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					             Sensitivity Index on Load Carrying Capacity of Framed Structures

                                                      to Member Disappearance
                                               Jae-hyouk CHOI1, Takumi Ito2 & Kenichi Ohi3
                              (1 Department of Architecture & Civil Engineering, Kobe University, Kobe 657-8501, Japan;
                                   2 Graduate School of Engineering, University of Tokyo, Tokyo 113-8656, Japan;
                              3 Department of Architecture & Civil Engineering, Kobe University, Kobe 657-8501, Japan)

Abstract: A sensitivity index of vertical load carrying capacity to member disappearance is proposed for a framed structure. The index is defined as the ratio of
the load carrying capacity after a member or set of adjacent members disappears to the original load carrying capacity. The member or the set of members that
has the highest index might be regarded as a key element or an Achilles heel of the frame. Countermeasures against accidental events or attacks might then be
multiple protection of such an Achilles heel. This study uses an efficient linear-programming technique Compact Procedure to evaluate the load carrying
capacities before and after the member disappearance. Planar and 3D frame models are used to evaluate the sensitivity index on vertical load carrying capacity
to member disappearance. These studies reveal basic properties of structural sensitivity to member disappearance: (1) For a strong floor-beam system, corner
columns are more important than central columns; (2) Violation of symmetric column layout usually results in high sensitivity
Keywords: sensitivity index, vertical load-carrying capacity; member disappearance; key element

1    Introduction
      The world was shaken at witnessing the progressive collapse of the World Trade Center (WTC) on September 11, 2001(Fig. 1).
Some other examples of building steel structures collapsing by gravity have been reported recently as outcomes of accidents or
excitation of inadequate structural designs: a steel beam was completely sheared off in turbine facilities in a thermal power plant by a
falling crane during the 2001 Atico earthquake in Peru (Fig. 2). As another example, a few school gymnasium roofs collapsed during
the unusually heavy 1998 snowfall in the Kanto-Koshin district in Japan.

                      Fig.1     World Trade Center (WTC), 2001                                        Fig.2     Damage caused by a falling crane

     If one or a set of structural members in a building disappears suddenly because of this kind of accidental action, the building
should remain standing against vertical gravity loads, dead or live, and not completely collapse. We therefore examine how to attain
such performance. A close-up view of new concepts, such as redundancy and key elements, has been taken from a perspective of
assuring structural robustness of a structure even if it receives unexpected disturbance. In such an existing code as the British
Standard[1], a design regulation for a component with high importance (key element) is incorporated for preventing progressive
collapse from accidents.
     Frangopol et al.[2], Wada et al.[3] studied how much the resistance of a structure would remain after structural components were
destroyed by accidental action, and compared it with the resistance at the original state. For instance, Frangopol et al. [2] proposed a
relevant index to the redundancy of a structure as the following equation.
                                                    R =Lintact / (Lintact-Ldamage) =  /(  -  *)                                                           (1)
where, Lintact,       (or  o in the next section) respectively represent the load carrying capacity and the corresponding collapse load
factor of the structure in its original state. Also, Ldamage,          *(or  damage in the next section) respectively represent the load carrying
capacity and the corresponding collapse load factor of the structure in its damaged state.
     Limit analysis using linear programming is used herein to evaluate the residual load carrying capacity of the framed structure to
a vertical gravity load after a certain member or a set of members disappear suddenly. This study evaluates the decreasing ratio
before and after the member disappearance. It is regarded as the sensitivity index of load carrying capacity to the member
disappearance. The index shows the member’s importance from the standpoint of reserving the vertical load carrying capacity. The
important member corresponding to the highest sensitivity index shall be regarded as the key element in the structure. A simple
means to increase the redundancy of a structure or to obtain high robustness is to protect that key element using multiple means. This

study demonstrates some examples of such a key element identification analysis on simple framed structures.
2 Sensitivity Analysis Method
2.1 Matrix Method of Limit Analysis
       This study makes use of the matrix method termed ‘Compact Procedure’ proposed by Livesley[4]; it is based on the lower-bound
theorem in limit analysis and linear programming for the optimization problem. The compact procedure solves the following
                                                   Maximize  (load factor)                                                          (2)
                                                    Subject to  {Po} = [ Con.] {M} (equilibrium)                                    (3)
                                                                       |Mj|<=MPj (plastic condition)                                 (4)
In those equations, {Po} is the non-factored nodal load vector, [Con.] is the connectivity matrix in the equilibrium equation, {M} is
the member force vector, and MPj represents the plastic resistance corresponding to member force Mj.
2.2 Sensitivity Index to Member Disappearance
       Similarly to the evaluation of Frangopol et al.[2] the present study evaluates a decreasing ratio of the vertical load carrying
capacity of the structural system before and after the disappearance of a certain member. It is regarded as the sensitivity index to the
member disappearance, denoted as S.I.
                                                   Sensitivity Index: S.I.=(  o-  damage)/  o                                     (5)
       The formula above is equivalent to the reciprocal of the index about the redundancy given by Eq. (1) introduced by Frangopol et
al.[2]. When the vertical load carrying capacity changes only slightly when a certain member disappears, the corresponding sensitivity
index is negligible: S.I.  0. Such a member does not determine the load carrying capacity of the whole structural system, and would
be regarded as less important from the standpoint of preserving the load carrying capacity.
       On the other hand, when a member with large sensitivity, e.g. S.I.  1, disappears, a part of the frame or the whole frame would
collapse immediately. Such a member with a high sensitivity index would be regarded as a key element of the structural system.
2.3 Modification of Equilibrium Equation in the Member Disappears
       In the case that a certain member in the frame disappears, the constraints represented by Eqs. (3) and (4) could be modified
thusly: the member force that disappears is merely removed from the member force vector. In addition, the corresponding column of
the connectivity matrix is also removed. A computer program can accomplish these tasks systematically.

                                                                                             M1 
                                                                                                
                                                               C11  Ci1         Cn1    
                                                                                            
                                           damage    P0                            Mi    Removed
                                                                                                                               (6)
                                                              Cm1  Cim
                                                                               Cnm          
                                                                              Removed       M m 
                                                                                                

3 Analysis Model to Low-Rise Planar Frame
3.1    Analysis Model of Single-Story Planar Frame
      The single-story four-bay frame shown in Fig. 3 is studied for its sensitivity to column-member disappearance. As a vertical
loading condition, the uniformly distributed load, the intensity of which is denoted by w (not factored), is applied to the beam; this
distributed load is replaced by a set of concentrated loads,
the magnitudes of which are Po on the center of each beam                    P0     P0 P0          P0     P0     P0 P0
                                                                    P0/2                                                     P0/2
and the mid-column tops, and 0.5 Po on the side-column tops,                                           
where Po=0.5wH. Two types of member proportions are
studied: a strong-column type, and a strong-beam type.                                      Strong columns
      The plastic condition of the member is illustrated in Fig.                                                                 H
4. No interaction between flexural and axial plastic
resistances is considered herein. The theoretical interaction     Side        Midpoint        Center       Midpoint       Side
based on the four-spring approximation for the member
                                                                             H            H               H            H
cross-section would be as that shown in solid lines of Fig.4.
In this approximation, the yield axial force NY and the
                                                                             P0     P0 P0          P0     P0     P0 P0
fully-plastic moment MP are equal to 4pY and 1.5pYd,                P0/2                                                    P0/2
respectively, where pY and d respectively denote the yield
resistance of each spring and the depth of member                                                Strong beam
cross-section. Adopting the assumption of depth d of as
much as one-eighth of the member length, we obtain the
relationship NY=64MP/3H. This relationship is also presumed
in the following analysis, which is performed under no            Side        Midpoint        Center       Midpoint       Side
interaction, as shown in broken lines of Fig. 4. The member
flexural and axial failures are considered only as                           H            H               H            H
simply-plastic; no buckling is considered.
3.2 Analysis Results of Single-Story Planar Frame                          Fig.3 Single-story four-bay frames analyzed
      Fig.5 shows the ultimate vertical load carrying capacity

of two frames in their original states. One column                                                                                          N
                                                                                                             No Interaction in                   pY
member is removed from the frames, and
                                                                                                                                         4                   NY  4 pY
the collapse load factor  damage and the sensitivity               N            M                   CP (broken lines)
                                                                                          Forces in CP                                                       M P  1.5 pY d
index S.I. after one column disappearance are shown
in Figs. 6 and 7. Although the sensitivity index for                                                                                     2
the strong-column frame reaches as much as about                           d
                                                                                2                                                                                M
80%, the sensitivity index for the strong-beam frame                           d                                                                                     pY d
is smaller, slightly larger than the reciprocal of the          p1 p2            p3 p4                           -1.5           -1               1           1.5
number of columns. That is, for the strong-column
type, if one column disappears, collapse of a local
floor beam will take place to form a collapse                      Forces in Multi-spring                                                -2
mechanism. On the other hand, for the strong-beam                                              pi  pY                                           Interaction in Multi
type, the whole floor collapses. Therefore, on the                                                                                               -spring      (solid )
strong-beam type frame, the remaining columns                                                                                            -4
resist the collapse of the whole floor and thereby              Fig.4     Interacted bending and axial resistances of members
produce a supportive effect among columns.
4     Analysis model to 3D frame with a soft                       bM p                bM p

or damaged story
4.1   Analysis model of 3D frame
      Consider a multi-story 3D frame with a strong
floor-beam system, as shown in Fig. 8. When a certain
accidental event occurs at one story and softens or
                                                                                                        0  8.0           (P0=cMp/H)
damages the inter-story structure, the first approximation
of the 3D frame might be a rigid body model for the
upper-story components to the extent that the floor-beam
system is sufficiently strong to be sound and suffer no                 cN                    cN                     cN                     cN                       cN
                                                                                1.0                  1.0                  1.0                      1.0                   1.0
damage. A single-story 3D framework of 2 × 2 bays,                      c NY                  c NY                  c NY                    c NY                     c NY

which has a rigid body in the upper part, as shown in Fig.
9, is examined herein. Sensitivity analysis to column                                                0  13.33            (P0=cMp/H)
disappearance is performed. In the equilibrium equation
of this frame, only the force balance in the vertical                    Fig.5         Ultimate load carrying capacity of original states
direction and two horizontal directions and moment
                                                                                                                    bM p          0.75 b M p     bM p
balances about the X, Y, and Z-axis are examined. In                                                                                                           0.25 b M p
addition, a rigid and strong floor-beam system is assumed;
the beams do not yield but the columns yield. No
                                                                                  0.5 b M p
interaction is considered among bi-axially flexural and
axial resistances. When the plastic region of the column is
approximated by an eight-spring model for a box type
cross-section, the actual interaction surface would be that                     damage  1.0                                                      S .I .  0.875
shown in Fig. 10. Similarly to the previous planar case,
                                                                                                                   bM p
the plastic resistances are assumed to satisfy the relation                                                                                                               bM p

cNY=16 cMp X/L=16 cMp Y/L (d/L=1/6)
                                                                                                             0.5 b M p          0.5 b M p                    0.5 b M p
4.2 Analysis results of 3D frame
                                                                                 bM p                                                       bM p
        The cases analyzed for column disappearance are
summarized in Fig. 11. One column, two columns, and
three columns disappear in these nine cases. Fig.12 shows
the sensitivity results. The results shown in Fig.12                            damage  2.0                   (P0=bMp/H)                         S .I .  0.75
indicate that one column’s disappearance does not cause
                                                                                                     bM p                                bM p
immediate structural degradation or a decrease the                           0.33 b M p                                                                        0.33 b M p
structure’s vertical load carrying capacity because it has as
many as nine columns. As shown in the example of planar                                                0.5 b M p                     0.5 b M p
frame solved in the previous section, sensitivity to side                                                                bM p

column disappearance sometimes becomes very high.
Similarly, when two or more columns disappear together
with corner or side columns, the decrease in the load                           damage  2.0                   (P0=bMp/H)                         S .I .  0.75
carrying capacity is considerable, in contrast to the center
column disappearance. When disappearance of three
columns occurs, it is apparent that the disappearance of            Fig.6        Reduced capacity after one-column disappearance
side columns engenders a considerable decrease of the
                                                                                                         (Strong-column case)
vertical load carrying capacity.

                             cM p                   cM p                   cM p                cM p

                                          0.094 c NY                    c NY               c NY                  c NY

                             cM p                   cM p                   cM p                cM p

                                               damage  8.25                   (P0=cMp/H)                      S .I .  0.381

                                       c NY                   c NY                c NY                               c NY

                                       damage  10.67                    (P0=cMp/H)                  S .I .  0.2

                                       c NY                   c NY                                      c NY             c NY

                                      damage  10.67                     (P0=cMp/H) S .I .  0.2
          Fig. 7          Reduced capacity after one column’s disappearance(Strong-beam case)

                                                                                                                                Vertical Load W
                                                                                                                                                  Moment about Y-axis MY
                                                                                                                         Lateral Load HX

                                                                                             Z                                                      Moment about Z-axis MZ

                                                                                                                     Lateral Load HY
                                                                                                                                             Moment about MX

                                                                                                            X                                                              L
                                                                                       Coordinate System                                    Rigid
                                                    Rigid Body

      Multi-story Frame                   Single Soft Story

Fig.8 Single soft-story frame analyzed                                         Fig.9       Single-story nine-column frame supporting a rigid body

                                      NY  8 pY , M PX  M PY  3 pY d                                          N/pY


                                                    Spring force   pi
                                                                   X                  -3

           -d/4                                                                                                                             Mx

                  -d/2     -d/4               d/4       d/2                                                                       MY/pY d
              Yield Condition          pi  pY                                    MX/pY d

                             Box Section                                                   Plastic Failure Surface

                            Fig.10            Plastic failure surface considered by multi-spring

                                                   ⑦                           ⑧                   ⑨                                       Columns
                                                                                                                      One                   ①
                                                                                                                     Column                 ②
                                                   ④                          ⑤                    ⑥                  Lost                  ⑤
                                                   ①                          ②                    ③                  Lost
                                                                                                                    Three                  ①②③
                                                              Column ID

                                                                         Fig.11          Overview of analyzed cases

   0.0               0.0            0.0                                                                   1.0                1.0            1.0              0.0               0.0           0.0
                                                              : Reached to Ny
   1.0         0.0   1.0     0.0 1.0        0.0                                                         0.719        0.0     1.0 0.0 0.719         0.0       1.0       1.0     1.0    1.0    1.0     1.0
                                                              : Not Reached to Ny

   0.0               0.0            0.0                                                                   1.0                1.0            1.0                                              0.0
                                                  Moment Ratios to cMp
   1.0      0.0      1.0    0.0     1.0     0.0                                                           1.0        0.0     1.0     0.0    1.0    0.0                                      0.438    1.0
                                                                                        c MY
   0.0               0.0            0.0                                                         cM p                         1.0                             0.0              1.0            0.0
                                                  Axial Force Ratio to Ny
   1.0      0.0      1.0    0.0     1.0     0.0
                                                                  NY                                                         1.0     0.0                     1.0      1.0     1.0     0.0    1.0     1.0
         Original State                                                                                         Two Columns Lost                                   Two Columns Lost

    0  144
                                                              Explanatory                                damage  87                S .I .  0.396          damage  103            S .I .  0.285
   1.0               1.0            1.0            1.0                  1.0              1.0              0.0                 0.0           0.0               1.0              1.0             1.0

   1.0      1.0      1.0    1.0     0.5     1.0    0.75       0.0       1.0    0.0       0.75    0.0      1.0         0.0     1.0 0.0       1.0       0.0    0.188     0.0     0.0    0.0 0.188      0.0

   1.0               1.0            1.0            1.0                  1.0              1.0                                 0.0                              1.0             1.0              1.0

   1.0      1.0      1.0    1.0     1.0     1.0    1.0        0.0       1.0       0.0    1.0     0.0                         1.0     0.0                      1.0      0.0    1.0     0.0      1.0   0.0

                     1.0            1.0             1.0                                  1.0              0.0                0.0            0.0

                     1.0    1.0     1.0     1.0     1.0       0.0                        1.0     0.0      1.0        0.0     1.0     0.0    1.0       0.0
         One     Corner Column                            One Side Column Lost                                  Two Columns Lost                                    Three    Side    Columns
 damage  120               S .I .  0.139       damage  120                   S .I .  0.139         damage  112               S .I .  0.222          damage  54              S .I .  0.625
         Lost                                                                                                                                                       Lost

   0.0               0.0            0.0              1.0                 1.0             1.0                  0.0              0.0           0.0

   1.0       0.0     1.0   0.0      1.0     0.0      1.0          0.0 0.438       0.0    0.0      0.0         1.0      0.0     1.0    0.0    1.0      0.0

   0.0                              0.0             1.0                 1.0              1.0

   1.0      0.0                     1.0     0.0     1.0           0.0   1.0       0.0    1.0      0.0

   0.0               0.0            0.0                                                  0.0                  0.0             0.0            0.0

   1.0      0.0      1.0    0.0     1.0     0.0                                          1.0      0.0         1.0      0.0    1.0     0.0    1.0       0.0

         One     Center    Column                          Two Columns Lost                                         Three Middle Columns

 damage  128             S .I .  0.111          damage  87                   S .I .  0.396              damage  96
                                                                                                                                      S .I .  0.333

                                                     Fig. 12             Reduced capacity after column disappearance

     Sensitivity to the disappearance of three middle columns is less than the sensitivity to two side columns’ disappearance.
Generally, the decrease in the load carrying capacity is predictable from the change of column number that takes place, insofar as the
remaining column arrangement is symmetrical. On the other hand, when it is asymmetrical, the remaining columns can not function
effectively. For that reason, the load-carrying capacity decreases greatly compared with the amount predicted from the remaining
column number.

5     Concluding remarks
     When a certain member in a frame disappears suddenly, the ratio of the decreased amount of the load-carrying capacity after the
disappearance to the original load carrying capacity is defined as the structure’s sensitivity to that member’s disappearance. A few
examples of analyses on planar frames and 3D frames were illustrated.
     Sensitivity evaluation was performed by removing the corresponding column to the member forces that disappear from the
original connectivity matrix in the equilibrium equation. Subsequently, the sensitivity index is evaluated easily with an efficient
matrix method of limit analysis based on linear programming.
     Sensitivity can be calculated easily to identify a structural system’s key elements. From case studies presented herein, the
following conclusions can be derived about the redundancy of a framed structure:
     1) For a strong-column and weak-beam planar frame, the sensitivity to the column member disappearance is large. In the case of
strong-beam weak-column type, the sensitivity is small. This relationship implies that a strong-beam or strong-floor system is
important to raise a structure’s redundancy. A weak-beam or weak-floor system tends to cause a local premature collapse to vertical
     2) For a 3D frame with as many as nine columns with a strong-beam strong-floor system, the disappearance of one column did
not sharp degrade the vertical load carrying capacity. However, when two or more columns disappeared, including side-columns, the
load carrying capacity decreased much more than in the case of mid-column disappearance.

    This research was supported by the Japan Society for the Promotion of Science, Grant in Aid for Scientific Research (Category
B, No. 17404014), which made this work possible. The authors are grateful appreciated.

[1]   BS CODE: BS5950, Part 1, Section 2, 1990. Author K, Coauthor T. Displacement based design methodology for steel frame structures. Journal of Steel
      Structures, 2001, 1 (1): 77-82
[2]   Frangopol D M, Curley J P. Effects of Damage and Redundancy on Structural Reliability. Journal of Structural Engineering, ASCE, 1987,113(7):
[3]   Livesley R K. Matrix Method of Structural Analysis. 2nd ed. Pergamon Press, 1976
[4]   Wada A. A Study on Strength Deterioration of Indeterminate Double-layer Space Truss due to Accidental Member Failure. Journal of Structural and
      Constructional Engineering (Transactions of AIJ), 1989, 402: 89-99 (in Japanese)


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