i e

i e

L

L

N L

L

N L

L

L N L L L

L

L N L L L

L

L

L



 

 





    

  

   

       

   

           

   

            



B

B

B

B

B

B

(1)

,

( 1, 2)

0 0 i

i





















 















  

  

(B4)

(3) (3) (3)

, , , 2

(3) (3) (3)

, , , 3

(3) (3) (3)

(4) (4) (4)

4 , , ,

(3)

(4)

5 , ,

2

0 0 0

2

0 0 0

2

0 0 0

2 2 2

2

0

2

q r s c e

q r s c e

e e e

q r s

c q r s e

e

q

c q r e

N N N

L

N N N

L

L N L N L N L L L

L

L N L L

L

  

  

  

 

  

   

  

   

       

           

       

 

       

B

B

B

B

(3) (3)

(4) (4)

,

0 0

2 2

e e

r s

s

L N L N L 

             

     

(B5)
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Dynamic Shape Reconstruction of Three-Dimensional Frame Structures 

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Gherlone, Marco; Cerrachio, Priscilla; Mattone, Massimiliano; Di Sciuva, 

Marco; Tessler, Alexander

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14. ABSTRACT

A robust and efficient computational method for reconstructing the three-dimensional displacement field of truss, beam, and frame structures, using measured 

surface-strain data, is presented. Known as “shape sensing”, this inverse problem has important implications for real-time actuation and control of smart 

structures, and for monitoring of structural integrity. The present formulation, based on the inverse Finite Element Method (iFEM), uses a least-squares 

variational principle involving strain measures of Timoshenko theory for stretching, torsion, bending, and transverse shear. Two inverse-frame finite elements 

are derived using interdependent interpolations whose interior degrees-of-freedom are condensed out at the element level. In addition, relationships between 

the order of kinematic-element interpolations and the number of required strain gauges are established. As an example problem, a thin-walled, circular 

cross-section cantilevered beam subjected to harmonic excitations in the presence of structural damping is modeled using iFEM; where, to simulate 

strain-gauge values and to provide reference displacements, a high-fidelity MSC/NASTRAN shell finite element model is used. Examples of low and 

high-frequency dynamic motion are analyzed and the solution accuracy examined with respect to various levels of discretization and the number of strain 

gauges. 

15. SUBJECT TERMS

Frame finite element; Inverse Finite Element Method; Structural Health Monitoring; Timoshenko beam theory; Variational 

principle

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